MATH
A partnership to improve mathematics education in Nebraska
Jim Lewis
University of Nebraska-Lincoln
Aaron Douglas Professor of Mathematics

• U.S. competitiveness depends on dramatic improvements in the math and science education of its K-12 students.
• Rising Above the Gathering Storm,
(National Research Council report
2007), identified four strategies for strengthening U.S. competitiveness.
The first is:
– Increase America’s talent pool by vastly improving K-12 science and mathematics education.
• The National Math Panel report asserts:
– To compete in the 21 st century global economy, knowledge of and proficiency in mathematics is critical.

• The National Assessment of Educational
Progress reports these data:
4 th Grade 8 th Grade
Top (Mass)
Indiana
252
243
299
287
Nebraska
Nation
Bottom (Miss)
239
239
227
284
282
265

4 th Grade 8 th Grade
Indiana Neb Indiana Neb
All students 18 th 33 rd 18 th 27 th
White
Black
28 th 38 th 28 th 27 th
24 th 44 th 11 th 38 th
Hispanic 24 th 40 th 13 th 41 st
* NAEP data has 52 “states” including DC and DoD schools. Data for Black students are reported only for 46 states and only for 47 states for Hispanic students.

In 2009, Nebraska reported that 92% of
8 th grade students met or exceeded Nebraska’s proficiency standards.

There is much evidence that teachers are the most important variable in student learning
– Studies have found that the effect size of good teaching is greater than any other variable, including students’ socioeconomic status.
But
Mathematics teaching is an extraordinarily complex activity involving interactions among teachers, students, and the mathematics to be learned in real classrooms. (National Math Panel, 2008)

• What Mathematics do Teachers “Need to
Know” and How Should They “Come to Know”
Mathematics?
– What does it mean to offer challenging courses and curricula for math teachers?
– How do we help teachers translate the mathematics they come to know into classroom practice that leads to improved student learning?

Of course:
49 x 25
245
+ 980
1,225
Why might a fourth grader think the answer is 1,485?*
5*(4+4)=40
49 or
5*4 + 4 x 25
405
+ 1080
1,485
* On May 4, 2010, Deborah Ball, Dean of the University of Michigan School of
Education used this question as part of her testimony before the U.S. House of
Representatives Education and Labor Committee.

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, Professor, University of California, Berkeley

What is so difficult ….?
• 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 teachers 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 teachers develop flexibility in their thinking, to be able to reason about elementary mathematics.
Roger Howe, Professor, Yale University

Educating Teachers of Science,
Mathematics, and Technology
New Practices for the New Millennium
This National Research Council report 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.”

Recommendations
• Teachers need mathematics courses that develop a deep understanding of the math they teach.
• Mathematics courses should
– focus on a thorough development of basic mathematical ideas.
– develop careful reasoning and mathematical ‘common sense’.
– develop the habits of mind of a mathematical thinker and demonstrate flexible, interactive styles of teaching.
• The mathematics education of teachers should be based on
– partnerships between mathematicians, mathematics education faculty and school mathematics teachers.

At the University of Nebraska-Lincoln, our work is supported by:
• The Center for Science, Mathematics and
Computer Education (permanent infrastructure)
• Math & Science Teachers for the 21 st Century
Program of Excellence ($350,000/yr 2002-2015)
• $18,100,000 in NSF support
– Math in the Middle Institute Partnership (2004-2011)
– Nebraska MATH Partnership (2009 – 2013)
– NebraskaNOYCE (2010 – 2016)

• Principal Investigators
– Jim Lewis, Mathematics
– Ruth Heaton, Teaching, Learning and Teacher
Education (TLTE)
– Carolyn Edwards, Psychology and Child, Youth and
Family Studies
– Walt Stroup, Statistics
– Ira Papick, Mathematics
– Tom McGowan, TLTE
– Barb Jacobson, Lincoln Public Schools

• Grand Island Public Schools
• Lincoln Public Schools
• Omaha Public Schools
• Papillion-La Vista Public Schools
• Nebraska’s Educational Service Units
– Our work involves teachers from 101 school districts and 216 schools.

NebraskaMATH
Professional Development
• Math in the Middle Institute
– A master’s program for middle level (5-8) teachers
• Primarily Math
– An 18-hour certificate program for K-3 teachers
• Nebraska Algebra
– A 9-hour program for Algebra 1 teachers
• New Teacher Network
– A 24-hour PD and mentoring program for new teachers
• Robert Noyce NSF Master Teaching Fellowships
– A program for extraordinary master teachers
• Robert Noyce NSF Teaching Fellowships
– A postbac master’s and certification program

• A 25-month, master’s program that educates and supports outstanding middle level teachers who will become intellectual leaders in their schools, districts, and ESUs.
• A major research initiative to provide evidence-based contributions to research on learning, teaching, and professional development.
• A special focus on supporting rural teachers, schools and districts

M 2 courses focus on these objectives:
• enhancing mathematical knowledge
• enabling teachers to transfer mathematics they have learned into their classrooms
• leadership development and
• action research

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

SUMMER
• Offer 1 and 2 week classes.
• Class meets from 8:00 a.m. -
5:00 p.m.
• 35 teachers – 5 instructors in class at one time.
• Substantial homework each night.
• End-of-Course problem set
– Purpose – long term retention of knowledge gained.
ACADEMIC YEAR
• Two-day (8:00 – 5:00) oncampus class session.
• Course completed as an online, distance education course using Blackboard and Adobe
Connect.
– Major problem sets
– End-of-Course problem set
– Substantial support available for teachers

• Focuses on strengthening the teaching & learning of mathematics in grades K-3
• Six course, 18-credit hour program leading to a K-3 Mathematics Specialist certificate
– 3 mathematics courses
– 3 pedagogy courses
• Optional 7 th course focusing on leadership
• On-going support in the form of study groups lasting 2 years after coursework

• What happens to student achievement as elementary buildings employ math specialists as coaches, to departmentalize math instruction, or to continue as general classroom teachers?
• What happens to teachers’ classroom practices during and after their participation in Primarily Math?

• Research design led us to recruit three cohorts of participants during 2008-2009
– First cohort began summer 2009 (classes met in Lincoln)
– Second cohort began summer 2010 (classes in Omaha)
– Third cohort begins summer 2011
(two institutes, one in Lincoln and one in Grand Island)
• Collect data from all three cohorts beginning summer 2009, continuing through summer 2013

• School building climate and degree of teacher networking
• Teacher Network Survey administered to elementary buildings in four core partner districts
2010, 2011, 2013
• Survey allows principals to see where teacher collaboration & isolation are occurring
• Partnership with colleagues at Northwestern
University

Lewis Middle School

• 9 hours of graduate coursework
– Math 810T: Algebra for Algebra Teachers
– EDPS 991: Cognition and Instruction for High School
Algebra Teachers
– TEAC 991: Field Studies in Mathematics
• Some districts are able to provide participants with an algebra coach
– If a coach is not available, we provide a teacher mentor
• All teachers have a university mentor

• Objectives
– To help teachers better understand conceptual underpinnings of school algebra
– To leverage new understanding into improved classroom practice
• Pedagogy
– Combines collaborative learning with direct instruction
– Provide teachers with dynamic learning & teaching models
• Assessment
– Individual & group presentations, written & historical assignments, mathematical analyses of curricula, extended projects, and a final course assessment

• Teachers need specialized content knowledge:
– Deep understanding of content
– Representations and connections
– Understand student thinking
– Assess student learning
– Make curricular decisions
• This type of knowledge is not typically gained through most pre-service mathematics programs
(i.e., Ball, Thames & Phelps, 2008; NCTM, 2000)

Mathematical Knowledge for Teaching Algebra
• Teachers need knowledge of mathematics that enables them to address a wide range of mathematical ideas and questions. For example, here are some questions that school algebra teachers might be asked.
1) My teacher from last year told me that I whatever I do to one side of an equation, I must do the same thing to the other side to keep the equality true. What am I doing wrong when I add 1 to the numerator of both fractions in the equality 1/2 =2/4 and get 2/2 = 3/4?
2) My father (who is very smart) was helping me with my homework last night and he said the book is wrong. He said that

4=2 and

4=–2, because 2 2 =4 and (–2) 2 =4, but the book says that

4≠–2. He wants to know why we are using a book that has mistakes.

Nebraska Math & Science
Summer Institutes

Nebraska Math & Science
Summer Institutes
• In 2010 we offered 13 courses for math teachers.
Examples include:
– Algebraic Thinking in the K-4 classroom
– Functions for Precalculus Teachers
– Concepts of Calculus for Middle Level Teachers

The habits of mind of a mathematical thinker
Have you observed two people who appear to know the same
“facts” but for whom there is a marked difference in their ability to use that information to answer questions or solve problems?
Why?
• Do mathematical thinkers approach problems differently?
• And, if so, how do we develop the “habits of mind of a mathematical thinker” in teachers and assist them in cultivating this knowledge among their students?
• To study this question, we developed a working definition based on experience and the work of other mathematics educators (e.g., Cuoco, et al., Driscoll)

Our Courses Emphasize
Problems that Develop Teachers’
Mathematical Habits of Mind
Goals: Give teachers experiences to develop their:
• Strategies for solving problems
• Flexibility in thinking
• An appreciation for the importance of precise mathematical definitions and careful reasoning
• Ability to explain solutions to others
• Persistence and self-efficacy

The Chicken Nugget Conundrum
• There’s a famous fast-food restaurant where you can order chicken nuggets. They come in boxes of various sizes. You can only buy them in a box of 6, a box of 9, or a box of 20. Using these order sizes, you can order, for example, 32 pieces of chicken if you wanted. You’d order a box of 20 and two boxes of 6. Here’s the question: What is the largest number of chicken pieces that you cannot order? For example, if you wanted, say 31 of them, could you get 31? No. Is there a larger number of chicken nuggets that you cannot get? And if there is, what number is it? How do you know your answer is correct?
A complete answer will: i) Choose a whole number “N” that is your answer to the question.
ii) Explain why it is not possible to have a combination of “boxes of 6” and “boxes of 9” and “boxes of 20” chicken nuggets that add to exactly N pieces of chicken.
iii) Explain why it is possible to have a combination that equals any number larger than N.

Problematic Answers
• Explain why it is not possible to order exactly 43 pieces.
Argument #1: You can not have any combination that adds to 43 because it can’t evenly divide by 6, 9, or 20. It is not a multiple of 15 and it can’t be evenly divided in half.
Argument #2: You are not able to get the number 43 because none of the numbers add equally into that number.
• Explain why it is possible to have a combination that equals any number larger than N.
Argument: It’s possible to have a combination greater then 43. This is because you can buy all the multiples of the numbers. For example, if you buy 18, you can buy 36 and 70. Or if you by 20 you can buy
40, 60, 80, 100, etc.

(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 an Equal
Side Sum Solution.)
– Is it possible to have two different Equal Side Sum
Solutions?
– Which Equal Side Sum Solutions are possible?
– How can you generalize this game?

Side Sum Solutions for Hexagons
Side Sum 17: 3, 8, 6, 4, 7, 9, 1, 11, 5, 10, 2, 12
Side Sum 18: None
Side Sum 19: 6, 2, 11, 5, 3, 9, 7, 4, 8, 10, 1, 12
And 4, 10, 5, 8, 6, 2, 11, 1, 7, 9, 3, 12
And 5, 11, 3, 9, 7, 4, 8, 10, 1, 6, 12, 2
And 3, 9, 7, 11, 1, 10, 8, 6, 5, 2, 12, 4
Side Sum 20: 7, 11, 2, 8, 10, 4, 6, 9, 5, 3, 12, 1
And 9, 3, 8, 5, 7, 11, 2, 12, 6, 4, 10, 1
And 8, 2, 10, 4, 6, 9, 5, 3, 12, 7, 1, 11
And 10, 4, 6, 2, 12, 3, 5, 7, 8, 11, 1, 9
Side Sum 21: None
Side Sum 22: 10, 5, 7, 9, 6, 4, 12, 2, 8, 3, 11, 1

Patterns with Minimums & Maximums
Polygon
Triangle
Square
Pentagon
Hexagon
Heptagon
Octagon
Minimum
Side Sum
14
17
9
12
19
22
+3
+2
+3
To find the next
Minimum
Maximum
Side Sum
12
+2 15
+3
19
22
26
29
+3
+4
+3
To find the next
Maximum
+4
+3

A Solution for an n-sided polygon, n odd
• General solution for an n-gon where n = 2k + 1, n odd
• For a Heptagon Solution, n = 7; k = 3
1
12
5
13
2
11
6 10
14
4
8
7
9
3
To find the vertices begin with 1, move clockwise by k each time, and reduce mod n. The midpoints begin with 2n between 1 and 1+k and move counterclockwise, subtracting 1 each time. For a heptagon, the
Side Sum = 5k + 4.

Many people are involved in NebraskaMATH teaching and research
UNL faculty
Other collegiate faculty
23
7
UNL graduate students 45
UNL undergraduates 18
Master teachers 38

• Please visit the Products section of our web site: http://scimath.unl.edu/MIM/ for information on Math in the Middle:
• Course Materials
• Teachers’ Expository Math Papers
• Teachers’ Action Research Papers