One characteristic of a good friend is that they have the ability to challenge your thinking. Without people like this in your life, it is easy to start thinking that you pretty much know everything. I am fortunate to have a lot of people in my life who challenge my thinking. Recently, a friend asked me about this new math being taught at his son's high school.

The teaching and learning of math is a pet peeve of mine. My biggest complaint about math is the common belief that because the kid who makes $7.50 an hour at the local Gas and Grub got my change wrong, schools have failed at math instruction. If you haven't said it yourself, someone has said it to you.

Let's think about the state of what kids in Kansas know and are able to do compared to the good old days when we could all make change. Folks my age can remember the "advanced kids" were tracked into algebra class in the freshman year. Thirty years later, my kids took algebra in the 7th grade. In my large high school, a handful of kids graduated with four years of math. Now many students go to college with calculus credit already on their transcript. Kids today take and learn more math than they did in the good old days.

The bigger issue is that it is still not enough. Our kids need to have a deeper understanding of math and how to use it. We have to expand our definition of math, and at the same time go deeper with our understanding of math. My friend's concern was that kids were not spending enough time on memorizing math facts, and too much time on learning new and different ways to solve math problems. That concern lays bare education's fundamental math problem.

In the good old days, we memorized the multiplication tables. We worked on speed addition and subtraction. A good memory made you a good math student. As an educator, I have heard countless parents talk about the fact that their kids were straight A math students until Algebra, and suddenly they were getting D's and F’s. Once a student gets to algebra, statistics, physics, chemistry, geometry, finance, economics, etc., memorization of the multiplication tables isn't the advantage that it once was. A deep understanding of math skills is far more valuable.

I often challenge people by asking what 1/3 divided by 7/16 is. Most everyone knows how to find the answer is to invert and multiply. Duh. Then I ask them to explain what 1/3 divided by 7/16 means. Most stare at me blankly. We memorize algorithms and formulas, but we don't understand what the numbers mean. When letters start to take the place of numbers, memorized applications have less power.

So how do we teach deeper math understanding? That is a question for experts in math instruction, but we don't do it by teaching math the way we always did it. My friend sent a problem that involved asking the student to solve the following: 86-12=? An explanation of one way to find the answer was to round to the nearest five, subtract, then add what your rounded back in. (I shortened it, but you get the gist.) The idea is that fives are easier to visualize (tens even more so).

What struck me about this "new math" is that my grandfather (not the same one who counted pinheads on the bus) taught me this method when I was a kid. He could do lengthy arithmetic problems in his head by using strategies like rounding and finger math. He also had a strange way of working long division that I have seen in some YouTube videos being described as an "ingenious new method."

My grandfather fought in WWI, and received his terminal degree (8th grade diploma) over a century ago. He didn't know calculus or statistics but he understood the concepts of math. The barn he built 100 years ago is still standing, so we can assume he understood geometry and angles. The family farm is still the family farm, so we can assume he understood finance.

So I started out by arguing that kids today know more math than ever, and then I tell you that my grandfather understood math better than many of our students do now? My point is that we have moved away from understanding math and more toward just following a formula and getting an answer. If we want to move beyond where we are, we will have to help kids know and be able to do both.

That is the conclusion my friend and I reached anyway. Maybe you will reach a different one, but please be open-minded enough to think that the way we have always done it may not be the best way. Oh, and as far as making change? My grandfather wasn't very good at that, because he always just rounded up to the nearest dollar on our wages when we worked for him. That's the kind of guy he was.

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