What’s Really Important to be Successful in Mathematics

The most important thing you can do for your children’s math education is to never let them know if you hated math or were bad at math. Instead, help them to build a number sense, and at the same time, you might get more comfortable with numbers yourself.

I am part of a generation with more developed computational mental math skills. Now that everyone carries a cell phone with a calculator, I think too many people have lost comfort with the basics of computation and approximation. After thirty-five years of teaching advanced mathematics courses in high school, I have come to believe that a well-developed number sense is one of the keys to success in mathematics. I’ve taught too many bright young people whose well-meaning parents get them tutoring to advance them in traditional content too quickly.

I frequently tell stories of distressing encounters with people who have a limited understanding of fractions and percents, both my students and people who work in retail.

I was in a store advertising a 30% off sale, looking at an item priced at $148. A store clerk came up to me and offered to tell me what the sale price would be. I had already rounded up to $150, multiplied by 0.7 in my head (actually my thinking was 70+35) which would be $105, and approximated to $104 because I had rounded up. I said to her, “Oh, I’ve got it, it’s about $104.” The clerk used her calculator to multiply $148 by 0.3 and then subtract that amount ($44.40) from $148 to get $103.60. She was shocked that I was so close. I was shocked that she didn’t understand that 30% off meant 70% of the original price and had to use two steps rather than one. If reading this paragraph has your head spinning or thinking I knew I wasn’t good at math, just try reading again and think a bit about your math education.

I was working with a high school sophomore on some fractions and asked him what ¼ of 40 was. He got that one, 10. But when I asked him what ¼ of 60 was he reached for his calculator. I stopped him and instead I asked him what ½ of 60 was, and then asked him to take half of that and he got 15. Then I realized that he didn’t truly understand that half of a half was a quarter. I then moved on to ask him what 33.33% of 60 was and he set up a beautiful proportion of 33.33/100 = x/60, used his calculator to solve it, and got 19.998. When I asked him if he knew what ⅓ of 60 was he paused and then said 20, but he truly didn’t understand why he got 19.998. There are too many students who can imitate algorithms but do not understand anything about what they are doing and why. Help yourself, your children, and their teachers using the following suggestions:

For parents:

• Have at least one analog clock in your home. Use it to teach fractions and angles. Talk about why 1:45 and a quarter to 2 mean the same thing. Teach them what clockwise and counterclockwise mean.
• Cook with your child using plastic measuring cups. For little ones let them play with filling the ⅓ cup 3 times and pouring it into the one cup.
• Approximate and estimate. How far do you think we’ve walked? More or less than a football field? How tall is that tree? Three times taller than I am? How long do you think we’ll have to wait in this line?
• Play with coins. How many different ways can you make 27 cents? It’s interesting that if I ask a student what 7 times 25 is he will start doing multiplication (or look for that calculator) but if I ask how much money 7 quarters is I get a quick answer.
• Play games. Card games like Uno and Take 5 are great; I also highly recommend Prime Climb.

For elementary and middle school teachers:

• Do NOT allow elementary school children to use a calculator, even to check their work.
• Emphasize variety to help with the flexibility of mind. How many different kinds of triangles can you draw? How many different ways can you add 2 numbers to get 10? What about subtracting 2 numbers to get 10? (What an interesting way to introduce the concept of infinity)
• Try not to say “you can’t” as in “you can’t subtract a bigger number from a smaller number,” or “you can’t take the square root of 3.” You can even introduce the idea of negative numbers. What if there’s a pile of dirt two feet high and you dig down three feet, or what if you owe me 10 dollars but you only have 9?
• Talk about really big numbers. A million, a billion, a trillion. Discover for yourself how to explain the differences and share that will your child. Or use my favorite. If you had a million dollars and spent a thousand dollars a day that money would last you 1000 days or about 2.7 years. If you had a billion dollars and spent a thousand dollars a day that money would last you a million days or about 2740 years!
• The first time students encounter decimals insist that they say them aloud. Not “point 6,” but “6 tenths.” I can’t tell you how many times a student has asked me how to convert a decimal to a fraction, or not understood that when they use the “flipper” button on their calculator to turn 0.6 into ⅗ that it came from 6/10 being reduced.
• Introduce the difference between linear and exponential growth as early as possible without ever using those words. Adding 2 repeatedly gives the sequence 2, 4, 6, 8 which doesn’t seem that different from multiplying by 2 and getting the sequence 2, 4, 8, 16 until you get to a few larger numbers (10, 12, 14, 16, 18, 20, 22 vs. 32, 64, 128, 256, 512, 1024, 2048) The game 2048 is the best thing ever!

Many people worry about illiteracy in our society I worry about innumeracy. To make math meaningful you have to start young and develop a number sense that will enable deeper understanding and maybe even get you to Calculus.