This is my sports


I’m not a big sports fan (is an understatement). I couldn’t tell you the difference between the Chicago Bears and the Chicago White Sox (in fact, I had to Google them to make sure they exist). But you gotta entertain yourself somehow, right?

Last week – as I’m sure you all know by now – an article by Katharine Beals and Barry Garelick appeared in The Atlantic bemoaning “Common Core-era rules” that “force” kids to explain their thought processes in math class (emphasis mine). The article opens with a middle-schooler subject to these cruel, inhumane Common Core laws “throwing her arms up in frustration” and asking:

Why can’t I just do the problem, enter the answer and be done with it?

In fairness, I’m sure most kids would also prefer to skip broccoli and just get to dessert. But I’m not sure I’d like living in a world where middle-schoolers decide what’s good for us…

The gist of the article was that teachers across America are needlessly taxing their students’ brains by asking them to explain their thinking for each and every problem they solve in math class (not true), often forcing them to use needlessly complicated graphic organizers that constrain their thinking (see the single example in article). My main question after reading was: why focus on a false dichotomy (always explain answers vs never explain answers) instead of talking about when explaining your thinking is useful/necessary?

Dan Meyer wrote an excellent response to this article on his blog (“Understanding is the goal… Let’s be inflexible in the goal but flexible about the many developmentally appropriate ways students can meet it.”) But the most interesting discussion took place somewhere I generally try to avoid if I’m not in the mood for pulling my hair out: the comments section.

Yet what I had expected to disintegrate into shameless mud-slinging from the trenches of the Math Wars actually turned into a (mostly) civilized and thought-provoking discussion about the value and purpose of having students explain their thinking when solving problems. So much so that I quickly found myself glued to the screen, and checking back daily, to follow the most exciting tennis match I’ve ever had the opportunity to watch (read). This is my sports.

Here are some of the highlights (in my shamelessly biased opinion):

Ze’ev Wurman suggests that teachers who ask their students to explain their thinking are performing illegal experiments:

The primary purpose of school is to educate the kids at hand, not to train the teacher. This doesn’t mean that teachers do not learn from experience, but if gaining experience and insight is the primary reason for what the teacher does, he’d better get approval from an IRB and a waiver from each individual student or parent that attend his class.

There is nothing wrong with having a student occasionally explain his thinking when s/he is called on in the class. There is a lot of wrong with having students explain their thinking in a voluminous verbal/written manner on every test item and every home assignment.

Michael Paul Goldenberg refutes the idea that explaining your thinking is some kind of rigid rule that teachers apply each and every situation:

No thoughtful, reflective mathematics educator advocates the sort of rigid application of “explain your thinking” to every problem, particularly those which are trivial for students at a particular developmental and grade level.

IF you’re teaching a classroom of students where you have adequate evidence that for those students the problems you’re asking for explanations on are trivial. If not, then good teachers are obligated to try to probe where students are having difficulties, confusion, etc. And one way that proves effective is asking for explanations.

The conversation takes a turn as Katherine Beals reminds us of America’s abysmal (my tongue is firmly in my cheek) results in PISA:

The internationally more successful approach has been to build conceptual understanding through teacher-directed instruction and individualized practice in challenging math problems. Finland is just one example. From the latest PISA ( consider the 20 other countries that outcompete us. Most use the teacher-directed, mathematically challenging approach. I’m particularly familiar with the Singapore, Russian, and French curricula, but they are largely representative what’s happening in continental Europe and East Asia.

Brett Gilland responds:

Pointing to the top 20 performing countries doesn’t actually support a given system unless you demonstrate that this system is distinct from the lower scoring nations. Moreover, it would help if you demonstrated that those countries with the best scores were more in line with this system than those who did worse. So, for instance, it would help to know if Sweden’s mathematical education system was significantly different than Finland’s. Since they are culturally quite similar, that might eliminate some confounding variables. Without this, you are cherry picking and not establishing significant impact from differences in pedagogical methods.

Similarly, though it has been pointed out that NCTM/CCSS methods (as I take you to be using the phrase) are incredibly rare in USA schools, you simply reassert that this is what you mean by “an American Approach”. The problem here is that this is not actually the approach taken in most American schools.

(See Michael Paul Goldenberg’s support of the latter claim here.)

Skipping ahead to Michael Paul Goldenberg on misrepresentations of the NCTM Standards, which was possibly my favorite play of the match (should I stop using sports metaphors?):

I’ve read so much from people who claim that NCTM called for eliminating everything and anything “traditional” from US mathematics classrooms (and how similar organizations did the same in other countries, including England, Canada, and Australia). Never mind that the people I’ve studied under, worked with, gotten to know through their writing online, listened to at professional conferences, etc., seem pretty consistent in making modest suggestions and calling for shifts, not revolutions, in how teaching is done and classroom time is spent (and I’m speaking about people from many countries besides the United States): the claims just will not cease that supporters of progressive mathematics education are all mad professors experimenting on innocent children, always wrongheadedly and to the utter detriment of the nation’s and world’s youngsters.

How, exactly, so many thoughtful, highly educated, dedicated professionals manage to get EVERYTHING wrong is never explained. It seems to suffice to make the claim that our mathematics classrooms are in a shambles, all due to progressive reform ideas and thinkers, while in those “other countries,” all is marvelous, all kids are above average, all teachers teach just the way that those who oppose NCTM, et al. were taught once upon a time in the good old days.

Well, maybe that’s true. But it pretty well defies credulity, as well as runs counter to my experience over the last quarter century. I think it runs counter to the experience of a lot of very smart, creative, thoughtful mathematics teachers who contribute to this blog and other practitioner blogs I’ve been following for going on a decade.

This was only the 67th comment, followed by 24 more (before Dan closed the comments yesterday evening), but somewhere around this time the discussion began to stray rather far from the original “explaining vs. understanding” debate into arguments about cheating in China, the inferiority of American textbooks, and everything in between. Regardless, I followed this game right until it’s surprising end, in which Katherine Beals announced her intent to blog about problems from American textbooks that demand high quality critical thinking. (“Happiest ending, as far as I’m concerned.” -Dan Meyer)


And to think that nobody used the terms “airy-fairy”, “fuzzy”, or “drill-and-kill” even once! Could this be the dawn of a new era where ‘traditionalists’ and ‘reformists’ join hands and sing Kumbaya?






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