One of the most humbling experiences in my time as an undergraduate student was when I took a first year mathematics course in my fifth year. It had been listed in the course calendar as a statistics course, and I enrolled hoping to get a better understanding of number crunching for the vast amounts of data I was compiling in my electrochemical experiments. As it would turn out however, the professor for the course dealt strictly with number theory in his research, and we soon found ourselves not doing Q-tests, but developing proofs for an infinite set of primes.
Up until that point I had conquered five senior courses in atomic theory and quantum mechanics, and I thought a first year course on number theory was going to be laughable. I was not prepared however to have everything I had ever learned in math put into question. Mathematical reasoning, as it turned out, was a language of its own, and did not look like anything I had ever done before. That is not to say that I did not
understand it, but that I struggled with the strict rules and conventions that at times seemed almost arbitrary to me.
Euclidean division algorithm
My experience in this course made me reflect on my entire understanding of mathematics. Yes, I knew what even, odd, and prime numbers were, but when it came to the rules around them I was not so certain. I felt like I was starting from scratch---that there was some hidden truth underlying mathematics that had been hidden away from me all along. And this, I suppose, is how most students feel in the math classroom. Lost, confused, and as if the inner-workings of math are always obfuscated behind cryptic rules and language.
Steven says that when you add two odd numbers with an even number, the answer is always even. Is Steven correct?
If like me you learned your even and odd rules by rote, you're probably thinking that
yes, it's always even---because the sum of two odd numbers is even, then the sum of two even numbers is even. But while we quickly come to Steven's defense, many of us probably don't know
why this is true. And this is where the way number sense has traditionally been taught is failing us.
Consider some arbitrary set of numbers, S = { 0, 1, 2, 3, 4, 5, 6, ... n }
We all probably know that numbers are either even or odd, and that even numbers are divisible by 2. This is what most of us learned by rote! But what about zero? Is it even or odd? Students will probably wonder this, and I must admit to having never thought much about it until now. Can we divide zero by two? Yes, we can, therefore it is even.
Our even and odd numbers therefore alternate (i.e. even, odd, even, odd, etc.) if we list out all of our integers, and each odd number is always greater than and less than an even number by 1. If we represent an even number using A and an odd number using B, we could say that B = A + 1, for all odd numbers. But what does this have to do with Steven? We haven't forgotten about him, have we?
What Steven is saying is that the sum of two odd numbers and an even number is always even, so let's take a look at the general expression:
A = B + B + A
A = (A + 1) + (A + 1) + A
A = A + A + A + 2
What we end up with is that we can rewrite our two odd numbers so that they are equal to even numbers plus 1. We now have three even numbers, which have an even sum, and two extra 1s. That extra 2 is also even! Would this work with three odd numbers? You would have three 1s, or 3, and....no, it's not even! And thus is the beauty of number theory, because these rules apply to
all of the integers.
Elizabeth says that if you add a negative to a positive integer, the sum is zero or less. Is Elizabeth's statement true for all integers?
Elizabeth's predicament is not as clear as Steven's. It might seem obvious that it can't
always be zero or less, because it's easy to find two numbers with a positive sum, but can we prove it? Coming at the problem from a number sense perspective, what she is really saying is that for two integers, one positive and one negative:
a + b ≤ 0
Where a is positive and b is negative. However, adding a negative integer is the same as subtracting a positive integer, so let's use some math rules to rewrite the problem a little more clearly:
a + b ≤ 0
a + (-1)|b| ≤ 0
a - |b| ≤ 0
a - b ≤ 0
For all positive integers, a and b.
What we now have is an expression that states that for any two positive integers, the difference is always less than or equal to zero. That can't be! And this line of reasoning is how an understanding of number sense can enable students to succeed in the math classroom, rather than just passing off rules they had learned previously.
