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How do you teach “understanding”?

In music, when someone learns to play an instrument, it’s very important to learn what the proper position for the hands is (When playing the piano, for example, your wrist is not supposed to be very low or very high; the fingers should be bent and not straight, etc…) You can play easy pieces even if your hands are not set up correctly, but as the music becomes harder, it’ll be hard and then impossible to play fluently. Professional music teachers are very critical of “amateurs” who don’t pay attention to hand position when they teach the beginners. It turns out that it’s very hard, next to impossible, even for children, to unlearn.
Something very similar happens when learning mathematics, I think. Sometimes I want to ask my students to forget all math they have “learned” before. It seems to be easier to teach them “from scratch” than to correct the misconceptions they have.
The biggest misconception of all is that mathematics is learned by memorization. (I know I am not saying anything new here.) A student told me once that I am the first instructor who told them that they have to understand the material, not to memorize it. Really. (To be fair to my CUNY colleagues, that happened when I was an adjunct somewhere else.) Students are surprised when I tell them that the multiplication tables are probably the first and the last thing I learned by rote, memorized purposefully, rather than “accidentally”. “Accidentally”, of course, is not a good word here. I didn’t memorize the Pythagorean Theorem by accident; I memorized it because I used it a lot.
It is a well-known problem of the secondary school curriculum (and our Algebra courses, yes) – being a mile wide and an inch deep. Students don’t get a chance to master the material before moving on to something new. Why, oh why did my daughter have to start learning statistics in Kindergarten? (No kidding. Everyday Mathematics curriculum in action here…) I would prefer her to do more addition/subtraction word problems in order to learn (yes, to learn, not to memorize per se) the “number facts”, rather than draw boxes for histograms.
Since the students don’t have enough practice to learn by doing, they are encouraged to memorize things. Memorization of the facts and formulas does not cause too much harm, though. Much worse is the memorization of concepts, which also seems to be encouraged. Students are instructed, for example, to use addition or multiplication when they see the word “more” in a word problem, and subtraction or division when they see the word “less”. “More” and “less” are the concepts, and they can mean different things in different situations. Look for example at the following problem, fresh from this year’s 4th grade test:
Twenty eight students attended the first meeting of the club. Four times more students attended the first meeting than the second meeting. How many students attended the second meeting?
It is a multiple-choice problem, and 7 and 112 are among the answer choices, of course. So what do the well-trained students do when they see the word “more”? They are prepped to recognize the word, not to think what it means for each particular problem. Under the pressure of the moment, who would think about which meeting really had more students in attendance? “More” means multiply, period.
On the other hand, may be the issue here is not memorization, or even mathematics in general. Could it be that children are having a problem processing the language here, and their trouble is not really of the “mathematical” kind? Elementary school teachers are teaching their students language skills, too, so they are supposed to be able to prepare them both to understand the problem and to do the calculations.
I, a college math instructor, am another story, though. I think it is a good problem. It is formulated that cumbersomely on purpose, obviously, but it’s OK, the students should be doing some thinking, too, not just mindless calculations. Unfortunately, this problem is not very different from the ones we ask our Algebra students to be able to solve. They also know that “more” means to multiply. I can tell them a million times to read the problem carefully, but if they don’t understand that “more” “belongs” to the first meeting, not the second, what do I do? Suddenly it is me who has to teach them the language processing skills. What’s more, I realize that many of my students don’t even know what it means to UNDERSTAND. They really think it is the same as to remember! How do I teach them to internalize, to absorb, to make their own? Isn’t it something we learn little by little since we are infants? How do I teach that to adults? I just don’t know.

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2 Responses to How do you teach “understanding”?

  1. It’s going to be ending of mine day, except before finish I am reading this enormous piece of writing to improve my know-how.

  2. Arjen says:

    Dear Asya,

    Thank you for your thoughts. Really good to see you write on understanding rather than just memorizing. When I look at children here in The Netherlands, lots of time is spend on drilling the multiplication tables instead of learning the ideas/concept behind it.

    Although it is good to use memory when needed, understanding is more powerful. That is what I did when I was studying physics. It saved me lots of time and stress! Understand the concepts and you can apply it in nearly any situation.

    When you do have to memorize new information, I would like to suggest people learn the memory techniques which are thousands of years old, but still deliver amazing results.

    I hope teachers think more about the usage of UNDERSTANDING and MEMORIZING. It will help children (and adults) a lot!

    (and yes, reading carefully is also important. blame it on my age or on the time we live in… but I feel that people seem to need a fast paced life. the first impression is not even processed… but the next one is already searched for. stop, think, evaluate/understand/remember, and then move on)

    Thanks for your thoughts!

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