Thoughts on “Math is not linear” by Alison Blank

We looked again, in a department meeting, at Alison Blank’s Prezi Math is not linear. I thought I would write down some detailed point-by-point reactions/responses. I’m very sympathetic to the piece overall, so the following is not meant to be overly critical. But I found it useful to think carefully about her statements to clarify my views.

“Math is not linear” — I really think “sequential” is a better word here, since in mathematics, “linear” has a very different meaning. I have seen many popularizations make the mistake that, for example “nonlinear analysis and chaos” means “non-sequential analysis and chaos.” But this is nitpicking in this context, I admit.

“There are so many directions you can choose”: This picture makes it look like there is no directionality, no hierarchical structure at all. In our meeting, we talked about how there is a middle ground between the strict ladder structure she is positing for the old-fashioned approach, and this opposing picture. A tree was suggested as a better structure: there are choices in direction, but there is some directionality.

Part of that comes from content dependencies. You can’t really get into the meat of calculus (as opposed to just touching the surface) without understanding functions and graphs (Alg II/Precal material). Another aspect of the directionality comes from levels of abstraction: topology, group theory, set theory all are rather abstract (if you do them for real), so they need to wait, for the most part, until students are ready for that level of abstraction.

One major issue with the whole Prezi is this: how much does one want to change the curriculum? One proposal would be a total revamp, in a much less sequential structure, with very significant, lengthy explorations in different directions, that would leave students with lasting content knowledge. Much less drastic would be to keep much of the usual sequence, but bring in diverse content areas as enrichment or motivation. I have a hard time imagining the total revamp as being practical or desirable. But applying the ideas and philosophy on a smaller scale is very appealing to me.

“Mastery of one subject is a prerequisite for dabbling in another”: Like her, I would definitely disagree with this statement, but I think that the word “dabbling” is key. If the goal is to master calculus, or even study it significantly, you really do need to have a strong knowledge (if only I could count on “mastery”!) of many prerequisites. But if the goal is to bring in selected ideas from calculus—or even better, selected questions that will eventually motivate the calculus ideas—then I think the earlier the better.

“It’s not motivating”: We had a good discussion about what really motivates students. It’s important to remember that different students are motivated by different aspects of the material. Some people are highly motivated by real-world applications, some aren’t. Some enjoy the “puzzle” side of math, some find it unappealing. And students definitely respond differently to rich, open-ended, challenging problems. So one can’t just say that a certain curricular change is certain to be more motivating. However, I do think that on balance, the kinds of changes she is advocating are more motivating to most students. More important, they motivate better kinds of thinking—higher-order, more connected (between topics in math and between math and reality) and more independent. Some students may not be too comfortable (especially at first) with that kind of challenge. That’s OK with me—we’re not in this business to make them comfortable if that comfort is at the expense of a great educational opportunity. And it’s not like they’re doing it without a net—a responsible teacher will support them in their engagement with the challenges they face.

“You need to walk before you can run/No, you need something to run towards” (Lockhart): we agreed that this is a false opposition. You do need some fundamentals, and you want to know what the goals are and what the future will hold. The issue is how much of the fundamentals do you need, and how much exposure to challenging questions and diverse topics would constitute “trying to run” too soon.

“Too much time studying skills simply because they will need to use them in a later class”: I agree strongly with this whole paragraph. Too often we follow the textbook’s presentation, which just drops a piece of mathematics in front of the student, says “learn this”, and only after the fact comes up with applications. And for some topics, no real applications are presented. I can’t believe that’s a good strategy for retention. There’s too little connection to anything, too little genuine motivation.

“…give a student a problem that required rational expressions, wait for her to discover that she needed more technique, and then introduce the topic.” That is a strategy I am trying to implement more and more on a small scale—not in way that radically changes the entire curriculum, but in a way that motivates individual topics better and leads to more discovery.

For example, right now in Trig/Precal we are doing triangle trig. They already know how to deal with right triangles, so I gave them a word problem (very loosely stated) that will require them (when they formulate it generally enough) to derive (some version of) the law of cosines. I’m hopeful that not only will they be able to come up with some (not very efficient) ad hoc method and apply it to specific cases, but also to go back, put variables in everywhere, and derive the actual law as it usually appears. I think that will be a very good experience for them.

I would say, however, that while she states it as if it is about one student figuring it out alone, it makes much more sense to do that kind of discovery in a group situation, where there is more collective brainpower and give-and-take.

“…high schoolers are capable of taking in a bit of topology, or set theory, or real analysis here and there, and they like it”: Yes, up to a point, and in limited ways. There’s a huge barrier involved, namely the abstraction that is necessary to actually do real problems in those subjects. I love telling my students about gluing rectangles to make tori, for example. And giving them a hint about the different kinds of infinity, or Fourier series, is very exciting too. But it’s hard to get beyond dabbling in such things, for most students. (And naturally they like it, since they’re not likely to get tested on it.)

For me, the meaningful curricular innovations are the ones that involve concrete problems that connect with known material. That’s why I like the law of cosines investigation I’m doing right now (assuming it will work!). And it’s why I like introducing maxima and minima as early as humanly possible (waaay before calculus, which you don’t need to find most maxes and mins). Maybe I just haven’t been creative enough yet to see how a topology problem could be tightly connected with, say, Algebra II material. It’s food for thought, but it’s not trivial. And of course, if she’s trying to suggest a widespread curricular innovation, including those topics in high school is a non-starter, since most teachers aren’t familiar with them.

“a slower pace, or a different route”: Practically speaking, we just can’t offer individualized instruction for everyone. That’s one huge reason for the standard model. If most everyone takes the same courses, putting them in a sequence makes the most sense. Again, this comment seems to be about the total revamp idea, which is just not practical.

“get to calculus by 12th grade”: Many wiser people than me have weighed in about how unnecessary the rush to calculus is. We’re not training most students to be mechanical engineers or physicists, so putting calculus in as the keystone is not necessary. But so many subjects depend on a lot of material from the rest of the sequence, that it doesn’t necessarily call into question the whole structure. Certainly more on discrete math and data analysis and less on Alg II/precalculus would be defensible. But the further down you go in giving people options, the harder it is to teach them all, practically speaking, and the harder it is for students to change their minds.

“prodigious young children…smug about it”: I happen to teach many such children, and I see very few of them being smug about it, so I think that word in particular is a bit ad hominem. But the point she raises hits home for me. In designing enrichments or extra courses for such kids, I oscillate between wanting to offer offbeat courses such as non-Euclidean Geometry or Combinatorics, and wanting to offer core courses such as Differential Equations or Linear Algebra.

I often do the latter, not out of a desire to push them along a narrow track, but simply because (a) they’ll want to take those courses soon anyway, (b) there actually are many other courses that really do depend on them as prerequisites, and (c) they are really good math. I would definitely not agree with someone who said that traditional core courses are inherently boring. However, I have also offered more offbeat options, since there’s no huge hurry for such students, and the diversity of thought processes and topics encountered that way can be really beneficial.

I certainly think that as such students fly through the standard curriculum, if that is all they see, they are doing themselves a disservice. But such students very often do other enrichments, particularly contests, which have a strong broadening and deepening effect. (Even though traditional contest problems have their own different sort of narrowness.) I suppose in an ideal world, very strong, advanced students would take a completely different set of courses, emphasizing interconnection and diverse mathematical ideas (as well as really deep approaches to the core material, real proofs, and very challenging problems) even more than the revamped standard courses she is imagining. But this is not an ideal world. For most highly advanced students, going quickly through the usual courses is far better than going slowly through them, which is the only other practical option.

“Connections between topics”: Yes, we should emphasize these connections as much as we can, without watering things down. We shouldn’t spend all of our time making connections and no time doing deep mathematics and holding students to high standards. I don’t think that the two are mutually exclusive, but I think it takes effort to have an integrated, connection-full curriculum that doesn’t gloss over lots of material too lightly. That ends up with teachers re-teaching material that should have been learned already.

“…learn…what it means to be a mathematician”: as a former professional mathematician myself, I sympathize. But I think it’s far more important for them to know how mathematics is used by non-mathematicians. I think students should have some feel for mathematics as a meaningful whole, and a living one at that, but with an emphasis on applications, not on research per se, and certainly not on pure mathematical research. That said, it’s not hard to interest most students in some of the ideas of pure research, at least at the superficial level—which is fine.

Well, that’s already a really long post, so I’ll respond to the rest later.

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One Response to Thoughts on “Math is not linear” by Alison Blank

  1. Jean says:

    The “rush to calculus” is because of the national standards, because of the national college expectation that kids will finish a year of calc before college. College admissions is driving this aspect of our “curricular goals.” I find this quite distressing. As a young teacher, I wrote to some chairs of math and econ departments (econ bc that was my college major), and asked what they thought were the mathematical strengths and weaknesses of incoming freshmen/women. The result of my highly unscientific poll was that the profs were discouraged by the lack of algebra (read: precalculus kind of algebra) skills. Their desire was to have us spend more time with Algebra. My school at the time adopted a 2 year precal program based on the Chicago series (Functions, Statistics and Trig was one text and the other was Precal and Discrete Math). Gosh it was fantastic. Our kids graduated without calculus, but with amazing skills. This program was abandoned after several years because of the push to get calculus into their lives before graduation. The irony is painful: admissions wants calculus, professors want algebra. The big question: can we have both? I think we can; I think we do here for many kids.

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