My first teaching experience was as a substitute teacher in Chicago assigned to an 11th grade Algebra 2 class for ELL Polish students. I began by giving students an assignment their teacher had left for them. But no one attempted it, so I asked a boy who understood English if he and his classmates needed help. He laughed and, after he translated my question for his classmates, they laughed too. He then let me in on the joke: “We learned this in 7th grade.”
To me, however, it was appalling rather than amusing: 11th grade here = 7th grade there?! Yet what I later discovered was even more appalling: 11th grade here = 9th grade here. In fact, Algebra 2 was such a rehash of the district’s Algebra 1 course that some teachers called it “Algebra T-o-o.” And really, the same point could be made about math curriculum as a whole in the U.S., since most content for any given year is a review of content from previous years. (The Common Core State Standards may help change this, but I’ll believe it when I see it.)
This approach, where we touch on lots of topics each year--rather than go deep with fewer topics--and then revisit them in subsequent years is often called spiraling. But what it is for many students is stifling. And this is as true for kids who’ve yet to master a skill as it is for those who nailed it right away. I first noticed this when I taught 9th grade Algebra classes where every student was performing at least two years below grade level.
“Meet them where they are,” fellow math teachers advised me. Makes sense, I thought, since I couldn’t imagine teaching Algebra to kids who didn’t know basic arithmetic. But what I soon learned is that perception matters more to students than performance. For many kids, having seen something is akin to having learned something. “Man, we already know this,” students said, as I presented lesson after lesson on fractions, decimals, and percents.
Other students, meanwhile, knew they didn’t understand the material, but had given up hope of ever understanding it. The implication was therefore the same for all students: encore presentations on previous years’ topics were pointless. And though I was able to engage a few students when I found new ways to present old topics, one group of students was always slighted: those who really did “already know this.”
I’ve seen this same scene play out in dozens of math classes: teachers presenting material as though students had never seen it when they had actually seen it early and often. Consider, for example, area and perimeter, which students are first exposed to in third or fourth grade, and see again in middle school. Yet when area and perimeter come up in high school, most teachers--including me at first--teach them from scratch.
The problem, of course, goes back to the disconnect between kids seeing something and actually learning--and retaining--it. But if it didn’t sink in for them the first, second, or third time a teacher presented it, why should we present it again?
We shouldn’t. At some point the focus needs to be on students practicing math rather than teachers presenting it. And to me, that point begins right after students are first introduced to a concept or skill and continues for the rest of that year and subsequent years. Instead of limiting assignments to recent content from the current course, we should also include problems on earlier content from that course AND previous courses.
In other words, we should provide students spiraled practice, not spiraled instruction. When I did this in 10th grade Geometry classes, students said they learned more Algebra than they had learned in their 9th grade Algebra course. And, as a result, they were ready for more advanced math--starting with Algebra T-w-o.
Image provided by Phillip Martin with permission
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