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 ___?
As your children try to put their thoughts into words, keep in mind this truth:
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very 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.
In a new blog post, Christopher Danielson warns us about a question to avoid and suggests a great one to ask instead:
CREDITS: Photo (top) by walknboston via flickr (CC BY 2.0). This post 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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