### And then, on sixth down, the Eagles gained 4 yards

“It has nothing to do with football,” [Mildred] Bazemore said. “It has to do with the mathematical concepts that you’re studying.”

GRRRRAAAAAGGGHHH!

That’s approximately how I reacted to the above quote, taken from a news report about a particularly boneheaded standardized test question devised by the geniuses at the North Carolina Department of Public Instruction (hat tip: *FO*). The question asks students to determine a football team’s average gain on the first six plays of some hypothetical football game. Unfortunately, this hypothetical game doesn’t abide by the most basic of football’s rules:

The team opened with a 6-yard loss, a 3-yard gain and a 2-yard loss, which would have made it fourth down with 15 yards to go for a first down. The team’s fourth play was just a 7-yard gain, yet it maintained possession for a 12-yard gain and a 4-yard gain on two additional plays.

Now, it doesn’t particularly bother me that the test question is badly written (and pretty much guaranteed to confuse anybody with an ounce of football awareness); these things, though unfortunate, do happen, no matter how much editorial oversight there is,^{1} as anybody with an ounce of teaching experience will tell you.

No, what gets my blood boiling is the nonchalant response on the part of Ms. Bazemore, the chief of the DPI‘s test development section. This notion that such a subjunctive test question “makes sense mathematically” and “has nothing to do with football” is, I submit, symptomatic of the educational institution’s generalized and deplorable mistreatment of mathematics at both the primary and secondary levels.

Okay, admittedly, I’m being a bit hyperbolic here, but the basic point is this: Bazemore’s comments suggest that she believes that there is a disjunction between “mathematics” and “the real world” (here embodied by football), that the platonic ideal of *(-6+3–2+7+12+4)/6 = 2.67* is only sullied by the interference of words and ambiguous readings. In other words, she seems to think that mathematics is (or, at least, should be) purely abstract, purely computational and, as a result, utterly boring to anybody that isn’t autistic.

Again, interpolating all of this from some throwaway comment to some undoubtedly bored reporter is a bit extreme, except for the fact that virtually every public school teacher and administrator I’ve had the extremely mitigated pleasure to interact with holds this exact view (I went to public school K-12, so I couldn’t tell you about private school teachers or administrators). This is especially true of elementary school teachers, who either secretly hate math or are exactly the sort of detail-oriented obsessive-compulsives who loved memorizing their multiplication tables as a kid but hated word problems and philosophy classes, but it also tends to hold among middle- and high-school math teachers (somewhat more surprising, since these people teach math *exclusively*, in contrast to their primary-school counterparts).

This all derives, I think, from a poor understanding of what mathematics really *is*, which is certainly understandable, but the end result is that the misunderstanding is propagated to the next generation for pretty much the same reason it got propagated to the last generation: teachers make math classes miserable, so students not unreasonably conclude that math is miserable.

So what’s the misunderstanding? Basically, the notion that math is conceptually equivalent to memorizing formulas and plugging numbers into them. Certainly, this is the bulk of the content of your average math class in both primary and secondary schools and even in most college math classes below about the 300 level (which range, needless to say, encompasses the totality of the majority of the population’s experience with formal mathematics education). Rare indeed is the math teacher who seems to understand and, more importantly, can communicate that mathematics is fundamentally not about plugging numbers into formulas but rather about coming up with those formulas in the first place. No matter which branch of mathematics we look at, from the purely theoretical to the applied, the mathematicians or scientists working in that branch are, fundamentally, taking what they know and trying to synthesize it in some original and creative way to produce some new theorem or formula that better describes the situation. The data that goes into this synthesis may range from the completely abstract to the completely concrete, but the basic process is pretty much the same and totally at odds with the plug-and-chug process, which produces nothing conceptually original.

And yes, I know the traditional objection of the public schools: “That all sounds great in theory, but you can’t even get to that point without memorizing your multiplication tables or simple integrals.” Which is all true, in a sense, but also completely false. It’s probably true that you won’t ever prove the Riemann Hypothesis if you don’t know that *8×9=72* or that *∫cos x dx = sin x + C* (though there’s no *theoretical* impediment), but such a perspective ignores the fact that, at some point in the course of human history, such “elementary” questions were just as mysterious, even to the intelligentsia, as the Riemann Hypothesis currently is and their solutions were just as exciting as a proof of the Riemann Hypothesis would be today.^{2} Whether the actual history of such problems is formally introduced into the course of instruction or not (and, despite generally being in favor of such an approach, I do have mixed feelings about it), there’s certainly no reason not at least to try to impart the same sense of mystery and discovery into the proceedings that the original discoverers/inventors of the material experienced. In other words, rather than taking the attitude that “I have a bunch of facts which I will try to cram into your head,” one would like to see more math teachers take the attitude that “I am going to try to give you the support and the tools you need to discover a bunch of interesting and useful facts for yourself” (with the additional side benefit that the students may discover *more* of those facts than appear on the curriculum). Admittedly, this is supposedly what “New Math” was (partially) about, but the methodology there was (or at least became) entirely wrong; a student’s feelings about math are pedagogically null to his fellow students.

The first step in this path, needless to say, is to try to view “word problems” less as particularly inefficiently coded messages (wherein we encode the “real problem,” which is something like *(-6+3–2+7+12+4)/6=2.67*, into this ambiguous cipher we call the English language) to be decrypted by the student and more as examples of actual, conceivable problems that might arise in the student’s experience and which can be attacked using various mathematical tools and tricks which he has (or, at least, should have) at his disposal.^{3}

1. Though Colby Cosh makes an interesting point in the context of journalism that too much editorial oversight may actually be a bad thing. His entire perspective is extremely interesting, especially since he is both a professional journalist and an experienced and widely-read blogger.

2. Although there’s some question as to whether anybody would actually recognize a proof of the Riemann Hypothesis even if it slapped him in the face, an issue addressed, more or less, in the provocatively-titled “Definitional Drift: Math Goes Postmodern.”

3. And yes, I know I’ve addressed this issue several times before (see “From politics to mathematics and back,” “A beginner’s guide to producing new results in mathematics,” “…you just get used to them” and, tangentially, “Mathematics and sex” for four of the more recent examples), but, as something of a math teacher myself, this is an issue that I think a lot about and, more importantly, I think I’m getting closer and closer to actually expressing myself clearly on the subject.

May 25th, 2005 at 3:32 pm

Semi-off-topic comment: I took a course on probability and statistics in college, and the texbook writer definitely had a well-developed sense of the macabre, which made its way into his story problems:

“A bus crash kills, on average, 32% of the occupants…” “A disease, which kills on average 10% of those who contract it…” “An ax-murderer picks houses at random from a phone book…”

(okay, I made up the examples as the book isn’t in front of me, but I’m not far off.)

May 26th, 2005 at 4:16 am

Nevertheless, one would have to interpret “actual, conceivable problems that might arise in the studentâ€™s experience” pretty loosely to include (correct me if I’m wrong) what I imagine to be the bulk of problem-solving in university mathematics departments.

May 27th, 2005 at 12:36 am

Nevertheless, one would have to interpret â€œactual, conceivable problems that might arise in the studentâ€™s experienceâ€? pretty loosely to include (correct me if Iâ€™m wrong) what I imagine to be the bulk of problem-solving in university mathematics departments.Well, maybe I should have said “actual, conceivable problems” without the qualifier to include those liberal arts majors taking Calc. I. Aside from that particular case, though, most students taking university math classes are either math majors or people that will (or at least could) use math in their fields (engineers, pre-med students, students in the hard sciences, etc.), so my stipulation isn’t entirely unreasonable.

May 28th, 2005 at 12:17 pm

But isn’t that, i.e. university classes, pretty much by definition then (or at least the way you’ve defined it) different in kind from the situation facing middle and high school teachers who are dealing with students who don’t necessarily have any common ground or interests?

May 28th, 2005 at 1:42 pm

I don’t dispute your premise that instruction should be more problem-oriented but I do question your assertion that: “No matter which branch of mathematics we look at, from the purely theoretical to the applied, the mathematicians or scientists working in that branch are, fundamentally, taking what they know and trying to synthesize it in some original and creative way to produce some new theorem or formula that better describes the situation.” I don’t think that that’s true in large swaths of engineering or science, let alone more remidal uses of math like accounting, where it seems to me that I lot of the work does consist of plugging in formulas. I’m not really sure if you are criticizing university math departments or middle- and high schools, but if the former the standarized-test example doesn’t seem particularly apposite, and if the later I’m not sure if the point of those classes is, or need be, to teach people to think like mathematicians.

May 29th, 2005 at 4:30 am

The situation stipulated in the problem could be an actual case if penalties weren’t included (and they wouldn’t be, if the object was to compile stats on the offense).

That in no way negates your points, though, which are spot on. (Spot on? Maybe I’ve been reading too much Billy Beck.)

Rock on,

Bill