03/31/2010 | Maggie Martin Connell
Math Moments With Maggie Martin Connell
Lies We Tell...
Who, us? I’m afraid so, but who knew? The good news is we can forgive ourselves, since we can only teach what we know. Here are some very common lies we can stop telling, beginning tomorrow (because we’ll know better).
There are lots of lies we tell about zero, mostly because it’s a complex idea. Here is one of my favorites.
What happens when you add zero to 27? (You get 27) Add zero to 346? (You get 346). Add zero to six million? (You get six million). So, what are you telling me about adding zero to a number? (Nothing happens) Now think of the rule’ we invariably give our students for multiplying any number by 10. (Add zero)! Hmmm.
You can’t take a bigger number from a smaller number.
Well, whomever came up with that little gem has never seen my bank account. And what of a northern winter, when it’s 25 degrees outside and a cold front hits, dropping the temperature by 30 degrees? And, what of a football game? Hmmm.
Taking a ‘bigger’ number from a ‘smaller’ number is entirely possible! No small wonder that negative integers are so challenging for so many.
Speaking of Bigger and Smaller
The following is an actual student response on a test:
Test Item: Write a number bigger than four.
Student Response: four
Even more telling is the teacher’s remedial language. “But, bigger doesn’t mean bigger, I mean not in this case.” Hmmm.
What the teacher really wants to know is whether or not the student understands basic quantitative ideas, not physical size. How would you modify the test item?
Write a number greater than four would work, but only if that language is consistent with the language used during instruction.
You know 380 ÷ 4, but “4 guzinta 3, doesn’t go”. Hmmm.
This one is a beauty! First of all, it isn’t three at all, it’s 300. And it does ‘go’ quite nicely — 75 times in fact. Now, all we need to know is how many times 4 ‘goes into’ 80 — 20 times. So, altogether, 4 ‘goes into’ 380... 95 times.
The ‘goes into’ part isn’t really a lie but, it’s a rather bizarre way to express the concept of dividing. A learner might make more meaning by thinking “380 shared by 4.”
Two forty-five (245)? Hmmm.
In the stage of learning where concepts are just developing, calling numbers by their real names forces a shift from digit-oriented thinking to number-oriented thinking, the root of all number sense. For example, asking students to turn to page “two hundred 45” rather than “two forty-five” subtly reinforces that 245 represents more than 200 pages. It’s a simple thing to do but, if done consistently, brings lasting benefits.
Of course, at some point, for the sake of efficiency, students need to understand the implicit value of each digit in a number, according to where it is placed. That understanding will most assuredly evolve with sufficient meaningful experiences over time. The rush toward digit-oriented procedures is typically a false economy in the end, as we find ourselves returning to pick up missing pieces in a learner’s understanding.
Raise your right hand and repeat after me, “I promise to call all numbers by their real names (for now).” This one habit will serve you well on your quest toward number sense for your students.
Some things we say about mathematical ideas are not exactly lies, but neither are they the whole truth. For example, consider a rectangle. If I asked you to draw one for me, there’s a very good chance that this is what I would see and you would be absolutely correct!
The problem is... so is this.
Many would insist on calling it a square and not a rectangle. That it is a square is true of course, but the notion that is not a rectangle is a misconception. But how can it be both? Just what is a rectangle anyway?
Four straight sides
Opposite sides are equal and parallel
Four square corners
That’s it. The misconception that it needs to have two long sides and two short sides stems from its traditional presentation — a half-truth. Since each of these attributes also applies to a square, it must also be a rectangle; but it is a special rectangle because all of its sides are equal in length. If a square is a rectangle, is a rectangle also a square? You do the math!
As teachers, we are crafters of words; nothing could be more important to making meaning, than the words we choose to talk about the math we see. Stay tuned for part two. We will continue our quest for pointed parlance. In the meantime, watch your language. What you say really does make all the difference.