The Mathematical Education of Teachers Lessons Learned from Math in the Middle Michelle Reeb Homp...

The Mathematical Education of Teachers

Lessons Learned from Math in the Middle

Michelle Reeb HompUniversity of Nebraska – Lincoln

Sandi Snyder Shickley Public Schools, NE

Math in the Middle Institute Partnership (M2)

• The information we’ll be sharing is based on our experiences with the Math in the Middle Institute Partnership

• Primary focus of M2 is professional development for middle level teachers, but the courses we will address have also been offered (with adjustments) to H.S. teachers

• Many of the components of the M2 curriculum are also part of the teacher preparation program at UNL

Math in the Middle Institute Partnership

A Brief Summary

Principal Investigators• Jim Lewis, Department of Mathematics• Ruth Heaton, Department of Teaching, Learning &

Teacher Education

M2 Partnership Vision• Create and sustain a University, Educational Service Unit

(ESU), Local School District partnership• Educate and support teams of outstanding middle level

mathematics teachers who will become intellectual leaders

• Place a special focus on rural teachers, schools, and districts (Funding began August 1, 2004)

M2 Major Components• The M2 Institute, a 25-month institute that offers a coherent

program of study to deepen mathematical and pedagogical knowledge and develop leadership skills

• Mathematics learning teams, led by M2 teachers and supported by school administrators and university faculty, which develop collegiality

• A research initiative for educational improvement and innovation

M2 Institute Courses• Eight new mathematics and statistics courses

designed for middle level teachers• Special sections of three pedagogical courses• An integrated capstone course

M2 courses focus on these objectives:• enhancing mathematical knowledge • enabling teachers to transfer mathematics

they have learned into their classrooms • leadership development• action research

Math in the Middle Institute

We will focus on three courses in particular:• Number Theory & Cryptology• Using Mathematics to Understand Our World• Concepts of Calculus

Some objectives which significantly influenced the development of these courses

We want our students to:• Know that effort matters • See connections to school math

Presentation Outline

1. Effort Matters

2. Connections to School Mathematics• Number Theory & Cryptology • Using Math to Understand Our World • Concepts of Calculus

3. Sandi’s Comments

Effort Matters

The National Mathematics Advisory Panel commissioned by the president in April 2006. Draft #42 of Final Report states:

“Children’s goals and beliefs about learning are related to their mathematics performance. Experimental studies have demonstrated that changing children’s beliefs from a focus on ability to a focus on effort increases their engagement in mathematics learning and improves mathematics outcomes. Effort matters.”

Effort Matters One of the five strands of mathematical

proficiency described in Adding it Up is

National Research Council 2001

Productive dispositionHabitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one’s own efficacy.

Effort Matters

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.

Effort Matters

“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.

Effort Matters

Benefits of using Habits of Mind problems• They typically have a variety of solution

strategies – allowing teachers to see the benefits of having students present different solutions

• The focus is on PROCESS – thus encouraging persistence and promoting the value of EFFORT

• Serve as a resource for teachers; adaptations of the Habits of Mind problems consistently find their way into teachers’ classrooms

Effort Matters

Examples of Habits of Mind Problems• Crossing the River• Locker Problem• Handshake Problem• Pentominoes • Rice on a Checkerboard• Others involving combinatorics, patterns,

repeated applications of the Pythagorean Theorem, geometric series, etc.

A problem to get started: Making change

What is the fewest number of coins that it will take to make 48 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.

Extension: 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.

Student

work

sample

(grade 7)

Studentwork sample(grade 8)

Habits of Mind Problems Sandi’s Response

• Allow students to see how others think and organize thoughts

• Reinforce the idea that there is not just one way to approach a problem

• Serve as a bridge between arithmetic, algebra and geometry

• Encourage cooperative learning


1. Effort Matters

2. Connections to School Mathematics• Number Theory & Cryptology • Using Math to Understand Our World • Concepts of Calculus


Making Connections

Why is it so important?

“Most teachers see very little connection between the mathematics they study as an undergraduate and the mathematics they teach. … a consequence is that, because individual topics are not recognized as things that fit into a larger landscape, the emphasis on a topic may end up being on some low-level application instead of on the mathematically important connections it makes”.

Al Cuoco, AMS Notices 2001

Making Connections

What should be done?

“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.”

Roger HoweAMS/MAA Joint Meetings 2001

Number Theory and Cryptology

Primary Goal:

Learn the concepts of Number Theory which are necessary to understand RSA cryptography, making connections to the classroom whenever possible.


Secondary goals:• Provide an example of a modern, real-world

application of mathematics• Convince teachers they are mathematicians,

capable of understanding some complex ideas

Fear of factors: Cracking prime-number case raises online security doubts, Lee Dembart, International Herald Tribune, March 10, 2003

“… the details of the mathematics (used in online security) may be understood by relatively few people…”


Connections

1.Congruences and boxes

Today is Wednesday. What day of the week will it be in 11 days, in 95 days?

Sun

4

Mon

5

Tues

6

Wed

0

Thurs

1

Fri

2

Sat

3

0

14

11

95

8

-6


Connections

2. Modular arithmetic and divisibility rulesDivisibility rules for: 3, 4, 8, 9, 11– Write the rules as you would explain them to your

students. – How do you write that an integer n is divisible by 9

using congruence statements?– How does the statement n 0 mod 9 translate into a

divisibility rule?

Number Theory and CryptologyConnections

3. Definitions and Proofs• Arrive at definitions of even and odd and use them to

prove: even x odd = even

odd x odd = odd (etc.)

Demonstrates, in a very tangible way, the value of precise definitions and their importance in mathematical proof

Number Theory and CryptologyConnections

4. Proof and the counter example• Does a display of 100 green frogs prove that all frogs are green? What about

1000 frogs?

To disprove, all that is required is

• Prove or disprove: If d|ab then d|a or d|b. (Where d, a & b are integers)

one blue frog

Number Theory and CryptologySandi’s Response

• Use of prime factorization to determine all factors of a number

• Numbers that are relatively prime

• The infinitude of primes—they keep going and going!

Using Math to UnderstandOur World

Course Description• Students will examine the mathematics underlying

socially-relevant questions from a variety of academic disciplines

• Students will construct and study mathematical models of the problems

• Sources will include original documentation whenever possible (such as government data, reports and research papers) to provide a sense of the very real role mathematics plays in society, both past and present


Primary course Goals: broaden students’ perspective by applying mathematical concepts to various interdisciplinary settings

Additional course goals:

(1) develop mathematical modeling and problem solving skills

(2) improve ability to read technical reports and research articles

(3) refine written mathematical communication skills


A few project titles:• Measuring Temperature & Newton’s Law of

Cooling• Using Body Temperature to Estimate Time Since

Death• Building Up Savings and Debt• Childhood Growth Charts


Project III: Containing Infectious Diseases

(a.k.a. School Pox)

Using Math to Understand Our World

Sandi’s Response

• Answers the question, “When am I ever going to use this?”

• Extensive writing about the mathematics

• Real life data that shows calculations aren’t always going to come out perfectly

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.

M2 Innovations:Learning & Teaching Projects

Learning & Teaching Project

H.S. Teacher sample(page 1)


H.S. Teacher sample(page 2)


H.S. Student A3 work sample


H.S. Student A1 work sample


Gr 5 Teacher sample(page 1)

Learning & Teaching ProjectGr 5 Teacher sample (page 2)

Learning & Teaching ProjectGr 5 Teacher sample (page 3)


1. Effort Matters

2. Connections to School Mathematics• Number Theory & Cryptology• Using Math to Understand Our World • Concepts of Calculus


Sandi’s Comments

• Action Research project—a wonderful learning

experience

• Need to know where our students came from mathematically AND where they will go in the future

• M2 has changed how I teach

• The Heart of Mathematics An invitation to effective thinking Burger & Starbird Key Curriculum Press, 2000

Concepts of CalculusPrimary Goals: • develop a fundamental understanding of key

mathematical ideas utilized in calculus

• develop conceptual knowledge of the processes of differentiation and integration, along with their applications

• help teachers gain insight into topics found in school mathematics which are related and foundational to the development of calculus

Concepts of Calculus

Features:

• Course topics progress from limits to the Fundamental Theorem of Calculus

• Rigorous treatment of various integration and differentiation techniques (such as the product rule for middle level teachers) are excluded

• Graphing calculators are used extensively


Features:

• Very little lecture: course consists of a series of problem sets students work through to explore calculus concepts

• Connections to school mathematics topics are made consistently throughout the course


Connections

The problem sets on which the course is based connect to school mathematics; many teachers take them directly to their classrooms

• Sample problem set

Note: The problem sets are adaptations of worksheets first developed by Kristin Umland and Matt Bardoe, University of New Mexico, La META project


ConnectionsOne vehicle for classroom connections:

homework assignments.

Calculus ConnectionsAsma Harcharras, University of MissouriDorina Mitrea, University of MissouriPublisher: Prentice Hall, 2007


Connections (taken verbatim out of M.L. or H.S. texts)

Examples•Fix area and minimize perimeter using integer dimensions

•Fix perimeter and maximize area using integer dimensions

•Estimate area of an irregular shape using grids; find lower and upper estimates; improve estimates with finer grids

Calculus ConnectionsHarcharras & Mitrea Prentice Hall, 2007

Concepts of calculus “As I took the Calculus class, I was reminded over and

over of the importance of Algebra in preparing students for upper level mathematics. One particularly important aspect was linear graphing. I know my students need a good understanding of slope and rate of change in order for them to understand instantaneous rate of change/derivatives.

Along with linear and quadratic graphing, Algebra students need to have experiences with a variety of graphs with many dips and turns. They also need to be able to describe these using words like increasing and decreasing. Polynomials is another concept of which my Algebra students need a good grasp in order to work with a broader range of mathematical situations. I also saw the importance of solving equations and inverse operations for them to also understand the idea of integrals and derivatives.” M. Bornemeier, 2006 M2 graduate

Linear and Polynomial Functions• Interpret slope. What does it mean? • Celsius and Fahrenheit are related by the equation

F(C)= C + 32 Interpret slope and identify the units for the slope.

• Define, distinguish, create polynomial functions• Given a standard 8.5 X 11 inch piece of paper, determine

a function which gives the volume of a box (no lid) made by cutting squares from each of the corners and folding up the sides. Let x be the length of side of the square.

5

9

Concepts of CalculusSandi’s Response

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