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 Mind
A 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 Education
& Human Sciences
Department of
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
Spring
Yr 1
Wk1
Wk2&3
M800T Teac800 & M802T
Stat892
Yr 2
M806T Teac801 & M905T
Teac888 M807T
Yr 3
M808T Teac889/M809T
and the Masters Exam
- A 25-month, 36-hour graduate program.
M804T
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.
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–
–
–
–
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 Language
This 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 Learning
Teac 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 Inquiry
This 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 Teachers
The 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 Practitioner
This 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 World
This 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 Mathematics
This 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 Innovations
Professional 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 Innovations
Learning 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.