Mathematicians Ask Questions
[Photo by walknboston via flickr. Part 2 of my Homeschooling with Math Anxiety Series.]
Wise mathematicians are never satisfied with merely finding the answer to a problem. If they decide to put effort into solving a math puzzle, then they are determined to milk every drop of knowledge they can get from that problem. When mathematicians find an answer, they always go back and think about the problem again.
- Is there another way to look at it?
- Can we make our solution simpler or more elegant?
- Does this problem relate to any other mathematical idea?
- Can we expand our solution and find a general principle?
Homeschooling with Math Anxiety
As math teacher Herb Gross says, “What’s really neat about mathematics is that even when there’s only one right answer, there’s never only one right way to do the problem.”
And other times, when you think there is only one right answer, your children may surprise you. I found this out when playing a pre-algebra puzzle game with my daughter: “What number am I? If you take away one fourth of me and then add two, you get 17.”
I was surprised when her answer didn’t match mine. In fact, it was triple the answer that I expected!
I asked my daughter, “How did you figure it out?” and discovered that the answer depends on how you understand the words in the question. When you “take away one fourth”, are you taking it as your own share, or are you throwing it away and keeping the rest? I saw subtraction as the latter, but my daughter thought the first way, as if she had taken a share of pizza.
Let this be a warning: If your child’s answer is not the same as yours, don’t automatically assume she is wrong! Language is a complicated thing, and even a math problem may be open to different interpretations.
But if you think like a mathematician and ask the right questions, you’ll learn something new.
Thinking like a Mathematician
School textbooks only ask questions for which they know the answer. When homeschoolers learn to think like mathematicians, we will ask a different type of question.
Try asking your children (and encouraging them to ask) questions to which you don’t know the answer, questions like:
- What do you think?
- What do you see?
- How did you figure that out?
- Is there another way to look at it?
- Will this always be true?
- Can we predict what will happen next?
- Is there a pattern?
- Will the pattern continue, or will it run out?
- How can we be sure?
- What would happen if ___?
- Why?
This Homeschooling with Math Anxiety Series is an excerpt from my new book Let’s Play Math: How Homeschooling Families Can Learn Math Together, and Enjoy It!
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Of course you are right, by “lazy” I mean something like “discerning the best use of limited time and energy” — or, as you said, “economical.” I explained this more thoroughly in the previous post, which introduces the series. By using the 4-letter word, I hope to take advantage of shock value to break through the wall of math anxiety so many homeschooling parents have.
As for the word problem, students do need to learn the standard expectations of test makers, but that is more of a language lesson than math. It is MUCH more important that we recognize and acknowledge the logic of the child’s response.
Incidentally, the daughter in question graduated from the University of Illinois this spring with an engineering major and math minor — yet when I asked her the same word problem the other day, she gave me the same unorthodox answer. Of course, an equation for the problem would only have one right answer, but the problem is that the words “take away” leave open for interpretation exactly what the correct equation should be:
(1/4)N + 2 = 17
or
[1 - (1/4)]N + 2 = 17
If the word problem had used “subtract” instead of “take away,” then there would not be the same ambiguity.
I think there still would’ve been the same ambiguity if the word “subtract” had been used. It’s just a poorly worded problem.
You may be right, Jay. If you read the original post, you’ll see this problem was part of an off-the-cuff conversational game, so I’m not surprised that I came up with a poorly-worded problem in a situation like that.
But I want parents to realize there can be poorly-worded problems in their textbooks, too. We’ve run into several of them over the years (and I wish I had thought to copy them down at the time!), so I’ve learned to listen to the student’s explanation. If her answer makes sense according to how she interpreted the words, then I count it as right.
We may also talk about the standard interpretation, as Michael suggested. But my first priority is to encourage the child’s logical reasoning.
Denise: Don’t get sidetracked by the wording of the problem. Yes, students need to be mind-readers to get high test scores, but this is NOT truly a useful skill in real life so it’s very good to make this distinction, as you have. Mathematicians and scientists are NOT like that. They are precise, but they also think about possibilities, which is how discoveries are made.
The best would have been for your daughter to have pointed out the ambiguities she saw in it.
I would say that almost every real life problem has most of the work in problem understanding and definition, which is not developed enough in school where the focus is in guessing what the meaning of “right answer” is. This is a valuable post.
I think you’re right, Alex. But my daughter was nine at the time, and even as an adult (I asked her the question again recently) she didn’t see the ambiguity. I think it’s our job as parents and teachers to:
(1) Be aware that language is ambiguous, and be on the lookout for it. Don’t assume the student is wrong, just because that’s not the way we understood the problem.
(2) Point ambiguity out to our children when we notice it, so we can help them learn to recognize it as well.
(3) (And least important) Help our children learn the standard expectations of math word problems. All rates stay constant unless otherwise noted, workers do not get in each others’ way during rate problem projects, etc.