The Overboard Partitioner

Why Should K-12 Teachers Learn Abstract Algebra?

Quality mathematics instruction necessarily includes explanation of why mathematical processes work as they do. This is most relevant at the foundational levels, when students first encounter arithmetic, algebra, and geometry. However, mastery of this understanding is taught at the third-year college level. In short, abstract algebra equips a teacher of K-12 mathematics with the understanding required to answer many of the tricky questions their own students will be coming up with.

The following is a non-exhaustive list of concepts from abstract algebra as they relate to exposition of the K-12 curriculum.

Abstraction

• Why do we treat quantity separately from the type of object being quantified?
• A mathematical expression (e.g., 3 + 4) may be treated either as a process (addition of two integers) or as the result of that process (the sum is 7). How do you identify which perspective is being used at any given time? When is each perspective fruitful?
• What does it mean to treat the variable x as an arbitrary number? How do we know what operations are valid, and how does this relate to our domain of discourse?
• An equation may be treated either as a process (an unkown is made equal to a known) or a relationship (two representations of the same quantity). When is each perspective fruitful?
• Simillarly, a function may be treated either as a process or the output of that process. When is each perspective fruitful?
• Why would a person think to add the expression (a-a) to an unknown? Or multiply by (a/a)? Or multiply by a conjugate radical?
• Why is there addition of counts, addition of lengths, addition of areas, addition of functions, addition of equations? How do we create a concept of addition that describes all of these processes?
• What does it mean for a count to be zero? Why is there a zero function, a zero matrix, a zero vector? What properties do these share that warrants the common name?

Commutivity

• Why should commutivity of addition and multiplication not be taken for granted?
• Why does order of symmetry composition matter?

Assosiativity

• How does the addition algorithm work? Why are there carries?
• What are all the ways that the addition (w + x + y + z) can be grouped, and why do they all produce the same result?

Distributivity

• How does the multiplication algorithm work? Why are there carries?
• Why is (x + y)^2 not equal to (x^2 + y^2)? Which other operations are nonlinear?

Closure

• Why are rational numbers closed under addition and multiplication, but irrational numbers are not?
• Why do rational numbers have a standard form, but irrational numbers do not?
• What does it mean to say that infinity is not a number?

Inverses

• Why is subtracting the number x equivalent to adding by its negative? Why is dividing by x equivalent to multiplying by its reciprocal?
• Why is the product of two negative numbers positive?
• Why doesn't zero have a reciprocal? Why can't we divide by zero?
• Why is (a+b)/c equal to a/c + b/c, but a/(b+c) is not equal to (a/b + a/c)?

Subgroups/Normality

• Why do two reflections always make a rotation?
• What is the relationship between the natural numbers, the integers, the rational numbers, and the real numbers?
• Why do we use divisibility and factoring when working with integers instead of division and fractions?

Group Quotient

• Why does mirror symmetry occur in relation to a line? What happens when we fold on the mirror line?
• What happens when we substitute x = a into a polynomial?
• Why can this be applied to polynomials without real roots?

Homomorphism

• When does the concept of ordering agree with the concept of magnitude, and when are they in conflict? How is this resolved?
• Why do manipulations of (a + b), (ab), and exp(a + b) follow simillar rules?
• What are inverse functions, and when do they exist?
• What is the difference between sqrt(x^2) and [sqrt(x)]^2? Bewtween exp(log(x)) and log(exp(x))?
• What are the properties of linear functions that make them a rich subject for study?
• What properties does the absolute value function preserve, and what properties does it negate?

Group Action

• Why are plane symmetries categorized by translation, reflection, and rotation? Why can symmetries be composed and decomposed?
• How can we take advantage of symmetry to simplify calculations?
• What does it mean to scale a geometric figure, a function, or a vector, and why are these all quantified by the real numbers?
• How does negation relate to reflection?
• Why do length, area, and volume scale at different rates?
• The process of solving an equation involves many algebraic steps. Which of these are reversible, and how does that influence our reasoning?
• How does the geometric interpretation of vectors relate to the group action of shifting geometric figures?
• How does multiplication and division of unit labels work? Why can't units of differing types be added or subtracted?

Rings

• Why do polynomials factor? How does this relate to factoring of integers?
• What is the relationship between polynomial arithmetic and base 10 arithmetic?