My husband, a school superintendent, always had trouble with math classes when he was in school. He didn’t do all that great during his high school math classes, and almost didn’t graduate from college during his undergraduate days because of a required college math class he took 4 times before finally passing it right before his graduation. He always said that the way they taught math in school from books, with those big sheets of problems, just never made sense to him. Which is funny, because a) he has his Chief School Business Official designation and is a master of school finance, and b) he taught drafting and construction and small gas engines and other career and technical education classes that require a good amount of math. Whenever I remind him of those things, he just shrugs and says, “Well, that math made sense because it’s hooked into something I am actually doing; it’s not a big sheet of problems.”
I passed all of my high school math classes with at least a B, sometimes As if it involved any geometry, which I loved. I even passed the required math test my college gave before starting my undergraduate degree so I could opt out of the freshman math class everyone usually had to take. The way they taught math in schools made perfect sense to me-find the formula or series of steps, follow them, and whammo–right answers magically appeared and you got good grades.
So who’s better at math? My husband is, hands-down.
I’ll never forget what happened one day soon after we were married. We were out shopping together, and we were looking at an item for sale in a store that was 25% off. Wondering what the “real” price was, I immediately dug into my purse for my calculator…while my husband spit out the price of the item out of his head before that calculator even saw the light of day.
I asked him how he got the answer without having to use a calculator…did he multiply by the decimal (o.25) and the original price and then subtract like I had always been taught? Nope – he just took 10 percent of the pre-sale price (easy to calculate mentally), added that amount to it (another 10%), and then added half for the last 5%.
What he did involved a lot more steps than my method, but made a lot more sense. What it boils down to is that he has better number sense than I do. While he could mentally break down the numbers and put them back together to get to the answer we needed, I was mentally dependent on the standard algorithm and couldn’t move forward without it.
I was more successful in school, getting good grades because I could follow those logical series of steps they taught us. But my husband was more successful in applying math in a real-life situation because he understand how to use the numbers flexibly. All I knew was my series of steps.
My lack of number sense was thrust in my face again about two years ago when I was teaching AP Environmental Science. Students cannot use a calculator on the APES test, so I was showing them how to do a problem on the board (just like I was taught…) using long division. The first thing that stunned me was that about 80% of the class had no idea what I was doing, telling me they had never seen long division before (!!). The second complete stunner came when I was marching through the steps of long division, and my students kept asking me why I was doing what I was doing with the numbers.
And I couldn’t tell them why. At all. I had absolutely no idea why I was doing anything with those numbers – all I knew was that these were the steps you took to get to the right answer.
This number sense, this “why” of using numbers is one of the things this recent Scientific American article said that students really need to be taught concerning math – letting students develop their number sense by tackling problems from multiple angles, experiencing more visual approaches to math learning, and much less emphasis on speed.
To build number sense, students need the opportunity to approach numbers in different ways, to see and use numbers visually, and to play around with different strategies for combining them.
You know what the approach to math learning described above sounds like to me? The approach that’s needed for students to master the oft-maligned Common Core Math Standards. Instruction of these standards looks different because the standards are different from the “learn the steps to get to the right answer quickly” approach that has been the norm in math education for 100 years or so. So yes, it may take a student longer to get at the answer…but the answer is no longer really the point. The process students take to get to that answer is the focus these days, and that students understand why they’re doing what they’re doing with the numbers and can use them fluidly, flexibly, and are able to apply those patterns and relationships we see in numbers when they’re out and about in the real world.
That understanding is even called for in terms of students learning their math facts:
In 2005 psychologist Margarete Delazer of Medical University of Innsbruck in Austria and her colleagues took functional MRI scans of students learning math facts in two ways: some were encouraged to memorize and others to work those facts out, considering various strategies. The scans revealed that these two approaches involved completely different brain pathways. The study also found that the subjects who did not memorize learned their math facts more securely and were more adept at applying them. Memorizing some mathematics is useful, but the researchers’ conclusions were clear: an automatic command of times tables or other facts should be reached through “understanding of the underlying numerical relations.”
So I guess the question we need to ask ourselves is this – do we want students who are mathematically capable at the times they need to be in life, or do we want to stick with the traditional approach to math instruction that allows students to get the right answers but with no idea why they’re doing what they’re doing?
Do we want students who understand and can use numbers in meaningful ways, or do we want students who, like me in the store fumbling for my calculator, are trapped into following those rigid steps we were taught?
Let’s teach students to be capable, not dependent.