One thing that keeps coming up in these books and discussions is a reference to Bloom's Taxonomy of Learning, a six-step hierarchy of tasks in cognitive development. Each step comes with a description, examples, and "key verbs". Here is a summary similar to what I've been seeing. Now, I'm perennially skeptical of these kinds of "N Distinct Types of P!" categorizations, as they've always struck me as at least somewhat flawed and intellectually dishonest in a real, messy world. But for argument's sake, let's say that we engage with the people who find this useful and temporarily accept the defined categories as given.

In every instance that I've seen, the discussion seems to turn on the following critique: "We are failing our students by perpetually being stuck in the lower stages of simple Knowledge and Comprehension recall (levels 1-2), and need to find ways to to lift our teaching into higher strata of Application, Analysis, etc. (levels 3-4 and above)". To a math instructor this sounds almost entirely vapid, because we never have time to test on levels 1-2 and entirely take those levels for granted without further commentary. In short, if Bloom's Taxonomy holds any weight at all, then I claim the following:

Math is hard because by its nature it's taught at TOO HIGH a level compared to other classes.

For example: I've never seen a math instructor testing students on simple knowledge recall of defined terms or articulated procedures. Which in a certain light is funny, because our defined terms have been hammered out over years and centuries, and it's important that they be entirely unambiguous and essential. I frequently tell my students, "All of your answers are back in the definitions". Richard Lipton has written something similar to this more than once (link one, two).

But in math education we basically don't have any friggin' time to spend drilling or testing on these definitions-of-terms. We say it, we write it, we just

*assume*that you remember it for all time afterward. This may be somewhat exacerbated by the math and computer scientist's custom of knowing to remember those key terms, and maybe our memory being trained in that way. I know in my own teaching I was at one time very frustrated with my students not picking up on this obvious requirement, and I've evolved and trained myself to constantly pepper them with side-questions on what the proper name is for different elements day after day to get these terms machine-gunned into their heads. They're not initially primed for instantaneous recall in the ways that we take for granted. At any rate: the time spent on

*testing*for these issues is effectively zero; it doesn't exist in the system. (Personally, I have actually inserted some early questions on my quizzes on definitions, but I simply can't find time or space to do it thereafter.)

So after the brief presentation of those colossally important defined terms, we will take for granted simple Recall and Comprehension (levels 1-2), and

*immediately*launch in to using them logically in the form of theorems, proofs, and exercises -- that is, Application and Analysis (levels 3-4). Note the following "key verbs", specific to the math project, in Bloom's categorization: "computes, operates, solves" are among Applications (level 3), things like "calculates, diagrams" are put among Analysis (level 4). These of course are the mainstays of our expected skills, questions on tests, and time spent in the math class..

And then of course we get to "word problems", or what we really call "applications" in the context of a math class. Frequently some outside critic expects that these kinds of exercises will make the work easier for students by making it more concrete, perhaps "real-world oriented". But the truth is that this increases the difficulty for students who are already grappling with higher-level skills than they're accustomed to in other classes, and are now being called upon to scale even higher. These kinds of problems require: (1) high-quality English parsing skills, (2) ability to translate from the language of English to that of Math, (3) selection

*and*application of the proper mathematical (level-3 and 4) procedures to solve the problem, and then (4) reverse translation from Math back to an English interpretation. (See what I did there? It's George Polya's How-To-Solve-It.) In other words, we might say: "Yo dawg, I heard you like applications? Well I made applications of your applications." Word problems boost the student effectively up to the Synthesis and Evaluation modes of thought (levels 5-6).

So perhaps this serves as the start of an explanation as to why the math class looks like a Brobdingnagian monster to so many students; if most of their other classes are perpetually operating at level 1 and 2 (as per the complaints of so many writers in the humanities and education departments), then the math class that is immediately using defined terms and logical reason to

*do stuff*at level 3 to 4 does look like a foreign country (to say nothing of word problems a few hours after that). And perhaps this can serve as a bridge between disciplines; if the humanities are wrestling with being stuck in level 1, then they need to keep in mind that the STEM struggle is not the same, that inherently the work demands reasoning at the highest levels, and we don't have time for anything else. Or perhaps this argues to find some way of working in more emphasis on those simple vocabulary recall and comprehension issues which are so critically important that we don't even bother talking about them?

I teach computer science (intro to programming) and just yesterday a student with a weak midterm performance came by, saying that he mainly studied the wrong way: he tried to memorize the book and terms like in his Business classes but what was tested was problem-solving skills.

ReplyDeleteWe try to work in some simple vocab/recall in class and in the small true/false portion of exams. Sometimes it takes some effort but it seems always like a good idea.

I agree! It seems like the experience with your student is pretty common (I've seen that as a trope in several books). On the one hand, I do claim that knowing special defined terms are a necessity, but only the beginning of the learning process.

DeleteI provide sample/practice tests to my students in advance of real tests, with the exact same directions for each problem ("Solve by completing the square", "Factor using a special formula", etc.; not too much work by using TestGen)... but even that's only a partial solution, because many students expect it will be the identical procedure but "only change the numbers" (e.g., change from square-of-binomial to sum-of-cubes and they didn't prepare/can't handle that).

The statement "I just studied it wrong" sounds like a cop-out but it is not. I've now come to realize that a lot of the early educational system just throws information at kids without teaching them first how to learn. Even something as straightforward as memorizing facts has a whole slew of techniques to make that easier and more efficient.

DeleteThis kid is ahead of the game if he recognized that "studying it the right way" matters. In a problem-solving type class, one key is to practice solving problems of a similar type that would be found on the exam.

My personal view that learning math is a lot like learning a language. There are various components needed to be fluent in a language -- character recognition, spelling, vocabulary, grammer, pronunciation, comprehension.. Learning history is generally not equivalent (at least not in the way it is commonly taught, by memorizing names and dates).

ReplyDeleteLearning English is very difficult but students learn it naturally at a young age through exposure. Anyone who has a chance to work with young kids will be likewise amazed at how easily they can grasp advanced mathematical concepts. Yet most kids are not exposed to this outside of isolated math classes. This is a societal problem.

I couldn't agree with that any more strongly. In fact (in my basic algebra classes) I explicitly announce that the principal goal of the course is to "Read and write math properly with variables". The very first mini-exercise is to look at a mangled English sentence and discuss what's wrong with it; and to think about transferring those ideas of grammar to mathematical-language statements (particularly in regards to the verb "to be", the equals). And within the last year I've eliminated all multiple-choice tests specifically so I can give individual writing feedback to each student on every line (which other instructors think I'm insane to do when I mention it... but I also greatly reduce the number of questions to make that grading feasible.)

DeleteNow I do get a few students who, when I say I look at the class as equivalent to a foreign-language class, groan and say they hate learning languages. Which you also have to respect and be willing to work through anyway.

This crystallizes something that I've dealt with in my Humanities survey - I test students' comprehension of why certain things occurred, or what significance the events had, and so on, rather than just on what happened and when. They like it, but a lot of them find it challenging. And yet, when I tell them on the first day that I will never ask them a name or a date on a test, they seem happy. They hate memorization, but they have no idea, it seems that this is a signal that we'll be working on a higher level of Bloom's Taxonomy (so to speak). They are excited about the extra labor of comprehension that they are about to take on... if only they knew!

ReplyDeleteI like it!

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