1. The Mathematical Education of K-8 Teachers
at the University of Nebraska-Lincoln,
a Mathematics – Mathematics Education
Partnership
Jim Lewis and Cheryl Olsen
University of Nebraska-Lincoln

2. The Mathematics Semester
for future elementary school teachers
and
Math in the Middle Institute Partnership a
professional development program for
middle level teachers

3. The Mathematics Semester
Math Matters, an NSF-funded CCLI grant began in 2000,
and became The Mathematics Semester in Fall 2003
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

4. 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)
(a field experience in a local elementary school)
– Students are in an elementary school on Mondays and
Wednesdays (four hours/day)
– Math 300 & TEAC 308 are taught as a 3-hour block on Tuesday
and Thursday
– TEAC 351 is taught on site at a participating elementary school

5. A look inside The Mathematics Semester
• Curriculum materials
• Homework to develop mathematical habits of
mind
• Professional writings
• The curriculum project
• Learning and Teaching Project
• Activities at the elementary school
– Teaching a Math Lesson
– Child Study

6. Curriculum Materials
• Sowder, J. et al. (2007). Reasoning about Numbers and
Quantities. W. H. Freeman. (prepublication copy)
• Schifter, D., Bastable, V., & Russell, S.J. (1999).
Number and operations, part I: Building a system of
tens. Parsippany, NJ: Dale Seymour.
• Reys, Lindquist, Lambdin, Smith, & Suydam (2007).
Helping Children Learn Mathematics. John Wiley &
Sons.
• Lampert, M. (2001). Teaching problems and the
problems of teaching. New Haven, CT: Yale University.

7. A problem to get started
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?
Note: We first encountered this problem in a conversation with
Deborah Ball.

8. A Typical Weekly Homework All Shook Up
Five couples met one evening at a local restaurant for
dinner. Alicia and her husband Samuel arrived first. As
the others came in some shook hands and some did not.
No one shook hands with his or her own spouse. At the
end Alicia noted that each of the other 9 people had
shaken the hands of a different number of people. That
is, one shook no one's hand, one shook one, one shook
two, etc., all the way to one who shook hands with 8 of
the people. How many people did Samuel shake hands
with?
Note: This problem is a slightly modified version of Exercise #14,
page 32, in The Heart of Mathematics, by Burger and Starbird, Key
Curriculum Publishing, 2005.

9. Professional Writings
Dear Math Professors,
We are 1st and 2nd graders in Wheeler Central Public School in
Erickson, Nebraska. We love to work with big numbers and have
been doing it all year! Every time we read something with a big
number in it we try to write it. Then our teacher explains how to write
it. We are getting pretty good at writing millions and billions!
We have a problem that we need your help with. We were reading
amazing ‘Super Mom’ facts in a Kid City magazine. It told how many
eggs some animals could lay. We came across a number that we
don’t know. It had a 2 and then a 1 followed by 105 zeros!! We
wrote the number out and it stretches clear across our classroom!
We know about a googol. We looked it up in the dictionary. A googol
has 100 zeros. Then what do you call a number if it has more than
100 zeros? Is there a name for it?

10. Another problem is that we learned about using commas in large
numbers. In the magazine article they used no commas when
writing this large number. That confused us. Also, if you write a
‘googol’ with 100 zeros, how do you put the commas in? It doesn’t
divide evenly into groups of 3 zeros. There will be one left over.
We appreciate any help you can give us solving this “big” problem.
Thank you for your time.
Sincerely,
Mrs. Thompson’s 1st & 2nd graders
Megan Kansier, Mark Rogers
Marcus Witt, Ashley Johnson

11. clipping from Kid City magazine
Apple Of My Eye
The tiny female apple aphid is a champ
as an egg-layer. This insect can lay as
many as 21000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
0000 eggs in 10 months.

12. 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. 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. If Clemens came to your classroom and watched you teach
math, how would he answer his question: Can this teacher teach it
[math] with conviction, and with some feeling for its essence?
Explain.
* Published in 1991 in Teaching academic subjects to diverse
learners (pp. 84-96).

13. Curriculum Project
The goal is to investigate a new mathematical area of the
elementary curriculum and consider what teachers need to know as
well as what children need to learn the topic in the deep and
meaningful ways suggested by the NCTM Standards (2000).
1) Pick a topic: data analysis and probability, geometry, reasoning and
proof, or algebra.
2) Read, analyze, and synthesize the mathematical topic in the NCTM
Standards.
3) Analyze and synthesize the topic in a set of reform curriculum
materials (Everyday Math, Trail Blazers, Investigations, local
curriculum).
4) What do teachers need to know to teach this?
5) Create 5 math problems that would help teachers learn. Create 5
math problems that would help children learn.

14. Learning and Teaching Project
Area and Perimeter
The task began with a homework problem. It is taken
from Reconceptualizing Mathematics: Courseware for
Elementary and Middle School Teachers, Center for
Research in Mathematics and Science Education, 1998.
Is there a relationship between the area and the
perimeter of a polygonal shape made with congruent
square regions? (For fixed area, find the minimum and
maximum perimeter. For fixed perimeter, find the
minimum and maximum area.) Squares must be joined
complete-side to complete-side. The outside “boundary”
should be a polygon. In particular, this would not permit
a shape with a “hole” in the middle.

15. A couple of weeks later we told our students:
We want to revisit the “Area and Perimeter” problem. This is
to be the basis for a mathematics lesson that you will
videotape yourself teaching to one elementary school student.
How can you present this task to the student you will teach?
How can you set the stage for the student to understand the
problem? How far can the student go in exploring this
problem? Remember that you want your student to discover
as much as possible for himself (or herself). But there may be
some critical points where you need to guide the student over
an intellectual “bump” so that he (she) can move on to the
next part of the problem.
Finally, produce a report analyzing the mathematics and your
teaching experience.

16. Activities at the elementary school
Math Lesson. You will teach a math lesson that
connects to the curriculum in the classroom in which you
are working. You should make use of the textbook and
other resources from your cooperating teacher and what
you are learning and reading in the Mathematics
Semester.
Child Study. This assignment will give you some
experience watching, listening to, probing, and assessing
one child’s understanding of several math problems
focused on a particular area of mathematics. You will
write a report of your interview and suggest instruction
for the child based on the information you gathered in
the interview session.

17. Math in the Middle
Institute Partnership
Principal Investigators
Jim Lewis, Department of Mathematics
Ruth Heaton, Department of Teaching,
Learning & Teacher Education
Barb Jacobson, Lincoln Public Schools
Tom McGowan, Chair, Department of
Teaching, Learning & Teacher Education
(Funding began August 1, 2004)

18. M2 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.

19. M2 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.
• Place a special focus on rural teachers, schools,
and districts.

20. M2 Partnerships
People and Organizations
• All 14 rural Educational
Service Units plus LPS
• 65 Local School Districts
• 91 Schools
• 130 Teachers (4th cohort
will begin summer 2007)
– 29 teachers earned their
Masters Degree 2006

21. M2 Major Components
• The M2 Institute, a multi-year (25-month) institute that
offers participants a coherent program of study to
deepen their mathematical and pedagogical 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.

22. 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

23. Math in the Middle
Institute Design
Summer Fall Spring
Wk1 Wk2&3
Yr 1 M800T Teac800 & M802T M804T M805T
Yr 2 M806T Teac801 & Stat892 Teac888 M807T
Yr 3 M808T Teac889/M809T
and the Masters Exam
- A 25-month, 36-hour graduate program.

24. M2 Summer Institute
• Combination of 1 week and 2 week classes.
• Teachers are in class from 8:00 a.m. - 5:00 p.m.
• 32-35 teachers – 5 instructors in class at one time.
• Substantial homework each night.
• Substantial End-of-Course problem set
– Purpose – long term retention of knowledge gained.
– Presentation of solutions/celebration of success at start of next
class.

25. M2 Academic Year Courses
• Two-day (8:00 – 5:00) on-campus class session.
• Course completed as an on-line, distance
education course using Blackboard and Breeze.
– Major problem sets
– Professional Writings
– Learning and Teaching Projects
– End-of-Course problem set
– Substantial support available for teachers

26. M2 Institute Courses
• Eight new mathematics and statistics courses designed
for middle level teachers (Grades 5 – 8) including:
– Mathematics as a Second Language
– Experimentation, Conjecture and Reasoning
– Number Theory and Cryptology for Middle Level Teachers
– Using Mathematics to Understand our World
• Special sections of three pedagogical courses:
– Inquiry into Teaching and Learning
– Curriculum Inquiry
– Teacher as Scholarly Practitioner
• An integrated capstone course:
– Masters Seminar/Integrating the Learning and Teaching of
Mathematics

27. Math 800T - 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.
• Innovations (i.e. additions)
– Habits of Mind problems
– Learning and Teaching Project

28. M2 Innovations
“Habits of Mind” Problems
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.

29. M2 Innovations
“Habits of Mind” Problems
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.

30. E
The Triangle Game
D A “Habits of Mind”
F
Problem
C
A B
(Paul Sally, U. Chicago) 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?

31. M2 Innovations
Learning & Teaching Projects
Select a challenging problem or topic that you have studied in MSL
and use it as 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? You want your
students to discover as much as possible on their own, but there
may be a critical point where you need to guide them over an
intellectual “bump.”
Produce a report analyzing the mathematics and your teaching
experience.

32. Action Research
Each teacher takes TEAC 888, Teacher as Scholarly
Practitioner, an action research course. Each teacher
then conducts their own research project and writes a
report about their findings.
– “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.
– In 2006, 31 teachers had 29 different research projects involving
29 IRB documents.
– Each teacher posed 3 research questions, used 3 forms of data
collection and used at least 5 from their literature review.

33. M2 Masters Degrees
• Two options for the Masters Degree
– MAT (Specialization in the teaching of middle level
mathematics (Mathematics Department)
– MA (Teaching, Learning and Teacher Ed.)
• Masters exam in mathematics
– Take home exam (two math questions)
– Write 5-7 page report on action research project
– Write an 8-10 page expository paper
– Give an oral presentation about the paper

34. A Sample Masters Exam Question
A math class with “n” students sits in a circle to play
mathematical chairs. The students choose an
elimination number “d” and then count off in order, 1, 2,
3, … . When the count gets to d, that student is
eliminated from the game. The next student starts the
count over and the students count 1, 2, 3, … . Again,
when the count gets to d, that student is eliminated.
Continue in this manner until only one student is left.
That student wins the game.
Where should you sit in order to win the game?
Hint: Solve the problem first for elimination number 2 or
3 and then try to solve it for elimination number d.
Note: This is a version of The Ring of Josephus problem.

35. M2 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?

36. What are we learning?
• Integrate content and pedagogy courses.
• Keep expectations of students or teachers high.
• Emphasize learning how to learn and offer
continued opportunities.
• Build on existing relationships.
• Commitment to the partnership need to be long
term.