The Mathematical Education of Teachers

Jim Lewis

University of Nebraska – Lincoln

The Challenge

• What Mathematics do Elementary and Middle Level Teachers “Need to Know” and How Should They “Come to Know” Mathematics?

– What does it mean to offer challenging courses and curricula for middle level math teachers?

– How do we help teachers translate the mathematics they come to know into classroom practice that leads to improved student learning?

What’s the big deal?

Can’t any good K-8 school teacher teach children 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.

• 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 understanding– Comprehension of

mathematical concepts, operations, and relations

• Procedural fluency– Skill in carrying out

procedures flexibly, accurately, efficiently, and appropriately

mathematical proficiency …

• Strategic competence– Ability to formulate, represent, and solve

mathematical problems

• Adaptive reasoning– Capacity for logical thought, reflection, explanation,

and justification

• Productive disposition– Habitual inclination to see mathematics as sensible,

useful, worthwhile, coupled with a belief in diligence and one’s on efficacy.

Educating Teachers of Science, Mathematics, and Technology

New Practices for the New Millennium

A report of the National Research Council’s

Committee 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 Teachers

funded by the U.S. Department of Education

Themes• the intellectual substance in school mathematics; and• the special nature of the mathematical knowledge

needed for teaching.

Recommendations• Teachers need mathematics courses that develop a

deep understanding of the math they teach.• Middle grades teachers (5-8) should take at least 21

semester-hours of mathematics including 12 hours on fundamental ideas of school mathematics appropriate for middle grades teachers.

The MET Recommends

• 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 math education of teachers should be based on– partnerships between mathematicians,

mathematics education faculty and school mathematics teachers.

Mathematical Habits of MindA person with the habits of mind of a mathematical thinker can use their knowledge to make conjectures, to reason, and to solve problems. Their use of mathematics is marked by great flexibility of thinking together with the strong belief that precise definitions are important. They use both direct and indirect arguments and make connections between the problem being considered and their mathematical knowledge. When presented with a problem to solve, they will assess the problem, collect appropriate information, find pathways to the answer, and be able to explain that answer clearly to others.

While an effective mathematical toolbox certainly includes algorithms, a person with well developed habits of mind knows both why algorithms work and under what circumstances an algorithm will be most effective.

Mathematical habits of mind are also marked by ease of calculation and estimation as well as persistence in pursuing solutions to problems. A person with well developed habits of mind has a disposition to analyze situations as well as the self-efficacy to believe that he or she can make progress toward a solution.

This definition was built with help from Mark Driscoll’s book, Fostering Algebraic Thinking: A guide for teachers grades 6-10.

A Mathematics – Mathematics Education Partnership at the

University of Nebraska-Lincoln College of EducationDepartment of

& Human Sciences Mathematics

The Math Matters Vision

• Create a mathematician – mathematics educator partnership with the goal of improving the mathematics education of future elementary school teachers

• Link field experiences, pedagogy and mathematics instruction

• Create math classes that are both accessible and useful for future elementary school teachers

The Mathematics Semester(For all Elementary Education majors starting Fall 2003)

MATH• Math 300 – Number and Number Sense (3 cr)

PEDAGOGY• TEAC 308 – Math Methods (3 cr)• TEAC 351 – The Learner Centered Classroom (2 cr)

FIELD EXPERIENCE• TEAC 297b – Professional Practicum Exper. (2 cr)

– Students are in an Elementary School classroom on Mondays and Wednesdays

– Math 300 & TEAC 308 are taught as a 3-hour block on Tuesday and Thursday

– TEAC 351 is taught by master teachers on Wednesdays

Sample Habits of Mind Problems for Elementary Teachers

Making Change

What is the fewest number of coins that it will take to make 43 cents if you have available pennies, nickels, dimes, and quarters? After you have solved this problem, provide an explanation that proves that your answer is correct?

How does the answer (and the justification) change if you only have pennies, dimes, and quarters available?

The Rice Problem • Recall our discussion about the game of chess and how

a humble servant for a generous king invented it. The king became fascinated by the game and offered the servant gold or jewels in payment, but the servant replied that he only wanted rice—one grain for the first square of the chess board, two on the second, four on the third, and so on with each square receiving twice as much as the previous square. In class we discussed how the total amount of rice was 264 grains of rice. (To be completely precise, it is this number minus one grains of rice.) Suppose it was your job to pick up the rice. What might you use to collect the rice, a grocery sack, a wheelbarrow, or perhaps a Mac truck? Where might you store the rice?

Math in the Middle Institute Partnership

Principal Investigators

Jim Lewis

Ruth Heaton

Barb Jacobson

Tom McGowan(Funding began August 1, 2004)

Special Thanks To

Vermont Mathematics Initiative

In developing our proposal and gearing up to work with the first cohort of teachers in Math in the Middle, we have benefited greatly from our interactions with the leadership of VMI.

(VMI has been jointly funded by the U.S. Department of Education and NSF)

Math in the Middle Institute Partnership

Vision• Create and sustain a University, Educational Service

Unit (ESU), Local School District partnership• educate and support teams of outstanding middle

level (Grades 5 – 8) mathematics teachers who will become intellectual leaders in their schools, districts, and ESUs.

• provide evidence-based contributions to research on learning, teaching, and professional development.

• special focus on rural teachers, schools, and districts.

Math in the Middle Institute Partnership

Goal

Invest in high-quality teachers

* To improve K-12 student achievement in mathematics and to significantly reduce achievement gaps in the mathematical performance of diverse student populations.

Math in the Middle Institute Partnership

Math in the Middle major components

• The M2 Institute, a multi-year institute that offers participants a coherent program of study to deepen their mathematical knowledge for teaching and to develop their leadership skills;

• Mathematics learning teams, led by M2 teachers and supported by school administrators and university faculty, which develop collegiality, help teachers align their teaching with state standards, and assist teachers in examining their instructional and assessment practices; and

• A research initiative that will transform the M2 Institute and the M2 mathematics learning teams into laboratories for educational improvement and innovation.

Math in the Middle Institute Partnership

M2 courses focus on these objectives:

• enhancing mathematical knowledge

• enabling teachers to transfer mathematics they have learned into their classrooms

• leadership development and

• action research

Math in the Middle Institute Design

Summer Fall SpringWk1 Wk2&3

Yr 1 M800T Teac800 & M802T Stat892 M804T

Yr 2 M806T Teac801 & M905T Teac888 M807T

Yr 3 M808T Teac889/M809T and the Masters Exam

- A 25-month, 36-hour graduate program.

Math in the Middle Summer Courses

• Combination of 1 week and 2 week classes.• Teachers are in class from 8:00 a.m. - 5:00 p.m.• 32 students – 5 instructors in class at one time.• Substantial homework each night.• Substantial End-of-Course problem set

– Graded for purpose of helping teacher learn to work the problems.

– Presentation of solutions/celebration of success at start of next class.

Math in the Middle Academic Year Courses

• Two-day (8:00 – 5:00) on-campus class session.• Course completed as an on-line, distance

education course.– Major problem sets– Professional Writings– Learning and Teaching Projects– End-of-Course problem set– Substantial support available for teachers

Math in the Middle Institute Courses

• Math 800T: Mathematics as a Second LanguageThis course lays the foundation for developing the “habits of mind of a mathematical thinker,” a theme that is further developed in subsequent M2 courses. The approach is to understand arithmetic (number) and (introductory) algebra as a means of communicating mathematical ideas (i.e. as a language). The course will stress a deep understanding of the basic operations of arithmetic, as well as the interconnected nature of arithmetic, algebra and geometry.

• Math 802T: Functions, Algebra and Geometry for ML Teachers A careful study of fractions, ratios and rational numbers will lay the foundation for a study of algebra. Participants will gain a deep understanding of the concept of function and gain a deeper understanding of the algebra and geometry taught in the middle grades. Participants will also study measurement with an emphasis on length, area and volume.

• TEAC 800: Inquiry into Teaching and LearningTeac 800 focuses on inquiry into mathematics teaching and learning. Participants will be introduced to the field of educational inquiry through a study of various designs and methods of doing educational research. Participants will develop knowledge, skills, and dispositions of educational inquiry through a study of artifacts of mathematics teaching and learning.

Math in the Middle Institute Courses

• STAT 892: Statistics For Middle Level Teachers This academic year course is taught using distance learning approaches together with one 2-day, on-site classroom experience. The course offers an introduction to statistics and probability with an emphasis on teaching statistics and probability to middle level students. The course provides the foundation for later study of how data is used in education and for school-based research.

• Math 804T: Experimentation, Conjecture and Proof(Problem Solving for the Middle Level Mathematics Teacher) This academic year course is taught using distance learning approaches together with one 2-day, on-site classroom experience. Focus will be placed on problem solving, reasoning and proof and communicating mathematics. With the support of the American Mathematics Competitions, which is housed in Lincoln, NE, this course will utilize the extensive resources of the AMC to help middle level mathematics teachers develop problem solving skills.

Math in the Middle Institute Courses

• Math 805T: Discrete Mathematics for Middle Level Teachers Discrete mathematics topics introduced in this class will include social decision making, vertex-edge graph theory, counting techniques, matrix models, and the mathematics of iteration. The unifying themes for these topics will be mathematical modeling, the use of technology, algorithmic thinking, recursive thinking, decision making, and mathematical induction as a way of knowing.

• TEAC 801: Curriculum InquiryThis course is taught in partnership with Math 805T and will focus on gaining a deeper understanding of mathematics curriculum development, including historical and contemporary issues that influence curriculum planning and educational change. Participants will consider current curricular issues in relationship to their own mathematics teaching and learning and how the mathematics learned in other M2 courses transfers into the planned and enacted curriculum of one’s own teaching practice.

• Math 806T: Number Theory for Middle Level TeachersThe inspiration for the course is the very successful honors seminar “The Joy of Numbers: The Search for Big Primes.” A Socratic approach will guide the teachers in a development of key ideas in number theory that have many applications to the middle level classroom.

Math in the Middle Institute Courses

• TEAC 888: Teacher as Scholarly PractitionerThis academic year course is taught using distance learning approaches together with one 2-day, on-site classroom experience. The course introduces participants to the theory and practice of teacher-led inquiry into effective practice. The course prepares teachers to engage in a school-based action research project that will be conducted during the following spring semester.

• Math 807T: Using Mathematics to Understand our WorldThis academic year course is taught using distance learning approaches together with one 2-day, on-site classroom experience. It will explore what kind of processes in students’ daily lives can be modeled by simple equations or distributions. The course will also address measurement of various natural phenomena, from weather to soil and water makeup to structure of living organisms.  The first goal of this course is to provide teacher leaders with the capacity not only for utilizing a broad range of resources for teaching mathematics, but also for adjusting their techniques and strategies to match their clientele.  The second goal is to provide these teacher leaders with the techniques for helping other teachers develop their own capacities for resource utilization, adjustments of techniques, and adapting to a changing clientele.

Math in the Middle Institute Courses

• Math 808T: Concepts of Calculus The design of this 1-week course is similar the corresponding courses in the first and second summer. The course will begin with a development of trigonometry from the perspective of the middle school classroom and then shift to an introduction to the “concepts of calculus.” Emphasis will be on understanding the concepts of limit, derivative, and the definite integral (and how the mathematics in the middle school curriculum can lay a foundation for the study of continuous mathematics). Participants will also be introduced to series, vectors and multivariable functions.

• TEAC 889/Math 809: Capstone Course Integrating the Learning and Teaching of MathematicsThis course is the capstone experience of the M2 Institute. Considerable time will be devoted to discussing how the mathematics learned in M2 courses can enrich the middle level classroom. This course will be a fully integrated mathematics and pedagogy course whose goal is to enable the teacher to be a better teacher of mathematics because of the mathematics and pedagogy that they have learned. Concurrently with this course, teachers will be working on satisfying the requirements for their Masters Degree.

Math in the Middle Course 1 - Mathematics as a Second Language

• The “text” was written by Kenneth and Herbert Gross of the Vermont Mathematics Initiative.

• Ken helped us “kick off” our first weekend.• Our Innovations (i.e. additions)

– Habits of Mind problems– Professional Writings– Learning and Teaching Project

M2 Innovations “Habits of Mind” Problems

The Triangle Game: (Thanks to Paul Sally) Consider an equilateral triangle with points located at each vertex and at each midpoint of a side. The problem uses the set of numbers {1, 2, 3, 4, 5, 6}. Find a way to put one of the numbers on each point so that the sum of the numbers along any side is equal to the sum of the numbers along each of the two other sides. (Call this a Side Sum.) – Is it possible to have two different Side Sums? – What Side Sums are possible?– How can you generalize this game?

M2 InnovationsProfessional Writings

What do Math Teachers Need to Be?

Read “What do Math Teachers Need to Be?” by Herb Clemens, a mathematics professor at The Ohio State University. The article was published in 1991 in Teaching academic subjects to diverse learners (pp. 84-96). In this article, Clemens lists what he thinks teachers of mathematics need to be. Where does your own practice of teaching mathematics stand in relationship to what Clemens says mathematics teachers need to be:  unafraid, reverent, humble, opportunistic, versatile, and in control of their math. On p. 92, Clemens lists four fundamental questions about mathematics teaching that matter to him. If he came to your classroom and watched you teach math, how would he answer his own last question about your practice: Can this teacher teach it [math] with conviction, and with some feeling for its essence? Explain.

M2 InnovationsLearning and Teaching

Projects

Select a challenging problem or topic that you have studied in MSL. This is to be the basis for a mathematics lesson that you will videotape yourself teaching to your students.

How can you present this task to the students you teach? How can you set the stage for your students to understand the problem? How far can your students go in exploring this problem? Remember that you want your students to discover as much as possible on their own. But there may be some critical points where you need to guide your students over an intellectual “bump” so that they can move on to the next part of the problem.

Finally, produce a report analyzing the mathematics and your teaching experience.

TEAC 800 Action Research

“Action research is research done by teachers for themselves; it is not imposed on them by someone else” (Mills, 2003, p. 5, italics in original).

In conducting action research, drawing conclusions isn’t about making generalizations for others but about deciding on a course of action for one’s own teaching.

Action Research 2005-2006

• Course for planning action research projects in the fall semester; implementation in the spring semester

• 31 teachers• 29 different research projects tied into individual

problems of practice• 3 research questions each• 3 forms of data collection (both quantitative and

qualitative) for each research question• 31 literature reviews of at least 5 research articles each,

connected to individual problems of practice• 29 sets of IRB documents

Teachers as Leaders

• Question: Are there ways we can use Learning Teams to build the capacities of M2 teachers as leaders?

Developing teachers’ capacities as leaders is a central challenge as well as an enabling feature of sustainability.

• How does M2 support these learning teams in such a way that they will evolve into active professional networks?

Leadership Academy

The Leadership Academy’s goal is to encourage and support the mission of Math in the Middle teachers, their administrators, and ESU administrators and staff development officers as they work together to develop plans to strengthen mathematics teaching and learning in the middle grades.

Leadership Academy, September 2005

• The state of the schools, teacher expertise (and how to gain it), surviving in a high-stakes testing world and becoming a leader in your school, Dr. David Berliner, Regent’s Professor and Dean Emeritus in the College of Education at Arizona State University.

• The Power of Professional Learning Communities, Dr. Tim Kanold, superintendent of Stevenson (IL) High School and mathematics textbook author

Math in the Middle Research Questions

• What are the capacities of teachers to translate the mathematical knowledge and habits of mind acquired through the professional development opportunities of M2 into measurable changes in teaching practices?

• To what extent do observable changes in mathematics teaching practice translate into measurable improvement in student performance?

What are we learning?

• Integrate content and pedagogy courses.

• Keep expectations of teachers high.

• Emphasize learning how to learn and offer continued opportunities.

• Build on existing relationships.

• Commitment to the partnership need to be long term.