How do we effectively effect remediation for students who are desperately behind state and districts grade level expectations? Or behind reasonable developmental expectations, which may not necessarily be the same thing? Regardless of issues of tracking, there are always going to be students who need remediation in mathematics as things currently stand (I tend not to make very long-range predictions, but I don't see this situation changing in my lifetime).
In my experience observing K-12 mathematics teaching (as well as community college and university), the most common approach is what I have come to call "louder and slower." [L&S] Rather than attempt something different, many teachers seem to believe that teaching mathematics to those who aren't getting it can be best done in the same way many people try to communicate with foreigners: say the same exact thing you just failed to communicate, but do it louder and slower.
Of course, in the real world, even many incredibly dense people eventually realize that this approach is likely doomed to failure. So they begin to employ alternatives. Non-verbal things like gestures and drawings are often employed, often with far more success than L&S.
I would think that such experiences might lead some of those same people to realize that L&S isn't a big winner in the classroom. If it were, would there be such a sizable number and percentage of our students over the past century who are sooner or later in remedial mathematics classes - in grade school, middle school, high school, or college?
As I've noted frequently on this blog, the critics of progressive reform contend both that things are far worse today than they were during some amorphous Golden Age of US mathematical competence, and that the main culprits in the alleged fall from grace are NCTM, NSF, "fuzzy math," "dumbed-down" curricula and state standards, constructivism, discovery learning, student-centered teaching, and, of course, calculators. (This list is not meant to be exhaustive, of course, but it's pretty indicative of the nature of their candidates for blame).
Against that, I have to put my observations of typical remedial and at-risk mathematics. In the vast majority of cases, L&S is the order of the day. And not surprisingly, it doesn't work for most kids. They're turned off, defeated, depressed, resentful, and completely pessimistic. And why wouldn't they be? What could be less engaging than being told for the umpteenth time that "when you subtract a negative number, change the sign to positive and then add" if you aren't clear on your addition table in the first place, have made so many errors with integer arithmetic in the past that you are convinced that you'll never get it, and are pretty sure that the whole exercise is pointless and irrelevant?
The above example is but one of dozens from the basic elementary curriculum that students have to go over repeatedly once they're caught in the remediation cycle. (Of course, there are students who are allowed to avoid all this in certain school districts, but not to their advantage, and with NCLB still the law of the land, as well as state-mandated high stakes standardized tests, such students are increasingly unlikely to be able to graduate high school, for good or for ill). The hurdles for students who struggle with basic mathematics are legion: subtraction, division, signed numbers, fractions, decimals, percents, and, of course, basic algebra, to name but some of the most obvious.
Are there reasonable alternatives to the L&S remedial merry-go-round and the destructive turning a blind eye approaches? I think there are, though I offer no magic bullets or panaceas. In this regard, I've had a unique and most unexpected opportunity to work with a teacher who faces serious challenges in her first full year of teaching mathematics. Becca is working at a charter school that currently serves at-risk students in grades 7-10 (the plan is to add 11th & 12th grades over the next two years). The school is in a small city in northeastern Michigan with a significant Latino and African-American population. Becca's students are about about 50% black, 48% Latino, and 2% white. They are low-income kids who are significantly below grade level, particularly in literacy and mathematics, as is generally the case in alternative schools.
Becca contacted me in September of 2007, about a month into the school year, because she knew I had prior experience teaching mathematics at an alternative high school. I did not view my teaching there as particularly successful and felt it to be far from exemplary, so I was skeptical that I would have much to offer her. However, as she grew frustrated with the sorts of traditional remedial materials and methods she was using, she was also developing bonds with her students that made it possible for her to try something different. In particular, Becca became increasingly interested in trying to make problem-solving a central focus of her instruction.
My initial response was mixed. Internally, I looked back on my own less-than-glowing results with this sort of approach, which I had tried several times. Generally, I found that there were a number of serious obstacles. First, the vast, vast majority of students in at-risk or low-achieving classrooms were highly averse to any sort of mathematics instruction or content. Yet ironically, they were sure that the problem-solving activities I asked them to engage in were "not real math." Time and again, they made clear to me that "real math" was book exercises and work sheets. Of course, the books and worksheets they had in mind focused strictly on computation, and even though they were extremely week in this area (few knew their whole number arithmetic facts solidly, let alone signed numbers or anything having to do with fractions, decimals, or percents), they made noises indicating that they "wanted" book work.
Inevitably, when I gave them exactly the sorts of things they said they wanted, however, whether in the form of a diagnostic test, or the exercises and follow-up worksheets, with or without prefacing any of this with direct instruction on the relevant content, things went no better. The same sorts of completely inappropriate social interactions went on: screaming, insulting peers, insulting me, throwing thing at me and each other, talk about the most inappropriate topics (sex, drugs, and violence topping the list, though not necessarily in that order or, for that matter, separately), and very few attempts at doing the "bookwork," let alone successful completion of the assignments.
Of course, the second concern I had is likely obvious from the first: most of these students were operating so far below grade level that finding problems in which they could successfully engage if they were willing to try would take a great deal of thought. My students were 9-12 graders, and while most were at the 4th to 5th grade level in mathematics at best, they deeply resented anything that looked "too babyish." Thus, I suspected that it would take rewriting problems to give them contexts that were more relevant and less like what might appeal to elementary school kids, while maintaining an appropriate level of mathematical content so that students actually could engage with and solve the problems.
Given my track record, I was hardly sanguine about someone trying to go in the other direction: changing from more familiar sorts of "comfortable failure" to more challenging and unfamiliar ground that would pose greater difficulties than simply finding the answer to a straight-forward computational exercise.
Nonetheless, Becca was very intent on at least trying a problem-solving orientation in her class. So I invited her to meet with me so we could share ideas. I showed her many resources I had for teaching, content, problem-solving methods, as well as some of the social/emotional education materials I'd consulted in early 2004 while doing work with middle school math teachers in very challenging schools in Region 2 of New York City.
In the next part of this piece, I will report on what Becca has been doing this semester and how she has been reflecting on the successes and challenges, as well as how we have continued to communicate about her practice.
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