The backward approach roughly follows the same standard scheme, as exemplified in every textbook ever (and far too many classrooms): introduce an idea, term, or a definition, and then explain what it means, how it's relevant, and how to do "problems" or answer "questions" using it.

For example, most high school math textbooks come to a chapter titled "trigonometry" or something like that, give definitions of the trigonometric functions (sine, cosine, tangent, etc.), and then proceed to show how useful they are in solving problems involving triangles.

Later in the chapter, in the section titled "the sum and difference formulas," the book gives the sum and difference formulas for sine and cosine, before giving the proofs, and before even stating a relevant question that would require the sum and difference formulas.

Unfortunately, this approach completely eliminates any and all creative insight into the very real problems at hand, and leaves the student with no reason or desire to acquire that insight. It's as if the definitions of the trigonometric functions were handed down by God, followed by a set of problems to solve that require them.

In the real world, of course, everything happens in the

*opposite order*: definitions don't lead to problems. Problems and questions lead to insights, which lead to generalizations, which eventually lead to generalizations and definitions. Someone tries to find the distance between two points knowing their respective distances to a third point and one of the angles. After studying geometry (or even before), most students have an intuitive sense of how to approach this problem, and with some care and coaxing, you can get them to "discover" the law of sines. But if they learn trigonometry from a standard textbook, they'll never even get the chance, because there it is before the problem is given!

Most students can learn to use and manipulate trigonometric functions fine by the backwards method, but its flaws aren't merely aesthetic. If a student has no sense for the scope of a problem he or she is trying to solve, what reason does he or she have to remember the trigonometric functions beyond the next test, or the SATs? For most students, the sum and difference formulas are something to memorize and then forget, rather than a beautiful solution to a seemingly intractable problem.

On the first day of my 4th grade science class, I handed the students a sheet with just three questions: What is science?, Why do we care about/study science?, and How do we study science? Their answers were revealing and kind of depressing (also somewhat hilarious). How do we study science? Why, from science books of course! Why do we study science? So that if we need something to fall back on, we can be science teachers!

These students, like so many, have completely missed the point (so far at least!). Maybe some of them will miss the point either way. But it seems more likely that the things they learn will leave enduring memories if they have to confront the same problems that the people who actually discovered and developed them had to confront. At least that way, they get some sense of what math and science are really about!

I should add that it's easier to teach the backwards way, as it requires less student involvement, less struggle, and less flexibility. You can't ask your students how to find the sine of 75 degrees and then expect them to find it in the next 15 minutes. But I hope it's worth the sacrifice.

ReplyDeleteI should also clarify that I'm not some sort of expert teacher...far from it. But I can still see flaws in the system where they exist!

Have you heard my rant about how the New York State Regents exam in general chemistry requires students to learn a very specific small slice of organic chemistry? These poor, innocent 10th graders have to memorize the names of the main functional groups in organic molecules and learn the nomenclature system (ie be able to look at a drawing of a molecule and figure out what its name is).

ReplyDeleteThis process is a miserable slog because, with no context about what these groups are, how they interact with each other, or why anyone would remotely care about any of this, it's really just a (moderately difficult) exercise in visual recognition and rule memorization. It certainly doesn't teach anybody ANYTHING about chemistry (they can draw you an aldehyde, or know that some molecule should be called 1,3-diethylbenzene or whatever, but nothing about what that means), especially if they never go on to take a full class in organic chemistry-- which, why the hell would they, when it seems to be nothing but stressful busy work? The main lesson it imparts is basically: science is frustrating, yet very, very boring.

that sounds dumb! and, like a lot of school in general, unfortunately! the ability to memorize and then forget is not an altogether useless one, but, on the other hand, it doesn't seem like it needs so much attention throughout the education system

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ReplyDeleteI am very proud of the creative and caring Learn Trigonometry staff we have they really do go the extra mile to interest and reach kids. I remember being bored in school, some teachers are just that boring and that does exactly what you said. Thanks for giving your teachers perspective.

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