*MathAMATYC Educator*2.3 (2011): 4-18. (Link.)

They make the argument that prior instructors' emphasis on procedure has overwhelmed students' natural, conceptual, sense-making ability. Now, I agree that terrible, knowledge-poor, even abusive math teaching in the K-6 time frame is endemic. But I'm a bit skeptical that students in this situation have a natural number sense waiting to be un-, or re-, covered.

- In my experience, students in these courses commonly have no sense for numbers or magnitudes. Frequently they cannot even name numbers, decimals, places over a thousand, or know that multiplying by 10 appends a 0 to a whole number.
- Givvin, et. al. assume that students "Like all young children, they had, no doubt, developed some measure of mathematical competence and intuition... ". I'm pretty skeptical of this claim (and see little evidence for it.)
- One task is to check additions via subtractions, and quiz students on whether they know that either of the addends can be subtracted in the check. Admittedly, this is an unusual task: usually we take addition (or multiply or exponents) as the base operation, and later check the inverse via the more basic one -- for which the order definitely does matter. (Because addition is commutative but subtraction is not, etc.) So it's unsurprising that students' intuition is that the order matters in the check; as usually applied, it does.
- Another student multiplies two fractions together when asked to compare them (obviously nonsensical). But it's easy to diagnose this: many instructors teach equating the fractions and then
*cross-multiplying*them, and seeing which side has the higher resulting product; in this case the student scrambled the cross-multiplying of equations with multiplying fractions. Which highlights two things: teaching mangled mathematical writing as in this process leads to problems later; and the whole idea of*cross-multiplying*is so striking that it "sucks the oxygen out of the room" for other visually-similar concepts (like multiplying fractions). - The authors state that "These students lack an understanding of how important (and seemingly obvious) concepts relate (e.g., that 1/3 is the same as 1 divided by 3)." Not only is this not obvious, but I can repeat this about every day for a whole semester and still not have students remember it. Just this semester I had a student who literally couldn't
*repeat it*when I just said it about five times. - The importance of "combining like terms", which is essentially the
*only*concept under-girding the operation of addition (and subtraction and comparisons) -- in terms of like units, variables, radicals, common denominators, and decimal place values -- is highlighted here. I don't know how many times I express this, but I'm doubtful that*any*of my students have really ever understood what I'm saying. I wouldn't be surprised but some students could take a dozen years of classes and never understand this point. Which is dispiriting. - The authors have some lovely anecdotes of students making a small discovery or two within the context of the hour-to-two-hour interview. This they hold at as a hopeful sign that discovery-based learning might be an effective treatment. But I ask: How many of these students will remember their apparent discoveries outside the interview? I find it quite common for students to have "A-ha, that's so easy!" moments in class, and then have effectively no memory of it a day or two later. "I do fine when I'm with you, and then I can't do it on my own" is a fairly common refrain.
- Discussing concepts is Element Two (of three) in the authors' list of prescriptions. "A teacher might, for instance, connect fractions and division, discussing that a fraction is a division in which you divide a unit into n number of pieces of equal size. Alternatively, the teacher might initiate a discussion of the equal sign, pointing out that it means 'is the same as' and not 'here comes the answer.'". Sure, I offer both of those specific explanations regularly, almost daily -- they're essential and without them you're not really discussing real math at all. But many of my students can't remember those foundational facts no matter how much I repeat or quiz them on it.
- From my perspective, it almost as though most of my developmental students
*aggressively refuse to remember*the overarching, connecting definitions and concepts that I try to share with them, even when they're*immediately*put to use within the scope of each daily class session. - I can't help but feel that the distinction between "procedures" versus "reasoning" is an artificial, untenable one. The authors admit, "Even efforts to capitalize on students’ intuitions

(as with estimating) often quickly turn to rules and procedures (as in 'rounding to the nearest')". I think this argues, perhaps, for the following:*All reasoning is ultimately procedural*. The only question is knowing what definitions and qualities of a certain situation allow a given procedure to be applied (even so simple a one as comparing the denominators of 1/5 and 1/8, for example). Even*counting*is ultimately a learned procedure.

While I don't seem to have access to Part 1 of the same report, the initial draft report has a few other items I can't help but respond to:

- "'Drill-and-skill' is still thought to dominate most instruction (Goldrick-Rab, 2007)." This is a now-common diatribe (my French-educated partner is aghast at the term). But let's compare to, say, the #1 top scientifically proven method for learning, according to a summary article by Dunlosky, et. al. ("What Works, What Doesn't", Scientific American Mind, Sep/Oct 2013). "Self-Testing... Unlike a test that evaluates knowledge, practice tests are done by students on their own, outside of class. Methods might include using flash cards (physical or digital) to test recall or answering the sample questions at the end of a textbook chapter. Although most students prefer to take as few tests as possible, hundreds of experiments show that self-testing improves learning and retention." Which is a somewhat elaborate way of saying:
*Practice and homework*. - "The limitations in K-12 teaching methods have been well-documented in the research literature... An assumption we make in this report is that substantive improvements in mathematics learning will not occur unless we can succeed in transforming the way mathematics is taught." I would not so blithely accept that assumption. What it overlooks is the perennial decrepitude of mathematical understanding by K-6 elementary educators. My argument would be that it doesn't matter how many times you overhaul the curriculum or teaching methodology at that level; if the teachers themselves don't understand the concepts involved, there is no way that even the best curriculum or methods will be delivered or supported properly.
- "Perhaps most disturbing is that the students in community college developmental mathematics courses did, for the most part, pass high school algebra. They were able, at one point, to remember enough to pass the tests they were given in high school." But were they, really? A few years ago when I was counseling a group of about a dozen of my community-college students, as they left the exit exam and thought that they had failed, I stumbled into asking exactly this question in passing: "But this is totally material that you took in junior high school, right?" To which one student replied, "But there it was just about buttering up the teacher so he liked you enough to pass you," and the other students present all nodded and seemed to agree with this. The evidence that students are being passed through the high school system on effectively fraudulent grounds seems, to me, nearly inescapable.

Near the end of the Part 2 article, the authors appear to express a bit more caution concerning their hypothesis; a cautionary question which I'd be prone to answer in the negative:

For some students we interviewed, basic concepts of number and numeric operations were severely lacking. Whether the concepts were once there and atrophied, or whether never sufficiently developed in the first place, we cannot be certain. What we do know is that these students’ lack of conceptual understanding has, by the time they entered developmental math classes, significantly impeded the effectiveness of their application of procedures. (p. 14)

We hope that future work will seek to address questions such as whether community college is too late to draw upon students' intuitive concepts about math. Do those concepts still exist? Is community college too late to change students' conceptions of what mathis? (p. 16)

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