Number sense: adding understanding to math

As a former elementary teacher who’s now a teacher of teachers, Katie Schwartz offers more than a little sympathy to teachers trying to get their heads around math instruction where finding the right answer is no longer the only aim.

“It’s scary to jump in and do something completely different from the way you’ve always done it, and you have the pressure of test scores hanging over your head,” said Schwartz, an assistant professor of math education at East Carolina University. “Teachers were taught the same way the rest of us were. They were just taught to find the right answer, too, so there is this process of figuring out why things work and what kinds of activities do we do with our students.”

In Lenoir County, that process involves a three-year collaboration between LCPS and ECU for professional development, funded by a $481,518 grant from the N.C. Department of Public Instruction. It got started this month with a two-week “support institute” that brought six ECU professors from the departments of math, math education and engineering together with 27 of our elementary and middle school teachers. It will continue during the coming school year with a reconvening of that group each quarter and — provided the grant is renewed annually for each of the next two years — will incorporate about 30 more teachers into the training each year.

From the original group, 10 teachers will be selected as members of a leadership team. “Those 10 teachers will complete two extra graduate courses (in math ed) and get extra leadership training the next two years, so once the money leaves and we leave,” Schwartz said, “the knowledge is still here.”

Dr. Katie Schwartz of ECU's math ed department

Dr. Katie Schwartz of ECU’s math ed department

And what is that knowledge? Basically, it’s teaching techniques that can give students an understanding of numbers and math that goes beyond the mechanical process of multiplying and dividing and the rest of it. Being a good mathematical mechanic may help you find the right answer, but it doesn’t help you understand how numbers work. It doesn’t help you understand how math relates to the real world. It doesn’t give you what math people call “number sense.”

That’s a phrase that can be defined several different ways, depending on context, but generally it means having a conceptual understanding of numbers, seeing how they relate to each other and being able to use that number fluency to solve problems in nontraditional ways. Number sense helps students with mental calculations, problem solving and finding the math in real-world situations.

For many researchers, number sense is to math what phonological awareness — the knowledge of how sounds blend to form words — is to reading. It’s basic, and it’s the key to breaking the code. For Schwartz, the comparison to reading works perfectly.

“If we said we were just going to teach you to read out loud beautifully, but you don’t have to understand anything you read, nobody would go for that,” she said. “But that’s the way we do math. We teach you to get the answer, but we don’t teach you the understanding behind it. That’s what we’re focusing on.” (Long, but interesting, video on the subject here

“Students need to try out their own best way for solving a problem instead of me telling them how to do it,” said Kimberly Smith, a fifth-grade math and science teacher at Banks Elementary School and a support institute participant. “We all need to buy in to flexibility.”

Nicolette Morgan, a classmate of Smith’s in this first professional development group, says both teachers and students have work to do in that area. “The teachers have to become more comfortable with it,” Morgan, a middle school math teacher at Contentnea-Savannah K-8 School, said. “There’s more than one way to solve a problem. You want the kids to be flexible. You want the kids to be able to reverse explain how they got to their solution. It’s all about flexibility.”

It is at this point that parents’ eyes begin to cross. “Flexible” is not a word much heard in the math classes of old. There was one way of dividing whole numbers, for instance, and if you didn’t know why that digit had to move from there to there, it didn’t matter as long as you remembered to move it. No number sense required. But allowing room for creativity in problem solving — for employing math basics like multiplication and division — isn’t a revelation as much as a recognition of how people with some conceptual understanding of numbers work in the real world.

A carpenter who needs to halve a board that is 39 1/2 inches long could pull out his pencil or, instead, could divide 40 by 2 and subtract a quarter-inch from each end, a computation easy enough to do in your head. If you’re trying to figure a 15 percent tip on a restaurant bill of $21.83, you could fire up the calculator on your smartphone or, almost at a glance, could take 10 percent of $21.83 (a matter of simply moving a decimal) and then taking half of that (to compute the remaining 5 percent). Adding those numbers together gives you the total tip.

It takes longer to explain than it does to do, and probably less time than it takes to boot up your phone. But it takes some understanding of how numbers work and how they fit together, an understanding being eroded in a world of calculators and digital clocks.

“When you can make sense of what you’re doing and why you’re doing it instead of just memorizing the steps, you take it in in a much deeper way,” Schwartz said. “You do need to teach the steps involved, but you also need to understand what you’re doing and why you’re doing it; or when you get into the real world you’re not going to know when to use it. You’re just going to know the steps.”

That could be the difference between flipping burgers and making good money in manufacturing. As part of their two-week workshop, Morgan and the other middle school teachers spent afternoons touring seven local plants and businesses. The tours — of Spirit AeroSystems, Barnet, West Pharmaceutical, Lenoir Memorial Hospital and Chef and the Farmer restaurant, among others — were arranged by STEM East coordinator Steve Hill to tie together math education and math application, a basic tenet of STEM’s emphasis on science, technology, engineering and math.

(Here are a couple of reports on that aspect of the workshop, from WNCT and The Free Press

Teachers learned what math concepts were required in the working world and got ideas for making their lessons more relevant. They also learned, according to Morgan, that the students coming out of school today don’t know enough math. “The majority of the businesses use middle-school math, but most of the businesses said they have trouble recruiting people with that level of math,” Morgan said. “They’re struggling with those basic skills.”

In most workplaces, math is everywhere. It could be as simple as figuring profit and losses or setting the price of a dinner entre to ensure a reasonable return. It could be as essential as medical personnel in a hospital calculating exponential decay, the rate at which a particular medicine leaves the body, in order to know when to safely administer more.

Whatever the requirements, most students in middle school need to catch up to them, Morgan said. “They don’t have the math strategies. They don’t understand the basics of fractions, decimals and percent,” she said. “They need to have a conceptual knowledge of it, not just the rote memory. That’s something we have to make sure we work on more with our kids.”

She left the support institute not only with homework assignments, but with ideas that she thinks will improve her teaching. “I saw some different methods I’m excited about showing the kids this year and we got a lot of resources that we can take back and try.”

Schwartz also expects they got a good answer to what she calls the “age-old question” put to math teachers: When am I ever going to use this?

“Well, guess what,” Schwartz said. “You’re going to use this if you go to work at DuPont or if you go to work at Spirit. You’re going to use this. We’re helping teachers make those connections and bring that into their classroom.”