Commentary on Theories of Mathematics Education

442 J.E. Jacobs
ways of knowing. The availability of high speed computers enables mathematicians
to prove things by exhaustion. Computers have demonstrated that which had only
been a conjecture is actually a theorem, as was done with the Four Color Theorem.
In addition to computer proofs, there can be other demonstrations that are sufficient
and valid ways of knowing in mathematics that play to the connected style of
knowing.
The following example demonstrates how a mathematics instructor can use experiential
and intuitive knowing to prove a mathematical theorem. In this example
a proof is something other than a deductive argument, yet is convincing and generalizable.
The theorem is one which any first grader knows, yet traditionally we wait
until individuals can use algebra to prove it. Here are three ways of knowing that
The sum of two odd numbers is even, the last of which uses feminist pedagogy.
The first way generates the conjecture. This is done by induction. By examining
the sum of two odd numbers, once one knows the definitions of odd and even
numbers, one easily concludes that their sum is always even.
3 + 5 = 8, 7 + 5 = 12, 43 + 31 = 94
Though looking at these examples may be convincing and is certainly necessary to
generating a hypothesis, this is not a proof. The second way of knowing is traditional
mathematics. The proof is done algebraically, involving the manipulation of
symbols.
Let the first odd number = 2a + 1, where a is a whole number.
Let the second odd number = 2b + 1, where b is a whole number.
(2a + 1) + (2b + 1) = 2a + 2b + 1 + 1 = 2a + 2b + 2 = 2(a + b + 1)
and 2(a + b + 1) is even for it is 2 times a whole number.
(This proof requires algebraic manipulation, using the closure and associative
laws for addition and the distributive law for multiplication over addition.) Generally,
this is what is expected of individuals who are asked to prove the stated
theorem.
An alternative proof involves the students modeling even and odd numbers using
egg cartons. Any even number will look like in Fig. 1.
No matter what even number we want, we can picture, or create, that number by
stapling together as many cartons for a dozen eggs as needed. This generalizability
of the model is essential. Similarly, odd numbers are modeled by taking an even
number and adding one, or in terms of egg cartons, leaving one “doohickey” or egg
compartment on.
To add two odd numbers, use the model in Fig. 2 for odd numbers.
Figure 3 shows the rearrangement of the models of two odd numbers that portray
the sum as an even number. This is just like what happens in the deductive proof. The
two “l’s” make a 2 or the “doohickeys” match up. This last approach lets learners
Fig. 1 Egg carton model of
even numbers