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Curricular Touchstones for Secondary (Mathematics)
Methods Courses
SMTI 2015
Sean P. Yee, University of South Carolina
Samuel Otten, University of Missouri
Megan W. Taylor, Center to Support Excellence in Teaching,
Stanford University
STaR Fellows (amte.net/star)
Megan W. Taylor, Center to Support Excellence in Teaching, Stanford University
Samuel Otten, University of Missouri [email protected]
Sean P. Yee, University of South Carolina [email protected]
Agenda
- (10 min.) Background and Research Design - (10 min) Discussion of Quantitative Data - (10 min.) Discussion for Future Development
Background ● Teacher preparation programs are diverse and involve many complex,
interrelated features. (Sowder, 2007)
● In mathematics teacher education, there is “no shared professional curriculum” and so PSTs’ experiences “reflect the orientations and expertise of their instructors and cooperating teachers” (Ball et al., 2009, p. 259)
● Methods content and structure are inconsistent (Kidd, 2008; Taylor &
Ronau, 2006) ● Inconsistency prevents our field from making substantial progress on a
broad scale (Ball et al., 2009)
Achieving Systematic Improvement (Arbaugh and Taylor, 2008)
Phase 1
Studying a Single Course or Single
Teacher Preparation Program at a Single
Institution
Phase 2
Studying a Single Course or Single
Teacher Preparation Program at Multiple
Institutions
Phase 3
Comparing Programs with
Varying Features Across Multiple
Sites
Background
● Guiding Question
o What does the field believe all prospective secondary mathematics teachers learn in methods courses?
● Purpose
o To gather input from secondary mathematics teacher educators from across the country with regard to what topics/foci they value in methods courses
Background
● “Touchstone” (n.) a piece of fine-grained dark schist or jasper formerly used for testing alloys of gold by observing the color of the mark that they made on it.
● Used here to refer to a framework of comparison, identified by the
professional community, that could be touched upon or addressed in secondary methods courses generally.
Research Design
Collect touchstones from courses or
Suggest touchstones and have a rating?
Research Design
● 41 potential “touchstones” developed from existing research, collected syllabi, researcher experience o e.g., “Knowledge of written curriculum materials,” “Understanding of
content standards,” “formative assessment,” “productive classroom discourse”
● balancing comprehensiveness and specificity
Research Design
● survey of mathematics teacher educators o value of each touchstone to a secondary mathematics methods course: 5-
point Likert rating (1=not at all important; 5=very important) o professional information: title, department, methods course teaching
experience
● piloted with Service, Teaching and Research (STaR) Fellows in math education, summer 2013
● sent digitally to AMTE members (N>1000), winter 2014, with responses solicited from those involved in secondary mathematics education (N=116)
Research Design
Meet and Greet
•Take 5 minutes, introduce yourself to some people near you, tell them your institution and your role there. •Discuss which touchstones you find are valuable to
you.
Results: Ordered Averages
● Most touchstones were rated between 4.00 and 4.71 (5=most important)
● This indicates o validity to the set of touchstones proposed o a broad set of values and foci within secondary methods o On free response, 5/116 people or fewer suggested any specific additional
touchstone. Only 32/116 people suggested anything at all.
Coded Topic
(Number of
Suggestions)
Examples
Trajectories
(4 Suggested)
The trajectory of mathematics concepts from middle school to high school.
I'm not sure what "curriculum vision" means. I would add "progressions of development of
key ideas within standards."
Big Ideas
(3 Suggested)
Examining the 'big ideas' of mathematics
Identifying "big ideas" in a unit of a study.
Handling Errors
(3 Suggested)
How do you work with high school students who really don't understand fractions? or
integers? or variables?
Learning that students' incorrect ways of reasoning may actually still make sense from the
students' points of view.
Interdisciplinarity
(2 Suggested)
Knowledge of the Next Generation Science Standards since mathematics is addressed in
the Dimensions of Practice.
I think we should be teaching in every methods course (math, science, English, modern
languages, etc.) the ability to value and learn from other teachers in the various other
fields—that is, helping our candidates know how to look at an activity/approach in the
English classroom and modify/use it in the mathematics classroom. We're too stuck in our
mathematics silo and need to work the culture of this field away from that myopic
approach.
Results: Ordered Averages
Top Four
Touchstones
Understanding of
practice/process
standards
(M=4.71, SD=0.56)
Multiple
representations of
math. ideas
(M=4.68, SD=0.57)
Attending to
student thinking
and understanding
(M=4.68, SD=0.58)
Mathematical
knowledge for
teaching
(M=4.68, SD=0.64)
Bottom Four
Touchstones
Teaching theories
and applications
(M=3.54, SD=0.88)
Read educational
research
(M=3.38, SD=0.90)
History and nature
of mathematics
(M=3.09, SD=0.94)
Doing educational
research
(M=2.78, SD=1.07)
Touchstone TS3 TS9 TS10 TS14 TS15
Description
understanding of
content standards (e.g.,
CCSS, state, district,
school)
enacting mathematical
tasks
informal assessment
(e.g., observation,
conversations with
students)
issues of equity,
status, fairness,
and social justice
needs of
underrepresented
populations
Equal Variance Assumed
or Not Not Assumed Not Assumed Assumed Not Assumed Not Assumed
t-score 3.399 -2.288 -3.258 -3.205 -3.601
Deg. of Freedom 96.557 62.510 104 53.573 55.199
Significance (Two-
Tailed) 0.001 0.026 0.002 0.002 0.001
Education Dept. Mean
(N=70) 4.429 4.686 4.529 4.271 4.271 Education Dept.
Standard Deviation 0.627 0.627 0.583 0.779 0.779
Mathematics Dept.
Mean (N=36) 4.778 4.361 4.028 3.611 3.556
Mathematics Dept.
Standard Deviation 0.422 0.723 1.000 1.103 1.054
Touchstone TS12 TS34
Description
summative
assessment to
assess student
understandings
mathematical
content
knowledge
Levene Statistic and Significance (N=95)
(Equal Variance Assumed for both TS12 and TS 34)
F(94)=2.405
p=0.096
F(94)=1.649
p=0.198
F-score F(94)=3.869 F(94)=3.219
Significance 0.024 0.045
Mean and Standard Deviation for Assistant Professors M=3.688
SD=0.965
M=4.438
SD=0.716
Mean and Standard Deviation for Associate Professors M=4.200
SD=0.714
M=3.867
SD=1.042
Mean and Standard Deviation for Full Professor/Emeritus M=4.121
SD=0.650
M=4.061
SD=0.933
Post-Hoc LSD Test Between Assistant and Associate
Professors
Mean Difference
Standard Error
Significance
Significant
Mdiff=-0.513
SE=0.200
p=0.012
Significant
Mdiff=0.571
SE=0.230
p=0.015
Post-Hoc LSD Test Between Associate and
Professor/Emeritus
Mean Difference
Standard Error
Significance
Not Significant
Mdiff=0.079
SE=0.199
p=0.693
Not Significant
Mdiff=-0.194
SE=0.228
p=0.397
Post-Hoc LSD Test Between Assistant and
Professor/Emeritus
Mean Difference
Standard Error
Significance
Significant
Mdiff=-0.434
SE=0.196
p=0.029
Not Significant
Mdiff=0.377
SE=0.224
p=0.096
Discussion
● We see significant variety in certain touchstones between departments of faculty.
● We see significant variety in certain touchstones with levels of experience.
● We have a general rating format to share with the field. ● What do you see as future implications for this research and
where it should go next? ● How can our research better support and inform SMTI?
Thank You!
Megan W. Taylor, Center to Support Excellence in Teaching, Stanford University
Samuel Otten, University of Missouri [email protected]
Sean P. Yee, University of South Carolina [email protected]
References
● Ball, D. L., Sleep, L., Boerst, T. A., & Bass, H. (2009). Combining the development of practice
and the practice of development in teacher education. Elementary School Journal, 109, 458-
474.
● Kazemi, E., Franke, M., & Lampert, M. (2009). Developing pedagogies in teacher education to
support novice teachers’ ability to enact ambitious instruction. In Crossing divides:
Proceedings of the 32nd annual conference of the Mathematics Education Research Group of
Australasia (Vol. 1, pp. 12-30).
● Kidd, M. (2008). A comparison of secondary mathematics methods courses in California. In P.
M. Lutz (Ed.), California Association of Mathematics Teacher Educators Monograph (Vol. 1, pp.
1-5).
● Taylor, P. M., & Ronau, R. (2006). Syllabus study: A structured look at mathematics methods
courses. AMTE Connections, 16(1), 12-15.
Curricular Touchstones for Secondary (Mathematics) Methods Courses
Sean Yee, University of South Carolina, [email protected]
Samuel Otten, University of Missouri, [email protected]
Megan W. Taylor, Stanford University, [email protected]
116 Higher Education Instructors of Methods Courses Participated
31% Mathematics Dept., 60% Education Dept., 7% Joint, 2% Other
28% Full Professor, 26% Associate Professor, 28% Assistant Professor, 10% Adjunct, 4% Graduate Student
Touchstone and Description
TS1 curriculum vision
TS2 knowledge of written curriculum materials
TS3 understanding of content standards (e.g, CCSS, state, district, school)
TS4 understanding of practice/process standards (e.g., CCSS, NCTM, NRC)
TS5 choosing and writing instructional goals
TS6 lesson and unit planning
TS7 cognitive features of mathematical tasks
TS8 adapting, choosing, and generating mathematical tasks
TS9 enacting mathematical tasks
TS10 informal assessment (e.g., observation, conversations with students)
TS11 formative assessment (on-going assessment)
TS12 summative assessment to assess student understandings
TS13 expectations, purposes, and design of homework
TS14 issues of equity, status, fairness, and social justice
TS15 needs of underrepresented populations
TS16 multiple representations of mathematical ideas
TS17 the relationship between conceptual and procedural knowledge
TS18 pedagogies that address different types of knowledge and skills (e.g., procedural, conceptual, strategic,
declarative)
TS19 relationship between participation structures (e.g., pair work, complex instruction) and cultural and learning
Goals.
TS20 productive classroom discourse
TS21 positive classroom culture
TS22 sociomathematical norms
TS23 roles of the mathematics teacher (e.g., teacher as guide, teacher as lecturer)
TS24 mathematical applications or mathematics in context
TS25 digital tools and technologies (e.g., calculators)
TS26 analog tools and technologies (e.g., manipulatives)
TS27 classroom management that supports cultural and learning goals
TS28 attending to student thinking and using student ideas to push understandings forward
TS29 motivating students to persevere and take risks
TS30 nature of problem-solving
TS31 students’ metacognitive skills
TS32 history and nature of mathematics
TS33 personal and societal beliefs about teaching and learning mathematics
TS34 mathematical content knowledge
TS35 mathematical knowledge for teaching
TS36 reflection on practice and development as a professional educator
TS37 repertoires of effective mathematical teaching practices and pedagogical tools
TS38 read educational research
TS39 teaching theories and applications to practice
TS40 learning theories and applications to practice
TS41 do educational research (e.g., Action Research)
