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Abstract Algebra for Teachers:
Challenges and Opportunities
Geoff Birky, Jim Fey, Tim Fukawa-Connelly,
Kadian Howell, and Carolina Napp-Avelli
University of Maryland
“Mr. Joshi, why is it that the product of
two negatives is always a positive?”
0
-3(0)
0
0
So (-3)(-5)
=
=
=
=
=
-5 + 5
-3(-5 + 5)
(-3)(-5) + (-3)(5)
(-3)(-5) + (-15)
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.
y
x
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:
Question 15: Consider division and fractions in Zn.
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?
Analyze Situations and Look for Patterns:
Symmetries of an Equilateral Triangle
O
R0
R120
R240
D1
D2
D3
R0 R120 R240 D1 D2 D3
R0 R120 R240 D1 D2 D3
R120 R240 R0 D3 D1 D2
R240 R0 R120 D2 D3 D1
D1 D2 D3 R0 R120 R240
D2 D3 D1 R240 R0 R120
D3 D1 D2 R120 R240 R0
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:
C
A
B
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  b
a  bi  

a 
b
cos  sin  
r(cos   i sin  )  r 

cos
sin 
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
a(0) 
a(0) 
0
0 0
a(0 0)
a(0) a(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) a
 a  a
0
2

2
2
(0)a  (1 (1))a
 (1)a  (1)a
 a  (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 8element 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:
f(x):
R0
R90
R180
R270
V
H
D1
D2
Follow up conversations with
students about their experience:
 He never really gave his thoughts, he always made us
give our thoughts. I don’t think anyone felt
uncomfortable asking a question or saying wait a minute,
you lost me back there…
 He’s engaging the students and I 100% agree with that
and I hope to be doing that when I’m a teacher.
 It was very student directed. Dr. Fey would try
and keep a low profile, he was basically just
there to ask some questions and clear things up.
He wouldn’t lecture and he would give us the
notes and expect us to discover the math.
Follow up cont’d:
 We worked in groups a lot. That happened way more
than him lecturing or, not just lecturing, but he didn’t
promote discussion as people did it with each other. I
feel like it was really beneficial. I feel like some of the
other math classes are helpful but not completely
necessary. … It was nice to have a unique way of
teaching 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:
 Tim Fukawa-Connelly
[email protected]
 Jim Fey
[email protected]
 Other (former) graduate students also
available