Geoff Birky, Jim Fey, Tim Fukawa-Connelly, Kadian Howell, and Carolina Napp-Avelli

University of Maryland

Abstract Algebra for Teachers:

Challenges and Opportunities

“Mr. Joshi, why is it that the product of two negatives is always a positive?”

0 = -5 + 5-3(0) = -3(-5 + 5)

0 = (-3)(-5) + (-3)(5)0 = (-3)(-5) + (-15)

So (-3)(-5) = 15

Challenges

• Engage students actively in doing mathematics

• Develop understanding and skill in proof

• Develop mathematical habits of mind

• Make relevant to future secondary teachers

• Develop insight into school algebra

• Provide models of teaching practice

Opportunities: Selection and Organization of Course Content

• Present problems rather than results

• Develop from examples to theory

• Explicit focus on mathematics as process

• Emphasize connections to school algebra

• Less might be more

Opportunities: Restructuring Teacher Roles and Use of Class

• Asking questions more than giving answers

• Encourage discourse among students

• Include discussion of mathematical habits of mind as well as mathematical results

• Encourage reflection on thinking

• Refocus assignments, examinations, and grading

What Might It Look Like?

Course Content and Materials

A set of 84 questions that asked students to:

• Analyze situations and look for patterns

• Formulate definitions and construct examples

• Formulate and test conjectures

• Prove theorems

Analyze Situations and Look for Patterns:

Question 12: Compare the familiar properties of number systems to those of Z4, Z7, and Z12.

a. Are there properties that hold in each of the finite number systems?

b. Are there properties that hold in some finite number systems and not in others?

c. Are there properties that hold for some numbers in a finite number system but not for others?

d. What is so special about Z7 and why do you think this finite number system has special properties?

Analyze Situations and Look for Patterns:

Question 14: Explore linear functions in (Z4, +4, •4).

a. Find all solutions to y = 2x, y = 2x + 1, and y = 2x + 2

b. Plot the solutions on a 4 x 4 coordinate grid.

x

y

c. Repeat the work of (a) and (b) with Z7 and Z12.

d. How do the results of your work in (a) - (c) compare with what you might have expected?

Analyze Situations and Look for Patterns:

a. Find standard names for results of these calculations in Z6:

2 – 3 2 ÷ 3 3 ÷ 2 1/3 – 2/3

b. Find standard names for results of these calculations in Z7

2 – 3 2 ÷ 3 3 ÷ 2 3/5 – 4/3

c. Find standard names for results of these calculations in Z12

7 ÷ 8 8 ÷ 7 3/7 – 4/5

d. What causes division and fractions to behave differently in different Zn? What causes division to behave differently than subtraction in Zn?

Question 15: Consider division and fractions in Zn.

Analyze Situations and Look for Patterns:

O R0 R120 R240 D1 D2 D3

R0 R0 R120 R240 D1 D2 D3

R120 R120 R240 R0 D3 D1 D2

R240 R240 R0 R120 D2 D3 D1

D1 D1 D2 D3 R0 R120 R240

D2 D2 D3 D1 R240 R0 R120

D3 D3 D1 D2 R120 R240 R0

Symmetries of an Equilateral Triangle

Analyze Definitions and Construct Examples:

Question 21: Which of these mathematical systems are groups?

a. The rational numbers with operation a * b = (a + b)/2.

b. The points of a plane with operation A * B = C as shown in this sketch:

A B

C

c. The whole numbers with operation a * b = LCM (a, b).

d. The 2 x 2 matrices with non-zero determinant and operation of multiplication.

Formulate and Test Conjectures:

Question 65: Consider the arithmetic systems (Zn, +n, n).

a. Which of these modular arithmetic systems are: (1) rings; (2) rings with unity; (3) commutative rings; (4) integral domains; (5) fields.

b. In any particular Zn, how can you predict the elements that will or will not be divisors of 0?

c. In any particular Zn, how can you predict the elements that will or will not have multiplicative inverses?

Prototypical examples?

Question 44: The items here give two familiar functions that are isomorphisms between familiar number system groups. In each case determine the groups that are involved and explain how you know that the proposed

correspondence is an isomorphism. a. The function f(x) = 2x is an isomorphism from ___ to ____.

[Hint: What are the domain and range of the function f(x) ?]

b. The function g(x) = log(x) is an isomorphism from ____ to ____. [Hint: What are the domain and range of the function g(x)?]

Formulate and Test Conjectures:

Question 46: Which of the following functions describe automorphisms of the indicated group?

a. In the group (Z,+), is the function f(x) = -x an automorphism?

b. In the group (R >0 , ) of positive real numbers under multiplication:

1. Is f(x) = an automorphism? 2. Is g(x) = x2 an automorphism? 3. Is h(x) = | x | an automorphism? 4. Is j(x) = 7x an automorphism? 5. Is k(x) = 1/x an automorphism?

Which of those functions are automorphisms if the group of positive real numbers is replaced by the group of non-zero real numbers?

Analyze Definitions and Construct Examples:

a bi a b

b a

r(cos isin) r cos sinsin cos

Prove Theorems:

Question 24: Evaluate the following claims about properties of all groups and provide counterexamples or proofs.

a. If a, b, and c are elements of group G and ab = cb,then a = c.

b. If a, b, and c are elements of group G and ab = bc, then a = c.

c. If a and b are elements of group G, the equation ax = b has a unique solution.

d. If a and b are elements of group G, then (ab)2 = a2b2.

How would your answers change if G is commutative?

What Might It Look Like?

Activities of a Typical Class Meeting

• Students present and defend results of independent work.

• New concepts and problems introduced with examples.

• Students work in small groups to explore ideas—analyze and produce examples, evaluate conjectures, develop proofs.

• Students present successes from immediate work on the new ideas and problems.

Students Work in Small Groups to Explore Ideas

Analyze and produce examples, generate and evaluate conjectures, and begin to develop proofs.

In any ring (R, , )

a 0 = ...

(-a) b = ...

(-a) (-b) = ...

Students Present Successes and Ask Questions About New Ideas and Problems Based on Their Work.

0 0 0

a(0) a(0 0)

a(0) a(0) a(0)

0 a(0)

Students Present Successes and Ask Questions About New Ideas and Problems Based on Their Work.

a(0) a( a a)

a( a) a2

a2 a2

0

(0)a (1 ( 1))a(1)a ( 1)aa ( 1)a

So ( 1)a a

Students Present Successes and Ask Questions About New Ideas and Problems Based on Their Work.

A(0) = (0 + 0 + 0 + … + 0) [a addends of 0]

Patience with Students Teaches Us

How might we define a function from the 8-element group of symmetries of a square onto the 2-element group (T, •) where T = {-1, 1} so that the resulting map is a homomorphism?

x: R0 R90 R180 R270 V H D1 D2

f(x):

Follow up conversations with students about their experience:

He never really gave his thoughts, he always made us He never really gave his thoughts, he always made us give our thoughts. I don’t think anyone felt give our thoughts. I don’t think anyone felt uncomfortable asking a question or saying wait a minute, uncomfortable asking a question or saying wait a minute, you lost me back there…you lost me back there…

He’s engaging the students and I 100% agree with that He’s engaging the students and I 100% agree with that and I hope to be doing that when I’m a teacher. and I hope to be doing that when I’m a teacher.

It was very student directed. Dr. Fey would try It was very student directed. Dr. Fey would try and keep a low profile, he was basically just and keep a low profile, he was basically just there to ask some questions and clear things up. there to ask some questions and clear things up. He wouldn’t lecture and he would give us the He wouldn’t lecture and he would give us the notes and expect us to discover the math. notes and expect us to discover the math.

Follow up cont’d:

We worked in groups a lot. That happened way more We worked in groups a lot. That happened way more than him lecturing or, not just lecturing, but he didn’t than him lecturing or, not just lecturing, but he didn’t promote discussion as people did it with each other. I promote discussion as people did it with each other. I feel like it was really beneficial. I feel like some of the feel like it was really beneficial. I feel like some of the other math classes are helpful but not completely other math classes are helpful but not completely necessary. … It was nice to have a unique way of necessary. … It was nice to have a unique way of teaching since that’s what I’m gonna be doingteaching since that’s what I’m gonna be doing

I think he basically wanted us to explore things instead of just him teaching them to us… he wanted us to explore so that we could understand them more.

Questions?

Contact information:Contact information:

Tim Fukawa-ConnellyTim Fukawa-Connelly

Tim_fukawaconnelly@yahoo.comTim_fukawaconnelly@yahoo.com

Jim FeyJim Fey

Jimfey@umd.eduJimfey@umd.edu

Other (former) graduate students also Other (former) graduate students also availableavailable