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 notice?
- What do you wonder?
- What does it remind you of?
- Is there another way to look at it?
- Will this always be true?
- If it’s only true sometimes, what are the conditions that make it true? And what conditions make it false?
- Could part of it be true, and part of it be false?
- If this is false, then is something else true?
- Can you predict what will happen next?
- How did you figure that out?
- Is there a pattern?
- Will the pattern continue, or will it run out?
- How can we be sure?
- How would you change it?
- What would happen if ___?
- Why?
As your children try to put their thoughts into words, keep in mind this truth:
Most remarks made by children consist of correct ideas badly expressed. A good teacher will be wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.
Don’t worry if you can’t find the answers to all of the questions you or your children ask. Some mathematical questions have taken centuries to answer and led to entirely new branches of study. In the quest of learning math, wondering can be its own reward.
Update
In a new blog post, Christopher Danielson warns us about a question to avoid and suggests a great one to ask instead:
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“Mathematicians Ask Questions” is the second of three in my Homeschooling with Math Anxiety Series, which is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, now available at your favorite online book dealer.
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“Mathematicians Ask Questions” copyright © 2012 by Denise Gaskins. Image at the top of the post copyright © Depositphotos / walknboston via Flickr (CC BY 2.0).
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I get why you say mathematicians are ‘lazy’ but of course it’s quite the reverse. A better, less misleading word would be ‘economical.’
As for your daughter, her calculation is sensible (given her interpretation of the words, she sets up a reasonable equation and solves it properly); does that mean you’ve both gotten right answers? I don’t believe so. The standard translation of the words into math symbols would lead to 20, not 60, as THE answer.
Granted, all ambiguity could be removed with wording that was more precise, but were this problem to appear on a standardized test, my bet would be heavily on 20 as the “best answer.” Our friendly testing giants don’t ask for the ‘right’ answer; just the best one. CYA has long been their watchword.
Students need to learn that unless you are in a brain-teaser situation, there is usually only one “reasonable” reading of word problems. We can debate this issue all we like, but the wise student, mentor, teacher, parent, or tutor will want to know how to get the “best” answer. When your daughter writes her own math problems (as she should), ask her to think about alternate interpretations of her words that someone might make and be led to different results using proper mathematics. This may help her be more precise as she does math and science, and to improve her writing skills in general. Worth a shot.
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.