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Program Report for the Preparation of Middle Level Mathematics Teachers

National Council of Teachers of Mathematics (NCTM)Option C

NATIONAL COUNCIL FOR ACCREDITATION OF TEACHER EDUCATION

COVER SHEET

1. Institution NameGrand Valley State University

2. StateMichigan

3. Date submitted

MM DD YYYY

08 / 02 / 2010

4. Report Preparer's Information:

Name of Preparer:

Phone: Ext.

(

E-mail:

5. NCATE Coordinator's Information:

Name:

Phone: Ext.

(

E-mail:

6. Name of institution's programMathematics Major (Elementary Teaching Emphasis)

7. NCATE CategoryMathematics Education

8. Grade levels(1) for which candidates are being prepared

(1) e.g., 5 - 9, 7 - 9, 6 - 8

K-8

9. Program Type

nmlkji First teaching license

10. Degree or award level

nmlkji Baccalaureate

nmlkj Post Baccalaureate

nmlkj Master's

nmlkj Post Master's

nmlkj Specialist or C.A.S.

nmlkj Doctorate

nmlkj Endorsement only

11. Is this program offered at more than one site?

nmlkj Yes

nmlkji No

12. If your answer is "yes" to above question, list the sites at which the program is offered

13. Title of the state license for which candidates are preparedMathematics - Elementary

14. Program report status:

nmlkji Initial Review

nmlkj Response to One of the Following Decisions: Further Development Required or Recognition with Probation

nmlkj Response to National Recognition With Conditions

15. State Licensure requirement for national recognition:NCATE requires 80% of the program completers who have taken the test to pass the applicable state licensure test for the content field, if the state has a testing requirement. Test information and data must be reported in Section III. Does your state require such a test?

nmlkji Yes

nmlkj No

SECTION I - CONTEXT

1. Description of any state or institutional policies that may influence the application of NCTM standards. (Response limited to 4,000 characters INCLUDING SPACES)(Please note, the following information is the same as what was presented in our previous report submitted in September 2007.)

The State of Michigan grants teacher certification only for teachable subject matter majors (e.g., mathematics, science, English, etc.). Furthermore, the State grants only two levels of teacher certification in each subject area, elementary and secondary, with middle level certification incorporated into both levels. Thus, candidates who receive initial elementary teacher certification in mathematics through Grand Valley State University (GVSU) are certified to teach all subjects in Grade K-5 and in Grade 6-8 self-contained classrooms and are certified through an endorsement to teach mathematics in Grade 6-8.

Due to the structure of Michigan’s levels of teacher certification, the College of Education (COE) at GVSU requires all candidates seeking initial elementary teacher certification to complete one semester of field experience and one semester of student teaching in a Grade K-5 classroom. The COE and State further require candidates seeking elementary mathematics certification to demonstrate strong mathematical content knowledge for topics addressed in Grade 6-8.

The Elementary Mathematics Program at GVSU is housed in the Department of Mathematics with secondary admission to the COE. As such, the program addresses a combination of indicators from NCTM’s Elementary Mathematics Specialist Report and Middle Level Mathematics Teachers Report. Per a June 1, 2006 agreement with Margaret D. Crutchfield, NCATE Associate Vice President, and Sue Wittick, Michigan Department of Education, the Elementary Mathematics Program is required to meet the following.

1. The program needs to address the mathematical process, pedagogy, and content indicators in Standards 1-15 of the Middle Level Report.

2. Candidate’s student teaching experience is allowed to occur in a Grade K-5 classroom (Standard 14.2 - Elementary Mathematics Specialist) rather than being required in a middle level mathematics classroom (Standard 16.2 - Middle Level Report).

3. During specific program courses, candidates will engage in brief structured field experiences with Grade 6-8 students and in Grade 6-8 mathematics classrooms to address Standard 16.1 - Middle Level Report.

2. Description of the field and clinical experiences required for the program, including the number of hours for early field experiences and the number of hours/weeks for student teaching or internships. (Response limited to 8,000 characters INCLUDING SPACES)

(Please note, the following information is the same as what was presented in our previous report submitted in September 2007.)

Field and clinical experiences are associated with specific courses offered by the Department of Mathematics and the COE that are part of the required program of study.

Three mathematics courses (MTH 322 Geometry for Elementary Teachers, MTH 323 Probability and Statistics for Elementary Teachers, and MTH 324 Algebra for Elementary Teachers) combine to provide candidates with approximately 16-20 hours of structured field experiences involving mathematics with middle grade students and classrooms. These field experiences are part of the required graded work in MTH 322, 323, and 324 and are arranged for and supervised by course instructors who are faculty members in the Department of Mathematics. Additionally, the field experiences for MTH 322 and 324 are part of two key assessments in this report. The MTH 322, 323, and 324 field experiences occur prior to candidates’ admission to the COE.

The purposes of the MTH 322, 323, 324 field experiences are to develop candidates’ ability to plan and teach mathematics lessons focused on developing students’ conceptual understanding, to effectively communicate mathematics to students, to assess students’ mathematical knowledge, to provide evidence illustrating growth in students’ mathematical knowledge, and to further develop candidate’s mathematical content knowledge needed to teach Grade K-8. Candidates engage in the following activities:

• Observe at least two middle grade mathematics lessons taught by an experienced and highly qualified classroom teacher. Plan and teach a minimum of one mathematics lesson to a subgroup of the students in this class and reflect on the experience (completed in MTH 322)

• Tutor individual or small groups of middle school students in mathematics for a minimum of eight hours. The goals are to develop students’ conceptual understanding in a mathematical area where they have experienced difficulty and for the candidate to develop the ability to provide evidence illustrating growth in students’ mathematical knowledge (completed in MTH 323)

• Write rich, mathematically meaningful problems for middle school students, pilot test the problems with middle school students, and revise the problems based on students’ solutions and feedback (completed in MTH 324)

After admission to the COE, candidates engage in a two-semester sequence of courses involving approximately 900 hours of supervised field and clinical experiences in Grade K-5 classrooms. The Student Information and Services Center in the COE makes all arrangements for these placements. The field and clinical placements are required to occur at different grade ranges (K-2 or 3-5), and at least one placement must have the candidate work with diverse student populations.

The first COE course (field experience), ED 330 Methods and Strategies of Elementary Teaching, has the candidate serve as a teacher assistant in an assigned classroom half a day five days a week for 15 weeks (approximately 300 hours). The candidate’s responsibilities gradually increase from assisting individual students and small groups of students to teaching whole group lessons. The candidate is also guided in lesson planning, classroom management, and evaluation procedures. The candidate is expected to work collaboratively with the cooperating classroom teacher during this experience. The candidate must satisfactorily complete ED 330 before enrolling in the second course.

The second COE course (clinical experience), ED 430 Student Teaching, involves the candidate in full day student teaching five days a week for 15 weeks (approximately 600 hours). The candidate is required to teach mathematics, as well as all other subjects normally taught by her/his cooperating classroom teacher. The candidate gradually assumes responsibility and works up to solo teaching and assuming total responsibility for the classroom for at least three weeks (approximately 120 hours).

In both ED 330 and 430, the candidate is jointly supervised by the cooperating classroom teacher and the university supervisor. University supervisors are faculty in the COE who have several years of

elementary school teaching experience. The COE provides training for all university supervisors and cooperating teachers through a variety of activities.

3. Please attach files to describe a program of study that outlines the courses and experiences required for candidates to complete the program. The program of study must include course titles. (This information may be provided as an attachment from the college catalog or as a student advisement sheet.)

Program of Study (GVSU Elementary Mathematics)

See Attachments panel below.

4. This system will not permit you to include tables or graphics in text fields. Therefore any tables or charts must be attached as files here. The title of the file should clearly indicate the content of the file. Word documents, pdf files, and other commonly used file formats are acceptable.

Easy to Read Format - Section I Contextual Information Narrative


5. Candidate InformationDirections: Provide three years of data on candidates enrolled in the program and completing the program, beginning with the most recent academic year for which numbers have been tabulated. Report the data separately for the levels/tracks (e.g., baccalaureate, post-baccalaureate, alternate routes, master's, doctorate) being addressed in this report. Data must also be reported separately for programs offered at multiple sites. Update academic years (column 1) as appropriate for your data span. Create additional tables as necessary.

(2) NCATE uses the Title II definition for program completers. Program completers are persons who have met all the requirements of a state-approved teacher preparation program. Program completers include all those who are documented as having met such requirements. Documentation may take the form of a degree, institutional certificate, program credential, transcript, or other written proof of having met the program's requirements.

Program:Mathematics Major (Elementary Teaching Emphasis)

Academic Year# of CandidatesEnrolled in the

Program

# of ProgramCompleters(2)

2009-2010 118 20

2008-2009 118 23

2007-2008 135 20

6. Faculty InformationDirections: Complete the following information for each faculty member responsible for professional coursework, clinical supervision, or administration in this program.

Faculty Member Name

Highest Degree, Field, & University(3)

Assignment: Indicate the rolof the faculty member(4)

Faculty Rank(5)

Tenure Track YESgfedc

Scholarship(6), Leadership in Professional Associations, andService(7):List up to 3 major contributions in the past 3 years(8)

Grand Rapids, MI

Teaching or other professional experience in P-12 schools(9)

31 years of P-12 teaching experience

Faculty Member Name


Assignment: Indicate the role of the faculty member(4)

Faculty Rank(5) Associate Professor

Tenure Track YESgfedcb

Scholarship(6), Leadership in Professional Associations, and Service(7):List up to 3 major contributions in the past 3 years(8)

Coffey, David & Billings, E. (December 2008/January 2009). Teachers as Lifelong Learners The Role of Reading. Teaching Children Mathematics, 15(5), 267-274. Billings, E. (2008). Exploring Generalization through Growth Patterns. In C. Greenes and R. Rubenstein (eds.) NCTM 70th Yearbook (2008): Algebra and Algebraic Thinking in School Mathematics (pp. 279-294), Reston, VA: The National Council of Teachers of Mathematics. Billings, E., Tiedt, T., and Slater, L. (December 2007/January 2008). Algebraic Thinking and Pictorial Growth Patterns. Teaching Children Mathematics 14 (5) 302-308.


Provided In-service professional development training for Grade K-8 teachers for 5 years

Faculty Member Name Bomgaars, Wayne

Highest Degree, Field, & University(3) MA in Elementary Administration, Michigan State University

Assignment: Indicate the role of the faculty member(4) Field Coordinator

Faculty Rank(5) Adjunct Professor

Tenure Track YESgfedc



(3) e.g., PhD in Curriculum & Instruction, University of Nebraska. (4) e.g., faculty, clinical supervisor, department chair, administrator (5) e.g., professor, associate professor, assistant professor, adjunct professor, instructor (6) Scholarship is defined by NCATE as systematic inquiry into the areas related to teaching, learning, and the education of teachers and other school personnel. Scholarship includes traditional research and publication as well as the rigorous and systematic study of pedagogy, and the application of current research findings in new settings. Scholarship further presupposes submission of one's work for professional review and evaluation. (7) Service includes faculty contributions to college or university activities, schools, communities, and professional associations in ways that are consistent with the institution and unit's mission. (8) e.g., officer of a state or national association, article published in a specific journal, and an evaluation of a local school program. (9) Briefly describe the nature of recent experience in P-12 schools (e.g. clinical supervision, inservice training, teaching in a PDS) indicating the discipline and grade level of the assignment(s). List current P-12 licensure or certification(s) held, if any.


Faculty Member Name


Assignment: Indicate the role of the faculty member(4)

Faculty Rank(5)

Tenure Track



SECTION II - LIST OF ASSESSMENTS

In this section, list the 6-8 assessments that are being submitted as evidence for meeting the NCTM standards. All programs must provide a minimum of six assessments. If your state does not require a state licensure test in the content area, you must substitute an assessment that documents candidate attainment of content knowledge in #1 below. For each assessment, indicate the type or form of the assessment and when it is administered in the program.

1. In this section, list the 6-8 assessments that are being submitted as evidence for meeting the NCTM standards. All programs must provide a minimum of six assessments. If your state does not

require a state licensure test in the content area, you must substitute an assessment that documents candidate attainment of content knowledge in #1 below. For each assessment, indicate the type or form of the assessment and when it is administered in the program.(Response limited to 250 characters each field).

Type and Number of Assessment

Name of Assessment

(10)

Type or Form of Assessment

(11)

When the Assessment Is

Administered (12)

Since the previous

submission is this assessment

New

Since the previous submission is this

assessment Substantially

changed

Since the previous submission is this assessment Not

Substantially changed

Assessment #1: Licensure assessment, or other content-based assessment (required)

Michigan Test for Teacher

Certification Mathematics (Elementary)

(Test 89)

State licensure

mathematical content exam for Grade 6-8

mathematics teaching

endorsement that is part

of initial elementary teaching

certification

Candidates are

encouraged to take the

exam during their final year of the elementary

mathematics program, but

they may take it at any time during

the program.

No. No.Yes, it has

not changed.

Assessment #2: Assessment of content (required)

Course Grades from

Selected Core

Mathematics Courses

Course Grades

Combination of course

grades from the following core courses – MTH 210

Communicating in

Mathematics, MTH 227

Linear Algebra, MTH 310 Modern Algebra, MTH 341 Euclidean Geometry, MTH 345 Discrete

Mathematics, & MTH 495 The Nature of Modern

Mathematics

Yes. No No.

Assessment #3: Assessment of candidate ability to plan(required)

Unit Planning and

Assessment Project

Course Project

No. No. Yes, it has not changed.

Required assignment

for the course

MTH 323 Probability

and Statistics for

Elementary Teachers

Assessment #4: Assessment of clinical practice (required)

General Student Teaching Evaluation

Form & Student Teaching Evaluation Form Major

Program Addendum: Mathematics

Performance Rating

Used in ED 430 for the mid-term and final student teaching

evaluations by University Supervisor

and Cooperating Teacher to evaluate

candidate’s student

teaching, in general, and

in mathematics, in particular

No. No.Yes, it has

not changed.

Assessment #5: Assessment of candidate effect on student learning (required)

Lesson Observation,

Planning, and

Reflecting Project

Course Project

Required assignment

for the course

MTH 322 Geometry for Elementary Teachers

No. No.Yes, it has

not changed.

Assessment #6: Additional assessment (required)

Menu of Problems Project

Course Project

Required assignment

for the course

MTH 324 Algebra for Elementary Teachers

No. No.Yes, it has

not changed.

Historical & Cultural

Perspectives in

Mathematics Assignment

Course Assignments Yes, in part. Yes, in part. No.

(10) Identify assessment by title used in the program; refer to Section IV for further information on appropriate assessment to include. (11) Identify the type of assessment (e.g., essay, case study, project, comprehensive exam, reflection, state licensure test, portfolio). (12) Indicate the point in the program when the assessment is administered (e.g., admission to the program, admission to student teaching/internship, required courses [specify course title and numbers], or completion of the program).

Assessment #7: Additional assessment that addresses NCTM standards (optional)

Combination of required assignments

for three courses –MTH 341 Euclidean Geometry, MTH 345 Discrete

Mathematics, and MTH 495 The Nature of Modern

MathematicsAssessment #8: Additional assessment that addresses NCTM standards (optional)

None.

SECTION III - RELATIONSHIP OF ASSESSMENT TO STANDARDS

1. For each NCTM standard on the chart below, identify the assessment(s) in Section II that address the standard. One assessment may apply to multiple NCTM standards.

#1 #2 #3 #4 #5 #6 #7 #8

Mathematics Preparation for All Mathematics Teacher Candidates. gfedc gfedc gfedc gfedc gfedc gfedc gfedc gfedc

1. Knowledge of Problem Solving. Candidates know, understand and apply the process of mathematical problem solving. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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2. Knowledge of Reasoning and Proof Candidates reason, construct, and evaluate mathematical arguments and develop appreciation for mathematical rigor and inquiry. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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3. Knowledge of Mathematical Communication. Candidates communicate their mathematical thinking orally and in writing to peers,faculty and others. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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4. Knowledge of Mathematical Connections. Candidates recognize, use, and make connections between and among mathematical ideas and in contexts outside mathematics to build mathematical understanding.

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[Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]5. Knowledge of Mathematical Representaion. Candidates use varied representations of mathematical ideas to support and deepen students' mathematical understanding. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]


6.Knowledge of Technology. Candidates embrace technolgy as an essential tool for teaching and learning mathematics. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]


7. Dispositions. Candidates support a postive disposition toward mathematical processes and mathematical learning. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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8. Knowledge of Mathematics Pedagogy. Candidates possess a deep understanding of how students learn mathematics and of the pedagogical knowledge specific to mathematics teaching and learning. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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9. Knowledge of Number and Operations. Candidates demonstrate computational proficiency, including a conceptual understanding of numbers, ways of representing number, relationships among number and number systems, and the meaning of operations.[Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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10. Knowledge of Different Perspectives on Algebra.Candidates emphasize relationships among quantities including functions, ways of representing mathematical relationships, and the analysis of change. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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11. Knowledge of Geometries. Candidates use spatial visualization and geometric modeling to explore and analyze geometric shapes, structures, and their properties. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]


12. Knowledge of Calculus Candidates demonstrate a conceptual understanding of limit, continuity, differentiation, and integration and a thorough background in techniques and application of calculus. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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13. Knowledge of Discrete Mathematics. Candidates apply the fundamental ideas of discrete mathematics in the formulation and solution of problems. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]


14. Knowledge of Data Analysis, Statistics and Probability. Candidates demonstrate an understanding of concepts and practices related to data analysis, statistics, and probability. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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15. Knowledge of Measurement. Candidates apply and use measurement concepts and tools. [Indicators are listed at http://www.nctm.org/about/ncate/secondary_indic.htm]

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2. 16. 1 Field-based Experiences. Engage in a sequence of planned opportunities prior to student teaching that include observing and participating mathematics classrooms under the supervision of

experienced and highly qualified teachers.

Information should be provided in Section I (Context) to address this standard.

3. 16.2 Field-based Experiences. Experience full-time student teaching middle level mathematics that is supervised by experienced and highly qualified teacher and a university or college supervisor with mathematics teaching experience.

Information should be provided in Section I (Context) to address this standard.

4. For the NCTM standard on the chart below, identify the assessment(s) in Section II that address the standard. One assessment may apply to multiple NCTM standards.

#1 #2 #3 #4 #5 #6 #7 #816.3 Field-Based Experiences. Demonstrate the ability to increase students' knowledge of mathematics. gfedc gfedc gfedc gfedc gfedcb gfedc gfedc gfedc

SECTION IV - EVIDENCE FOR MEETING STANDARDS

DIRECTIONS: Submit the following documentation for any assessments listed in Section II as new or substantially changed since previous submission. Submit data on all assessments.

The 6-8 key assessments listed in Section II must be documented and discussed in Section IV. Taken as a whole, the assessments must demonstrate candidate mastery of the SPA standards. The key assessments should be required of all candidates. Assessments and scoring guides and data charts should be aligned with the SPA standards. This means that the concepts in the SPA standards should be apparent in the assessments and in the scoring guides to the same depth, breadth, and specificity as in the SPA standards. Data tables should also be aligned with the SPA standards. The data should be presented, in general, at the same level it is collected. For example, if a rubric collects data on 10 elements [each relating to specific SPA standard(s)], then the data chart should report the data on each of the elements rather that reporting a cumulative score..

In the description of each assessment below, the SPA has identified potential assessments that would be appropriate. Assessments have been organized into the following three areas to be aligned with the elements in NCATE’s unit standard 1:• Content knowledge (Assessments 1 and 2)• Pedagogical and professional knowledge, skills and dispositions (Assessments 3 and 4)• Focus on student learning (Assessment 5)

Note that in some disciplines, content knowledge may include or be inextricable from professional knowledge. If this is the case, assessments that combine content and professional knowledge may be considered "content knowledge" assessments for the purpose of this report.

For each assessment, the compiler should prepare one document that includes the following items:

(1) A two-page narrative that includes the following:a. A brief description of the assessment and its use in the program (one sentence may be sufficient);b. A description of how this assessment specifically aligns with the standards it is cited for in Section III. Cite SPA standards by number, title, and/or standard wording.c. A brief analysis of the data findings;

d. An interpretation of how that data provides evidence for meeting standards, indicating the specific SPA standards by number, title, and/or standard wording; and

(2) Assessment Documentatione. The assessment tool itself or a rich description of the assessment (often the directions given to candidates);f. The scoring guide for the assessment; andg. Charts that provide candidate data derived from the assessment.

The responses for e, f, and g (above) should be limited to the equivalent of five text pages each , however in some cases assessment instruments or scoring guides may go beyond five pages.

Note: As much as possible, combine all of the files for one assessment into a single file. That is, create one file for Assessment #4 that includes the two-page narrative (items a – d above), the assessment itself (item e above), the scoring guide (item f above, and the data chart (item g above). Each attachment should be no larger than 2 mb. Do not include candidate work or syllabi. There is a limit of 20 attachments for the entire report so it is crucial that you combine files as much as possible.

1. State licensure tests or professional examinations of content knowledge. NCTM standards addressed in this entry could include all of the standards 1-7 and 9-15. If your state does not require licensure tests or professional examinations in the content area, data from another assessment must be presented to document candidate attainment of content knowledge. (Assessment Required)

Provide assessment information (items 1-5) as outlined in the directions for Section IV

Assessment 1 - State Licensure Exam


2. Assessment of content knowledge in mathematics. NCTM standards addressed in this entry could include but are not limited to Standards 1-7 and 9-15. Examples of assessments include comprehensive examinations, GPAs or grades, and portfolio tasks(13). For post-baccalaureate teacher preparation, include an assessment used to determine that candidates have adequate content backgroud in the subject to be taught.(Assessment Required)


(13) For program review purposes, there are two ways to list a portfolio as an assessment. In some programs a portfolio is considered a single assessment and scoring criteria (usually rubrics) have been developed for the contents of the portfolio as a whole. In this instance, the portfolio would be considered a single assessment. However, in many programs a portfolio is a collection of candidate work—and the artifacts included

Assessment 2 - Course Grades


3. Assessment that demonstrates candidates can effectively plan classroom-based instruction. NCTM standards that could be addressed in this assessment include but are not limited to Standard 8. Examples of assessments inculde the evaluation of candidates' abilities to develop leasson or unit plans, individualized educational plans, needs assessments, or intervention plans. (Assessment Required)


Assessment 3 - Planning


4. Assessment that demonstrates candidates' knowledge, skills, and dispositions are applied effectively in practice. NCTM standards that could be addressed in this assessment include but are not limited to standard 8. An assessment instrument used in student teaching or an internship should be submitted. (Assessment Required)


Assessment 4 - Student Teaching


5. Assessment that demonstrates candidate effects on student learning. NCTM standards that could be addressed in this assessment include but are not limited to Standard 8. Examples of assessments include those based on student work samples, portfolio tasks, case studies, follow-up studies, and employer surveys. (Assessment Required)


Assessment 5 - Effect on Student Learning


6. Additional assessment that addresses NCTM standards. Examples of assessments include evaluations of field experiences, case studies, portfolio tasks,licensure tests not reported in #1, and follow-up studies. (Assessment Required)


Assessment 6 - NCTM Standards


7. Additional assessment that addresses NCTM standards. Examples of assessments include evaluations of field experiences, case studies, portfolio tasks,licensure tests not reported in #1, and follow-up studies. (Optional)


Assessment 7 - Historical and Cultural Perspectives (NCTM Standards)


8. Additional assessment that addresses NCTM standards. Examples of assessments include evaluations of field experiences, case studies, portfolio tasks,licensure tests not reported in #1, and follow-up studies. (Optional)


SECTION V - USE OF ASSESSMENT RESULTS TO IMPROVE PROGRAM

1. Evidence must be presented in this section that assessment results have been analyzed and have been or will be used to improve candidate performance and strengthen the program. This description should not link improvements to individual assessments but, rather, it should summarize principal findings from the evidence, the faculty's interpretation of those findings, and changes made in (or planned for) the program as a result. Describe the steps program faculty has taken to use information from assessments for improvement of both candidate performance and the program. This information should be organized around (1) content knowledge, (2) professional and pedagogical knowledge, skill, and dispositions, and (3) student learning. In addition, for each assessment listed in Section II, describe why or why not the assessment has been changed since the program was submitted previously.

(Response limited to 24,000 characters INCLUDING SPACES)

The systematic program assessment process of designing and implementing key assessments, collecting data from the assessments, and using results from the data to strengthen and improve the program has been an enriching experience for the Elementary Mathematics Program at Grand Valley State University. In particular, the experience engaged program faculty in deep discussions about what we want candidates to learn from specific courses and activities and what it means for candidates to meet our expectations in regards to the different indicators that comprise the Standards for Middle Level Mathematics Teachers. The experience also challenged faculty to use data from our previous report to design efforts focused on ways to better prepare candidates with the knowledge, skills, and dispositions they need to be effective mathematics teachers in Grades K-8.

Changes to Assessments Since Our Prior Report

Changes that were made to key assessments since submitting our prior report in September 2007 were discussed in the narrative for each assessment presented in Section IV. It is important to note that Assessment 2 and 7 changed significantly in response to data from our 2007 report.

Mathematical Content Knowledge

Data from the key assessments suggested one strength of the Elementary Mathematics Program is that it consistently prepares candidates with strong knowledge of a variety of mathematical content domains and mathematical processes. Results from this report and our 2007 report generally showed candidates’work on key assessments demonstrated the expected level of competency in the mathematical content domains number concepts and operations, algebra, geometry, measurement, calculus, discrete mathematics, and data analysis, statistics, and probability. Both reports also generally showed candidates effectively utilized the mathematical processes Problem Solving, Representation, Reasoning and Proof, as well as aspects related to Communication and Connections in their work on key assessments. Candidates’ strong knowledge of mathematical content and processes was likely influenced by the broad range of required courses in the program, which focus on all mathematical content areas addressed in NCTM’s Middle Level Standards. It was also likely influenced by instructional methods widely used by faculty in the department, which focus on developing candidates’ conceptual understanding of mathematical ideas and guiding candidates in constructing mathematical knowledge for themselves. Given the consistent results related to candidates’ knowledge of mathematical content and processes, program faculty plan to continue their efforts to help candidates learn mathematics with understanding in the broad range of courses required for the program.

Candidates’ work on several key assessments in our 2007 report consistently demonstrated a lower level of competence in two primary areas, Communication and Connections. In particular, their work showed weaknesses in the precise use of mathematical notation and terminology to communicate ideas (Indicator 3.2) and in demonstrating how mathematical ideas interconnect and build on one another to produce a coherent whole (Indicator 4.3). These weaknesses appeared not only in Elementary Mathematics Program candidates’ work, but also in the work of candidates in the Secondary Mathematics Program and those candidates in the non-teaching program. In response to these results, faculty concentrated on strengthening all departmental programs, which included the Elementary Mathematics Program, in the areas Communication and Connections.

In order to design and make informed decisions regarding initiatives that would support candidates’work in the areas Communication and Connections, faculty wanted more information about candidates’ability to use the language of mathematics to express ideas precisely and their ability to interconnect ideas and see mathematics as a coherent whole. Faculty also wanted to be able to track candidates’ work in these two areas over time to see changes that occurred in the overall percentage of candidates who demonstrated competency in Communication and Connections. Thus, faculty created new versions of Assessment 2 and Assessment 7 to gather the desired information.

In response to the results from our 2007 report, faculty undertook another initiative focused on improving candidates’ performance specifically in the area Communication. Faculty in the Department of Mathematics at Grand Valley State University have high expectations for candidates’ mathematical communication. Therefore, in an effort to help candidates in the Elementary Mathematics Program, as well as those in the other two departmental programs, meet expectations for mathematical communication, faculty created a document titled Guidelines for Writing Mathematical Proofs in the Department of Mathematics at Grand Valley State University. The document provides guidelines and examples to help candidates understand what it means to express ideas precisely in the language of mathematics, for proofs to be fully developed, and what overall expectations are for mathematical writing whether the writing is for a particular course or for the profession at large. The document is also intended to serve as a resource candidates can use throughout their program to remind them of the departmental expectations for mathematical communication. The writing guidelines were completed early in the Winter 2010 semester and approved by departmental faculty in April 2010. Their use with candidates in all three departmental programs was encouraged in all courses beginning in May 2010. It is

currently too soon to determine what influence the writing guidelines may have on candidates’demonstrated competency in the area Communication. However, faculty plan to examine data from the key assessments at regular intervals in coming semesters to see if after becoming familiar with the writing guidelines more candidates in each departmental program demonstrate competency in using the language of mathematics to express ideas precisely.

Results in this report related to Communication were similar to those in our 2007 report and showed several candidates’ work consistently demonstrated weaknesses in using the language of mathematics to express ideas precisely. Therefore, faculty plan to continue working on efforts to improve the performance of Elementary Mathematics Program candidates, as well as that of candidates in the other two departmental programs, in the area Communication. As a next step, faculty intend to take a close look at important factors involved in candidates’ first in-depth programmatic experience with mathematical communication. To do so, faculty will examine various aspects of the course MTH 210 Communicating in Mathematics, which is a rigorous introductory proof course that is prerequisite to most of the mathematics courses candidates complete in the program. Course grade data from Assessment 2 showed several candidates in all departmental programs struggled with MTH 210. Additionally, many candidates’ work on key assessments illustrated they often did not apply ideas from MTH 210 to their work on these assessments. Therefore, faculty plan to examine factors such as the course goals and objectives for MTH 210, the pace of topics, the amount of work required of candidates, the number of credit hours for the course, and the semesters that the course is offered to gain insights into how these factors might contribute to candidates’ learning, retention of material, and struggles with MTH 210 and with courses they take later in the program. Following this examination, faculty plan to recommend ways MTH 210 could be strengthened to better prepare all candidates to develop strong knowledge and abilities in mathematical communication that they are able to use throughout the program.

Faculty also plan to continue working on improving the performance of Elementary Mathematics Program candidates, as well as the performance of candidates in the other two departmental programs, in the area Connections. Data from key assessments showed a higher overall percentage of candidates demonstrated competence in illustrating how mathematical ideas interconnect and build on one another to produce a coherent whole in this report than in our 2007 report. However, these recent results also suggested more needs to be done to help all candidates demonstrate competency in this area. The next step faculty plan to take involves carefully examining the course MTH 227 Linear Algebra and how it prepares candidates for courses that come later in their program, particularly with respect to how it prepares candidates to make mathematical connections and view mathematics as a coherent whole. MTH 227 is candidates’ first structured, in-depth experience exploring mathematical ideas from multiple perspectives (i.e., algebraic, geometric, and function theoretic). Course grade data from Assessment 2 suggested several candidates in the Elementary Mathematics Program, as well as those in other departmental programs, struggled with ideas in MTH 227. It is not currently clear if and how these struggles influenced candidates’ work on key assessments or how these struggles may have influenced candidates’ overall work in courses they took later in the program. Therefore, faculty intend to take a critical look at MTH 227 Linear Algebra and how it currently prepares candidates for upper level mathematics courses. Faculty also plan to explore what could be done in MTH 227 to help more candidates in all departmental programs perceive connections between mathematical ideas and view mathematics as a coherent whole.

Professional and Pedagogical Knowledge, Skills, and Dispositions

The data from key assessments suggested the Elementary Mathematics Program prepares candidates

with sound knowledge and abilities in the area Professional and Pedagogical Knowledge, Skills, and Dispositions. Results from this report and from our 2007 report generally showed candidates met program expectations on key assessments by demonstrating positive dispositions towards a variety of pedagogical issues, such as all students learning mathematics with understanding and using concrete materials, technology, and a variety of resources to promote conceptual learning. Results from both reports also generally showed candidates demonstrated competency in the ability to use assessment and a variety of materials and resources to guide students in developing a conceptual understanding of mathematics. Thus, the Elementary Mathematics Program has been consistent in preparing candidates with the productive dispositions and professional and pedagogical knowledge and skills they need to effectively teach mathematics to Grades K-8 students.

Data from key assessments in our 2007 report showed several candidates’ initial work related to planning lessons and units did not quite meet expectations. However, candidates appeared to grow in their ability to plan effective lessons and units as they progressed through the program. Consequently, candidates’ strongest area of demonstrated competence during their student teaching experience was their ability to plan lessons and units. In light of this data, faculty decided to focus on improving candidates’ performance in the area Planning, particularly with respect to their planning work that occurred early in the program. Faculty concentrated on providing more structure for and feedback on mathematics lesson plans (e.g., goals, activities, assessments, etc.) that candidates developed for Grades 1-3 students as part of the required coursework in MTH 222 Mathematics for Elementary Teachers II and MTH 223 Mathematics for Elementary Teachers III. MTH 222 and MTH 223 are part of the Elementary Distributed Minor Program rather than part of the Elementary Mathematics Program. However, these courses are a prerequisite for MTH 322 Geometry for Elementary Teachers, MTH 323 Probability and Statistics for Elementary Teachers, and MTH 324 Algebra for Elementary Teachers, which are all part of the Elementary Mathematics Program. Furthermore, MTH 222 and MTH 223 are taught by faculty in the Elementary Mathematics Program. Additionally, faculty intentionally provided candidates with more planning experiences and more feedback on their planning of mathematics lessons and units for middle grades students that they developed as part of the required coursework in MTH 322 and MTH 323. The data related to planning on key assessments suggested these efforts had a positive influence on candidates’ planning work. More specifically, the data showed a higher overall percentage of candidates demonstrated competence in planning mathematics lessons and units in this report than in our 2007 report. Additionally, results of this report, like those of our 2007 report, showed Planning to be candidates’ strongest area of demonstrated competency during their student teaching experience. In fact, it was the only area where 100% of the candidates completed work that met program expectations. Faculty intend to continue their efforts focused on candidates’ early work with Planning to further strengthen candidates’ ability to plan appropriate and effective mathematics lessons and units for Grades K-8 students.

Effect on Student Learning

Data from key assessments suggested the Elementary Mathematics Program does a good job of preparing candidates to demonstrate the ability to increase students’ knowledge of mathematics. Results from both this report and from our 2007 report showed candidates demonstrated this ability most strongly during their student teaching experience as they led classes in problem solving and developing conceptual understanding. Thus, the Elementary Mathematics Program has been consistent in preparing candidates who effectively help students grow in their knowledge of mathematics.

Shortly before our 2007 report was submitted, program faculty decided it would be beneficial for candidates to engage in more instructional experiences with Grades 6-8 students in mathematics and for

them to have more opportunities to provide evidence that illustrated growth in Grades 6-8 students’mathematical knowledge. Faculty provided these experiences by redesigning the field experience component of MTH 323 Probability and Statistics for Elementary Teachers. This field experience currently involves candidates in tutoring middle grades students in mathematics for eight hours over the course of the semester. Candidates also regularly provide evidence that shows how the middle grade students’ knowledge grew during their work together and receive feedback on the evidence presented. Since submitting our 2007 report, faculty worked on strengthening the MTH 323 field experience to make it even more beneficial for candidates. This involved communicating with middle grade classroom teachers to find mutually beneficial ways to structure the field experience, providing candidates with more guidance and feedback on lessons they develop for their work with middle grades students, and providing more guidance and feedback on evidence candidates presented showing growth in student knowledge. Program faculty plan to continue efforts to strengthen the MTH 323 field experience to help all candidates continue to grow in their pedagogical content knowledge and their ability to effect student learning of mathematics.

Summary

The data from key assessments illustrated that the Elementary Mathematics Program at Grand Valley State University consistently and effectively prepares candidates to teach mathematics in Grades K-8. Since submitting our 2007 report, faculty concentrated on strengthening the program primarily in the areas Communication, Connections, Planning, and Effect on Student Learning. Data from key assessments showed candidates’ work in each of these areas was typically stronger in this report than in our 2007 report. However, the data also suggested more could be done to improve candidates’performance in each area. As future initiatives are developed and implemented, faculty will carefully examine the data from key assessments to determine whether more program candidates demonstrate competence in the areas Communication, Connections, Planning, and Effect of Student Learning. Our goal is for all Elementary Mathematics Program candidates to demonstrate they possess the knowledge, skills, and dispositions expected of effective mathematics teachers in Grades K-8.

SECTION VI - FOR REVISED REPORTS OR RESPONSE TO CONDITIONS REPORTS ONLY

1. For Revised Reports: Describe what changes or additions have been made to address the standards that were not met in the original submission. Provide new responses to questions and/or new documents to verify the changes described in this section. Specific instructions for preparing a Revised Report are available on the NCATE web site at http://www.ncate.org/institutions/resourcesNewPgm.asp?ch=90

For Response to Conditions Reports: Describe what changes or additions have been made to address the conditions cited in the original recognition report. Provide new responses to questions and/or new documents to verify the changes described in this section. Specific instructions for preparing a Response to Conditions Report are available on the NCATE web site at http://www.ncate.org/institutions/resourcesNewPgm.asp?ch=90

(Response limited to 24,000 characters, including spaces)

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NCA TE Report (GVSU Elementary Mathematics Program - 2010) 1

Program of Study

Grand Valley State University Mathematics Department

Mathematics Major with Elementary Teaching Certification Emphasis Checklist

Name: ________________________________ GVSU ID#: ______________________ Phone: ________________________________ Email: __________________________ General Education Program Requirements:

Foundations:

Natural Sciences (required: 2 courses, 1 from each category, 1 course must contain a lab) Category Course # & Title Lab

Course Sem Taken

Sem Hours

Grade

Physical Sciences

Life Sciences

Arts (required: 1 course)

Course # & Title Sem Taken

Sem Hours

Grade

Mathematical Sciences (required: 1 course)


Sem Hours

Grade

Philosophy and Literature (required: 1 course)


Sem Hours

Grade

Historical Perspectives (required: 1 course)


Sem Hours

Grade

Social Sciences (required: 2 courses from 2 disciplines) Course # & Title Sem

Taken Sem Hours

Grade


Cultures: World Perspectives (required: 1 course)


Sem Hours

Grade

U.S. Diversity (required: 1 course)


Sem Hours

Grade

Themes: Select 1 Theme. (required: 3 courses from 3 different disciplines with the selected Theme.)

Theme Selected


Sem Hours

Grade

All College Requirements: Basic Skills (all courses required)


Sem Hours

Grade

MTH 110 Algebra 4 WRT 150 Strategies in Writing 4 WRT 305 Writing in the Disciplines or passing Junior essay exam 3

Supplemental Writing Skills (required: 2 courses after WRT 150)


Sem Hours

Grade

MTH 210 Communicating in Mathematics 3

B.S. Degree Cognate (required: 3 courses)


Sem Hours

Grade

MTH 201 Calculus I 5 MTH 202 Calculus II 4 STA 312 Probability and Statistics 3


Mathematics Major (Elementary Teaching Certification Emphasis) Program Requirements: Core Requirements (required: all courses listed)


Sem Hours

Grade

MTH 201 Calculus I 5 MTH 202 Calculus II 4 MTH 210 Communicating in Mathematics 3 MTH 227 Linear Algebra I 3 MTH 310 Modern Algebra 3 MTH 495 The Nature of Modern Mathematics (capstone course) or MTH 496 Senior Thesis (capstone course)

3

Elementary Certification Emphasis (required: all courses listed)


Sem Hours

Grade

MTH 322 Geometry for Elementary Teachers 3 MTH 323 Probability and Statistics for Elementary Teachers 3 MTH 324 Algebra for Elementary Teachers 3 MTH 341 Euclidean Geometry 3 MTH 345 Discrete Mathematics 3

Cognate Courses (required: 2 courses, one of which must be STA 312 and the other must be chosen from selected courses in biology, chemistry, computer science, economics, engineering, geology, physics, psychology, social sciences, or statistics)


Sem Hours

Grade

STA 312 Probability and Statistics 3 3

* Elementary Distributed Teaching Minor Program Requirements: Core Requirements (required: all courses listed)


Sem Hours

Grade

ENG 308 Teaching Reading: The Necessary Skills 4 One of: MTH 221 Mathematics for Elementary Teachers I (4 credits)

and MTH 222 Mathematics for Elementary Teachers II (3 credits)

MTH 223 Mathematics for Elementary Teachers III (5 credits)

7/5

One of: ART 230 Art for the Classroom Teacher (4 credits) MUS 350 Music for Classroom Teachers (3 credits)

4/3

One of: ENG 307 Teaching Writing: Elementary ENG 309 Teaching Literature to Children WRT 219 Introduction to Creative Writing

Or a foreign language course

3/4

One of: PED 305 Movement Education CTH 366 Drama in Education

3

One of: Selected courses from economics, geography, history, political science, or sociology

3

Lab Science Course: 5


* Professional Program Requirements (required: all courses listed) Course # & Title Sem

Taken Sem Hours

Grade

ED 200 Introduction to Education 3 ED 205 Computers in Education 3 ED 225 Diversity in Education 3 PSY 301 Child Development 3 PSY 325 Educational Psychology 3 ED 310 Organizing and Managing Classroom Environments 3 ED 320 Reading: Assessment and Instruction 3 ED 330 Methods and Strategies of Elementary Teaching 5 ED 350 Current Practices in Elementary Education 1 ED 430 Student Teaching 10 ED 480 Professional Development in Teacher Education 2

*Please note: The above describes the Elementary Distributed Teaching Minor Program Core and Professional Program Requirements that have been in place until AY 2009-2010. The Core and the Professional Program requirements are currently being updated to better prepare candidates for the teaching profession. The updates to these programs are noted below along with a notation of when the change became or will become effective.

New Elementary Distributed Teaching Minor Program Requirements:

New Core Requirements (required: all courses listed): (Beginning Fall 2009 unless otherwise noted. Will be required of all candidates who

complete the program after December 2012.)


Sem Hours

Grade

ENG 308 Teaching Reading: The Necessary Skills 4 One of: MTH 221 Mathematics for Elementary Teachers I (4 credits)

and MTH 222 Mathematics for Elementary Teachers II (3 credits)

MTH 223 Mathematics for Elementary Teachers III (5 credits)

7/5

MAT 300 Music, Art, and Theater for Elementary Education 3 ENG 302 Introduction to Language Arts (beginning Fall 2010) 3 Both of: PED 265 Teaching Health in Elementary Schools (2 credits)

PED 266 Movement and Dance (2 credits) 4

Both of: SCI 225 Integrated Life Sciences for K-8 (4 credits) SCI 226 Integrated Physical Sciences for K-8 (4 credits)

7

SST 309 Social Studies for Elementary Teachers (beginning Fall 2010) 3


New Professional Program Requirements (required: all courses listed): (Being phased in Winter 2010 – Winter 2011. Will be required of all candidates who

complete the program after December 2012.)


Sem Hours

Grade

Courses to Complete Prior to Admission to the College of Education (beginning Winter 2010)

ED 337 Introduction to Learning and Assessment 3 ED 315 Diverse Perspectives in Education 3 PSY 301 Child Development 3 Courses to Complete during Teaching Assisting Semester

(beginning Fall 2010 – take tw o semesters before Student Teaching Semester)

ED 310 Organizing and Managing Classroom Environments 3 ED 320 Reading: Assessment and Instruction 3 ED 330 Methods and Strategies of Elementary Teaching (Teacher

Assisting) 5

ED 370 Technology in Education 3 ED 378 Universal Design for Learning: Elementary 3 Courses to Complete during Student Teaching Semester

(beginning Fall 2010)

ED 430 Student Teaching: Elementary 10 ED 485 The Context of Educational Issues 3

NCATE Report (GVSU Elementary Mathematics Program - 2010) 1

Grand Valley State University Elementary Mathematics Education – Baccalaureate

NCATE Report Section I – Contextual Information Narrative 1. Description of any state or institutional policies that may influence the application of

NCTM Standards. (Please note, the following information is the same as what was presented in our previous report submitted in September 2007.) The State of Michigan grants teacher certification only for teachable subject matter majors (e.g., mathematics, science, English, etc.). Furthermore, the State grants only two levels of teacher certification in each subject area, elementary and secondary, with middle level certification incorporated into both levels. Thus, candidates who receive initial elementary teacher certification in mathematics through Grand Valley State University (GVSU) are certified to teach all subjects in Grade K-5 and in Grade 6-8 self-contained classrooms and are certified through an endorsement to teach mathematics in Grade 6-8. Due to the structure of Michigan’s levels of teacher certification, the College of Education (COE) at GVSU requires all candidates seeking initial elementary teacher certification to complete one semester of field experience and one semester of student teaching in a Grade K-5 classroom. The COE and State further require candidates seeking elementary mathematics certification to demonstrate strong mathematical content knowledge for topics addressed in Grade 6-8. The Elementary Mathematics Program at GVSU is housed in the Department of Mathematics with secondary admission to the COE. As such, the program addresses a combination of indicators from NCTM’s Elementary Mathematics Specialist Report and Middle Level Mathematics Teachers Report. Per a June 1, 2006 agreement with Margaret D. Crutchfield, NCATE Associate Vice President, and Sue Wittick, Michigan Department of Education, the Elementary Mathematics Program is required to meet the following.

1. The program needs to address the mathematical process, pedagogy, and content indicators in Standards 1-15 of the Middle Level Report.

2. Candidate’s student teaching experience is allowed to occur in a Grade K-5 classroom (Standard 14.2 - Elementary Mathematics Specialist) rather than being required in a middle level mathematics classroom (Standard 16.2 - Middle Level Report).

3. During specific program courses, candidates will engage in brief structured field experiences with Grade 6-8 students and in Grade 6-8 mathematics classrooms to address Standard 16.1 - Middle Level Report.

2. Description of the field and clinical experiences required for the program, including the number of hours for early field experiences and the number of hours/weeks for student teaching or internships.

(Please note, the following information is the same as what was presented in our previous report submitted in September 2007.) Field and clinical experiences are associated with specific courses offered by the Department of Mathematics and the COE that are part of the required program of study.

Section 1 – Contextual Information Narrative


Three mathematics courses (MTH 322 Geometry for Elementary Teachers, MTH 323 Probability and Statistics for Elementary Teachers, and MTH 324 Algebra for Elementary Teachers) combine to provide candidates with approximately 16-20 hours of structured field experiences involving mathematics with middle grade students and classrooms. These field experiences are part of the required graded work in MTH 322, 323, and 324 and are arranged for and supervised by course instructors who are faculty members in the Department of Mathematics. Additionally, the field experiences for MTH 322 and 324 are part of two key assessments in this report. The MTH 322, 323, and 324 field experiences occur prior to candidates’ admission to the COE. The purposes of the MTH 322, 323, 324 field experiences are to develop candidates’ ability to plan and teach mathematics lessons focused on developing students’ conceptual understanding, to effectively communicate mathematics to students, to assess students’ mathematical knowledge, to provide evidence illustrating growth in students’ mathematical knowledge, and to further develop candidate’s mathematical content knowledge needed to teach Grade K-8. Candidates engage in the following activities:

• Observe at least two middle grade mathematics lessons taught by an experienced and highly qualified classroom teacher. Plan and teach a minimum of one mathematics lesson to a subgroup of the students in this class and reflect on the experience (completed in MTH 322)

• Tutor individual or small groups of middle school students in mathematics for a minimum of eight hours. The goals are to develop students’ conceptual understanding in a mathematical area where they have experienced difficulty and for the candidate to develop the ability to provide evidence illustrating growth in students’ mathematical knowledge (completed in MTH 323)

• Write rich, mathematically meaningful problems for middle school students, pilot test the problems with middle school students, and revise the problems based on students’ solutions and feedback (completed in MTH 324)

After admission to the COE, candidates engage in a two-semester sequence of courses involving approximately 900 hours of supervised field and clinical experiences in Grade K-5 classrooms. The Student Information and Services Center in the COE makes all arrangements for these placements. The field and clinical placements are required to occur at different grade ranges (K-2 or 3-5), and at least one placement must have the candidate work with diverse student populations. The first COE course (field experience), ED 330 Methods and Strategies of Elementary Teaching, has the candidate serve as a teacher assistant in an assigned classroom half a day five days a week for 15 weeks (approximately 300 hours). The candidate’s responsibilities gradually increase from assisting individual students and small groups of students to teaching whole group lessons. The candidate is also guided in lesson planning, classroom management, and evaluation procedures. The candidate is expected to work collaboratively with the cooperating classroom teacher during this experience. The candidate must satisfactorily complete ED 330 before enrolling in the second course. The second COE course (clinical experience), ED 430 Student Teaching, involves the candidate in full day student teaching five days a week for 15 weeks (approximately 600 hours). The candidate is required to teach mathematics, as well as all other subjects normally taught by her/his cooperating classroom teacher. The candidate gradually assumes responsibility and works up to solo teaching and assuming total responsibility for the classroom for at least three weeks (approximately 120 hours). In both ED 330 and 430, the candidate is jointly supervised by the cooperating classroom teacher and the university supervisor. University supervisors are faculty in the COE who have several years of elementary school teaching experience. The COE provides training for all university supervisors and cooperating teachers through a variety of activities.

NCA TE Report (GVSU Elementary Mathematics Program – 2010) 1

Section IV Assessment 1 – Michigan Test for Teacher Certification Mathematics (Elementary) Test 89: State Licensure Mathematical Content Exam 1. Description of the Assessment

All candidates seeking elementary teacher certification with the Grades 6-8 mathematics teaching endorsement in the State of Michigan are required to pass the Michigan Test for Teacher Certification Mathematics (Elementary) Test 89 (MTTC Elementary Mathematics test). The test is designed to assess the mathematical content knowledge candidates need to teach Grades 6-8. The test consists of multiple-choice questions focusing on four mathematical subareas. Approximately 28% of the questions address the subarea Mathematical Processes and Number Concepts, 28% of the questions address Patterns, Algebraic Relationships, and Functions, 22% of the questions address Measurement and Geometry, and 22% of the questions address the subarea Data Analysis, Statistics, Probability, and Discrete Mathematics. The test also requires use of a graphing calculator. The MTTC Elementary Mathematics test is criterion referenced and a candidate’s passing status is based on her/his total test performance. The test is scored on a scale of 100 to 300 with a score of 220 representing the minimum passing score accepted by the State of Michigan. Candidates may take the MTTC Elementary Mathematics test at any time during the Elementary Mathematics Program. However, candidates are strongly encouraged to wait until they have completed at least 90% of their required mathematics courses before taking the test since it addresses topics covered in the required program courses.

2. Changes to Assessment 1 Since Our Prior Report

No significant changes occurred in the structure and focus of the MTTC Elementary Mathematics test since submitting our prior report in September 2007. The assessment description in #1 above and the assessment/indicator alignment noted in #3 below are the same as what appeared in our original report submitted in 2007.

3. Alignment with NCTM Standards and Indicators

The MTTC Elementary Mathematics test aligns with numerous standard indicators in NCTM’s Middle Level Mathematics Teachers Report. The alignment is summarized in the table below. According to the Michigan Department of Education, the alignment has been accepted by NCATE/NCTM. Please see our 2007 report for the detailed alignment and information on how the alignment was developed.

Assessment 1 – State Licensure Exam


Alignment of NCATE/NCTM Middle Level Mathematics Teachers Standard Indicators with Michigan Test for Teacher Certification (MTTC) state licensure exam

(Elementary Mathematics covering Grade 6-8, Test 89)

MTTC Test 89 (Elementary Mathematics covering Grade 6-8) Test Topic Sub Areas

NCTM Indicators Addressed by MTTC Test 89 (Elementary Mathematics covering

Grade 6-8)

Use of Technology 6.1

Mathematical Processes and Number Concepts

1.1, 1.2, 1.4, 2.1, 2.2, 2.3, 2.4, 3.1, 3.2, 4.1, 4.2, 5.1, 5.2, 5.3, 9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 10.1, 10.2, 10.3, 10.4

Patterns, Algebraic Relationships, and Functions

1.2, 1.3, 5.1, 5.2, 5.3, 6.1, 10.1,10.2, 10.3, 10.4, 10.5, 12.1, 13.1

Measurement and Geometry 1.1, 1.2, 2.3, 3.2, 4.1, 4.3, 5.1, 5.2, 5.3, 6.1, 11.1, 11.2, 11.3, 11.4, 11.5, 11.6, 15.1, 15.2, 15.3, 15.4

Data Analysis, Statistics, Probability, and Discrete Mathematics

1.2, 2.3, 3.1, 3.2, 5.1, 5.2, 5.3, 6.1, 10.2, 10.4, 10.5, 13.1, 13.2, 14.1, 14.2, 14.3, 14.4, 14.5

4. Analysis of Data Findings

The data for the past three years show 100% (n = 63) of the Elementary Mathematics Program completers who took the MTTC Elementary Mathematics test passed the test. Data related to each of the test’s four subareas consistently show all or nearly all program completers (90% - 100%) passed the subarea. The data further show 100% of program completers passed the subareas Mathematical Processes and Number Concepts and

Assessment 1 – State Licensure Exam


Patterns, Algebraic Relationships, and Functions all three years. Additionally, there were two years where 100% of completers passed the subarea Measurement and Geometry.

5. Data Interpretation

Data related to the MTTC Elementary Mathematics test show all completers of the Elementary Mathematics Program Fall 2007 – Winter 2010 demonstrated they possess the mathematical content and process knowledge the State requires as a minimum for mathematics teachers of Grades 6-8. The data also shows these program completers have demonstrated they possess the mathematical content and process knowledge needed to meet the NCTM Middle Level standard indicators aligned with the test. Based on the data related to the MTTC Elementary Mathematics test, it is logical to conclude that the Elementary Mathematics Program is adequately preparing its’ candidates with the mathematical content and process knowledge they will need to teach Grades 6-8. Program faculty plan to continue to teach mathematics in ways that help candidates develop their ability to think critically and develop their conceptual understanding of important mathematical ideas.

Assessment 1 – Attachment A: Data Tables for State Licensure Exam

NCA TE Report (GVSU Elementary Mathematics Program – 2010) 4

Michigan Test for Teacher Certification (MTTC) state licensure exam (Elementary Mathematics covering Grade 6-8, Test #89)

Elementary Mathematics Program Completer Data

Total Test Performance

Academic Year

Number of Program Completers Who Took Test

Number (Percent) of Program Completers Who Passed Test

2007-2008

2008- 2009

2009 - 2010

Total

Subarea Performance

Academic Year

Number of Program Completers Who Took Test

Number (Percent) of Program Completers Who Passed Subarea

Mathematical Processes &

Number Concepts

Patterns, Algebraic

Relationships, & Functions

Measurement & Geometry

Data Analysis, Statistics,

Probability, & Discrete

Mathematics

2007- 2008

2008 - 2009

2009 - 2010

Total


Section IV Assessment 2 – Course Grades: Mathematical Content 1. Description of the Assessment

All candidates in the Elementary Mathematics Program complete six required pure mathematics courses after completing calculus I (MTH 201) and II (MTH 202). Assessment 2 consists of candidates’ course grades from these six courses: MTH 210 Communicating in Mathematics, MTH 227 Linear Algebra I, MTH 310 Modern Algebra, MTH 341 Euclidean Geometry, MTH 345 Discrete Mathematics, and MTH 495 The Nature of Modern Mathematics. The instructional focus for all six courses is on developing candidates’ conceptual understanding of the mathematical ideas explored. Candidates are consistently encouraged to make connections between mathematical ideas explored within the course, as well as make connections to mathematical ideas explored in other courses. Successful completion of each course requires that candidates explore mathematical ideas in different ways from various perspectives and use a variety of strategies to solve problems. Throughout each course, candidates are expected to clearly communicate their mathematical thinking verbally and in writing by using precise mathematical notation and terminology. Thus, candidates’ grades in these six courses provide a view into their learning of important mathematical content and processes.


Assessment 2 changed significantly since our 2007 report and is considered to be a new assessment for this report. In our prior report, Assessment 2 consisted of a course assignment that examined candidates’ knowledge of mathematical content and processes primarily with respect to one area, discrete mathematics. Our 2007 report contained two other assessments that also examined candidates’ knowledge of mathematical content and processes in different areas; Assessment 7 focused on Euclidean geometry and Assessment 8 focused on solutions of quadratic and cubic equations. Assessment 2 for this report replaces portions of our previous Assessment 2, 7 and 8 that focused specifically on candidates’ knowledge of mathematical content and processes in particular areas. The new Assessment 2 uses candidates’ grades from six courses as a different way to gain general information about candidates’ knowledge of mathematical content and processes with respect to a variety of areas. Assessment 2 was changed for the following reasons.

a. Use of course grades as an assessment measure was not encouraged at the time of our 2007 report. However, shortly after our 2007 report was submitted, NCATE developed guidelines that facilitated using course grades as an assessment.

b. Faculty in the Department of Mathematics wanted a broader view of candidates’ knowledge of mathematical content and processes than was provided by our previous set of content-related assessments.

c. Our original Assessment 2 was mathematically rich but required a significant amount of faculty time to evaluate candidate’s work on the assessment.

d. Our previous set of content-related assessments stayed essentially the same from semester to semester. Because of this, we were beginning to have concerns about whether the work we received from candidates was their own work and not that of candidates from previous semesters. If the work on the assessments was not the work of the current candidates, then the data from each administration of the assessments did not provide us with an accurate reflection of the program.

Assessment 2 – Course Grades


3. Alignment with NCTM Standards and Indicators Course Name and Number

Standards & Indicators Addressed

University Catalog Course Descriptions

Description of How Course Meets Cited Standards & Indicators

MTH 210 Communicating in Mathematics

1.1, 1.2, 1.3, 1.4, 2.1, 2.2, 2.3, 2.4 3.1, 3.2, 3.3, 3.4, 4.1, 9.5, 10.1

A study of proof techniques used in mathematics. Intensive practice in reading mathematics, expository writing in mathematics, and constructing and writing mathematical proofs. Mathematical content includes elementary logic, congruence arithmetic, set theory, functions, equivalence relations, and equivalence classes. Offered fall and winter semesters. Prerequisites & Notes Prerequisites: MTH 201 and fulfillment of the composition requirement. Credits: 3

This is an introductory proof course that focuses on proof as problem solving. Candidates learn to use direct proof, proof by cases, indirect proofs, proof by contradiction, proof by contrapositive, and proof by mathematical induction. Candidates develop proof portfolios that require them to choose a proof technique, construct and revise proofs using the selected technique, and reflect on the proof as problem solving process. Candidates also have extensive experience proving whether conjectures are true or false.

MTH 227 Linear Algebra I

1.1, 1.2, 1.3, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, 6.1

Vectors in two and three dimensional space, systems of linear equations, matrix algebra, determinants, vectors in n dimensional space, subspace, dependence, bases, linear transformations, eigenvectors and applications. Offered fall and winter semesters. Prerequisites & Notes Prerequisites: MTH 202. Credits: 3

This course is candidates’ first formal structured experience thinking about mathematical ideas from multiple perspectives: algebraic, geometric, and function theoretic (e.g., representing ideas as systems of equations, sets of vectors, and injectivity/surjectivity of linear transformations). Candidates apply ideas to other mathematical content areas and realistic problems (e.g., weather patterns, population trends, etc.) Candidates are consistently challenged to determine how to solve problems – by hand, with a calculator, or with a computer algebra system. The course meets in a computer lab one day a week. When in the lab, candidates use a computer algebra system as well as applets specifically designed to illuminate mathematical ideas of



the course.

MTH 310 Modern Algebra

1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 4.1, 4.3, 5.2, 9.7, 10.2

Algebraic properties of the integers and the development of the rational, real, and complex number systems as algebraic structures. Topics from modern algebra include rings, integral domains, fields, and ring isomorphisms. Further study of algebraic structures using congruence arithmetic and factorization in the ring of integers and polynomial rings. Offered fall and winter semesters. Prerequisites & Notes Prerequisites: MTH 210, and either MTH 225 or MTH 227. Credits: 3

This proof-oriented course focuses on investigating and hypothesizing characteristics of number systems through the study of rings and fields. Candidates apply ideas from linear algebra to topics explored in this course.

MTH 341 Euclidean Geometry

1.1, 1.2, 1.3, 2.1. 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4, 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, 6.1, 11.1, 11.3, 11.4,

11.5, 11.6

Critical analysis of Euclidean geometry from transformational, algebraic, and synthetic perspectives in two and three dimensions. Coordinate and vector geometry relating transformational geometry to linear algebra. Informal study of historical development of Euclidean and non-Euclidean geometries and the questions relating to the parallel postulate to develop understanding of axiomatic systems. Offered fall and winter semesters. Prerequisites & Notes Prerequisites: MTH 210 and either MTH 227 or MTH 322. Credits: 3

This proof-oriented course focuses on Euclidean, spherical, and neutral geometries. Candidates investigate and prove conjectures as well as evaluate the proofs of others. Candidates apply ideas from linear algebra to topics explored in this course. The course meets in a computer lab one day a week. Candidates use Geometers Sketchpad® and other software for geometrical investigations.

MTH 345 Discrete Mathematics

1.1, 1.2, 1.3, 2.1. 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4,

Basic and advanced counting techniques, including the Pigeonhole Principle and inclusion-exclusion; recurrence relations; partial orderings; graph theory, special paths, planarity, chromatic number, networks, trees, traversals, digraphs.

This proof-oriented course requires candidates to apply mathematical ideas from other courses, such as using matrices to represent and analyze vertex edge graphs.



4.1, 4.2, 4.3, 5.1, 6.1, 13.1, 13.2

Algorithms and proof techniques. Offered fall and winter semesters. Prerequisites & Notes Prerequisites: MTH 210. Credits: 3

The course consistently engages candidates in solving realistic problems (e.g., scheduling, tournaments, etc.) and challenges candidates to determine appropriate technology to use to solve problems.

MTH 495 The Nature of Modern Mathematics

1.1, 1.2, 1.3, 2.1. 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4, 4.1, 4.3, 5.1, 5.2, 5.3

A study of mathematics as a human intellectual endeavor impacting our culture, history, and philosophy. Includes an in-depth investigation, including analyses from the mathematical, historical, and philosophical perspectives, of several significant developments from various fields of mathematics. The specific developments considered will vary from semester to semester. Offered fall and winter semesters. Prerequisites & Notes Prerequisites: MTH 210, MTH 227, MTH 310, and at least three other 300–400 level mathematics courses. Credits: 3

This is a capstone course designed to tie together ideas from all required courses in the major. The course has a special emphasis on evaluating the thinking of other mathematicians and on representing mathematical expressions in various ways (e.g., verbal, equation, geometric).




Course grades were implemented as Assessment 2 during five semesters: Fall 2008, Winter 2009, Spring 2009, Fall 2009, and Winter 2010. The total number of Elementary Mathematics Program candidates who completed each course during the five semesters varied from 30 – 57. A grade of 1.7 (C-) represented the minimum course grade expectation for candidates. The university’s grading policy and system are described in Attachment A. The data for the five semesters combined showed 82.5% - 100% of the candidates in the Elementary Mathematics Program met the minimum course grade expectation for each of the six required mathematics courses. The data for the five semesters combined also showed the mean grade for each course was equivalent to a C+ (2.3) or higher. The data further showed candidates earned a grade of 2.0 (C) or higher in MTH 341 Euclidean Geometry each semester and that the mean grade for this course for all five semesters combined (3.13) was considerably higher than the mean grade for any of the other courses. Additionally, the data showed the mean course grades for MTH 210 and MTH 227 for the five semesters combined were lower than the mean grade for other courses.


The fact that the overall mean and median course grades are all 2.3 (C+) or above and the high percentage of candidates who met the minimum course grade expectation in each of the six courses for the five semesters combined (at least 74.3%) suggested the Elementary Mathematics Program is preparing candidates with sound knowledge of mathematical content and processes associated with the NCTM Middle Level standards for which the courses are aligned. Elementary Mathematics Program faculty are currently trying to gain deeper insights into the data related to MTH 210 and MTH 227. In particular, faculty are trying to understand candidate’s lower grades in MTH 210 and MTH 227, as well as the decrease in the number of candidates who took courses after MTH 210 (i.e., MTH 227, MTH 310, MTH 341, MTH 345, & MTH 495). MTH 210 is the first proof-based course taken by candidates in the program and as such presents them with new challenges. Faculty in the Mathematics Department were not surprised the data reflected that candidates sometimes struggled in this course. We are further investigating the data related to MTH 210 and how we might help candidates be more successful in this course by asking questions such as the following. How can we make the transition from computation-based courses to proof-based courses smoother for candidates? How did candidates’ poor grades in MTH 210 influence their decision to continue to pursue the mathematics major or to change majors? What are our overall goals for MTH 210? Are we trying to cover too much material in this course and how is what we are trying to cover aligned with the course goals? Are we asking students to do too much in terms of course assignments and projects in this course and how might the amount of required work hinder candidate’s learning in MTH 210? What effect would increasing the number of credits for MTH 210 from 3 to 4 have on candidates’ learning in this course? Candidates’ lower grades in MTH 227 were a surprise for faculty in the Mathematics Department. Faculty think candidates’ struggles in MTH 227 may be related to their difficulties grasping definitions of mathematical ideas and applying these definitions to problem situations, as well as the fact that this course is candidate’s first formal structured experience where they are challenged to think about mathematical ideas from multiple perspectives. Faculty are trying to understand the situation related to MTH 227 by investigating questions such as the following. How did candidates’ poor grades in MTH 227



influence their decision to continue to pursue the mathematics major or to change majors? What are our overall goals for MTH 227? In what ways are the types of course assignments and projects in MTH 227 contributing to or hindering candidates’ learning in this course? How might we better advise candidates about the content and requirements of MTH 227, as well as when to take MTH 227 in their program of study? Faculty in the Mathematics Department are currently developing plans of action focused on gathering data related to when and why candidates change their major from elementary mathematics to a different academic area. Faculty are also beginning to carefully examine course goals for both MTH 210 and MTH 227, as well as the amount of material covered and course assignments for MTH 210 to find ways to help all candidates be successful in their learning and meet the minimum course grade expectation.

Assessment 2 – Attachment A: Grading System


Attachment A System of Grading

Candidates’ course grades in all six courses were based on their performance on a variety of assessment measures that included proof portfolios, problem sets, in-class presentations, quizzes, exams, and other projects. According to university guidelines, a grade of D is the minimum passing grade in all courses. However, Mathematics Department faculty recommend candidates earn a grade of C- (1.7) or higher before moving on to the next required mathematics course. In reality, candidates must earn grades higher than what is recommended because an overall gpa of 2.7 (B-) or higher is required in the mathematics major (elementary emphasis) for admission to the College of Education. Thus, if a candidate earns a grade lower than a B- in a particular mathematics course, it must be balanced by a grade higher than a B- in another mathematics course.

Grand Valley State University’s grading policy is outlined and defined in the university catalogue and on the back of official transcripts as shown in the table below.

Grade Quality Points

Grade Explanation

A 4.0 Excellent A- 3.7 B+ 3.3 B 3.0 Good B- 2.7 C+ 2.3 C 2.0 Fair C- 1.7 D+ 1.3 D 1.0 Poor F 0.0 Failure

Assessment 2 – Attachment B: Data Tables for Course Grades


Candidate Data for Assessment 2: Course Grades

MATH 210 - Communicating in Mathematics

Semester Number of Candidates

Mean Course Grade

Median Course Grade

Mode of Course Grade

Range of

Course Grades

Percent (number) of Candidates

meeting minimum course grade expectation

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

OVERALL *57 2.49 2.7 2.7 0.0 - 4.0 82.5% (47)

MATH 227 - Linear Algebra I


Mean Course Grade

Median Course Grade


Range of

Course Grades



Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

OVERALL 35 2.31 2.3 2.7 &

3.3 0.0 - 4.0 74.3% (26)

MATH 310 - Modern Algebra


Mean Course Grade

Median Course Grade


Range of

Course Grades



Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

OVERALL 42 2.67 2.85 3.0 0.0 - 4.0 92.9% (39)



MATH 341 - Euclidean Geometry


Mean Course Grade

Median Course Grade


Range of

Course Grades



Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

OVERALL 38 3.13 3.3 3.0, 3.3, & 3.7 2.0 - 4.0 100% (38)

MATH 345 - Discrete Mathematics


Mean Course Grade

Median Course Grade


Range of

Course Grades



Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

OVERALL 38 2.82 3.15 3.3 0.0 - 4.0 92.1% (35)

MATH 495 - The Nature of Modern Mathematics


Mean Course Grade

Median Course Grade


Range of

Course Grades



Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

OVERALL 30 2.64 2.7 3.0 1.7 – 3.7 100% (30)



Summary of Candidates’ Grades in Required Mathematics Courses Mathematics Major (Elementary Emphasis)

Grades Fall 2008, Winter 2009, Spring 2009, Fall 2009, & Winter 2010 Combined

Frequency by Course

Letter Grade

Numerical Grade

MTH 210 Communicating in Mathematics




MTH 345 Discrete

Mathematics

MTH 495 The Nature of

Modern Mathematics

A 4.0 6 4 3 2 7 0

A- 3.7 7 0 3 9 4 2 B+ 3.3 6 5 7 9 8 2 B 3.0 3 3 8 9 5 9 B- 2.7 13 5 7 4 3 4 C+ 2.3 3 2 4 0 2 6 C 2.0 3 4 3 5 3 6 C- 1.7 6 3 4 0 3 1 D+ 1.3 1 4 0 0 0 0 D 1.0 4 3 1 0 1 0 F 0.0 5 2 2 0 2 0

Total Number of Candidates *57 35 42 38 38 30

Percent (number) of candidates meeting

minimum course grade expectation of

C- (1.7) or higher

82.5% (47) 74.3% (26) 92.9% (39) 100% (38) 92.1% (35) 100% (30)

Mean Course Grade 2.49 2.31 2.67 3.13 2.82 2.64 Median Course Grade 2.7 2.3 2.85 3.3 3.15 2.7 Modal Course Grade 2.7 2.7 & 3.3 3.0 3.0. 3.3,& 3.7 3.3 3.0



Summary of Candidates’ Grades in Required Mathematics Courses Mathematics Major (Elementary Emphasis)

Course

Fall 2008, Winter 2009, Spring 2009, Fall 2009, & Winter 2010 Combined

Number of Candidates

**Percent (number) of Candidates

meeting minimum

course grade expectation

Mean Course Grade

Median Course Grade

Mode of Course Grades

Range of

Course Grades

MTH 210 Communicating in

Mathematics *57 82.5% (47) 2.49 2.7 2.7 0.0 - 4.0


35 74.3% (26) 2.31 2.3 2.7 & 3.3 0.0 – 4.0


42 92.9% (39) 2.67 2.85 3.0 0.0 – 4.0


38 100% (38) 3.13 3.3 3.0, 3.3, & 3.7 2.0 – 4.0

MTH 345 Discrete

Mathematics 38 92.1% (35) 2.82 3.15 3.3 0.0 – 4.0

MTH 495 The Nature of

Modern Mathematics

30 100% (30) 2.64 2.7 3.0 1.7 – 3.7

**Minimum course grade expectation is a grade of C- (1.7) or higher to move on to the next required mathematics class. An overall gpa of 2.7 or higher is

required in the mathematics major (elementary emphasis) for admission to the College of Education.

Letter & Numerical Grade explanation: A = 4.0, A- = 3.7, B+ = 3.3, B = 3.0, B- = 2.7, C+ = 2.3, C = 2.0, C- = 1.7, D+ = 1.3, D = 1.0, F = 0.0



*Discussion of Number of Candidates Who Completed MTH 210 Fall 2008-Winter 2010

The number of candidates who completed MTH 210 Fall 2008 – Winter 2010 was considerably higher than the number of candidates who completed each of the other five courses during this time period. Several factors may have contributed to this difference in numbers, as discussed below.

a. MTH 210 is a prerequisite for MTH 310, MTH 341, MTH 345, and MTH 495. As such, candidates take MTH 210 as soon as possible after completing Calculus II. It is logical to conclude that a large number of candidates proceeded directly from Calculus II to MTH 210 at the same time.

b. After completing MTH 210, candidates have options related to the order in which they

complete MTH 310, MTH 341, and MTH 345. c. Prior to taking MTH 210, candidates’ experiences with mathematics are largely

computation-oriented. MTH 210 is an introductory proof course. As such, the course extends and challenges candidates’ views of mathematics. Some candidates decided to change their major after completing MTH 210.

d. Prior to Fall 2009, undergraduate candidates who wanted to teach elementary school

had a choice of nine different majors. Beginning in Fall 2009, candidates had a choice of only four academic majors (i.e., mathematics, language arts, integrated science, and social studies). Thus, it is possible that the change in the number of choices of acceptable teachable majors has led to a recent increase in the number of candidates enrolled in the Elementary Mathematics Program. These candidates would currently be at the beginning of the program and may reflect the larger enrollments in MTH 210 than in the other courses during the semesters at the end of the time period covered by this program assessment.


Section IV Assessment 3 –Unit Planning and Assessment Project: Ability to Plan Instruction 1. Description of the Assessment

All candidates in the Elementary Mathematics Program complete Unit Planning and Assessment Project as part of the required coursework in MTH 323 Probability and Statistics for Elementary Teachers. The project, which spans the entire semester, asks candidates to plan a unit that could be used in a middle grades mathematics classroom. More specifically, the project asks candidates to use stimulating curriculum materials to plan a Data Analysis unit for middle grade students.


Overall, faculty thought Assessment 3 from our 2007 report provided good and helpful information about candidates’ dispositions and their knowledge of both mathematical content and pedagogy. Faculty also thought Assessment 3 was appropriate for candidates at this point in their program. Therefore, no significant changes occurred in the structure and focus of Assessment 3 or its accompanying rubric since our 2007 report. However, a few minor revisions were made to clarify expectations and to aid in gaining deeper insights into candidates’ mathematical content knowledge. The revisions are described below.

• On the assessment, the section Selecting a Curriculum Resource was revised to clearly communicate that candidates are required to use a stimulating curriculum resource as the basis for their unit and assessment plan.

• The rubric was revised in the following ways. • The Unit Planning portion was revised to more clearly evaluate candidates’

ability to select a stimulating curriculum as the basis for the unit plan. • The section Knowledge of Data Analysis, Statistics and Probability was

revised to more explicitly focus on mathematical correctness of ideas and expected student responses to gain deeper insights into candidates’ mathematical content knowledge.

The assessment description in #1 above and the assessment/indicator alignment noted in #3 below are largely the same as what appeared in our original report submitted in 2007. Minor revisions were made to both sections to include the requirement that a stimulating curriculum resource be used as the basis for the unit and assessment plan.

3. Alignment with NCTM Standards

Unit Planning and Assessment Project is designed to primarily assess candidates’ dispositions towards various aspects of mathematics learning, their knowledge of mathematics pedagogy with respect to planning, and their mathematical content knowledge in the area of data analysis. Unit Planning and Assessment Project addresses the following indicators from NCTM’s Middle Level Mathematics Teachers Report: 7.2, 7.3, 7.4, 7.5, 8.4, and 14.1. The indicators are addressed through activities that ask candidates to use stimulating curriculum resources and state and national mathematics standards to create the unit. The indicators are also addressed through activities that ask candidates to design an assessment plan for the unit and to address how the unit reflects NCTM’s Principles for teaching and learning mathematics. A detailed alignment of project activities with NCTM indicators is noted in the scoring rubric.

Assessment 3 – Unit Planning and Assessment Project



Unit Planning and Assessment Project was implemented as a key assessment in MTH 323 during the Fall 2007, Winter 2008, Fall 2008, Spring 2009, and Winter 2010 semesters. A total of 71 candidates completed all portions of this assessment. One additional candidate, for a total of 72, completed only the Unit Planning portion of the assessment. A score of “(2) Proficient” represents the level of competence expected of candidates in the areas assessed. The Unit Planning and Assessment Project data for all five semesters combined showed approximately ¾ (71.9% - 77.5%) or 51-55 of the candidates completed work that met the criteria for Proficient or Distinguished in all four areas assessed. The same data set also showed approximately ¼ (22.5% - 28.2%) or 16-19 of the candidates’ work on the unit planning project did not quite demonstrate the level of knowledge expected, which resulted in the score Progressing. In general, candidates’ work scored as Progressing typically showed an awareness of important issues related to effective teaching, learning, and assessment, as well as sound mathematical content knowledge in the area data analysis. However, the discussions and examples related to these issues were of a lesser quality and not as fully developed as what was needed for the score Proficient. The data for all five semesters combined further showed there were two semesters (Winter 2008 and Spring 2009) where one candidate completed Unsatisfactory work in the area Knowledge of Issues Related to Teaching and Learning and one semester (Spring 2009) where one candidates’ work was Unsatisfactory in the areas Unit Planning and Assessment of Student Learning. However, the data for individual semesters showed no candidates’ work was Unsatisfactory in any of the four areas assessed the last semester of the assessment’s implementation (Winter 2010) and no candidates’ work in the area Knowledge of Data Analysis, Statistics and Probability was Unsatisfactory during any of the five semesters.


Unit Planning and Assessment Project is the first structured experience in the Elementary Mathematics Program that asks candidates to develop a unit and assessment plan for middle grade students. It is also likely to be candidates’ first experience developing such plans for any grade level. The high percentage of candidates who demonstrated competency at or above the expected levels in the areas Unit Planning and Knowledge of Data Analysis, Statistics, and Probability suggested candidates were well-prepared with respect to their knowledge of mathematics pedagogy as it relates to planning and their mathematical content knowledge in the area data analysis. Similarly, most candidates’ work in the areas Unit Planning, Knowledge of Issues Related to Teaching and Learning, and Assessment of Student Learning suggested they had positive dispositions towards effective teaching, the use of stimulating curricula, the use of various assessments, and a commitment to learning with understanding. Thus, the Elementary Mathematics Program meets appropriate NCTM Middle Level standards with respect to knowledge of mathematics pedagogy, mathematical content knowledge, and dispositions. Unit Planning and Assessment Project was implemented as Assessment 3 only one time during the data collection phase for our 2007 report. Therefore, it is difficult to compare percentages of candidates whose work met or exceeded expectations in each area assessed for this report with those in our 2007 report. However, based on the results of both reports, faculty identified Planning as an area to address to strengthen the Elementary Mathematics Program..

Assessment 3 – Unit Planning and Assessment Project


Since our 2007 report, faculty focused on providing candidates with more experiences and feedback on their planning, both in MTH 323 and in courses that are prerequisite to MTH 323. Faculty plan to continue these efforts to support candidates’ in their planning of mathematics lessons, units, and assessments in order to help all candidates demonstrate proficiency in this area.

Assessment 3 – Attachment A: Unit Planning and Assessment Project


Unit Planning and Assessment Project MTH 323 Probability & Statistics for Elementary Teachers

Project Overview

In your previous classes you have spent considerable time thinking about and planning individual lessons. When you are teaching, you must also make sure the lessons you teach are related to each other and that you can combine them to form cohesive units for teaching and learning. In this project, you will use existing curriculum materials to plan a Data Analysis unit that could be used in a middle grades mathematics classroom. Good unit planning is a process that includes research, development, implementation, and reflection. This project provides you an opportunity to practice the first two phases of unit planning as you research and develop a unit in data analysis. The project includes three primary activities: (1) selecting a curriculum resource, (2) unit plan research and development, and (3) final unit plan.

Selecting a Curriculum Resource

The unit you develop must be between three and six lessons long and be based on existing curriculum materials. Your first task will be to consider several different stimulating curriculum sources and choose a portion of one of them to serve as the foundation for your unit. Once you choose the curriculum resource on which your unit will be based, you must obtain your instructor’s approval before you begin planning your unit.

Unit Plan Research and Development

Research and development for your unit plan should be on-going and occur simultaneously. Therefore, you should expect to consult external sources throughout the development of your unit and revise the unit as you reflect on ideas presented in those sources. The research portion of your unit plan must involve the following three activities.

1. Read research related to students’ statistical thinking. You need to read several different articles with at least one article focusing on students creating and interpreting data displays.

2. Examine and reflect on NCTM’s principles of curriculum, teaching, learning, and assessment, as presented in the Principles and Standards for School Mathematics (PSSM) (pp. 14-24).

3. Examine and reflect on NCTM’s PSSM and the State of Michigan’s Grade Level Content Expectations (GLCEs) that relate to the focus of your unit.

Final Unit Plan

Your final unit plan must include the following six components related to your research and development activities and the resulting unit plan.

1. A report on research related to students’ statistical thinking; this report must refer to

at least two different articles that you researched as part of your planning. At least one article must be related to students creating and interpreting data displays. Be sure that you not only describe the findings of each article but that you also indicate how those findings influenced your plan for the unit. This should be 2-3 pages long.

Assessment 3 – Attachment A: Unit Planning and Assessment Project


2. An analysis of how the unit plan and the lessons that make up the unit reflect the

PSSM principles on curriculum, teaching, learning, and assessment. You must include specific references to portions of your unit that reflect the ideas in each of these principles. This should be 2-4 pages long.

3. A list of the overall and individual objectives that are addressed in the unit and a

correlation of these objectives to both the state GLCEs and the PSSM. For each objective you should include a sentence or two indicating how the activities in the unit address that objective.

4. A detailed description of the prerequisite knowledge that students need to have in

order to be successful in the activities of the unit. Your description should include specific skills or conceptual understanding and should indicate why you think this knowledge is necessary for students to have before they begin the unit.

5. A day-by-day outline of the lessons that comprise the unit. This outline should contain

the specific objectives for each day, a brief description of the learning activities for each day, how assessment is embedded in each lesson, and any materials beyond the textbook that are needed for each lesson. Also, attach a photocopy of relevant pages from the curriculum resource that was used as a basis for your unit.

6. An overall assessment plan for the unit. This should include how you will assess

student progress toward the daily learning objectives and how you will assess each student’s understanding at the end of the unit. The assessment plan must include several different types of formal and informal assessment.

Assessment 3 – Attachment B: Scoring Rubric for Unit Planning and Assessment Project


Unit Planning and Assessment Project MTH 323 Probability and Statistics for Elementary Teachers

Assessment Evaluation Form

Directions: Use the following criteria to evaluate candidate’s work on the Unit Planning and Assessment Project

Unit Planning

Indicator 7.2: Use of stimulating curricula

Indicator 8.4: Plans lessons, units and courses that address appropriate learning goals, includin g those that address local, state, and national mathematics standards and legislative mandates.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s ability to plan a cohesive and stimulating unit of instruction focused on predetermined learning goals. This is demonstrated through the candidate’s written work on the data analysis unit plan. In particular, the candidate selects a stimulating middle grades mathematics curriculum to serve as the basis for the unit plan and identifies appropriat e learning goals for the unit that are aligned with state curriculum documents and with PSSM for both the grade level and mathematical content addressed. Additionally, the candidate develops a unit plan that is coherent and focused on important mathematical ideas relat ed to data analysis.

Candidate does not select an appropriat e stimulating curriculum to serve as the basis for the unit. Or Candidate presents a unit plan that does not represent a cohesive whole: significant mismatches exist between the various components or it does not address appropriate mathematical learning goals.

Candidate selects an appropriat e stimulating curriculum to serve as the basis for the unit. Candidate presents a sequence of lessons that address the learning the learning goals.

Candidate selects an appropriat e stimulating curriculum to serve as the basis for the unit. Candidate presents a coherent sequence of lessons that clearly lays out an effective development of the unit’s learning goals and objectives aligned with local, state and national mathematics standards for middle school mathematics in the area of data analysis.

Candidate selects one or more appropriat e stimulating curricul a to serve as the basis for the unit. Candidate presents a coherent and interconnected sequence of lessons that clearly lays out an effective development of the unit’s learning goals and objectives aligned with local, state and national mathematics standards for middle school mathematics in the area of data analysis. Additionally, candidate enhances the unit plan by utilizing the activities from outside the curriculum or an analysis of how s/he might improve the unit for future use.



Knowledge of issues related to teaching and learning

Indicator 7.3: Effective teaching

Indicator 7.4: Commitment to learning with understanding

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s ability to explain how the lessons in their unit allow for an environment that reflects best practices for teaching and learning. This is displayed through the candidate’s written analysis of the unit plan with respect to the PSSM principles, which includes a description of the teaching and learning environment envisioned for the unit and consideration of some instructional decisions a teacher would need to make when teaching the unit.

The candidate does not recognize the complexity of teaching and describes a unit that is primarily based on students developing procedural skills as opposed to learning mathematics with understanding.

The candidate recognizes the complexity of teaching and describes a unit in which there are activities that actively engage the students and are designed to help students develop deep understanding of the mathematics addressed.

The candidate recognizes the complexity of teaching and describes a unit in which there are activities that actively engage the students and are designed to help students develop deep understanding of the mathematics addressed. The candidate also acknowledges that he/she will need to be flexible in teaching the unit.

The candidate recognizes the complexity of teaching and describes a unit in which there are activities that actively engage the students and are designed to help students develop deep understanding of the mathematics addressed. The candidate also acknowledges that he/she will need to be flexible in teaching the unit. Additionally, the candidate provides speci fic evidence from his/her unit that shows different possible strategies that can be used while teaching the unit and describes how the activities will promote a deep understanding of the mathematics.

Assessment of Student Learning

Indicator 7.5: Use of various assessments

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s understanding of the need for assessment to be an integral part of classroom instruction and of the various means available for assessing students. This is demonstrated through the candidate’s written work on the assessment plan for the unit. In particular, the candidate designs an assessment plan that includes both formative and summative assessment that can be used to monitor student progress toward the learning objectives.

Candidate did not submit an assessment plan or most of the assessment items are not related to the mathematics of the unit.

Candidate’s assessment plan includes descriptions of both formative and summative assessments items. Some of the items may not be well connect ed to the objectives of the unit or fully developed. The candidate may not indicate an awareness of needing to employ a variety of types of assessments throughout the unit.

Candidate’s assessment plan includes appropriate formative and summative assessments items that are fully developed. The assessment items are clearly connect ed to the objectives of each lesson and of the unit. At least one assessment item requires the students to explain their mathematical reasoning. The candidate indicates awareness of the need to employ a variety of types of assessments during mathematical instruction and of how the assessments can be used to monitor student progress toward the objectives.

Candidate’s assessment plan includes appropriate and creative formative and summative assessments items. The assessment items are clearly connected to the objectives of the unit. At least one assessment item requires the students to explain their mathematical reasoning. The candidate indicates a clear understanding of why it is important to employ a variety of types of assessments during mathematical instruction and of how the assessments can be used to monitor student progress toward the objectives and make instructional decisions.



Knowledge of Data Analysis, Statistics and Probability Indicator 14.1: Design investigations, collect data through random sampling or random assignment to treatments, and

use a variety of ways to display the data and interpret data representations.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate displays knowledge of data collection, data representation, and interpretation. This is demonstrated through the candidate’s written work on the unit and assessment plans focused on data analysis.

The unit does not contain data analysis activities or the unit plan contains many significant conceptual errors.

The unit contains activities that involve the students in collecting, representing, and interpreting data. Portions of the unit plan contain at least one conceptual error in the mathematics and/or many errors in the mathematics, terminology, or use of symbols.

The unit contains appropriate activities that involve the students in collecting, representing, and interpreting data. All portions of the unit plan contain no conceptual errors in the mathematics, and no more than a few minor errors in terminology or use of symbols.

The unit contains creative, appropriat e, and engaging activities that involve the students in collecting, representing, and interpreting data. All portions of the unit plan contain no errors in the mathematics, terminology, or use of symbols.

Assessment 3 – Attachment C: Data Tables for Unit Planning and Assessment Project


Candidate Data for Unit Planning and Assessment Project

(7.2, 8.4) Unit Planning

Semester Taken

Number of Candidates Completing Project

Number (Percent) of Candidate Scores

(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2007

Winter 2008

Fall 2008

Spring 2009

Winter 2010

Total N = 72 1 (1.4%) 18 (25%) 41 (56.9%) 12 (16.7%) Overall percent of candidates scoring a 2 or 3 – 73.6% (7.3, 7.4) Knowledge of Issues Related to Teaching and Learning

Semester Taken


Number (Percent) of Candidates Scores

(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2007

Winter 2008

Fall 2008

Spring 2009

Winter 2010

Total N = 71 2 (2.8%) 17 (23.9%) 43 (60.6%) 9 (12.7%) Overall percent of candidates scoring a 2 or 3 – 73.3% (7.5) Assessment of Student Learning

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2007

Winter 2008

Fall 2008

Spring 2009

Winter 2010

Total N = 71 1 (1.4%) 19 (26.8%) 43 (60.6%) 8 (11.3%) Overall percent of candidates scoring a 2 or 3 – 71.9%

Assessment 3 – Attachment C: Data Tables for Unit Planning and Assessment Project


(14.1) Knowledge of Data Analysis, Statistics and Probability

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2007

Winter 2008

Fall 2008

Spring 2009

Winter 2010

Total N = 71 0 (0%) 16 (22.5%) 47 (66.2%) 8 (11.3%) Overall percent of candidates scoring a 2 or 3 – 77.5%


Section IV Assessment 4 – General Student Teaching Evaluation Form & Student Teaching Evaluation Form Major Program Addendum: Mathematics 1. Description of the Assessment

All Elementary Mathematics Program candidates’ student teaching experience is evaluated using both the (ED 430, ED 431) Common Course Assessment 2 Practicum Performance Evaluation (General Evaluation Form) and the Student Teaching Evaluation Form Major Program Addendum: Mathematics (Mathematics Addendum). The General Evaluation Form (see Attachment A1) was developed by the College of Education and is used to evaluate the student teaching experience of all GVSU candidates seeking teacher certification. For candidates seeking elementary teacher certification, the General Evaluation Form is used for the evaluation of candidates’ student teaching in all subject areas. The Mathematics Addendum (see Attachment A2) was developed by Elementary Mathematics Program faculty to specifically evaluate candidates’ student teaching in mathematics. The addendum is used only for program candidates and is completed collaboratively by the University Observer and the Cooperating Classroom Teacher for both the mid-term assessment and for the final evaluation of a candidate’s student teaching experience. Results from only the final evaluation are used for this assessment.


Overall, faculty thought the Mathematics Addendum in Assessment 4 from our 2007 report provided good and helpful information about candidates’ student teaching in mathematics. Therefore, no significant changes occurred in the structure and focus of the Mathematics Addendum in Assessment 4 since our 2007 report. The College of Education at Grand Valley State University revised its general student teaching evaluation form since our 2007 report. The revised general evaluation form, (ED 430, ED 431) Common Course Assessment 2 Practicum Performance Evaluation, is contained in Attachment A1. The assessment description in #1 above and the assessment/indicator alignment noted in #3 below are the same as what appeared in our original report submitted in 2007 except for the name of the evaluation form used by the College of Education


Mathematics Addendum is designed to primarily assess candidates’ knowledge of various aspects of mathematics pedagogy. The addendum addresses the following indicators from NCTM’s Middle Level Mathematics Teachers Report: 8.1, 8.2, 8.3, 8.4, and 8.8. The indicators are addressed through items focused on candidates’ use of resources to meet the learning needs of all students, effective use of concrete materials during mathematics instruction, use of multiple strategies to assess student knowledge of mathematics, ability to plan effective mathematics lessons, and ability to lead classes in mathematical problem solving and developing students’ conceptual understanding. A detailed alignment of mathematics student teaching activities with NCTM indicators is noted in the scoring rubric

Assessment 4 – Mathematics Student Teaching Evaluation Form



Mathematics Addendum was implemented as a key assessment during the Winter 2008, Fall 2008, Winter 2009, Fall 2009, and Winter 2010 semesters. A total of 50 candidates’ student teaching was assessed using the Mathematics Addendum during these five semesters. A score of “(2) Proficient” represents the level of competence expected of candidates in the areas assessed. The data for all five semesters combined showed 96% – 100% or 48-50 of the candidates demonstrated knowledge and abilities at or above the expected level in each of the six areas assessed and approximately ½ - 2/3 of all the candidates (or 24 – 31) completed work met the criteria for Distinguished in each area. The same data set also shows two candidates’ work in the area Assessing Student Knowledge and one candidates’ work in the areas Curricular Resources, Meeting Needs of All Students, and Use of Concrete Materials did not quite demonstrate the level of competence expected and resulted in the score Progressing. These candidates’ work met the criteria for Proficient or Distinguished in all other areas assessed. Furthermore, the data show no candidates completed Unsatisfactory work in any of the six areas assessed during the five semesters of the assessment’s implementation.


The high percentage of candidates who demonstrated knowledge and abilities at the expected level in all areas assessed for their mathematics student teaching suggested candidates were well-prepared with respect to their knowledge of mathematics pedagogy and their ability to effectively teach mathematics lessons. Thus, the Elementary Mathematics Program meets appropriate NCTM Middle Level standards with respect to knowledge of mathematics pedagogy. The data for all six areas assessed in this report was similar to the data in our 2007 report. Additionally, both reports showed 100% of the candidates completed work at or above the expected only in the area Planning, which suggests this is an area of particular strength in the Elementary Mathematics Program. Based on the strong, positive data related to candidates’ mathematics student teaching, Elementary Mathematics Program faculty plan to continue to teach mathematics in ways that model how to use concrete materials, assessments of student knowledge, and hands-on problem solving activities to develop all students’ conceptual understanding of important ideas. They also plan to continue to provide candidates with pre-student teaching field experiences that engage them in activities involve planning, delivering instruction, and gaining insights into students’ mathematical knowledge. These actions should continue to prepare candidates to demonstrate the expected level of knowledge and abilities for effective mathematics teaching during their student teaching experience.

Assessment 4 – Attachment A1: Mathematics Student Teaching Evaluation Form


(ED 430, ED 431) Common Course Assessment 2

Practicum Performance Evaluation Course Outcomes and Standards: INTASC Standards 1-10 INTASC Standard 1: Discipline Knowledge & Understanding INTASC Standard 2: Understanding Student Development INTASC Standard 3: Responsiveness During Lessons INTASC Standard 4: Instructional Variety INTASC Standard 5: Learning Environments INTASC Standard 6: Effective Communication Skills INTASC Standard 7: Instruction and Curriculum Planning INTASC Standard 8: Assessment: Learners & Strategies INTASC Standard 9: Reflective Practice INTASC Standard 10: Relationships: Colleagues and Community

Instructions to University Coordinator and Cooperating Teacher: Please evaluate this practicum student’s level of performance based on the indicators provided, both at the mid-term and final [whether the student is completing a half time or a full time practicum experience]. If a particular skill was not part of this experience, please mark NA for Not Applicable or Not Observed during this experience. (in either column). At the end of the document, a section is provided if you wish to write comments regarding the s tudent’s performance. Please be sure to share this evaluation with your s tudent and keep this form until the end of the s tudent’s experience. It should be noted that the Practicum Performance Evaluation Form is used primarily to assist the College of Education assess its program and not as a method of assigning a letter grade for the semester. The College of Education’s performance indicators are based upon INTASC Standards [Inters tate New Teacher Assessment and Support Consortium Standards. INTASC is comprised of ten s tandards with knowledge, dispositions, and performance indicators. Levels of Performance: 3 = Distinguished - The practicum students at this level have mastered the concepts. The classroom is a community of learners where s tudents are highly motivated and engaged and assume responsibility for learning. 2 = Proficient - The practicum student clearly understands the concepts underlying the components and implements it well. 1 = Progressing - The practicum student appears to understand the concepts underly ing the component and attempts to implement its elements. But implementation is sporadic, intermittent, or otherwise not entirely successful. Additional work is needed before the s tudent teacher will be proficient in this area. 0 = Unsatisfactory - The practicum student does not yet appear to understand the concepts under lying the component. NA = Not Applicable/Not Observed during this experience.



Some Guiding Principles: The “Distinguished” level is reserved for outstanding performance and therefore should not be selected commonly. A practicum student should not expect to receive the “Distinguished” rating unless his or her performance is or has been exceptional, at his/her particular level of practicum. This does not mean that you are prohibited from rating your practicum student, as you deem appropriate. Rubric for Practicum Performance Evaluation Element Distinguished - 3 Proficient - 2 Progressing – 1 Unsatisfactory – 0

INTASC: 1 Discipline Knowledge and Understanding Candidate dev elops/uses curricula that encourages students to see, question, and interpret ideas from div erse perspectiv es

Curricula developed encourage students to actively participate through discussion, questioning, and interpreting diverse perspectives.

Curricula used encourage students to participate through discussion, questioning, and sharing other perspectives.

Curricula used have students partic ipate on a moderate level through discussion.

Students discuss the curricula minimally .

INTASC: 2 Understanding Student Development Candidate assesses indiv idual and group performances to design instruction that meets learners’ current needs.

Candidate is highly effective assessing indiv idual and group performances and designing instruction that meet learners’ current needs.

Candidate is generally effective assessing indiv idual and group performances and designing instruction that meet learners’ current needs.

Candidate is moderately effective assessing group performances and designing instruction that meet the needs of the whole class.

Candidate is ineffective assessing group performances and in designing instruction.

INTASC: 3 Responsiveness During Lessons Candidate makes appropria te prov isions for indiv idual students w ho hav e particular learning differences or needs (i.e. time, circumstances for w ork, tasks assignedand communication and response modes).

Candidate consistently makes appropriate and effective prov isions for indiv idual students who have particular learning differences or needs (e.g. time, circumstances for work, tasks assigned and communication and response modes).

Candidate makes generally appropriate and effective prov isions for indiv idual students who have particular learning differences or needs (e.g. time, circumstances for work, tasks assigned and communication and response modes).

Candidate makes moderately appropriate prov isions for indiv idual students who have particular learning differences or needs sporadically (e.g. time, circumstances for work, tasks assigned, and communication and response modes).

Candidate makes minimal prov isions for indiv idual students who have particular learning differences.

INTASC: 3 Responsiveness During Lessons Candidate creates a learning community in w hich indiv idual differences are respected.

Candidate creates an effective learning community in which indiv idual differences are respected. Atmosphere is one of genuine caring and respect. Students exhibit respect for candidate and others.

Candidate creates a learning community in which indiv idual differences are respected. Atmosphere is warm, caring and respectful. Students exhibit respect for candidate.

Learning community is generally appropriate for indiv idual with differences. Students exhibit only minimal respect for candidate.

Learning community is inappropriate for indiv idual with differences. Students exhibit disrespect for candidate.



INTASC: 4 Instructional Variety Candidate uses multiple teaching/learning strategies to engage students in activ e learning opportuni ties and constantly monitors/adjusts strategies in response to learner feedback.

Candidate consistently and effectively uses multiple teaching and learning strategies to actively engage students in learning opportunities and constantly monitors/adjusts strategies in response to learner feedback. Candidate successfully makes a major adjustment to a lesson.

Candidate uses multiple teaching/learning strategies to engage students in active learning opportunities and monitors/adjusts strategies in response to learner feedback. Candidate makes a minor and smooth adjustment to a lesson.

Candidate uses different teaching/learning strategies to engage students in learning opportunities and makes some adjustments to strategies. Candidate attempts to adjust a lesson with mixed results.

Candidate uses very few teaching/learning strategies that engage students and infrequently makes any adjustments to strategies. Candidate adheres rigidly to the lesson plan, even when a change will improve a lesson.

INTASC: 5 Learning Environments Candidate organizes, allocates, and manages the resources of time, space, activ ities, and attention to prov ide activ e and equitable engagement of students in productiv e tasks.

Candidate successfully organizes, allocates, and manages the resources of time, space, activ ities, and attention to prov ide active and equitable engagement of all students in productive tasks.

Candidate is generally successful organizing, allocating, and managing the resources of time, space, activ ities, and attention to prov ide active engagement of students in productive tasks.

Candidate is minimally successful organizing, allocating, and managing the resources of time, space, activ ities, and attention. Students’ engagement in productive tasks is limited.

Candidate is not successful organizing, allocating, and managing the resources of time, space, activ ities, and attention. Students are not engaged in productive tasks.

INTASC: 6 Effective Communication Skills Candidate models effec tiv e communication strategies and know s how to ask questions that stimulate discussion.

Candidate’s spoken and written language is correct and expressive, using well-chosen vocabulary . Candidate’s questions are of high quality and stimulate class discussions by assuring all voices are heard in the classroom.

Candidate’s spoken and written language is generally clear and correct. Vocabulary is appropriate to age group. Questions are of high quality and most students are engaged in the discussion.

Candidate’s spoken language is audible and written language is legible. Both are used correctly . Vocabulary is limited or not appropriate to age level. Questions are of high and low quality . Some students are engaged.

Candidate’s spoken language is inaudible, or written language is illegible, both may contain grammar and/or syntax errors. Vocabulary is not age level appropriate. Questions are of poor quality with few students engaged.

INTASC: 7 Instruction and Curriculum Planning Candidate creates/plans learning oppor tunities, lessons and activ ities that address v arious learning sty les and meet the dev elopmental/indiv idual needs of div erse learners.

Candidate creates/plans learning opportunities, lessons and activ ities that are highly relevant to s tudents and address their various learning sty les. Instruction effectively meets the developmental and indiv idual needs of diverse learners.

Candidate creates/plans learning opportunities, lessons and activ ities that are mostly suitable to and supportive of instructional goals. Instruction meets the developmental and indiv idual needs of diverse learners.

Candidate creates/plans some learning opportunities, lessons and activ ities that are suitable and supportive of instructional goals. Instruction attempts to meet the developmental and indiv idual needs of diverse learners

Learning opportunities, lessons and activ ities that are not suitable to and supportive of instructional goals. Instruction does not address the indiv idual needs of diverse learners.



INTASC: 8 Assessment: Learners and Strategies Candidate uses formal and informal assessment techniques to enhance know ledge of learners, to ev aluate students' progress and performances, and to modify teaching and learning strategies.

Candidate displays extensive abili ty using and creating formal and informal assessment techniques to enhance knowledge of learners, to evaluate students' progress and performances, and to modify teaching and learning strategies. Candidate clearly communicates the assessment process and results to students.

Candidate displays the abil ity to nominally use formal and informal assessment techniques to enhance knowledge of learners, to evaluate students' progress and performances, and to modify teaching and learning strategies for indiv idual and small group instruction. Candidate communicates the assessment results to students.

Candidate displays the abil ity to use some formal and informal assessment technique, to evaluate students' progress and performances, and to adjust teaching for whole group instruction. Assessment results are not clear and are not clearly communicated to students.

Candidate displays lack of understanding of formal and informal assessment techniques to evaluate students' progress and performances. Assessment results are only minimally used for future planning.

INTASC: 9 Reflective Practice Candidate uses classroom observ ation, information about students and research as sources for ev aluating the outcomes of teaching and learning as a basis for ex perimenting w ith, reflecting on, and rev ising practice.

Candidate makes a thoughtful and accurate assessment of a lesson or unit’s effectiveness, and offers alternatives and suggestions for change. Candidate is highly effectively using observations and research to evaluate teaching and learning outcomes.

Candidate makes an accurate assessment of a lesson or unit’s effectiveness and makes a few specific suggestions of what may be tried the next time the lesson is taught. Candidate effectively uses observations and research to evaluate teaching and learning outcomes.

Candidate has an accurate impression of a lesson’s effectiveness and makes general suggestions about how a lesson may be improved. Candidate is moderately effective using observations to evaluate teaching and learning outcomes.

Candidate does not know if a lesson was effective and has no suggestions for how a lesson may be improved. Candidate does not use observations to evaluate teaching and learning outcomes.

INTASC: 10 Relationships: Colleagues and Community Candidate par ticipates in collegial activ ities to make the entire school a productiv e learning env ironment.

Candidate seeks out and participates in collegial activ ities to make the entire school a productive learning env ironment. Candidate seeks out opportunities for professional development and implements new ideas into the classroom.

Candidate participates in collegial activ ities to make the entire school a productive learning env ironment. Candidate seeks out opportunities for professional development and implements some new ideas into the classroom.

Candidate participates in collegial activ ities to a limited extent when convenient. Candidate attends professional development sessions also to a limited extent with minimal application in the classroom.

Candidate participates in no collegial activ ities and does not attend any professional development activ ities to enhance knowledge or skill.



Candidate Name ___________________________ Semester Observed _________________________

Student Teaching Evaluation Form Major Program Addendum: Mathematics

Directions: Please complete Part A and Part B as directed.

Part A: Background Information Form completed by (please select one):

_____ University Observer _____ Cooperating Teacher

Form completed for (please select one):

_____ Individual lesson observed _____ Mid-term Evaluation _____ Final Evaluation

Date form completed ___________________________

Part B: Assessment of NCATE/NCTM Standard 8: Knowledge of Mathematics Pedagogy Place an “X” on the line that best describes the candidate’s knowledge of mathematics pedagogy for

each element noted below.

Curricular Resources and Meeting Needs of All Students

Indicator 8.1: Selects, uses, and determines suitability of the wide variety of availabl e mathematics curricul a and teaching materials for all students including those with special needs such as the gifted, challenged and speakers of other languages.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Selects and uses available mathematics curricul a and resources

________ Candidate is unaware of or does not use a wide variety of suitable mathematics curricul a and teaching materials for both teachers and students.

________ Candidate displays limited awareness of a wide variety of suitable mathematics curricul a and teaching materials for both teachers and students. Candidate makes limited use of these resources for mathematics instruction.

________ Most of the time, candidate selects and uses a wide variety of suitable mathematics curricul a and teaching materials for both teachers and students, knows how to access these resources, and uses them regularly in the classroom for mathematics instruction and student resources.

________ On a consistent basis, candidate selects and uses a wide variety of suitable mathematics curricul a and teaching materials for both teachers and students, knows how to access them and use them, but also actively seeks and uses other resources and materials to enhance mathematics instruction and provide resources for students.

Selects and uses available mathematics curricul a and resources to meet the learning needs of all students

________ Candidate does not select, use, or adapt suitable mathematics curricul a and teaching materials to meet the needs of all students, including those with special needs such as the gifted, challenged, and speakers of other languages.

________ On a limited basis, candidate selects, uses, or adapts suitable mathematics curricul a and teaching materials to meet the needs of all students, including those with special needs such as the gifted, challenged, and speakers of other languages.

________ Most of the time, candidate selects, uses, and adapts suitable mathematics curricul a and teaching materials to meet the needs of all students, including those with special needs such as the gifted, challenged, and speakers of other languages.

________ On a consistent basis, candidate selects, uses, and adapts suitable mathematics curricul a and teaching materials to meet the needs of all students, including those with special needs such as the gifted, challenged, and speakers of other languages.



Use of Concrete Materials

Indicator 8.2: Selects and uses appropriate concrete materials for learning mathematics

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Selects and uses appropriat e concrete materials

________ Concrete materi als and their use for mathematics instruction are inappropriate or no materials are used when their use would be appropriat e and helpful to students during instruction.

________ Candidate makes limited use of concret e materials during mathematics instruction or only some use of concrete materials is appropriate during instruction. Student’s mental engagement by the use of concret e materials is moderate.

________ Candidate uses materials at appropriat e times during mathematics instruction and most use of concrete materials is appropriate. Almost all students are cognitively engaged by their use.

________ Candidate uses materials at appropriat e times during mathematics instruction and all use of concrete materials is appropriat e. Students initiate the use of materials and adapt materials to promote their own learning.

Assessing Student Knowledge

Indicator 8.3: Uses multiple strategies, including listening to and understanding the ways students think about mathematics, to assess students’ mathematical knowledge

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Uses multiple strategies to assess students’ knowledge

________ Candidate does not use multiple strategies to assess students’ mathematical knowledge or uses only low quality strategies to assess students’ mathematical knowledge.

________ Candidate makes limited use of multiple strategies to assess students’ mathematical knowledge, including listening to and understanding the ways students think about mathematics. These strategies are a combination of low and high quality.

________ Candidate uses multiple strategies to assess students’ mathematical knowledge, including listening to and understanding the ways students think about mathematics. Most of these strategies are of a high quality.

________ Candidate uses multiple strategies to assess students’ mathematical knowledge, including listening to and understanding the ways students think about mathematics. These strategies are uni formly of a high quality.

Planning

Indicator 8.4: Plans lessons, units and courses that address appropriat e learning goals, including those that address local, state, and national mathematics standards and legislative mandates

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Plans lessons and units that address appropriat e learning goals

________ Instructional goals for mathematics lessons and units are not valuable, or are not appropriat e for students. Goals have low expectations or require no conceptual understanding of student learning. Goals are imprecise, unclear, and not visibly measurable. Goals do not address district, state, and national mathematics standards.

________ Instructional goals for mathematics lessons and units are moderately valuable in expect ations, conceptual learning and student understanding. Some goals are unclear, imprecise and not adequately measurable. Some learning goals do not address district, state, and national mathematics standards.

________ Most instructional goals for mathematics lessons and units are clear, appropriat e, and measurable. Learning goals address district, state, and national mathematics standards.

________ Instructional goals are not only clear, appropriate, measurabl e and related to district, state and national mathematics standards, but are valuable and reflect high expect ations of all students.



Lead Classes in Mathematical Problem Solving

Indicator 8.8: Demonstrates the ability to lead classes in mathematical problem solving and in developing in-depth conceptual understanding, and to help students develop and test generalizations

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Ability to lead class in mathematical problem solving, developing conceptual understanding, and developing and testing generalizations.

________ Candidate does not demonstrates the ability to lead classes in mathematical problem solving and in developing in-depth conceptual understanding, and to help students develop and test generalizations.

________ In a few cases, candidat e demonstrates the ability to lead classes in mathematical problem solving and in developing in-depth conceptual understanding, and to help students develop and test generalizations.

________ In most cases, candidate demonstrates the ability to lead classes in mathematical problem solving and in developing in-depth conceptual understanding, and to help students develop and test generalizations. Candidate uses students’ thinking and methods of solution to encourage the development and testing of generalizations.

________ Candidate consistently demonstrates the ability to lead classes in mathematical problem solving and in developing in-depth conceptual understanding, and to help students develop and test generalizations. Candidate stimulates students to take the initiative to examine their own thinking and methods of solution to develop and test generalizations.

Assessment 4 – Attachment B: Data Tables for Mathematics Student Teaching Evaluation Form


Candidate Data for Student Teaching Evaluation Form Major Program Addendum: Mathematics (Mathematics Student Teaching Evaluation Form)

(8.1) Curricular Resources

Semester Taken

Number of Candidates Completing Student Teaching


(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 50 0 (0%) 1 (2%) 23 (46%) 26 (52%) Overall percent of candidates scoring a 2 or 3 – 98% (8.1) Meeting Needs of All Students

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 50 0 (0%) 1 (2%) 19 (38%) 24 (60%) Overall percent of candidates scoring a 2 or 3 – 98% (8.2) Use of Concrete Materials

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 50 0 (0%) 1 (2%) 23 (46%) 26 (52%) Overall percent of candidates scoring a 2 or 3 – 98%

Assessment 4 – Attachment B: Data Tables for Mathematics Student Teaching Evaluation Form


(8.3) Assessing Student Knowledge

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 50 0 (0%) 2 (4%) 19 (38%) 29 (58%) Overall percent of candidates scoring a 2 or 3 – 96% (8.4) Planning

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 50 0 (0%) 0 (0%) 19 (38%) 31 (62%) Overall percent of candidates scoring a 2 or 3 – 100% (8.8) Lead Classes in Mathematical Problem Solving

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 50 0 (0%) 1 (2%) 19 (38%) 30 (60%) Overall percent of candidates scoring a 2 or 3 – 98%


Section IV Assessment 5 – Lesson Observation, Planning, and Reflecting Project: Effect on Student Learning 1. Description of the Assessment

All candidates in the Elementary Mathematics Program complete Lesson Observation, Planning, and Reflecting Project as part of the required coursework in MTH 322 Geometry for Elementary Teachers. The project, which spans the last half of the semester, asks candidates to observe middle grade mathematics lessons taught by an experienced and highly-qualified teacher, to plan and teach a mathematics lesson to a subgroup of students from the classroom observed, and to reflect on the lesson they taught and its effect on student learning.


No significant changes occurred in the structure and focus of Assessment 5 since submitting our prior report in September 2007. Assessment 5 was not changed for the following reasons.

• MTH 322 is the first of the MTH 322,323, 324 sequence of classes candidates complete in the Elementary Mathematics Program. As such, faculty thought the focus and requirements of the assessment were appropriate for candidates at this point in the program.

• Faculty thought Assessment 5 from the 2007 report provided the Elementary Mathematics Program with good and helpful information related to candidates’ knowledge of effective lesson planning and their early ability to effect student learning.

The assessment description in #1 above and the assessment/indicator alignment noted in #3 below are the same as what appeared in our original report submitted in 2007.


Lesson Observation, Planning, and Reflecting Project is designed to primarily assess candidates’ dispositions towards various aspects of mathematics learning, their knowledge of mathematics pedagogy, and their ability to effect student learning. The project is also designed to provide one component of the required field experience that involves candidates in working with middle grade mathematics students and classrooms. The project addresses the following indicators from NCTM’s Middle Level Mathematics Teachers Report: 7.6, 8.1, 8.2, 8.3, 8.4, 8.6, 8.9, 16.1, and 16.3. The indicators are addressed through various activities that ask candidates to observe middle grade mathematics lessons, plan a lesson for middle grade students and reflect on their planning, and teach a mathematics lesson to middle grade students and reflect on their teaching, especially with respect to its effect on student learning. A detailed alignment of project activities with NCTM indicators is noted in the scoring rubric.


Lesson Observation, Planning, and Reflecting Project was implemented as a key assessment in MTH 322 during the Fall 2007, Spring 2008, Winter 2009, Fall 2009, and Winter 2010 semesters. A total of 75 candidates completed all portions of this assessment. A score of

Assessment 5 – Lesson Observation, Planning, and Reflecting Project

NCA TE Report (GVSU Elementary Mathematics Program) 2

“(2) Proficient” represents the level of competence expected of candidates in the areas assessed. The Lesson Observation, Planning, and Reflecting Project data for the five semesters combined showed approximately 79% or 59 of the candidates completed work in the area Student Knowledge that met the criteria for Proficient or Distinguished. The same data set also showed approximately 80% - 87% or 60-65 of the candidates’ work in the areas Planning, Use of Tools for Exploring mathematical Ideas, and Sequence of Planned Field Experiences met the criteria for Proficient or Distinguished. The data for all five semesters combined further showed that for each area assessed, there were a few candidates (10.7% - 17.3%) whose work on the project did not quite demonstrate the level of knowledge expected, which resulted in the score Progressing. In general, candidates’ work scored as Progressing typically showed an awareness of important issues related to effective planning, use of materials to promote learning, and ways to assess and increase students’ knowledge. However, candidate’s discussions and rationales related to these issues were of a lesser quality and not as fully developed as what was needed for the score Proficient. Additionally, the individual semester data for each of Fall 2007, Spring 2008 and Winter 2009 showed there were 1-2 candidates whose work met the criteria for Unsatisfactory in the four areas assessed. However, this data also showed no candidates completed Unsatisfactory work in any of the four areas during the last two semesters of the assessment’s implementation, Fall 2009 and Winter 2010.


Lesson Observation, Planning, and Reflecting Project is the first structured experience in the Elementary Mathematics Program that asks candidates to plan and teach mathematics lessons to middle grade students. It is also the first experience that asks candidates to demonstrate the ability to increase students’ knowledge of mathematics. Approximately 4/5 (78.7% - 86.7%) of the candidates’ work met or exceeded expectations in each of the four areas assessed. This suggests most candidates demonstrated the ability to increase students’ knowledge of mathematics and the ability to use a variety of curricula, teaching materials, and research to plan lessons with appropriate learning goals. This also suggests most candidates successfully engaged in a sequence of planned instructional experiences with middle school students and demonstrated positive dispositions towards the use of a variety of tools during mathematics instruction. Thus, most Elementary Mathematics Program candidates appear to be well-prepared with respect to their knowledge of various aspects of mathematics pedagogy and have positive dispositions towards mathematics teaching and learning, which suggests the program is meeting appropriate NCTM Middle Level standards. Additionally, the percentage of candidates who completed work at or above the expected level in each of the four areas assessed was 11.9% - 25.8% higher in this report than in our 2007 report. The largest increases occurred in the areas Planning (2010 report – 80%, 2007 report 54.2% → 25.8% increase) and Student Knowledge (2010 report – 78.7%, 2007 report 60.5% →18.2% increase). Since our 2007 report, faculty focused on providing candidates with more experiences and feedback on their planning, particularly in courses that are prerequisite to MTH 322. (These prerequisite courses (MTH 221, 222, and 223 – Mathematics for Elementary Teachers I, II, and III) are taken by all candidates seeking elementary teacher certification and are part of the Elementary Distributed Minor Program rather than the Elementary Mathematics Program.) Faculty in MTH 322 and MTH 323 also

Assessment 5 – Lesson Observation, Planning, and Reflecting Project


provided more structure for candidates’ instructional experiences with middle school students, as well as provided more opportunities for candidates to present and receive feedback on evidence illustrating their ability to promote growth in students’ mathematical knowledge. Faculty plan to continue efforts to encourage candidates to grow in their ability to increase student knowledge and to support candidates’ in their planning of mathematics lessons in order to help all candidates demonstrate proficiency in these areas.

Assessment 5 – Attachment A: Lesson Observation, Planning, Teaching, and Reflecting Project


Lesson Observation, Planning, Teaching, and Reflecting Project MTH 322 Geometry for Elementary Teachers

The purpose of this project is to engage you in a sequence of planned field-based experiences related to the teaching and learning of mathematics in the middle school. The project is divided into the following four parts: (1) observing mathematics lessons taught by practicing middle school teachers, (2) planning a mathematics lesson for a group of middle school students, (3) teaching a mathematics lesson to a group of middle school students, and (4) reflecting on the lesson you taught. Each portion of the project is described below.

Part 1 - Lesson Observations You will be assigned to a middle school classroom where you will observe two mathematics lessons taught by the regular classroom teacher. Based on your observations, write a reflective paper that addresses the following two questions.

1. What primary curricular material does the teacher use? Please be specific and cite the title and publisher of the material.

2. What did you observe that could help you plan and teach a mathematics lesson to students in this class? Address issues such as, but not limited to: the classroom culture, classroom resources, classroom management, and any special learning needs of the students (e.g., English language learners)

Part 2 - Lesson Plan and Lesson Plan Elaboration Working collaboratively with the classroom teacher you observed in Part 1, decide on a specific topic in the area of measurement or geometry for which you will plan a lesson and teach it to a group of middle school students. Once you have decided on a specific content area, write a lesson plan and lesson plan elaboration according to the following guidelines. Lesson Plan

Use the lesson plan framework suggested by your GVSU instructor. The framework includes items such as the specific learning goal, materials needed, descriptions of activities, specific problems to pose, questions to ask, and etc. Make sure the lesson plan is clear enough for all parts so another qualified teacher could follow and teach from your lesson plan.

Lesson Plan Elaboration

The purpose of the elaboration is for you to discuss the specific goals and rationale for the activities and resources you used in the lesson plan. Use the questions noted below to frame your discussion. Be sure to address all the questions and issues noted. Submit your Lesson Plan Elaboration with your Lesson Plan.

1. Lesson Title:

2. Lesson Goal and Resources:

a. What are the mathematical learning goals of the lesson? Please state the specific goals. Explain why this lesson and its goals are important.

b. Explain how your lesson and goals relate to the following professional resources.

• NCTM’s Principles and Standards for School Mathematics • Michigan Content Standards and Benchmarks

Assessment 5 – Attachment A: Lesson Observation, Planning, and Reflecting Project


• Michigan Grade Level Content Expectations (GLCE’s) • Local school district standards • Other Resources (e.g., the Strands of Mathematical Proficiency)

c. How do you plan to adapt your lesson so it is accessible to all learners, but challenges all learners? Give specific modifications you will make so the lesson meets the needs of all students.

d. Explain how your students’ prior knowledge and your knowledge of how children learn geometry and/or measurement influenced your choice of activities for this lesson. Please give 3-4 specific examples and cite readings and/or class discussions to support your response.

3. Motivation for the Lesson

Consider your Lesson Plan and the desired student mathematical learning goals. a. Explain why you thought the lesson context and activities/problems would be

appropriate for your particular students. b. Discuss the progression of the activities/problems. In particular, explain why

you sequenced activities and problems in the order presented in your lesson plan.

c. Explain why you chose to use a particular manipulative or technology during the lesson or why no manipulative or use of technology was appropriate for your lesson.

Part 3 - Teach the Lesson

Teach the lesson to a group of middle school students from the classroom in which you observed the mathematics lessons described in Part 1.

Part 4 - Lesson Reflection and Analysis

After teaching the lesson, write a report that addresses the following elements. 1. General Reflection on the Lesson 2. Lesson Analysis with Examples of Students’ Work 3. Reflection on Instructional Decisions

The following is a detailed description of each part.

1. General Reflection on the Lesson Provide a general overview of the overall lesson. This overview will serve as an introduction/context for the Lesson Analysis, which is described in the following section. The general overview should include reflective notes/thoughts on the effectiveness of specific problems and/or activities, as well as the use of concrete materials and/or technological tools. The emphasis here is on how you are assessing the lesson and reflecting on the causes for what you saw happening. Use the questions noted below to frame your overview. a. How did the lesson go differently from your plan or from what you expected

to do? What factors influenced this? b. Were your students intellectually engaged in the problems and activities?

Describe how you know whether they were or were not, and reflect on why they were or were not.

c. Were the materials (manipulatives, technology, or other materials) selected effective in promoting student learning or engagement? Describe how you know the effectiveness, and why it was so.

Assessment 5 – Attachment A: Lesson Observation, Planning, and Reflecting Project


2. Lesson Analysis with Examples of Students’ Work

This section is a reflective analysis of specific problems and themes that arose during the lesson. From the conducted lesson, select one problem or activity and discuss what you were able to assess about the students’ mathematical knowledge and how you assessed it. Use the following questions to frame your discussion. Also, use examples of students’ work to support your discussion. a. Describe how the students solved the problem or what they did during the

activity. Be specific in your description. b. How did the problem/activity effect the students’ mathematical knowledge? c. How was the problem’s/activity’s effect different for different students? d. Why did the problem/activity have the effect it did on students’ knowledge? e. If you had the opportunity to reteach the lesson, how you might adjust the

problem/activity to change its effect on students’ mathematical knowledge?

3. Reflection on Instructional Decisions Reflect on instructional decisions you made during the lesson that effected the students’ intellectual engagement and/or knowledge of the mathematics. Your reflection should address issues such as the following. a. Describe an instructional decision you made during the lesson that you would

make differently when reteaching the lesson. Include how you are assessing the results of your decision, and what variation you might try the next time this lesson is taught.

b. Describe an instructional decision you made during the lesson that went well. Include how you are assessing the results of your decision, and what the implications are for the next time this lesson is taught.

Assessment 5 – Attachment B: Scoring Rubric for Lesson Observation, Planning, and Reflecting Project


Lesson Observation, Planning, and Reflecting Project MTH 322 Geometry for Elementary Teachers

Assessment Evaluation Rubric

Planning

Indicator 8.1: Selects, uses, and determines suitability of the wide variety of available mathematics curricula and teaching materials for all students including those with special needs such as the gifted, challenged and speakers of other languages.

Indicator 8.4: Plans lessons, units and courses that address appropriate learning goals, including those that address local, state, and national mathematics standards and legislative mandates.

Indicator 8.6: Demonstrates knowledge of research results in the teaching and learning of mathematics.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s ability and resourcefulness in planning mathemati cs lessons.

This is demonstrated through the candidate’s written work on the Lesson Plan and Lesson Plan Elaboration. In particular, the candidate identi fies appropriate learning goals for students, bases the learning goals, problems, and activities on students’ prior mathematical knowledge, local, state and national recommendations, as well as on research related to the learning and teaching of mathematics. Additionally, the candidate plans appropriate modifications for the lesson to meet the learning needs of all students, including those with special needs.

________

Candidate identifies learning goals and objectives that are inappropriate for students.

or Candidate plans for use of problems and activities but lesson does not show candidate’s consideration of students’ prior knowledge, current local, state, and national recommendations, and research related to the learning and teaching of mathematics. Candidate plans modifications for lesson that are inappropriate for addressing students’ diffi culties and/or need for extensions.

________ Candidate identifies appropriat e learning goals and objectives for students. Candidate plans for use of a variety of effective problems and activities. Lesson shows candidate’s consideration of students’ prior knowledge, current local, state, and national recommendations, and research related to the learning and teaching of mathematics. Candidate plans modifications for lesson that are inappropriate for addressing students’ difficulties and/or need for extensions.

________ Candidate identifies appropriat e learning goals and objectives for students. Candidate plans for use of a variety of effective problems and activities. Lesson shows candidate’s consideration of students’ prior knowledge, current local, state, and national recommendations, and research related to the learning and teaching of mathematics. Candidate plans appropriate modifications for lesson to address students’ difficulties and/or need for extensions.

________ Candidate identifies appropriat e learning goals and objectives for students. Candidate plans for use of a variety of effective problems and activities. Lesson shows candidate’s consideration of students’ prior knowledge, current local, state, and national recommendations, and research related to the learning and teaching of mathematics. Candidate plans appropriate modifications for lesson to address students’ difficulties and/or need for extensions. Candidate’s reasoning for choosing particular aspects of the lesson (e.g., context, problems, sequence of problems/activities, materials, etc) reflects deep level of thought based on research related to the learning and teaching of mathematics.



Student Knowledge

Indicator 8.3: Uses multiple strategies, including listening to and understanding the ways students think about mathematics, to assess students’ mathematical knowledge.

Indicator 16.3: Demonstrates the ability to increase students’ knowledge of mathematics.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s ability to understand, assess, and increase students’ mathemati cal knowledge.

This is demonstrated through the candidate’s written work on the Lesson Reflection and Analysis. In particular, the candidate accurately describes students’ solution strategies. The candidate also provides convincing evidence illustrating growth in students’ mathematical knowledge or deep insights into why growth was not observed. Additionally, the candidate communicat es deep insights into ways that problems, activities and instructional decisions influenced growth in students’ mathematical knowledge.

________

Candidate does not accurately describe students’ solution strategies. Candidate does not provide convincing evidence to illustrate growth in students’ mathematical knowledge or clearly and logically explains why no growth was observed.

or Candidate’s explanation related to ways that problems, activities and instructional decisions influenced growth in students’ mathematical knowledge is illogical and/or incomplete.

________ Candidate accurately and fully describes students’ solution strategies. Candidate provides convincing evidence to illustrate growth in students’ mathematical knowledge or clearly and logically explains why no growth was observed. Candidate’s explanation related to ways that problems, activities, and instructional decisions influenced growth in students’ mathematical knowledge is illogical and/or incomplete.

________ Candidate accurately and fully describes students’ solution strategies. Candidate provides convincing evidence to illustrate growth in students’ mathematical knowledge or clearly and logically explains why no growth was observed. Candidate clearly and logically explains ways that problems, activities, and instructional decisions influenced growth in students’ mathematical knowledge.

________ Candidate accurately and fully describes students’ solution strategies. Candidate provides strong convincing evidence to illustrate growth in students’ mathematical knowledge or clearly and logically explains why no growth was observed. Candidate clearly, logically, and extensively explains ways that problems, activities, and instructional decisions influenced growth in students’ mathematical knowledge.



Use of Tools for Exploring Mathematical Ideas

Indicator 7.6: Use of various teaching tools including technology.

Indicator 8.2: Selects and uses appropriate concrete materials for learning mathematics.

Indicator 8.9: Develop lessons that use technology’s potential for building understanding of mathematical concepts and developing important mathematical ideas.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s ability to develop lessons that incorporate various teaching tools for exploring mathemati cal ideas.

This is demonstrated through the candidate’s written work on the Lesson Plan, Lesson Plan Elaboration, and Lesson Reflection and Analysis. In particular, the candidate incorporates use of appropriat e concrete materials and/or technology into lesson plan at opportune times. Candidate identifies speci fi c and logical ways that teaching tools were effective in promoting student learning. Candidate’s discussion of the use of teaching tools is positive in nature and reflects a productive disposition about using various teaching tools to promote students’ mathematical learning.

________

Candidate does not incorporat e concrete materials and/or technology into lesson plan when these tools would have supported student learning.

or Candidate incorporat es inappropriate concret e materials and/or technology into lesson plan.

or Candidate incorporat es appropriat e concrete materials and/or technology into lesson plan at inopportune times or in inappropriate ways. Candidate’s discussion of ways that teaching tools were effective in promoting students’ mathematical learning is illogical. Candidate’s discussion of the use of teaching tools is negative in nature and suggests the candidate does not have a productive disposition about the use of teaching tools in promoting students’ mathematical learning.

________ Candidate incorporat es appropriat e concrete materials and/or technology into lesson plan. Candidate incorporat es concrete materials and/or technology into lesson plan at appropriat e times. Candidate’s discussion of ways that teaching tools were effective in promoting students’ mathematical learning is not clear and/or fully developed. Candidate’s discussion of the use of teaching tools is positive in nature and suggests the candidate has a productive disposition about the use of teaching tools in promoting students’ mathematical learning.

________ Candidate incorporat es appropriat e concrete materials and/or technology into lesson plan. Candidate incorporat es concrete materials and/or technology into lesson at appropriat e times. Candidate identifies speci fic and logical ways that teaching tools were effective in promoting students’ mathematical learning. Candidate’s discussion of the use of teaching tools is positive in nature and suggests the candidate has a productive disposition about the use of teaching tools in promoting students’ mathematical learning.

________ Candidate incorporat es appropriat e concrete materials and/or technology into lesson plan. Candidate incorporat es concrete materials and/or technology into lesson at appropriat e times. Candidate identifies speci fic and logical ways that teaching tools were effective in promoting students’ mathematical learning and supports her/his views with references to current local, state, and national recommendations and research related to the learning and teaching of mathematics. Candidate’s discussion of the use of teaching tools is positive in nature and suggests the candidate has a productive disposition about the use of teaching tools in promoting students’ mathematical learning.



Sequence of Planned Field Experiences

Indicator 16.1: Engage in a sequence of planned opportunities prior to student teaching that includes observing and participating in middle grade mathematics classrooms under the supervision of experienced and highly qualified teachers.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate actively participates in a sequence of planned experiences in a middle school mathematics classroom.

This is demonstrated by the candidate’s written work on the Lesson Observation, Lesson Plan, Lesson Plan Elaboration, and Lesson Refl ection and Analysis. In particular, the candidate writes a clear and det ailed description of the two mathematics classes observed. The candidate develops a lesson plan containing appropriate learning goals, problems and activities in suffici ent detail so another reader could replicate the lesson. The candidate writes a reflection and analysis of the lesson that is clear, coherent, and fully developed.

________

Candidate did not satisfactorily complete planned sequence of observation and teaching.

________ Candidate’s description of two mathematics lessons observed lacks clarity and detail.

or Candidate develops a lesson plan with insuffi cient detail to enable a reader to replicate the lesson.

________ Candidate’s description of two mathematics lessons observed is clear and detailed. Candidate develops a lesson plan with sufficient detail so a reader could replicat e the lesson.

________ Candidate’s description of two mathematics lessons observed is clear, detailed, and shows insight into student learning of mathematics. Candidate develops a lesson plan with sufficient detail so another reader could replicate the lesson and the reader gains insight into the student learning of this content. Candidate’s work on all aspects of the project is of a high quality and supported throughout by references to local, state, and national recommendations, as well as research rel ated to the learning and teaching of mathematics.

Assessment 5 – Attachment C: Data Tables for Lesson Observation, Planning, and Reflecting Project


Candidate Data for Lesson Observation, Planning, and Reflecting Project

(8.1, 8.4, 8.6) Planning

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2007

Spring 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 75 2 (2.7%) 13 (17.3%) 38 (50.7%) 22 (29.3%) Overall percent of candidates scoring a 2 or 3 – 80% (8.3, 16.3) Student Knowledge

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2007

Spring 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 75 3 (4.0%) 13 (17.3%) 45 (60.0%) 14 (18.7%) Overall percent of candidates scoring a 2 or 3 – 78.7% (7.6, 8.2, 8.9) Use of Tools for Exploring Mathematical Ideas

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2007

Spring 2008

Winter 2009

Fall 2009

Winter 2010

Total N = 68 1 (1.3%) 12 (16.0 %) 33 (44.0%) 29 (38.7%) Overall percent of candidates scoring a 2 or 3 – 82.7%

Assessment 5 – Attachment C: Data Tables for Lesson Observation, Planning, and Reflecting Project


(16.1) Sequence of Planned Field Experiences

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2007

Spring 2008

Winter 2009

Fall 2009

Winter 2010



Section IV Assessment 6 – Menu of Problems Project: NCTM Standards 1. Description of the Assessment

All candidates in the Elementary Mathematics Program complete Menu of Problems Project as part of the required coursework in MTH 324 Algebra for Elementary Teachers. The project, which spans the entire semester, asks candidates to examine research and published problems to determine characteristics of mathematically rich problems, to write three mathematically rich problems focused on algebra or algebraic thinking that would be appropriate for middle grade students, to pilot test the problems with at least one middle grade student, to revise their problems based on the student’s solutions and instructor feedback, and to submit their problems for consideration for publication in a practitioner-oriented journal.


No significant changes occurred in the structure and focus of Assessment 6 since submitting our prior report in September 2007. The assessment was not changed for the following reasons.

• MTH 324 is the last of the MTH 322,323, 324 sequence of classes candidates complete in the Elementary Mathematics Program. As such, faculty thought the focus and requirements of the assessment were appropriate for candidates at this point in the program.

• Faculty thought Assessment 6 from the 2007 report provided the Elementary Mathematics Program with good and helpful information related to candidates’ knowledge of mathematical processes, content, and pedagogy, as well as their dispositions towards issues related to the learning and teaching of mathematics.

The assessment description in #1 above and the assessment/indicator alignment noted in #3 below are the same as what appeared in our original report submitted in 2007.


Menu of Problems Project is designed to primarily assess candidates’ knowledge of the mathematical processes problem solving and communication, their mathematical content knowledge of algebra, their dispositions towards various aspects of mathematics learning, and their knowledge of various aspects of mathematics pedagogy. The project is also designed to provide one component of the required field experience that involves candidates in working with middle grade students in mathematics. The project addresses the following indicators from NCTM’s Middle Level Mathematics Teachers Report: 1.1, 1.4, 3.3, 3.4, 7.1, 7.3, 7.4, 8.3, 8.5, 8.6, 10.1, 10.2, 10.3, 10.4, 10.5, and 16.1. The indicators are addressed through various activities that ask candidates to examine published research and resources to determine characteristics of mathematically rich problems, to write three mathematically rich problems, to pilot test the problems with middle grade students, and to submit their problems for consideration for publication in Mathematics Teaching in the Middle School. A detailed alignment of project activities with NCTM indicators is noted in the scoring rubric.

Assessment 6 - Menu of Problems Project


4. Analysis of Data Findings Menu of Problems Project was implemented as a key assessment in MTH 324 during the Spring 2007, Winter 2008, Fall 2008, Winter 2009, and Fall 2009 semesters. A total of 64 candidates completed all portions of the assessment. Three additional candidates, for a total of 67, completed all but one portion of the assessment. A score of “(2) Proficient” represents the level of competence expected of candidates in the areas assessed. The Menu of Problems Project data for all five semesters combined showed 81.3% – 88.1% or 52-59 of the candidates completed work that met the criteria for Proficient or Distinguished in the areas Knowledge of Mathematics Pedagogy, Knowledge of Algebra Structures and Models, and Understanding and Evaluation of Student Knowledge. The same data set also showed approximately 76% or 51 of the candidates completed work that met the criteria for Proficient or Distinguished in the area Knowledge of Algebra Patterns, Equalities, and Change. Additionally, the data for all five semesters combined showed 67.1% - 71.6% or 45-48 of the candidates’ work met or exceeded the expected level of competence in the areas Dispositions and Problem Solving and Communication. The data for all five semesters combined further showed that for five of the areas assessed, approximately 1/10 – 1/5 (11.9% - 20.9%) of the candidates’ work on the project did not quite demonstrate the level of knowledge expected, which resulted in the score Progressing. In general, candidate’s work scored as Progressing typically was limited to activities focused on one problem rather than two or more problems and/or was not as fully developed as was needed for the score Proficient. Additionally, the individual semester data showed no candidates completed work that met the criteria for Unsatisfactory in four of the areas assessed during any semester and no candidates completed Unsatisfactory work in any of the six areas assessed during the last three semesters of the assessment’s implementation (Fall 2008, Winter 2009, and Fall 2009). 5. Data Interpretation The high percentage of candidates who demonstrated competency at or above the expected level in the areas Knowledge of Algebra Patterns, Equalities, and Change, Knowledge of Algebra Structures and Models, Knowledge of Mathematics Pedagogy, and Understanding and Evaluation of Student Knowledge suggested candidates were well-prepared with respect to their mathematical content knowledge of algebra and their knowledge of mathematics pedagogy. Most candidates’ work in the area Problem Solving and Communication suggested they were able to clearly show use of the mathematical problem solving process as they utilized a variety of strategies to solve problems. Similarly, most candidates’ work in the area Dispositions suggested they had positive dispositions towards equity, effective teaching, and a commitment to learning with understanding. Thus, the Elementary Mathematics Program meets appropriate NCTM Middle Level standards with respect to mathematical content knowledge, knowledge of mathematics pedagogy, and dispositions. The Menu of Problems Project was implemented as Assessment 6 only one time during the data collection phase for our 2007 report. Therefore, it is difficult to compare percentages of candidates whose work met or exceeded expectations in each area in our 2007 report with this report. However, both reports show a smaller percentage of candidates’ work met the criteria for Proficient or Distinguished in the area Problem Solving and Communication than in most other areas assessed. Based on results of the 2007 report, Elementary Mathematics Program faculty initiated efforts to strengthen the program in the area Problem Solving and Communication as it occurred in MTH 324. In particular, faculty were intentional about exploring with candidates what it means for solutions to be different. They also provided candidates with more focused feedback on their solutions to problems and their communication of solutions. These efforts appear to have

Assessment 6 - Menu of Problems Project


helped more candidates complete work at the expected level in the area Problem Solving and Communication because greater than 85% of the candidates’ work met or exceeded expectations in this area during the last three semesters of the projects’ implementation. Faculty plan to continue efforts to support candidates’ in their development and communication of solutions to problems.

Assessment 6 – Attachment A: Menu of Problems Project


Menu of Problems Project MTH 324 Algebra for Elementary Teachers

Overview of Project: The purpose of this project is to engage you in thinking deeply about the mathematical richness of problems in general, and of those written specifically for middle school students (grades 6-8). The project is divided into the following six parts: (1) examining what research says about what makes a problem mathematically rich and worthwhile, (2) analyzing published problems written for middle school students with respect to mathematical richness, (3) creating mathematically rich and worthwhile problems for middle school students, (4) revising problems based on pilot testing with middle school students, (5) reflecting on students’ mathematical knowledge and its relation to mathematically rich problems, and (6) submitting mathematically rich problems for publication. Each portion of the project is described more fully below.

***************************** Please type all aspects of this project work (except, of course, the student work that you submit in Part 4. The notes that you take as the student solves the problem may also be handwritten.) You will submit each of Parts 1-3 of the project at a different point in the semester, as well as submit the entire project at the end of the semester. Your instructor will provide feedback on your work for each part. You will find it helpful to use this feedback to inform your work on later parts of the project.

*****************************

Part 1: Reflecting on Research: What Makes a Task Mathematically Rich and Worthwhile? A. Read all of the following articles/chapters: Hiebert, J. et al (1997). Introducing the Critical Features of Classrooms. Making Sense:

Teaching and Learning Mathematics with Understanding (Chapter 1: pp. 1-16). Portsmouth, NH: Heinemann.

Hiebert, J. et al (1997). The nature of classroom tasks. Making Sense: Teaching and Learning Mathematics with Understanding (Chapter 2: pp. 17-28). Portsmouth, NH: Heinemann.

National Council of Teachers of Mathematics (2000). Process Standards. Principles and Standards for School Mathematics. Reston, VA: Author. Neyland, J. (1994). Designing rich mathematical activities. Mathematics Education: A

Handbook for Teachers Volume 1 (Chapter 11 pp. 106-122).Wellington, New Zealand, Wellington College of Education.

Smith, Margaret Schwan & Stein, Mary Kay. (Feb 1998). Selecting and Creating Mathematical Tasks: From Research to Practice, Mathematics Teaching in the Middle School, 3, 344-350.

Van de Walle, J. (2003). Designing and Selecting Problem-Based Tasks. Teaching Mathematics through Problem Solving: Prekindergarten-Grade 6. (Chapter 4: pp. 67-80). Reston, VA: National Council of Teachers of Mathematics.



B. Write a reflective, research-based paper. After reading the articles/chapters listed in Part 1A, write a 4-6 page reflective and analytic paper that addresses the following 3 questions. As you write this paper, incorporate and pull together relevant ideas from the readings in Part 1A to support your argument and cite these sources (i.e., if three readings all emphasize the same characteristic, cite all 3 sources). Include a bibliography (not included in page count). 1. Why are good problems/tasks so critical for learning mathematics with understanding?

Please explain how you define “understanding” as you address this question. 2. What are the characteristics of a good, “rich,” mathematically meaningful problem?

Identify these characteristics and then briefly define what is meant by each of these characteristics. You need to consolidate ideas from multiple readings, extract all of the main characteristics from the readings, and cite your sources accordingly. You can add additional characteristics you think should be included on your list not found in the readings. You may describe these characteristics in a list or in paragraph style but either way, be sure that you explain what is meant by each characteristic.

3. Looking back over your list of characteristics, which three do you think are the most critical? Why? Be explicit and specific in your explanation.

Part 2: Analyzing Problems Look through issues of Mathematics Teaching in the Middle School(MTMS) from the past 3 years and read through the “Menu of Problems.” Older versions of the Menu of Problems sort the problems into three categories (Appetizer -- fairly easy, short problems; Entrée/Main Course -- somewhat more substantial in nature in terns of the “thoughtful energy” required for an answer; Dessert -- more open-ended). Newer versions of the Menu of Problems do not subdivide the problems in any way. Select two problems from the MTMS Menu(s) of Problems. The problems need to fit the descriptions in A and B below. A. Find one example of a “good, rich” problem.

1. Provide a copy of the problem statement and a solution for the problem. Do NOT just copy the solution given in the journal. Solve the problem yourself and write the solution in your own words.

2. Specifically explain why this problem is a “rich” and “mathematically meaningful” problem (fitting the descriptions you discussed in Part 1B). In particular: • evaluate the mathematical purpose of the problem, (i.e. what makes this

problem “mathematically problematic?”) • identify the specific types of residue you might expect to find after solving

this problem. • identify at least 3 additional (general) criteria to evaluate why the problem

is rich (drawing from your response in Part 1B, the readings, class discussion, and your own ideas).

B. Find one example of a problem you do not think meets your standard as a good rich mathematical problem.

1. Provide a copy of the problem statement and a solution for the problem. Again, do NOT just copy the solution from the journal.

2. Specifically explain why you think this problem is lacking.



• Be specific in explaining why the problem is not mathematically problematic (if applicable) or why it is not as mathematically problematic as it could be.

• Identify at least 3 additional (general) criteria (drawing from your response in part 1, the readings, class discussion, and your own ideas). to evaluate why the problem is not rich.

3. Discuss how you would revise the problem to make it a “rich” and mathematically meaningful problem. Give specific recommendations (i.e., rewrite the problem) and then briefly elaborate on why these changes will make the problem more meaningful and “rich.” If you think the problem has no redeeming value and you are unable to adapt it in a satisfactory way, explain why not and then provide an example of a rich problem that would meet the mathematical goals of the original problem.

C. Attach a photocopy of the Menu(s) of Problems you are drawing your problems from.

Clearly indicate the problems you are critiquing. Part 3: Creating/Writing Your Own Problems A. Create/Write three good, rich mathematical problems that relate to algebra or

algebraic thinking (for example, patterns, qualitative-graphs, functions, or proportional reasoning problems). 1. Before you write your problems, be sure to identify your mathematical goals.

Also reflect on the criteria that you view as critical in making a problem a “rich” and mathematically problematic (from part 1) and incorporate these aspects into your problems.

2. Be sure to consider the process and content standards of the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (2000) as you create and adapt problems for this assignment.

3. Set the problem in a context (real-life, imaginary, etc) of interest to a middle-school-age student.

4. Connections to other areas of mathematics or contexts outside of mathematics are also encouraged.

5. At least 2 of the 3 problems you submit must be original. One of the problems may be adapted from another source. However, you MUST cite your sources and submit a copy of the actual problem you adapted. The adaptation should be significant. If you use an internet source, include the address as well as a print out/hard copy of the materials you used/adapted.

B. Submit a cover page for your 3 problems that lists specific mathematical goals for

each of the problems you create. C. Submit one “clean” copy of the 3 problems. This should be the version you will later

give to a student, with room to work the problem out on the page. D. On a separate copy, submit a solution for each of the problems. Include any tables,

graphs, or other representations you expect students to use to solve the problem. If the answer to your problem may vary, state this and provide a possible solution. If there are multiple ways to solve the problem, provide at least 2 different solution strategies. Students often can handle problems without formulating a



symbolic/algebraic solution. When providing an algebraically symbolic solution, also provide a solution that does not require the use of a symbolic manipulation or algorithm.

Part 4: Testing and Revising your Problems A. You may want to revise your problems based on the written/verbal feedback you

received for Part 3 from your instructor. If you think you should revise one or more of your problems before testing/piloting them with a school-age child, do so, and provide a written rationale that states why and how you have decided to alter certain problems.

B. Pilot/test your problems with a willing middle-school-age student (6th-8th grade). After

the student solves the problem, ask him/her to explain his/her thought processes in solving the problems. In addition, ask if the statement of the problem made sense and if s/he has any feedback on the problems themselves. Take notes on what the student says and turn in these handwritten notes. Also, submit the student’s written work.

C. Write a summary of how the student solved the problems (based on your notes,

observations, and the students’ written work). Be sure to state the name and grade of the student with whom you worked. Also, explain all strategies the student used.

D. After piloting your problems, reflect over your problems one last time. Do any

problem(s) need revision? 1. If so, revise your problem(s) and provide a written rationale that states why

and how you have decided to alter certain problem(s). 2. If you do not think your problems need to be altered/revised, provide a written

justification as to why you are not changing the problems. NOTE: If you make any significant revisions, you must test your “finally” revised problem on another middle-school-age student (and repeat Part 4B & C for the revised problem). If your revisions are minor, then you do not have to test them on another student. E. Submit all of the written work of the student(s) who have completed the problems,

your notes (Part 4B), your summary (Part 4C), and your revisions (see Part 4C). Part 5: Reflecting Over the Experience Here you are asked to engage in two types of reflection: reflection about the middle grader's understanding of mathematics based on his/her responses to your questions and reflection of the process of creating and evaluating problems. Please answer the following four questions: Part 5a: What have you learned about the middle grades student's understanding of mathematical ideas based on his/her responses to your questions? 1. What did you learn about your student’s mathematical understanding of concepts,

relationships and/or solution strategies as you watched him/her solve these problems and probed his/her thinking? (Be specific about your student’s mathematical understanding, connections, and strategies)



Part 5b: Reflect over the process you have gone through to create and evaluate problems. 2. What have you learned about writing/creating short, rich mathematically meaningful

problems? How might you change how you write and evaluate problems in the future?

3. Which of the 3 problems (that you wrote) do you like the best? Explain why you

chose this problem. 4. Reflect over the process you have gone through as you have evaluated and created

problems. Explain how you have personally grown in your ability to evaluate what makes a problem “rich” and mathematically meaningful since writing your response in Parts 1 and 2 of the project. (Be specific, please.)

Part 6: Preparation for Submitting Your Work to MTMS Please submit the copyright form provided by your instructor and one additional clean copy of your final version of the problems (with solutions). Your instructor will submit your problems to the journal MTMS for their consideration for publication. If your problem(s) are selected for publication, your name will be published along with the problem(s). Many students who took MTH 324 in previous semesters are reaping the benefits of their hard work and have had their problems published. Be creative!!! Write problems you would enjoy solving if you were a middle school student!

Assessment 6 – Attachment B: Scoring Rubric for Menu of Problems Project


Menu of Problems Project MTH 324 Algebra for Elementary Teachers

Assessment Evaluation Form

Knowledge of Mathematics Pedagogy

Indicator 8.5: Participates in professional mathematics organizations and uses their print and on-line resources.

Indicator 8.6: Demonstrates knowledge of research results in the teaching and learning of mathematics.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED

Candidate’s knowledge of mathemati cs pedagogy.

This is demonstrated through the candidate’s written summary and analysis of research related to rich, mathematically meaningful tasks. In particular, the candidate analyzes research to determine important characteristics of rich, mathematically meaningful tasks and uses print and on-line resources of a professional mathematics organization, the National Council of Teachers of Mathematics (NCTM) and other professional organizations in this analysis.

__________ Candidate fails to use professional mathematics organizations’ print and on-line resources in the completion of their summary and analysis of characteristic of rich, mathematically meaningful tasks. Candidate’s summary of important charact eristics of rich, mathematically meaningful tasks is incomplete or is not appropriat ely supported by relevant research. Candidate does not consolidate ideas from multiple readings to provide a comprehensive overview of the characteristics of rich, mathematically meaningful tasks.

________ Candidate uses professional mathematics organizations’ print an on-line resources in the completion of their summary and analysis of characteristics of rich, mathematically meaningful tasks. Candidate provides a partial summary of the important characteristics of rich, mathematically meaningful tasks that is appropriately supported by research.

or Candidate does not consolidate idea from multiple readings to provide a comprehensive overview of the characteristics of rich, mathematically meaningful tasks.

________ Candidate uses professional mathematics organizations’ print and on-line resources in the completion of their summary and analysis of characteristics of rich, mathematically meaningful tasks. Candidate’s summary of important charact eristics of rich, mathematically meaningful tasks is comprehensive, organized, clearly presented, logical, and appropriat ely supported by relevant research. Candidate consolidates ideas from multiple readings.

________ Candidate uses professional mathematics organizations’ print and on-line resources in the completion of their summary and analysis of characteristics of rich, mathematically meaningful tasks. Candidate’s summary of important characteristics of rich, mathematically meaningful tasks is comprehensive,, organized, clearly presented, logical, and appropriately supported by relevant research. Candidate consolidates ideas from multiple readings. Candidate demonstrates creativity and insights by making connections among the readings or incorporating helpful examples as s/he synthesizes and/or summarizes relevant research.



Dispositions

Indicator 7.1: Attention to equity.

Indicator 7.3: Effective Teaching.

Indicator 7.4: Commitment to learning with understanding.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s dispositions toward mathemati cs teaching and learning.

This is demonstrated through the candidate’s work on the written reflection and analysis of rich, mathematically meaningful tasks. In particular, the candidate uses research-generated characteristics to analyze mathematics problems published in the print and on-line resources of a professional mathematics organization, the National Council of Teachers of Mathematics (NCTM). The candidate selects problems for analysis that are focused on equity by having multiple entry points that make them accessible to all middle grades students with varying prior mathematical knowledge. Also, the candidate demonstrat es a productive disposition towards effective teaching and learning with understanding by selecting to analyze problems that are mathematically problematic and by appropriately justifying why and how the problems analyzed can promote mathematical learning.

________ Candidate does not identify/modi fy at least one previously published problem from NCTM resources that promotes equity by its accessibility to all middle grades students with varying prior mathematical knowledge. Candidate does not identify and/or modify at least one problem that is mathematically problematic. Candidate’s analysis of the two problems does not demonstrate consideration of research related to important characteristics of rich, mathematically meaningful tasks. Candidate’s justification for why and how the two problems promote mathematical learning is incomplete or inappropriate.

________ Candidate identifies/modi fies at least one previously published problem from NCTM resources that promotes equity by its accessibility to all middle grades students with varying prior mathematical knowledge. Candidate identifies and/or modifies at least one problem that is mathematically problematic. Candidate’s analysis of at least one problem shows consideration of research related to important characteristics of rich, mathematically meaningful tasks. Candidate’s justification for why and how at least one of the problems promotes mathematical learning is incomplete or inappropriate.

________ Candidate identifies/modi fies two previously published problems from NCTM resources that promote equity by their accessibility to all middle grades students with varying prior mathematical knowledge. Candidate identifies and/or modifies two problems that are mathematically problematic. Candidate’s analysis of the two problems shows consideration of research related to important characteristics of rich, mathematically meaningful tasks. Candidate’s justification for why and how the two problems promote mathematical learning is appropriat e.

________ Candidate identifies/modi fies two previously published problems from NCTM resources that promote equity by their accessibility to all middle grades students with varying prior mathematical knowledge. Candidate identifies and/or modifies two problems that are mathematically problematic. Candidate’s analysis of the two problems shows consideration of research related to important characteristics of rich, mathematically meaningful tasks. Candidate’s justification for why and how the two problems promote mathematical learning is appropriat e, clearly presented, and well-organized. Candidate creatively and appropriat ely modifies a task to create an exemplar problem that exhibits the characteristics of a rich, mathematically meaningful task.



Knowledge of Algebra Patterns, Equalities, and Change

Indicator 10.1: Explore, analyze, and represent patterns, relations, and functions.

Indicator 10.3: Investigate equality, equations, and proportional relationships.

Indicator 10.5: Analyze change in various contexts.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s knowledge of different perspectives on algebra.

This is demonstrated through the candidate’s written work involving the creation of three rich, mathematically meaningful problems related to algebra or algebraic thinking. In particular, the candidate creates problems that demonstrate her/her ability: explore, analyze and represent patterns, relations, and functions; investigate equality, equations, and proportional relationships; and analyze change in various contexts.

________ Candidate’s mathematical goals for at least two problems are not clearly stated or do not focus on appropriat e algebrai c ideas. Candidate does not demonstrate knowledge of different perspectives of algebra and does not create at least one rich, mathematically meaningful problem related to algebrai c thinking.

________ Candidate’s mathematical goals for at least one problem is clearly stated and focused on appropriate algebraic ideas. Candidate demonstrates knowledge of different perspectives of algebra by creating at least two rich, mathematically meaningful problems related to algebraic thinking. At least one problem involves exploring, analyzing, and representing patterns, relations, or functions

or At least one problem involves investigating equality, equations, or proportional relationships.

or At least one problem involves analyzing change in various contents.

________ Candidate’s mathematical goals for at least two problems are clearly stated and focused on appropriat e algebrai c ideas. Candidate demonstrates knowledge of different perspectives of algebra by creating three rich, mathematically meaningful problems related to algebraic thinking. At least one problem involves exploring, analyzing, and representing patterns, relations, or functions. At least one problem involves investigating equality, equations, or proportional relationships. At least one problem involves analyzing change in various contents.

________ Candidate’s mathematical goals for all problems are clearly stated and focused on appropriat e algebrai c ideas. Candidate demonstrates knowledge of different perspectives of algebra by creating three rich, mathematically meaningful problems related to algebraic thinking. At least one problem involves exploring, analyzing, and representing patterns, relations, or functions. At least one problem involves investigating equality, equations, or proportional relationships. At least one problem involves analyzing change in various contents. At least one of the creat ed problems makes creative or discerning connections between di fferent perspectives of algebra.

Knowledge of Algebra Structures and Models

Indicator 10.2: Recognize and analyze mathematical structures

Indicator 10.4: Use mathematical models to represent quantitative relationships

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s knowledge of different perspectives on algebra.

This is demonstrated through the candidate’s written solutions to his/her three sel f-generated problems. In particular, the candidate demonstrates his/her ability to recognize and analyze mathematical structures and use mathematical models to represent quantitative relationships in the solutions to these problems.

________ Candidate’s solutions to all of the problems are not clearly stated or are not mathematically correct. Candidate fails to refer to appropriat e algebrai c ideas in his/her solutions, uses inappropriate mathematical models to represent quantitative relationships, or fails to appropriately analyze the mathematical structure inherent in the problems.

________ Candidate’s solution to at least one problem is clearly stated, logical, and mathematically correct. Candidate refers to appropriat e algebrai c ideas in at least one solution, uses appropriat e mathematical models to represent quantitative relationships, and appropriat ely analyzes the mathematical structure inherent in at least one of the problems.

________ Candidate’s solutions to at least two problems are clearly stated, logical, and mathematically correct. Candidate refers to appropriat e algebrai c ideas in at least two solutions, uses appropriat e mathematical models to represent quantitative relationships, and appropriat ely analyzes the mathematical structure inherent in these problems.

________ Candidate’s solutions to all three problems are cl early stated, logical, and mathematically correct. Candidate refers to appropriat e algebrai c ideas in all solutions, uses appropriate mathematical models to represent quantitative relationships, and appropriat ely analyzes the mathematical structure inherent in each problem. In at least one of his/her solutions, the candidate draws connections between models, strategies, and/or structures.



Problem Solving and Communication

Indicator 1.1: Apply and adapt a variety of appropriate strategies to solve problems.

Indicator 1.4: Monitor and reflect on the process of mathematical problem solving.

Indicator 3.3: Organize mathematical thinking through communication.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s ability to problem solve and communicate mathemati cally.

This is demonstrated by the candidate’s written solutions to three rich, mathematically meaningful algebrai c problems s/he created. In particular, the candidate applies and adapts a variety of strategies to solve the problems. The candidate also presents solutions to problems in a clear and organized manner.

________ Candidate presents solutions that are not appropriate or are not clear. Candidate is unable to present two different solutions for any of the problems. Candidate’s lack of organization and clarity in his/her written work illustrates only minimal consideration of the problem solving process.

________ Candidate presents solutions to at least one problem in a clear, organized, and appropriat e manner. Candidate presents at least two different solutions for at least one of the problems. Candidate’s organization and clarity in his/her written work illustrates consideration of the problem solving process for at least one of the problems.

________ Candidate presents solutions to at least two problems in a clear, organized, and appropriat e manner. Candidate presents at least two different solutions for at least two of the problems. Candidate’s organization and clarity in his/her written solutions illustrates consideration of the mathematical problem-solving process for at least two of the problems.

________ Candidate presents solutions to all three problems in a clear, organized, and appropriat e manner. Candidate presents at least two different solutions for each of the three problems. Candidate’s organization and clarity in his/her written solutions illustrates consideration of the mathematical problem solving process for each of the three problems.

Understanding and Evaluation of Student Knowledge

Indicator 3.4: Analyze and evaluate the mathematical thinking and strategies of others.

Indicator 8.3: Uses multiple strategies, includin g listening to and understanding the ways students think about mathematics, to assess students’ mathematical knowledge.

Indicator 16.1: Engage in a sequence of planned opportunities prior to student teaching that includes observing and participating in middle grades mathematics classrooms under the supervision of experienced and highly qualified teachers.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s understanding of and ability to evaluate student knowledge.

This is demonstrated through the candidate’s written work related to engaging a middle school student in solving rich, mathematically meaningful tasks. In particular, the candidate engages one middle school student in solving her/his three self-generated algebraic problems. The candidate listens to and demonstrates understanding of the student’s solutions strategies by accurat ely describing these strategies. Additionally, the candidate’s discussion of what s/he learned about the students’ thinking through the problem solving interview is clear, coherent, logical, and focused on appropriat e mathematical ideas.

________ Candidate does not complete a problem-solving interview with a middle grades student. Or Most of the candidate’s summary descriptions of middle school student’s solution strategies lack clarity, detail, or are not supported by the student’s written work. Candidate’s explanatory discussions of the middle school student’s solution strategies are missing or are inappropriate based on the student’s written work. Candidate’s discussion related to what s/he learned about students’ mathematical thinking is missing or is not focused on appropriat e mathematical ideas.

________ Most of the candidate’s summary descriptions of middle school student’s solution strategies are clear, detailed, and appropriately supported by the student’s written work. Candidate’s explanatory discussions of the middle school student’s solution strategies are incomplete or lack detail or clarity. Candidate’s discussion related to what s/he learned about students’ mathematical thinking lacks clarity or is not focused on appropriat e mathematical ideas.

________ Candidate’s summary descriptions of all of the middle school student’s solution strategies are clear, detailed, and appropriately supported by the student’s written work. Candidate’s explanatory discussions of all of the middle school student’s solution strategies are clear, detailed, and appropriate. Candidate’s discussion related to what s/he learned about students’ mathematical thinking is clear, coherent, logical, and focused on appropriat e mathematical ideas.

________ Candidate’s summary descriptions of all of the middle school student’s solution strategies are clear, detailed, and appropriately supported by the student’s written work. Candidate’s explanatory discussions of all of the middle school student’s solution strategies are clear, detailed, appropriate, and fully developed. Candidate’s discussion related to what s/he learned about students’ mathematical thinking is clear, coherent, fully developed, logical, and focused on appropriat e mathematical ideas.

Assessment 6 – Attachment C: Data Tables for Menu of Problems Project


Candidate Data for Menu of Problems Project

(8.5, 8.6) Knowledge of Mathematics Pedagogy

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Spring 2007

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Total N = 67 0 (0%) 8 (11.9%) 47 (70.1%) 12 (17.9%) Overall percent of candidates scoring a 2 or 3 – 88% (7.1, 7.3, 7.4) Dispositions

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Spring 2007

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Total N = 67 0 (0%) 22 (32.8%) 35 (52.2%) 10 (14.9%) Overall percent of candidates scoring a 2 or 3 – 67.1% (10.1, 10.2, 10.3) Knowledge of Algebra Patterns, Equalities, and Change

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Spring 2007

Winter 2008

Fall 2008

Winter 2009

Fall 2009


Assessment 6 – Attachment C: Data Tables for Menu of Problems Project


(10.4, 10.5) Knowledge of Algebra Structures and Models

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Spring 2007

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Total N = 67 0 (0%) 8 (11.9%) 41 (61.2%) 18 (26.9%) Overall percent of candidates scoring a 2 or 3 – 88.1% (1.1, 1.4, 3.3) Problem Solving and Communication

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Spring 2007

Winter 2008

Fall 2008

Winter 2009

Fall 2009

Total N = 67 6 (9.0%) 13 (19.4%) 27 (40.3%) 21 (31.3%) Overall percent of candidates scoring a 2 or 3 – 71.6% (3.4, 8.3, 16.1) Understanding and Evaluation of Student Knowledge

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Spring 2007

Winter 2008

Fall 2008

Winter 2009

Fall 2009



Section IV Assessment 7 – Historical and Cultural Perspectives in Mathematics Assignment: NCTM Standards 1. Description of the Assessment

Historical and Cultural Perspectives in Mathematics Assignment examines three primary areas of candidates’ mathematical content knowledge: communication, connections, and historical development of particular mathematical topics including cultural contributions to the development. The assignment is a combination of three similarly-structured explorations all candidates in the Elementary Mathematics Program complete as part of the required coursework in MTH 341 Euclidean Geometry, MTH 345 Discrete Mathematics, and MTH 495 The Nature of Modern Mathematics. Each exploration involves candidates in reading about a mathematical topic from a historical perspective, researching and writing about the mathematical topic from both a historical and cultural perspective, and drawing on historical and cultural aspects to prove ideas and/or solve problems related to the mathematical topic. In MTH 341 Euclidean Geometry, the assignment asks candidates to explore Euclid’s Fifth postulate (the parallel postulate) while it involves candidates in investigating the Arithmetical Triangle from a discrete mathematics perspective in MTH 345 and solutions to equations, particularly quadratic equations, in MTH 495.


Assessment 7 changed significantly since our 2007 report and is considered to be a new assessment for this report. In our prior report, Assessment 7 consisted of a project where candidates wrote three short chapters of a Euclidean geometry textbook in their own words with their own examples and proofs. One of these chapters focused on the historical development of Euclid’s fifth postulate. A portion of this assessment examined candidates’ knowledge of historical and cultural aspects related to geometry, as well as their ability to communicate in the language of mathematics and to make connections between mathematical ideas. Our 2007 report also included an Assessment 2, part of which examined candidates’ knowledge of communication and connections as they solved a discrete mathematics problem involving tournaments. Additionally, our 2007 report included an Assessment 8 that investigated candidates’ knowledge of communication, connections, and historical and cultural contributions as they solved problems by using various historical methods and translated historical texts into modern algebraic notation. This new version of Assessment 7 replaces the portions of our previous Assessment 2, 7, and 8 that focused on communication, connections, and historical and cultural perspectives and requires more of candidates in each of these three areas to gain deep insights into their mathematical content knowledge. Assessment 7 was changed for the following reasons.

a. Data from our 2007 report showed candidates consistently completed work at a lower level of competence in two primary areas: (1) communicating in the language of mathematics by using precise notation and terminology and (2) demonstrating how mathematical ideas interconnect and build on one another to produce a coherent whole. Based on this data, Faculty in the Department of Mathematics concentrated on strengthening the Elementary Mathematics program in the areas of communication and connections. Faculty also wanted deeper insights into candidates’ knowledge in these areas to inform their efforts.

b. Our 2007 report met only one indicator (11.7) related to historical developments and cultural contributions, which focused on geometry. Faculty thought the Elementary Mathematics Program would be stronger if historical and cultural perspectives were integrated into more required courses in the program.

Assessment 7– History & Culture Assignment



Historical and Cultural Perspectives in Mathematics Assignment is designed to primarily assess candidates’ knowledge of the mathematical processes communication and connections, as well as their knowledge of historical developments and cultural contributions in the areas of geometry, discrete mathematics, algebra, and number and number systems. As such, the assessment addresses the following indicators from NCTM’s Middle Level Mathematics Teachers Report: 3.2, 4.3, 9.8, 10.6, 11.7, and 13.3. The indicators are addressed through activities that ask candidates to write historical texts in modern day mathematical notation, to prove ideas such as the summit angles of a Saccheri quadrilateral are congruent, and to apply and describe how Pascal’s Twelfth Consequence can be used to calculate ratios between entries in a given row of an Arithmetical Triangle. The indicators are also addressed through activities that ask candidates to research and write a historical and cultural perspective paper about a mathematical topic in the areas of geometry, discrete mathematics, algebra, and number and number systems. Historical and Cultural Perspectives in Mathematics Assignment is completed by all individuals enrolled in MTH 341, MTH 345, and MTH 495, which includes candidates in the Elementary Mathematics Program, candidates seeking secondary mathematics teacher certification, and individuals who are not seeking teacher certification. To insure consistency in evaluation, the same scoring rubric is used for all individual’s work. Thus, in three places the rubrics note indicators from NCTM’s Middle Level Mathematics Teachers Report and the Secondary Mathematics Teachers Report. These indicators relate to historical developments and cultural contributions in the areas of geometry, discrete mathematics, and number and number systems. These indicators have different identification numbers but comparable descriptions in the Middle Level and Secondary reports and are identified as (Middle Level) or (Secondary) to denote the specific report to which they refer.


Historical and Cultural Perspectives in Mathematics Assignment was implemented as a key assessment in MTH 341 Euclidean Geometry, MTH 345 Discrete Mathematics, and MTH 495 The Nature of Modern Mathematics during the Fall 2008, Winter 2009, Spring 2009, Fall 2009, and Winter 2010 semesters. A total of 30 candidates completed the MTH 495 exploration while a total of 39 candidates completed each of the MTH 341 and MTH 345 explorations during the five semesters. A score of “(2) Proficient” represents the level of competence expected of candidates in each area assessed. All three explorations in Historical and Cultural Perspectives in Mathematics Assignment examined candidates’ ability to “use the language of mathematics to express ideas precisely” (Communication, Indicator 3.2). The Communication data for all five semesters combined showed 71.8% or 28 of the candidates completed work that met the criteria for Proficient or Distinguished for the MTH 341 geometry exploration. The same data set also showed 69.2% or 27 of the candidates’ work on the MTH 345 discrete mathematics exploration met or exceeded expectations while 53.4% or 16 of the candidates completed work at or above the expected level on the MTH 495 exploration. Additionally, the Communication data for all five semesters combined showed approximately 1/5 – 1/3 (20.5% - 36.7%) of the candidates completed work on each of the three explorations that did not quite demonstrate the level of knowledge expected, which resulted in the score Progressing. Furthermore, the Communication data for individual semesters showed no candidates completed Unsatisfactory work on two of the three explorations during Spring 2009 and Winter 2010 while the data for Fall 2009 showed no candidates’ communication work was Unsatisfactory on any of the three explorations.



The MTH 341 geometry exploration and the MTH 345 discrete mathematics exploration investigated candidates’ ability to “demonstrate how mathematical ideas interconnect and build on one another to produce a coherent whole” (Connections, Indicator 4.3). The Connection data for all five semesters combined showed approximately 74% or 29 of the candidates completed work that met the criteria for Proficient or Distinguished on the MTH 345 exploration while 82% or 32 candidates completed work at these levels on the MTH 341 geometry exploration. This data set also showed approximately 1/5 (17.9% - 20.5%) of the candidates’ work on each of the two explorations was evaluated as Progressing, not quite the expected level. Additionally, the Connections data for individual semesters showed no candidates’ work on the MTH 345 discrete mathematics exploration was Unsatisfactory for four of the five semesters of the assessment’s implementation while no candidates completed Unsatisfactory work on the MTH 341 geometry exploration during any semester. All three explorations in this assignment examined candidates’ knowledge of the historical development of a particular mathematical topic and ways various cultures contributed to the development. The historical and cultural perspectives data for all five semesters combined showed approximately 85% or 33 of the candidates completed work that met the criteria for Proficient or Distinguished on the MTH 341 geometry exploration and approximately 92% or 36 of the candidates completed work at these levels on the MTH 345 discrete mathematics exploration. This data set also showed approximately 57% or 17 of the candidates’ work met or exceeded the expected level on the MTH 495 exploration in the area of number and number systems while approximately 53% or 16 of the candidates’ work was at these levels in the area of algebra. In addition, the same data set showed 15.4% - 33.3% or 6 - 11 of the candidates’ work on the MTH 341 geometry exploration and the two areas assessed in the MTH 495 exploration was a little less than expected, which resulted in the score Progressing. The historical and cultural perspectives data for each individual semester showed no candidates’ work on the MTH 345 discrete mathematics exploration and on the number and number systems portion of the MTH 495 exploration met the criteria for Unsatisfactory during four of the five semesters of the assessment’s implementation. The same data set further showed no candidates’ completed Unsatisfactory work on the MTH 341 geometry exploration during any of the five semesters. In general, candidates’ work that was evaluated as Progressing (not quite the expected level) on any portion of the historical and cultural perspectives assignment showed a conceptual understanding of the mathematical idea being explored but was lacking in one or more of the following ways: (1) several minor errors occurred in the use of mathematical symbols and terminology, (2) explanations did not go beyond what was presented in the sources they read, or (3) explanations were not fully developed for the mathematical idea explored. Candidates’ Unsatisfactory work was typically characterized by major conceptual errors related to the mathematical idea explored.


Historical and Cultural Perspectives in Mathematics Assignment was designed in response to data from our 2007 report. More specifically, the assessment was designed to address three particular areas (communication, connections, and historical and cultural aspects) faculty identified for strengthening in the Elementary Mathematics Program. As such, faculty set an initial goal that at least 2/3 of the candidates would complete work that met the criteria for Proficient or Distinguished in each area assessed through the assignment. The high percentage of candidates who demonstrated competence at or above the expected level on the MTH 341 geometry and the MTH 345 discrete mathematics explorations suggested the Elementary Mathematics Program met its goal in these areas. The data overall data for all three explorations further suggested most candidates were well-prepared



with respect to their knowledge of all areas assessed. Thus, the Elementary Mathematics Program meets the appropriate Middle Level standards focused on mathematical content knowledge. Faculty are puzzled by the data related to the three areas (communication, historical and cultural aspects of algebra and of numbers and number systems) assessed through the MTH 495 exploration. This data showed slightly more than half (53.3% - 56.7%) of the candidates completed work at or above the expected level in each area while approximately 1/3 (33.3% - 36.7%) of the candidates completed work at the Progressing level. The MTH 495 exploration data stands in sharp contrast to the data from the MTH 341 geometry and MTH 345 discrete mathematics explorations. Candidates complete MTH 341 and MTH 345 near the middle of the Elementary Mathematics Program and complete MTH 495 as the last required course in the program. Therefore, faculty initially expected candidates’ work on the MTH 495 historical and cultural perspectives exploration to be stronger. Faculty in the Department of Mathematics are currently trying to better understand the data related to the MTH 495 historical and cultural perspectives exploration. As a first step, faculty compared the MTH 495 exploration data for candidates in the Elementary Mathematics Program with the data for candidates in the Secondary Mathematics Program and those candidates who are not seeking teacher certification. All three populations take MTH 495 together and complete the exploration at the same time. The comparison showed a higher percentage of candidates in the Secondary Mathematics Program or in the non-teaching program completed work at or above the expected level (approximately 70% – 80%) on all areas assessed through the MTH 495 exploration. Faculty are now trying to gain deeper insights into the situation by asking good questions such as the following. Candidates complete MTH 495 as the last course in their program. In what ways does this influence their work on the historical and cultural perspectives assignment? In particular, are candidates already admitted to or enrolled in the College of Education and completing their practicum field experiences when they take MTH 495? If so, what influence might this have on candidates’ conscientiousness about completing high quality work on the assignment? Do candidates view the historical and cultural perspective exploration as relevant to their future teaching career and what influence might their view have on their work on the assignment? In what important ways does the nature of the MTH 495 exploration differ from the nature of the MTH 341 and 345 explorations and how might this influence candidates’ work on the exploration in each area assessed? When did candidates last take a course where they were asked to work extensively with algebra and what influence might this have on their work on the assignment? Is the writing instruction in MTH 495 sufficient or does the course need to include aspects of writing instruction that are not currently there? The next step is for faculty to explore possible answers to these questions to gain insights into factors that may help more candidates complete work that meets expectations on the MTH 495 portion of the historical and cultural perspectives assignment. These insights may also assist faculty in helping more candidates successfully complete work that meets or exceeds expectations on the MTH 341 geometry and MTH 345 discrete mathematics explorations in the assignment.

Assessment 7 – Attachment A: History & Cultures Assignment


Math 341 Euclidean Geometry Mathematical History & Culture Assignment:

Euclid’s Fifth Postulate

Assignment Overview:

Euclid’s fifth postulate has played an important role in the history of mathematics. Investigations of this postulate have highlighted significant contributions from diverse cultures to the historical development of Euclidean and non-Euclidean geometries. For this assignment, you will explore Euclid’s fifth postulate with respect to both its historical development and the contributions of diverse cultures to its development.

Assignment Requirements

Find the article “Euclid’s Fifth Postulate” by Renuka Ravindran. The citation for the article appears below. Journal: Resonance Publisher: Springer India ISSN: 0971-8044 (Print) 0973-712X (Online) Issue: Volume 12, Number 4 / April 2007 Pages: 26-36

Read the article carefully. Pay special attention to historical and cultural aspects that are addressed in the article. Consider the information presented in the article, as well as our class discussions on subject matter that can be related to Euclid’s fifth postulate. Write a paper that addresses the three items noted below. Be concise in your discussion and use proper mathematical language, terminology, and symbols throughout the paper. Additionally, your paper should be typeset or word processed.

1. Write a summary of the article by Ravindran. The summary needs to be stated in your own words and address the following items.

• Discuss similarities/ differences between spherical, hyperbolic and Euclidean Geometries, and how the attempts of proving Euclid’s fifth postulate, the parallel postulate, lead to the discovery of spherical and hyperbolic geometries.

• Do not list only theorems or postulates. Rather, state theorems and postulates as needed to support your discussion of similarities and differences.

2. Your instructor will select three of the mathematicians listed below who represent cultures that differ Euclid.

Al-Gauhary: Persian (9th centry) Omar Khayyám: Persian (1050-1123) John Wallis: English (1616-1703) Girolamo Saccheri: Italian (1667-1733) John Playfair: Scottish (1748 –1819) János Bolyai: Hungarian (1802 –1860) Carl Friedrich Gauss: German (1777-1855) Lobachevsky: Russian (1792-1856) Bernhard Riemann: German (1826-1866) Henri Poincaré: French (1854-1912)

http://en.wikipedia.org/wiki/Girolamo_Saccheri

http://en.wikipedia.org/wiki/1748


http://en.wikipedia.org/wiki/Hungary



http://en.wikipedia.org/wiki/Russian_language

Assessment 7 – Attachment A: History & Culture Assignment


Choose one of the three mathematicians to research and write about in your paper.

Use at least two sources for your research. The sources may be either print material from professional books and/or journals or they may be electronic sources. Provide the complete citation for each resource you used. In your paper, discuss the mathematician’s work in general and the mathematician’s contribution to Euclid’s fifth postulate, the parallel postulate. More specifically, briefly discuss the mathematician’s place in history and her/his main contribution(s) to the discipline of mathematics as a whole. Be sure to include a few important facts about the mathematician’s contributions and state at least one important finding of her/his work, such as a theorem or other significant proof. Follow the introduction with an in-depth discussion of the mathematician’s contributions to the story of the parallel postulate.

Your discussion needs to be written in your own words and reflect your understanding of the mathematician’s work and her/his contributions to Euclid’s fifth postulate.

3. The article by Ravindran presented several theorems related to Euclid’s fifth

postulate. One theorem, which was presented as Theorem 2, stated “The summit angles of a Saccheri quadrilateral are [congruent].” One proof of Theorem 2 was presented in the article.

In your paper, write a different mathematical proof for Theorem 2. Be sure to write the proof in neutral geometry according to your class writing guidelines.



Math 345 Discrete Mathematics Mathematical History & Culture Assignment:

The Arithmetical Triangle Assignment Overview

The arithmetical triangle has played an important role in the history of discrete mathematics. Investigations of this triangle have highlighted significant contributions from diverse cultures to the historical development of mathematical induction, combinations, and the binomial theorem. In this assignment, you will explore French mathematician Blaise Pascal’s Treatise on the Arithmetical Triangle and the historical and cultural development of a mathematical topic related to the arithmetical triangle. Assignment Requirements

The assignment consists of two related parts. In the first part, you will read an article on Pascal’s original work where he describes the arithmetical triangle and you will respond to specific items related to this work. The second part requires you to research a topic related to the arithmetical triangle from both a historical and cultural perspective.

Part 1 Read the excerpt from the article Pascal’s Treatise on the Arithmetical Triangle: Mathematical Induction, Combinations, the Binomial Theorem and Fermat’s Theorem by David Pengelley that is provided by your instructor. The full article can be found at http://www.math.nmsu.edu/hist_projects/pascalII.pdf. The excerpt from the article by Pengelley includes sections from Pascal’s original work Treatise on the Arithmetical Triangle. Read the excerpt carefully and respond to each of the items noted below. Be concise in your responses and use proper mathematical language, terminology, and symbols. 1. Describe, in your own words, how to construct an arithmetical triangle. Use your explanation

to create a triangle with 8 rows. 2. Pascal introduces the terms base, exponent, parallel row, and perpendicular row when

defining the arithmetical triangle. Explain in your own words what is meant by each of these four terms. Use specific examples from the triangle you created above to support your explanation.

3. Pascal’s Treatise on the Arithmetical Triangle presents several “consequences” and proofs of

these consequences. Pascal’s Twelfth Consequence is

“In every arithmetical triangle, of two contiguous cells in the same base the upper is to the lower as the number of cells from the upper to the top of the base is to the number of cells from the lower to the bottom of the base, inclusive.”

Apply Pascal’s Twelfth Consequence to the arithmetical triangle you created above by explaining how to calculate the ratio of the third and fourth entries in the eighth row.

http://www.math.nmsu.edu/hist_projects/pascalII.pdf



4. Pascal defines the arithmetical triangle recursively, but he also gives a method for computing

the entries in the triangle directly, which is found in the section titled “Problem” after the Twelfth Consequence. Apply Pascal’s formula to compute the fifth entry in the eighth row of the arithmetical triangle that you created.

Part 2 Your instructor will choose at least three of the mathematical topics noted below, all of which have connections to the arithmetical triangle. Choose one of the topics selected by your instructor and research the topic. Based on your research, write a paper that describes the mathematical topic, its historical and cultural development, and its connection to Pascal’s arithmetical triangle. Be concise in your discussion and use proper mathematical language, terminology and symbols throughout the paper. Additionally, cite the sources you used for your research. The audience for your report is other students who are taking MTH 345 Discrete Mathematics.

Fibonacci numbers Combinations and permutations Catalan numbers Narayana numbers Probability theory Triangular numbers Polygonal numbers Arithmetical triangle in China



Math 495: The Nature of Modern Mathematics Mathematical History & Culture Assignment:

Contributions of Diverse Cultures to Solving Equations Assignment Overview:

Methods and theorems about the solution of equations are interwoven throughout the history of mathematics. Investigations of equations and their solutions have highlighted significant contributions from diverse cultures to the historical development of algebra, number, and complex numbers. For this assignment, you will explore the development of solutions to equations from mathematical, historical and cultural perspectives.

Assignment Requirements

The assignment is divided into four parts, each of which focuses on aspects related to the historical development of and contributions of diverse cultures to solutions of equations. For each part, respond by providing a narrative in which you address the issues raised and questions asked. Be sure to use complete sentences and careful mathematical notation according to the standards established in Math 210 (Communicating in Mathematics). Your discussion should be typeset or word processed. 1. Quadratic Equations and the Quadratic Formula

Quadratic equations are studied in high school algebra. • Give a careful definition of a quadratic equation with real number coefficients. • There is a formula that gives the solutions of a quadratic equation in terms of the

coefficients of the equation. What is the quadratic formula? • Explain how to use a part of the quadratic formula (called the discriminant) to

determine if the quadratic equation has two real number solutions, one real number solution, or no real number solutions.

2. Reading: A Historical Development of the Solutions of Quadratic Equations

Read carefully the following narrative. You will respond to questions about the narrative in part 3 of this assignment. Some people have said that no new mathematics was created between the fall of the Roman Empire and the Renaissance. It is more correct to say that no new mathematics was created in Europe during this time. There were many mathematical concepts and ideas that were transmitted to the Europeans through their contact with the Arabic cultures. One such idea was methods of solving quadratic equations. We take it for granted that all quadratic equations can be solved by the use of the quadratic formula, but of course the quadratic formula is the product of the work of many pioneering mathematicians. We will begin our exploration of quadratic by looking at some of the work done by an Arabic mathematician named al-Khwarizmi, keeping in mind that none of his work used modern symbolic notation.

Abu Ja’far Muhammad ibn Musa al-Khwarizmi lived in Baghdad (in what is Iraq today) from about 790 to 840 C.E. He worked at the House of Wisdom translating mathematical texts from Greek to Arabic, and writing texts of his own. His works were so influential that his name and his work have made their way into our language! At least two important mathematical terms can be traced directly to Al-Khwarizmi.



Among al-Khwarizmi’s contributions to mathematics is his collection of methods for solving equations involving “squares, roots, and numbers,” what we would call quadratic equations.

Al-Khwarizmi’s most famous text is Al-kitab at-muhtasar fi hisab al-jabr was-I-muqabala or The Condensed Book on the Calculation of al- Jabr and al-Muqabala. In Arabic, al-jabr refers to restoring, or moving a subtracted quantity from one side of an equation to become an added quantity on the other side. An example of this is changing 3 2 4 2x x+ = − to 5 2 4x + = . On the other hand, al-muqabala means comparing, and refers to reducing a positive term by subtracting equal amounts from both sides of the equation, such as changing 5 2 4x + = to 5 2x = .

In his famous text, Al-Khwarizmi described how to solve linear equations, which were well known to the Greeks. He then went on to describe how to solve all types of quadratic equations. The algebra of al-Khwarizmi was done mostly in words — for example, instead of using 2x , he would refer to the square and would refer to x as the root.

Furthermore, Arabic mathematicians, like their Greek predecessors, avoided using negative numbers and zero. This led to five distinct types of equations involving squares and either roots, or numbers, or both, such as squares and numbers are equal to roots, which we would write as 2ax c bx+ = . Several of these types of equations had straightforward solutions, essentially reducing to either solving a linear equation or extracting a square root (a method for which was already well-known). Others, however, when they were first seen in Europe were novel. Here is a passage from al-Khwarizmi’s text that describes the solution for one of the non-trivial types:

Chapter IV: Concerning Squares and Roots Equal to Numbers: The following is an example of squares and roots equal to numbers: a square and 10 roots are equal to 39 units. So the question in this example can be stated as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39, (giving 64). Having taken then the square root of this, which is 8, subtract from it the half of the roots, 5, leaving 3. The number three therefore represents one root of this square, which itself, of course, is 9. Nine therefore gives that square.

Although Al-Khwarizmi didn’t prove his results, he did feel compelled to provide geometric justification for the procedures. For the example in the passage above, he starts with a square representing 2x , and then adds a rectangle with height 5 and width x to two adjacent sides of the square. So the combined area of the square and the two rectangles is 2 10x x+ , which is equal to 39. Because the missing square in the corner has area 25, we have that 2 10 25 64x x+ + = which quickly leads to 3x = .

3. Discussion of al-Khwarizmi

Research the work of al-Khwarizmi on your own to find out more about the mathematician and his contribution to the field of mathematics. Use at least two sources for your research. Write a discussion of al-Khwarizmi, based on the reading above and your own research. In your discussion:



• Include a brief biography of al-Khwarizmi and a brief description of the House of Wisdom.

• Find at least two important mathematical terms that were transliterated from either al-Khwarizmi’s name or works.

• Write down all possible types of equations, both in words and in symbols, of valid equations involving squares together with roots and/or numbers. (Note: Recall that al-Khwarizmi did not use negative numbers; neither zero nor negative solutions considered valid. In order for an equation to be considered valid it must have a positive solution.)

• Describe the solutions for the straightforward cases.

• The passage given above from al-Khwarizmi’s text describes one of the types of quadratic equations used by al-Khwarizmi. Rewrite this passage using familiar algebraic notation and describe the solution method.

• Draw the picture that goes with the solution and then explain why x must be 3.

• Discuss whether the solution method and picture can be generalized to the other non-trivial cases.

Provide the complete citation for each resource that you use. 4. Other Cultures and the Solutions of Equations

Cultures other than Arabic (above) and European (which we will study further in class) made significant contributions to the solving of equations. Some of these cultures are described below. Choose one of the cultures listed to research and write a short paper about, based on the directions below.

Use at least two sources for your research. The sources may be either print material from professional books and/or journals or they may be electronic sources. Provide the complete citation for each resource you used. In your paper, discuss the culture’s contribution to the solutions of equations. Be sure to include a few important facts about the culture’s contributions and state at least one important result, such as a theorem or other significant proof. Additionally, address specific questions that are noted in the description of a particular culture. Your discussion must be written in your own words and reflect your understanding of the culture’s contribution to the solution of equations. Babylonians Although methods of solving quadratic equations were transmitted from the Arabic cultures to Europe, mathematicians from other cultures and civilizations worked on quadratic equations prior to the work done by Al-Khwarizmi. As early as 1800 BCE, the ancient Babylonians could solve a system of equations of the form

and x y p xy q+ = = , where p and q are constants. The modern algebraic notation used in this system of equations was not used by the Babylonians.



In addition to the general discussion requirements above, be sure to address the following in your paper: • Who were the Babylonians? • Explain why solving this system of equations is equivalent to solving the quadratic

equation2x q px+ = .

• Give a description of how the Babylonians solved the specified system of equations.

(You may use modern algebraic notation to help in your description of the method.) India In the 8th century BCE, mathematicians in India used geometry to explore solutions of certain types of quadratic equations. Research the work of the mathematician Brahmagupta and at least one other Indian mathematician. (Some options are listed below.) In addition to the general discussion requirements above, be sure to address the following in your paper. • Give a brief biography of the Indian mathematician named Brahmagupta who

described how to solve certain quadratic equations. • Your paper should include descriptions, in words and in symbols, of at least one of

Brahmagupta's methods of solution for a quadratic equation. Compare and contrast Brahmagupta's work to that of Al-Khwarizmi.

Other Indian mathematicians to study to determine their contribution to algebra and number: Aryabhatta the Elder (476-550), Bhaskara II (1114-1185). China Jiuzhang suanshu (The Nine Chapters on the Mathematical Art) is the greatest of the Chinese classics in mathematics. This work consists of 246 problems separated into nine chapters. The problems deal with practical math for use in daily life. Additionally, the work contains problems involving the calculations of areas of all kinds of shapes, and volumes of various vessels and dams.

Liu Hui is the best known Chinese mathematician of the 3rd century, AD. In 263, he wrote a commentary on the Nine Chapters in which he verified theoretically the solution procedures, and added some problems of his own. He also presented Gauss-Jordan elimination and Calvalieri's principle to find the volume of a cylinder. Around 600 AD, Liu Hui’s work was separated from the Nine Chapters and published as the Haidao suanjing (Sea Island Mathematical Manual). Determine more information about the Sea Island Mathematical Manual when it comes to solving equations, as well as other Chinese contributions to this topic. Compare and contrast equation solving in China with the work of Al-Khwarizmi. For example, did China have solutions to quadratic equations before Al-Khwarizmi?



Italy Your paper needs to address the lettered items contained in the narrative below. Italian mathematicians were exploring the general solution to the cubic equation, while Italian explorers such as Christopher Columbus were sailing to the Western Hemisphere. In 1535, Tartaglia of Brescia, a.k.a. Niccolo Fontana (1499-1557), had discovered the general solution of the “depressed cubic”:

x3 + mx = n,

where m and n are real numbers. Tartaglia did not publish his result, and his solution was reported ten years later by Gerolamo Cardano (1501-1576) in his book Ars Magna, along with Cardano’s own solution to the general cubic equation:

x3 + bx2 + cx + d = 0,

where b, c, and d are real numbers. It should be noted that Cardano’s general solution relied on Tartaglia’s solution for the depressed cubic.

Cardano

1. Here, in Cardano’s own words (translated into English, of course), is the “Rule to

solve x3 + mx = n.”

Cube one-third of the coefficient of x; add to it the square of one-half the constant of the equation; and take the square root of the whole. You will duplicate [repeat] this, and to one of the two you add one-half the number you have already squared [the constant] and from the other you subtract one-half the same … Then, subtracting the cube root of the second from the cube root of the first, the remainder which is left is the value of x.

a. Write out Cardano’s method using symbolic notation. Explain your translation in

a step-by-step fashion with written commentary. b. Use your answer to “a” above to find a solution to x3 + 6x = 2. Then, explain

using a graph and important behavior of the function y = x3 + 6x (that is, use a key calculus idea) why your solution is the only real solution.

c. Use the quadratic formula to find the roots of x2 - 6x + 13 = 0. How many of

these roots are real? How could you determine the number of real roots from a graph?

d. Explain why every cubic equation has at least one real solution. In what situation

will a cubic equation have exactly two real solutions? Exactly three? Can a cubic equation have more than three different roots?

e. Use Tartaglia’s method (as described by Cardano) to get a solution to x3 – 15x =

4. Do not simplify your solution in any way. Does it appear that your root is real?



f. Use a graph to determine how many of the solutions to x3− 15 x=4 are real and how many are imaginary. What does this tell you about your answer to the item above?

g. Given that mathematicians at the time of Cardano were still leery of negative

numbers, how do you imagine they reacted to the solution that you discovered? h. Explain why it would have been possible for mathematicians to ignore square

roots of negative numbers that arise from the quadratic formula, but not from the cubic formula.

2. In his book Algebra, published in 1572, the Italian mathematician Rafael Bombelli

(1526-1572) took the radical approach of treating 1− like an acceptable number in order to get from the real cubic x3− 15 x=4 to its real solutions. We will follow his procedure in the following problems.

Bombelli

a. Simplify ( )12 −+ 3 b. Find a number of the form 1−b+a whose cube is 1212 −− + . c. Use your answers from “a” and “b” above to find a new expression for

3 12123 1212 −−−− ++=x .

Note: We should point out that it isn’t obvious how one might initially determine

that 3 1212 −+ is actually 12 −+ in disguise. In fact, it wasn’t until the 18th century that German mathematician Leonhard Euler discovered an algorithm for finding such roots.

d. What real number was Cardano’s formula really giving us? Use this information

to find the remaining solutions of x3− 15 x=4 . (Note: The description of the work from Italy is based on the discussion in William Dunham’s Journey Through Genius, 1991, Penguin Publishing.)

Assessment 7 – Attachment B: Scoring Rubrics for History & Cultures Assignment


MTH 341 Euclidean Geometry Mathematical History and Culture Assignment: Euclid’s Fifth Postulate


Communication

Indicator 3.2: Use the language of mathematics to express ideas precisely.


Candidate’s ability to use the language of mathemati cs to express ideas precisely and to communicate their thinking clearly, coherently, and in an organized manner.

This is demonstrated through the candidate’s written proof of the theorem “The summit angles of a Saccheri quadrilateral are equal”. It is also demonstrated through the candidate’s written discussion of the researched mathematician’s contributions to the proof of Euclid’s fifth postulate. In particular, the candidate presents a clear, concise, complete, mathematically correct, and valid proof of the theorem related to the summit angles of a Saccheri quadrilat eral. The candidate also articulates assumptions used in the proof and uses correct mathematical notation and terminology throughout the proof. Additionally, the candidate clearly discusses the major contribution of the researched mathematician. The candidate also clearly discusses the contribution of the researched mathematician to Euclid’s fi fth postulate and writes persuasively about fl awed reasoning in the mathematician’s work related to Euclid’s fi fth postulate.

__________ Candidate does not write a proof of the theorem related to the summit angles of a Saccheri quadrilateral, or the candidate writes a proof that is mathematically incorrect and not valid. The candidate’s use of mathematical notation and terminology in the proof is characterized by major error. OR Candidate does not discuss the major contributions of a researched mathematician to the field of mathematics and to Euclid’s fifth postulate,

________ Candidate writes a mathematically correct and valid proof of the theorem related to the summit angles of a Saccheri quadrilateral. The candidate does not clearly articulate assumptions used in the proof or makes several minor errors with respect to mathematical notation and terminology. The candidate may inadvertently use a result whose proof relies on this theorem, and therefore, uses invalid circular reasoning. AND* Candidate discusses the major contributions of the researched mathematician to the field of mathematics. The candidate also discusses the contributions of the researched mathematician to Euclid’s fifth postulate. The candidate identi fies the mathematician’s contributions to the story of Euclid’s fifth postulate. The candidate’s use of mathematical notation and terminology when discussing the researched mathematician’s work may contain several minor errors. .

________ Candidate writes a complete, mathematically correct and valid proof of the theorem related to the summit angles of a Saccheri quadrilat eral. The candidate also clearly articulates assumptions used in the proof. Additionally, candidate’s use of mathematical notation and terminology is largely correct throughout the proof, only a few minor errors may occur. AND* Candidate clearly discusses the major contributions of the researched mathematician to the field of mathematics. The candidate also clearly discusses the contributions of the researched mathematician to Euclid’s fi fth postulate and writes persuasively about the mathematician’s work related to Euclid’s fi fth postulate. The candidate’s use of mathematical notation and terminology when discussing the researched mathematician’s work may contain only a few minor errors. .

________ In addition to writing a complete, mathematically correct and valid proof of the theorem related to the summit angles theorem where assumptions are clearly articulat ed, the candidate’s proof is concise and free of errors. AND* In addition to clearly discussing the contribution of the researched mathematician to the field of mathematics and to the work of Euclid’s fifth postulate, the candidate’s discussion of the researched mathematician’s contributions for the story of Euclid’s fi fth postulate is highly detailed and free of errors.

Assessment 7 – Attachment B: Scoring Rubrics for History & Culture Assignment


Connections

Indicator 4.3: Demonstrate how mathematical ideas interconnect and build on one another to produce a coherent whole.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s ability to interconnect the mathematical ideas and see their progression into a coherent whole.

This is demonstrated through the candidate’s written summary of the article focused on historical and cultural aspects relat ed to Euclid’s fi fth postulate. It is also demonstrated through the candidate’s written discussion of the researched mathematician’s contributions to Euclid’s fifth postulate. In particular, the candidate discusses how various perspectives on geometry (i.e., Euclidean, neutral, & hyperbolic) connect to form a coherent whole. The candidate also connects one individual mathematician’s contributions to the larger story of Euclid’s fifth postulate in particular, and to the larger coherent whole of geometry, in general.

________ Candidate does not articulate and discuss various perspectives on geometry (i.e., Euclidean, spherical, & hyperbolic), or the candidate’s discussion of various perspectives on geometry is unorganized, nonsensical, or fails to address major ideas that connect the various perspectives.

________ Candidate articulates various perspectives on geometry (i.e., Euclidean, spherical, & hyperbolic). Candidate does not fully or completely accurately address how these various perspectives are connect ed to one another or how they collectively form a larger whole.

________ Candidate articulates various perspectives on geometry (i.e., Euclidean, spherical, & hyperbolic). Candidate also addresses how these various perspectives on geometry are connect ed to one another and collectively form a larger whole.

________ Candidate articulates various perspectives on geometry (i.e., Euclidean, neutral, & hyperbolic). Candidate also addresses how these various perspectives on geometry are connect ed to one another and collectively form a larger whole. Additionally, candidate’s discussion of the interconnectedness of various perspectives on geometry is not limited to ideas directly related to Euclid’s fifth postulate.

History of and Diverse Culture Contributions to Geometry

Indicator 11.7 (Middle Level): Demonstrate knowledge of the historical development of Euclidean and non-Euclidean geometries including contributions from diverse cultures.

Indicator 11.8 (Secondary): Demonstrate knowledge of the historical development of Euclidean and non-Euclidean geometries including contributions from diverse cultures.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s knowledge of the historical development of Euclidean and non-Euclidean geometries and contributions from diverse cultures.

This is demonstrated through the candidate’s written summary of the article describing the origin of Euclid’s Fifth Postulate in ancient Greek mathematics and subsequent attempts by mathematicians from diverse cultures to understand its relationship to the other four postulates. It is also demonstrated through the candidate’s discussion of a researched mathematician’s contributions to Euclid’s fi fth postulate. In particular, the candidate displays knowledge of the mathematical and philosophical implications of these developments, as well as the resulting interplay between culture and mathematics.

________ Candidate does not clearly describe the origin of Euclid’s axiomatic formulation of geometry and the distinct nature of Euclid’s Fifth Postulate within this formulation. OR Candidate does not clearly describe attempts by at least one other mathematician from a culture different than Euclid’s to prove or disprove Euclid’s Fifth Postulate as a consequence of the other four postulates. Candidate does not clearly describe how these attempts led to the discovery of non-Euclidean geometries.

________ Candidate describes, without complete clarity, the origin of Euclid’s axiomatic formulation of geometry and the distinct nature of Euclid’s Fifth Postulate within this formulation. AND* Candidate describes, without complete clarity, attempts by at least one mathematician from a culture different than Euclid’s to prove or disprove Euclid’s Fifth Postulate as a consequence of the other four postulates. Candidate describes, without complete clarity, how these attempts led to the discovery of non-Euclidean geometries.

________ Candidate clearly describes the origin of Euclid’s axiomatic formulation of geometry and the distinct nature of Euclid’s Fifth Postulate within this formulation. AND* Candidate clearly describes attempts by at least one mathematician from a culture different than Euclid’s to prove or disprove Euclid’s Fifth Postulate as a consequence of the other four postulates. Candidate clearly describes how these attempts led to the discovery of non-Euclidean geometries.

________ Candidate clearly describes the origin of Euclid’s axiomatic formulation of geometry and the distinct nature of Euclid’s Fifth Postulate within this formulation. AND* Candidate clearly describes attempts by at least one mathematician from a culture different than Euclid’s to prove or disprove Euclid’s Fifth Postulate as a consequence of the other four postulates. Candidate clearly describes how these attempts led to the discovery of non-Euclidean geometries. Additionally, candidate discusses the impact of these mathematical discoveries on philosophy, culture, and/or society.



MTH 345 Discrete Mathematics Mathematical History and Culture Assignment: The Arithmetical Triangle


Communication




This is demonstrated through the candidate’s written description of how to construct an arithmetical triangle and their ability to interpret and apply Pascal’s terminology and algorithms to their self-construct ed arithmetic triangle.

Candidate does not write a description of how to construct an arithmetical triangle or does not apply Pascal’s terminology and algorithms to their self-constructed arithmetical triangle. Candidate’s description of how to construct an arithmetical triangle or the application of Pascal’s terminology to their self-constructed arithmetical triangle lacks clarity or is characterized by major errors.

Candidate writes a mathematically correct description of how to construct an arithmetical triangle but the description is not fully developed. The candidate accurately applies and interprets Pascal’s terminology and algorithms to their self-constructed arithmetical triangle. Several minor errors may occur throughout the candidate’s written description and application of Pascal’s terminology and algorithms.

Candidate writes a complete and mathematically correct description of how to construct an arithmetical triangle. The candidate also accurately applies and interprets Pascal’s terminology and algorithms to their self-constructed arithmetical triangle. Additionally, candidate’s use of grammar, mathematical notation, and terminology is largely correct throughout, only a few minor errors may occur.

In addition to writing a complete, mathematically correct description of how to construct an arithmetical triangle and apply Pascal’s terminology and algorithms to their self-constructed triangle, the candidate’s description and applications are concise, clearly articulated, well-organized and free of errors.

Connections

Indicator 4.3: Demonstrate how mathematical ideas interconnect and build on one another to produce a coherent whole.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s ability to interconnect the mathematical ideas and see their progression into a coherent whole.

This is demonstrated through the candidate’s written discussion of their researched mathematical topic and its connection to the arithmetical triangle.

Candidate does not fully describe the researched mathematical topic or fails to address major ideas that connect the researched topic and the arithmetical triangle. Additionally, candidate’s discussion may be unorganized or nonsensical.

Candidate writes a complete and correct description of the researched mathematical topic. The discussion of appropriat e ways in which the researched topic is connected to the arithmetical triangle is not fully developed.

Candidate writes a complete and correct description of the researched mathematical topic. The discussion includes appropriate ways in which the researched topic is connect ed to the arithmetical triangle.

Candidate’s discussion of the researched mathematical topic and its connection to the arithmetical triangle is thorough, and thoughtfully developed. Additionally, the discussion may address connections between the researched topic and other important mathematical ideas.



History & Culture Area of Discrete Mathematics Indicator 13.3 (Middle Level): Demonstrate knowledge of the historical development of discrete mathematics

including contributions from other cultures.

Indicator 13.4 (Secondary): Demonstrate knowledge of the historical development of discrete mathematics including contributions from other cultures.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s knowledge of the historical development of discrete mathemati cs including contributions from diverse cultures.

This is demonstrated through the candidate’s written discussion of the historical and cultural development of the researched mathematical topic in the area of discrete mathematics.

Candidate does not discuss the historical and cultural development of the researched mathematical topic or the discussion is unorganized or nonsensical.

Candidate’s discussion of the historical and cultural development of the researched mathematical topic is not fully developed. The discussion does not fully address who or what culture is given credit for developing the mathematical topic, how the researched topic was used by the developing culture, or the significance of the topic to discrete mathematics.

Candidate clearly discusses the historical and cultural development of the researched mathematical topic. The discussion addresses who or what culture is given credit for developing the mathematical topic. The discussion also addresses how the researched topic was used by the developing culture and the significance of the topic to discrete mathematics.

In addition to clearly discussing the historical and cultural development of the researched mathematical topic, the candidate discusses the development with respect to multiple origins or discusses how the topic was introduced to or used by other cultures.



MTH 495 The Nature of Modern Mathematics Mathematical History and Culture Assignment: Contributions of Diverse Cultures to Solving Equations


Communication




This is demonstrated through the candidate’s use of correct mathematical notation and terminology when communicating about the quadratic formula, as well as when discussing the work of al-Khwarizmi and the contributions of a researched culture to solutions of equations.

Candidate does not write a description of how to use the determinant of the quadratic formula to determine the number of solutions to a quadratic equation. Candidate’s use of modern algebrai c notation to represent the solutions methods of al-Khwarizmi and that of the researched culture is characterized by major errors.

Candidate writes a mathematically correct description of how to use the determinant of the quadratic formula to determine the number of solutions to a quadratic equation but the description is not fully developed. The candidate accurately uses modern algebraic notation to describe the solution methods of al-Khwarizmi and of the researched culture. Several minor errors may occur throughout the candidate’s written description and use of modern algebraic notation.

Candidate writes a complete and mathematically correct description of how to use the determinant of the quadratic formula to determine the number of solutions to a quadratic equation. The candidate accurately uses modern algebrai c notation to describe the solution methods of al-Khwarizmi and of the researched culture. Additionally, candidate’s use mathematical notation and terminology is largely correct throughout, only a few minor errors may occur.

In addition to the criteria noted for “profici ent”, the candidate’s work throughout is concise, clearly articulated, well-organized and free of errors.

History & Culture Area of Algebra and of Number and Number Systems

Indicator 9.8 (Middle Level): Demonstrate knowledge of the historical development of number and number systems including contributions from other cultures.

Indicator 9.10 (Secondary): Demonstrate knowledge of the historical development of number and number systems including contributions from other cultures.

Indicator 10.6: (Middle Level & Secondary): Demonstrate knowledge of the historical development of algebra including contributions from diverse cultures.

ELEMENT (0) UNSATISFACTORY (1) PROGRESSING (2) PROFICIENT (3) DISTINGUISHED Candidate’s knowledge of the historical development of algebra and of number and number systems. This is demonstrated through the candidate’s written discussion of the work of al-Khwarizmi and that of the researched culture with respect to their contribution to solutions of equations and the development of numbers. It is also demonstrated through the candidate’s use of the historical methods to solve equations.

Candidate does not fully describe the work of Al-Khwarizmi and that of the researched culture with respect to solutions of equations and the development of numbers. Candidate’s discussion may be unorganized or nonsensical. The candidate’s use of the historical methods of Al-Khwarizmi and that of the researched culture to solve equations is characterized by major errors.

Candidate writes a mathematically correct description of the work of al-Khwarizmi and that of the researched culture. However, the description is not fully developed. It may not address contributions made to the solutions of equations and the development of numbers. The candidate uses the historical methods of Al-Khwarizmi and that of the researched culture to solve equations. Several minor errors may occur in the solutions.

Candidate writes a complete and mathematically correct description of the work of al-Khwarizmi and that of the researched culture. The discussion includes significant contributions made to the solutions of equations and the development of numbers. The candidate uses the historical methods of Al-Khwarizmi and that of the researched culture to solve equations. Only a few minor errors may occur in the solutions.

Candidate’s discussion of the work of al-Khwarizmi and that of the researched culture is thorough and thoughtfully developed. Additionally, the discussion may address connections between various cultures with respect to solving equations or the development of numbers. It may also address connections between historical methods of solving equations and other important mathematical ideas. The candidate’s use of the historical methods of Al-Khwarizmi and that of the researched culture to solve equations is free of errors.

Assessment 7 – Attachment C: Data Tables for History & Cultures Assignment


Candidate Data for Historical and Cultural Perspectives in Mathematics Assignment

(3.2) Communication (from MTH 341 Euclidean Geometry)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

Total N = 39 1 (2.6%) 10 (25.6%) 19 (48.7%) 9 (23.1%) Overall percent of candidates scoring a 2 or 3 – 71.8% (3.2) Communication (from MTH 345 Discrete Mathematics)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

Total N = 39 4 (10.3%) 8 (20.5%) 22 (56.4%) 5 (12.8%) Overall percent of candidates scoring a 2 or 3 – 69.2% (3.2) Communication (from MTH 495 The Nature of Modern Mathematics)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008 N = 5

Winter 2009 N =6

Spring 2009 N = 8

Fall 2009 N = 8

Winter 2010 N = 3


Assessment 7 – Attachment C: Data Tables for History & Culture Assignment


(4.3) Connections (from MTH 341 Euclidean Geometry)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

Total N = 39 0 (0%) 7 (17.9%) 25 (64.1%) 7 (17.9%) Overall percent of candidates scoring a 2 or 3 – 82.0% (4.3) Connections (from MTH 345 Discrete Mathematics)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

Total N = 39 2 (5.1%) 8 (20.5%) 21 (53.8%) 8 (20.5%) Overall percent of candidates scoring a 2 or 3 – 74.3% (11.7/11.8) Historical & Cultural Perspectives – Geometry (from MTH 341 Euclidean Geometry)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010


Assessment 7 – Attachment C: Data Tables for History & Culture Assignment


(13.3/13.4) Historical & Cultural Perspectives – Discrete Mathematics (from MTH 345 Discrete Mathematics)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

Total N = 39 2 (5.1%) 1 (2.6%) 30 (76.9%) 6 (15.4%) Overall percent of candidates scoring a 2 or 3 – 92.3% (9.8/9.10) Historical & Cultural Perspectives – Number & Number Systems (from MTH 495

The Nature of Modern Mathematics)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

Total N = 30 2 (6.7%) 11 (36.7%) 17 (56.7%) 0 (0%) Overall percent of candidates scoring a 2 or 3 – 56.7% (10.6) Historical & Cultural Perspectives – Algebra (from MTH 495 The Nature of Modern

Mathematics)

Semester Taken



(0) Unsatisfactory

(1) Progressing

(2) Proficient

(3) Distinguished

Fall 2008

Winter 2009

Spring 2009

Fall 2009

Winter 2010

Total N = 30 3 (10%) 10 (33.3%) 13 (43.3%) 3 (10%) Overall percent of candidates scoring a 2 or 3 – 53.3%

Documents

› eresources › pdf › nctmm_grandvalley.pdf · Program Report for the Preparation of Middle Level Mathematics Teachers National Council of Teachers of Mathematics (NCTM) Option