What Important Mathematics doElementary School Teachers Need to Know?
How Should Elementary School Teachers Come to Know Mathematics?
Jim LewisUniversity of Nebraska-Lincoln
Math MattersA Mathematics Mathematics Education Partnership at theUniversity of Nebraska-Lincoln
Ruth Heaton and Jim Lewis
Whats the big deal?
Cant any good elementary school teacher teach children elementary school mathematics?
What is so difficult about the preparation of mathematics teachers?Our universities do not adequately prepare mathematics teachers for their mathematical needs in the school classroom. Most teachers cannot bridge the gap between what we teach them in the undergraduate curriculum and what they teach in schools.We have not done nearly enough to help teachers understand the essential characteristics of mathematics: its precision, the ubiquity of logical reasoning, and its coherence as a discipline.The goal is not to help future teachers learn mathematics but to make them better teachers. H. Wu
What is so difficult .?For most future elementary school teachers the level of need is so basic, that what a mathematician might envision as an appropriate course is likely to be hopelessly over the heads of most of the students.
Forget what seems like a good thing to do in some idealized world, and get real: adapt the courses to the actual level and needs of the students.
The mathematics taught should be connected as directly as possible to the classroom. This is more important, the more abstract and powerful the principles are. Teachers cannot be expected to make the links on their own.
Get teacher candidates to believe, that mathematics is something you think about - that validity comes from inner conviction that things make sense, that mathematical situations can be reasoned about on the basis of a few basic principles.
The goal is to have them develop some flexibility in their thinking, to be able to reason about elementary mathematics.Roger Howe
Adding it Up argues that mathematical proficiency has five strands:Conceptual understandingComprehension of mathematical concepts, operations, and relationsProcedural fluencySkill in carrying out procedures flexibly, accurately, efficiently, and appropriately
mathematical proficiency Strategic competenceAbility to formulate, represent, and solve mathematical problemsAdaptive reasoningCapacity for logical thought, reflection, explanation, and justificationProductive dispositionHabitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and ones on efficacy.
Educating Teachers of Science, Mathematics, and TechnologyNew Practices for the New Millennium
A report of the National Research CouncilsCommittee on Science and Mathematics Teacher Preparation
Educating Teachers argues:1) Many teachers are not adequately prepared to teach science and mathematics, in ways that bolster student learning and achievement. 2) The preparation of teachers does not meet the needs of the modern classroom.3) Professional development for teachers may do little to enhance teachers content knowledge or the techniques and skills they need to teach science and mathematics effectively.
and recommends:a new partnership between K-12 schools and the higher education community designed to ensure high-quality teacher education and professional development for teachers.
The Mathematical Education of TeachersIs guided by two general themes:
the intellectual substance in school mathematics; and
the special nature of the mathematical knowledge needed for teaching.
The MET recommends:Prospective teachers need mathematics courses that develop a deep understanding of the mathematics they will teach.Prospective elementary grade teachers (K-4) should take at least 9 semester-hours on fundamental ideas of elementary school mathematics.
Mathematics courses should focus on a thorough development of basic mathematical ideas. develop careful reasoning and mathematical common sense in analyzing conceptual relationships and in solving problems. develop the habits of mind of a mathematical thinker and demonstrate flexible, interactive styles of teaching.
The mathematical education of teachers should be based onpartnerships between mathematics and mathematics education faculty.collaboration between mathematics faculty and school mathematics teachers.
The Math Matters VisionCreate a mathematician mathematics educator partnership with the goal of improving the mathematics education of future elementary school teachersLink field experiences, pedagogy and mathematics instruction Create math classes that are both accessible and useful for future elementary school teachers
MATH MATTERS Timeline Project began 01/01/2000 with NSF support.AY 2000/2001, first Math Matters cohort (16 students) participated in a year-long, 18-hour block of courses. Each semester had a math class, a pedagogy class, and a field experience.Significant progress in learning mathematics and in developing a positive attitude towards teaching mathSignificant growth in potential to be an outstanding teacherStrong bonding between students and faculty
MATH MATTERS Timeline Math Matters was repeated with a second cohort (16 students) during AY2001/2002Findings similar to first cohort AY 2002/2003 Two experiments with a one-semester, 12 hour block of courses (28 students, 19 students)Fall 2003 The Mathematics Semester beginsCurrent program is a one-semester, 10-hour, math, pedagogy, & field experience program.
Barriers to a Successful Partnership Math expectations seem to overwhelm students in Elementary EducationStudent evaluations critical of math facultyType of Course Faculty GPA#StudentsHonors class 3.20 1,367All faculty courses 3.0416,693Large Lectures2.88 6,060Education Majors2.48 726
Comments from a math class for elementary school teachers: (the course GPA was 2.93)This wasn't a course where we learn to teach math. Why do we have to explain our answers.I did not like getting a 0 on problems that I attempted. I could have just of left the problem blank then. tests are invalid. They ask questions we have never seen before. It would help if we knew more about the questions on the exams - If examples in class were used on the exams.
Comments from a math class for elementary school teachers: (the course GPA was 2.93)Her way of assessing her class aren't fair.
there was never partial credit. When 20 people drop a class ... there is an obvious problem. [Note: Only 3 of 33 students dropped the class.]
Test materials were not consistent or reliable with the material covered in class. Grading was very biased.
Comments from a Contemporary Math class (She) does a good job making the subject matter interesting. She always seems very enthusiastic about the class and and actual work. More teachers should be like her.(She) is a great teacher with a love for her subject that becomes addictive. It has really been my lucky pick to have gotten her as an instructor.(She) made the class exciting. It is obvious she enjoys math and teaching. She was always clear in her expectations and directions.
Comments from a Contemporary Math classThis was a very useful class. I also think that (she) is a great teacher.(She) was one of the best teachers I have had here at UNL. She was always available for questions!This was a very good class. I failed the class last semester with a different teacher but (she) did a much better job and I am doing great in the class!
Barriers to a Successful PartnershipStudents took math courses before admission to Elementary Education ProgramMath for Elementary Education was often taught by graduate students or part-time lecturersCultural differences in how instruction delivered and students assessedFall 2000 Undergraduate GPA by Dept.Math2.53(UNLs lowest)Curr & Inst3.64(among highest)
Beliefs of Math and El Ed Faculty1 2 3 4Strongly Disagree Disagree Agree Strongly Agree El EdMath Question 1.712.12 Stated from Traditional Viewpoint------------------------------------------------------------------------------2.002.92Algorithms are best learned through repeated drill and practice.1.572.55An advantage of teaching math is that there is one correct answer.2.003.22Frequent drills on the basic facts are essential in order for children to learn them.1.832.70Time should be spent practicing computational procedures before children are expected to understand the procedures.2.832.14The use of key words is an effective way for children to solve word problems.
Beliefs of Math and El Ed Faculty1 2 3 4Strongly Disagree Disagree Agree Strongly Agree3.273.07 Stated from Reform Viewpoint ------------------------------------------------------------------------3.292.25Teachers should let children work from their own assumptions when solving problems.3.862.78Mathematics assessment should occur every day.2.713.40Leading a class discussion is one of the most important skills for a math teacher.
What is the typical ability of a future elementary school teacher at UNL?How would our students do on two famous problems?1) How do you solve the problem 1 / ?Imagine you are teaching division with fractions. Teachers need to write story-problems to show the application of some particular piece of content. What would be a good problem for 1/ ?
In Knowing and Teaching Elementary Mathematics this question was asked of 23 US teachers. Liping Ma reports:9 of 23 US teachers could perform the calculation using a correct algorithm and provide a complete answer.Only 1 of 23 US teachers could create a valid story problem for the computation. It was pedagogically problematic in that the answer involved 3 children.
Surely our teachers and future teachers can do better? In the summer of 2002, Ruth interviewed a group of Lincoln Public School teachers who were participating in a summer workshop Jim had organized.14 of 17 (elem and middle level) LPS teachers tested could perform the division of fractions algorithm correctly.Only 4 out of 17 could create a valid word problem. 3 of these 4 were middle school teachersWe also asked our students to work this problem.AlgorithmWord ProblemFall 2003 Students23 of 25 8 of 25Year 2 (early in sem)10 of 16 6 of 16Year 1 (mid year)12 of 1610 of 16
Sample solutions from our class last Fall:Among the better answers we received were these:Bobby only ran the distance that he hoped to run in practice. If Bobby ran 1 miles and that was only of the amount he wanted to run, what was his original goal?Bill is running a race that is 1 miles. The director of the race wants to put markers every mile. How many markers does he need?
Sample solutions from our class this Fall:Other solutions included the following:Suzie has one and pies. Ed is hungry and decides he is going to eat of what Suzie has. When Ed is finished, how many halves of the pie does Suzie have left?Jane has 7 quarters. She wants to put the quarters into groups of 2 quarters each. How many groups can she make?You have 1 of a pie! That is not enough. You want more of what they started with. How much more will you need? Mom baked 2 pies and divided the pies in quarters. After giving away one slice shes left with 1 pies. She then divides each left over piece into halves. How many pieces does she have now?
Perimeter and Area2) A student comes to class excited. She tells you she has figured out a theory you never told the class. She says she has discovered that as the perimeter of a closed figure increases, the area also increases. She shows you a picture to prove what she is doing.
The square is 4 by 4Perimeter = 16Area = 16
The rectangle is 4 by 8Perimeter = 24Area = 32
The first day of our Fall 2002 class, we asked our (geometry) students to respond to the area perimeter question. 24 of 32 future elementary teachers believed the childs theory was correct and indicated that they would congratulate the child.Eight of 32 future elementary teachers questioned the child's theory about area and perimeter. Only six of the eight explained that the theory was definitely not true.
Mental Math Quiz (Work all problems in your head.)1) 48 + 39 =2) 113 98 =3)14 x 5 x 7 =4) is closest to what integer?5)4 x 249 =6) 6(37 + 63) + 18 = 7) .25 x 9 =8) 12.03 + .4 + 2.36 =9) 1/2 + 1/3 = 10) 90% of 160 =11) The sum of the first ten odd positive integers (1+3+5++17+19) is equal to what integer?12) If you buy items (tax included) at $1.99, $2.99 and $3.98, the change from a $10 bill would be?13) To the nearest dollar, the sale price of a dress listed at $49.35 and sold at 25% off is _____?
Mental Math Quiz (Work all problems in your head.)14)The area of a square of perimeter 20 is ___?The ratio of the area of a circle of radius one to that of a circumscribed square region is closest to? a) .5,b) .6, c) .7, d) .8, e) .916) The average (arithmetic mean) of 89, 94, 85, 90, and 97 is _____ ?17) If 4/6 = 16/x, then x = _____ ?18) If 2x + 3 = 25, then x = _____ ?19) The square root of 75 is closest to what integer?20) To the nearest dollar, a 15% tip on a restaurant bill of $79.87 is _____ ?
How do UNLs future elementary school teacher do on the Mental Math Quiz?GroupNumberMedian AverageCorrect CorrectMM Fall 20001615.5 14.8GW Fall 20032113 13.1LOK Fall 2003*1912 12.7SW Fall 2003*2312 12.4MM Fall 20011612 12.3BH Fall 20003212 11.9JL Fall 20032412 11.8MM Fall 2002*2812 11.6MG Fall 20013512 11.5WH Fall 20002411 11.0WH Fall 20013210 10.9TM Fall 2003*2010.5 10.4Total29012 11.9*LPS Sum Wks.2213.5 13.7* Have already taken the Arithmetic course.
The Mathematics Semester(For all Elementary Education majors starting Fall 2003)MATHMath 300 Number and Number Sense (3 cr)PEDAGOGYCURR 308 Math Methods (3 cr)CURR 351 The Learner Centered Classroom (2 cr) FIELD EXPERIENCECURR 297b Professional Practicum Exper. (2 cr)(at Roper Elementary School)Students are in Roper Elementary School on Mondays and Wednesdays (four hours/day)Math 300 & CURR 308 are taught as a 3-hour block on Tuesday and ThursdayCURR 351 meets at Roper on Wednesdays
A look inside Math Matters(and The Mathematics Semester)Early math assignments establish our expectationsProfessional/Reflective WritingsCurriculum MaterialsNumber and Number Sense ItemsWorking together as a teamThe Curriculum ProjectSome Geometry AssignmentsActivities at RoperTeaching a Math LessonChild StudyLearning and Teaching Project
A look inside Math Matters(and The Mathematics Semester)Early math assignments establish our expectationsProfessional/Reflective WritingsCurriculum MaterialsNumber and Number Sense ItemsWorking together as a teamThe Curriculum ProjectSome Geometry AssignmentsActivities at RoperTeaching a Math...