The Mathematical Education of Teachers Lessons Learned from

The Mathematical Education of Teachers Lessons Learned from Math in the Middle Michelle Reeb Homp University of Nebraska – Lincoln Sandi Snyder Shickley Public Schools, NE

Math in the Middle Institute Partnership (M 2) • The information we’ll be sharing is based on our experiences with the Math in the Middle Institute Partnership • Primary focus of M 2 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 M 2 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 M 2 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)

M 2 Major Components • The M 2 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 M 2 teachers and supported by school administrators and university faculty, which develop collegiality • A research initiative for educational improvement and innovation

M 2 Institute Courses • Eight new mathematics and statistics courses designed for middle level teachers • Special sections of three pedagogical courses • An integrated capstone course M 2 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 Productive disposition Habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one’s own efficacy. National Research Council 2001

Effort Matters M 2 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 selfefficacy 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)

Student work 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

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

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 Howe AMS/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.

Number Theory and Cryptology 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…”

Number Theory and Cryptology Connections 1. Congruences and boxes Today is Wednesday. What day of the week will it be in 11 days, in 95 days? 11 0 8 95 14 -6 Sun Mon Tues Wed Thurs Fri Sat 4 5 6 0 1 2 3

Number Theory and Cryptology Connections 2. Modular arithmetic and divisibility rules Divisibility 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 Cryptology Connections 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 Cryptology Connections 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 one blue frog • Prove or disprove: If d|ab then d|a or d|b. (Where d, a & b are integers)

Number Theory and Cryptology Sandi’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 Understand Our 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

Using Math to Understand Our World 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

Using Math to Understand Our World 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

Using Math to Understand Our World 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

M 2 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.

Learning & Teaching Project H. S. Teacher sample (page 1)

Learning & Teaching Project H. S. Teacher sample (page 2)

Learning & Teaching Project H. S. Student A 3 work sample

Learning & Teaching Project H. S. Student A 1 work sample

Learning & Teaching Project Gr 5 Teacher sample (page 1)

Learning & Teaching Project Gr 5 Teacher sample (page 2)

Learning & Teaching Project Gr 5 Teacher sample (page 3)

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

Sandi’s Comments • The Heart of Mathematics An invitation to effective thinking Burger & Starbird Key Curriculum Press, 2000 • Action Research project—a wonderful learning experience • Need to know where our students came from mathematically AND where they will go in the future • M 2 has changed how I teach

Concepts of Calculus Primary 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

Concepts of Calculus 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

Concepts of Calculus 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 Matt Bardoe, University of New Mexico, La META project

Concepts of Calculus Connections One vehicle for classroom connections: homework assignments. Calculus Connections Asma Harcharras, University of Missouri Dorina Mitrea, University of Missouri Publisher: Prentice Hall, 2007

Concepts of Calculus Connections (taken verbatim out of M. L. or H. S. texts) Examples • Fix area and minimize perimeter using integer dimensions Calculus Connections Harcharras & Mitrea Prentice Hall, 2007 • 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

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 M 2 graduate

Concepts of Calculus Sandi’s Response 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.