Current practice in math instruction assumes that students who meet learning standards in arithmetic will experience success in algebra. But that hasn’t always proven to be the case. We call this the “arithmetic to algebra gap.”
CEC’s new book, “Bridging the Gap Between Arithmetic and Algebra,” asks teachers to rethink the way we teach elementary math, to address this gap in learning.
I recently spoke with editor Brad Witzel about the book and what it says about the “big ideas” of algebra that we need to start teaching in kindergarten.
Editor and Manager, CEC Professional Publications
LS: One essay in the collection talks about “big ideas” in building skills for algebra. Can you tell us about those?
BW: Instead of focusing on ways to meet arithmetic standards, the authors identify algebraic skills that can be developed much earlier, during elementary school. These skills are the “big ideas.” They include things like teaching variables and doing a better job of explaining what “=” means.
LS: So early mastery of these “big idea” skills will mean greater success in algebra?
BW: Yes. And I’m glad you put it that way. Mastery is key. There’s a difference between mastering a skill and simply answering a question correctly. A correct answer does not necessarily reflect mastery.
LS: What does reflect mastery?
BW: In the context of the book, mastery means to understand both a math concept and its procedure. One area of debate in math instruction today is the question of whether teachers should emphasize teaching math concepts or teaching math procedures. If you must choose, choose procedures, but when we’re talking about mastery, we need to teach both.
LS: That makes a lot of sense, but how do you teach to mastery?
BW: There are multiple ways to help students master math content. Always show multiple examples of the same concept, for instance, and talk through the “why” of each step. If students don’t understand why they’re doing what they’re doing, they won’t be able to apply what they’ve learned in other situations. Also, I like to use “non-examples.” These are math problems that are completed incorrectly. The student is asked to find the error, explain it, and correct it. In the process, they learn to analyze different approaches to solving problems.
LS: So they are literally learning from mistakes!
BW: Yes. Everyone makes errors in math, including me, so it’s useful to practice with non-examples that can make students aware of potential errors, especially ones that are more common, and challenge their thinking.
LS: How about students with severe disabilities? Can we bridge the gap for them?
BW: Yes. I believe our expectations should remain high for all students. Too often, curricular expectations drop sharply when a student is identified as having a developmental or intellectual disability. In “Bridging the Gap,” we argue that there are too many highly effective strategies in mathematics to concede defeat so early. We can focus on the “A” in disAbility by setting math goals for individual students to help them grow as much as possible.
About the editor:
Brad Witzel is professor and program director of the MEd in Intervention Program at Winthrop University in South Carolina. He is an award-winning teacher and researcher who has taught math and science to high achieving students with disabilities and at-risk concerns. He has also authored dozens of research and practitioner articles and several books.