· Number magnitude
· Mental computation
· Mathematical properties
In this article, we will explore the effects of operations.
At a recent math training in Pacific Grove, California, I asked an audience if it is true that squaring a number results in a larger number than the original factor? When I pose this question in class, my students often quickly say, “Yes”. Then they hesitate a bit and begin to question themselves.
The math teachers correctly responded that no, squaring a number does not always result in a greater product than the factors. I asked them when the statement is not true.
Immediately someone in the audience said, “When it’s less than one.”
“So,” I countered, if I square negative three, the product is less than negative three. Now they began to question themselves.
“No, I meant a fraction because one half squared is only one fourth,” they corrected.
“Negative three is a fraction,” I responded. “It can be written with a denominator of one. Two and a half is also a fraction.”
“It has to be a fraction between zero and one,” they suggested. Now they were refining the domain of possible exceptions.
“If you say, between zero and one, then it can’t be equal to zero or one,” I again countered. “Does that mean that 02<0 or that 12<1?”
They finally said that it could be any number between zero and one and inclusive. I asked, “How would you write that domain mathematically.” We finally settled on the expression:
This illustrated how succinctly mathematics can capture a concept that is more difficult to express linguistically. This is just one example of how we can explore the effects of operations with students. It also illustrates the richness of the mathematical dialog that results and the importance of the teacher’s direction of this dialog through targeted questioning. We also developed the idea of proof by negation as that is how I tested each statement the audience gave. Lastly, if we as math teachers struggled with this question, what do we think our students might go through in their reasoning process?
Other questions we might ask could include:
When is the sum of two numbers less than either of the addends? (If one or both addends are negative, the sum is less.)
Can a sum be equal to one of its addends? (If one of the addends is zero, the additive identity, the sum is equal to the other addend.)
Is a number divided by itself always one? (This is not true for zero, as division by zero cannot be defined.)
If you subtract the same number from both sides of an inequality, will the inequality sign need to change? Illustrate this. (No, it won’t. 6<8 and 6–1<8–1 because 5<7.)
Illustrate that the commutative property doesn’t apply to subtraction. (6–3≠3–6)
What happens when you try to apply the commutative property to subtraction? (The two answers are additive inverses.)
If you keep adding one the numerator and one to the denominator of ½, how big can the fraction get? (It approaches one: n/n+1)
These questions force us to think mathematically and to use sophisticated reasoning processes to justify our thinking. Our brains thrive on these sorts of morsels and grow healthy and strong like a body that is fed nutritional foods.
As we approach the Common Core standards, we need to be moving our students in this direction. Future testing will not only require students to perform mathematical procedures correctly, but to reason correctly about numbers. Like exercise, proficiency with number sense is not achieved as a unit of instruction but is a part of an ongoing training program. Investing just a few minutes at the start of class with these types of warm ups will produce notable improvements. Adjusting our questioning and teaching strategies will achieve even greater results.