On Fuzziness

I have debated (with myself) on whether to blog about my (mis-)adventures in tutoring at the City Heights Farmer's Market/Library as part of SIO GDAWG's outreach efforts. I figure it will be ok to simply remark about both the (1) gaps in knowledge (conceptual or otherwise) of the kids and (2) weirdness of homework that these kids bring.

One girl (who came twice, but whom I haven't seen in a month or two) had trouble in her math class. She was taking Algebra I, and somehow had made it through most of the first semester (I'm guessing they started algebra in the Fall) without anyone noticing that she was having basic trouble with arithmetic. For the most part, she could do basic numerical operations, but had a very flaky grasp of order of operations, distributive properties, and negative numbers. (from what I could tell) Needless to say, it was a bit much to ask her to do operations involving linear equations.

One boy (4th grade?) hadn't yet memorized his multiplication tables. Irregardless of this (which isn't uncommon for his age), he didn't understand that multiplication represented repeated addition. The problem with not knowing this concept is that treating multiplication as a black box operation is going to be very bad news for this kid when he starts algebra (see above).

Last week I also helped a kid with some worksheets dealing with St. Patrick's Day (SPD). The first page was basically a crossword with a single clue running vertically that spelled out "leprechaun". Problem is, there were some really vague (and obscure) clues that depended on both awareness that it was SPD-themed as well as the cultural aspects associated with SPD. (Really, how many people will know of the top of their heads that "Emerald Isle" refers to Ireland?)

In the same packet, there was a word association page where one item would be given and two blanks needed to be filled with items from a word box. Unfortunately, some of the examples were synonyms ("wee" -> "little", "small") and some were same-category-membership ("Japan" -> "Mexico", "Japan"). What this demonstrates and what you were supposed to do if you didn't know some of the words (e.g. the kid didn't know what "wee" meant) is beyond me.

On a later page, there were a series of analogies. Besides the fact that analogies were not explained, a lot of the problems violated the standard guidelines for analogies (as they are used in the SAT and other "important" tests). For example, in A:B::C:D, the relationship between A and B should be the same as that between C and D. The analogy should NOT rely on any association between A and C or B and D. (which is what SAT writers like to do to trip people up)

On another page, there were some weird set exclusion problems (e.g. "What are the numbers that are in the square, but not in the triangle and circle?") in reference to a figure. Problematically, the use of language was imprecise: the questions sometimes used "and" and sometimes used "or", when in both cases, the desired conjunction was "nor"!

The busywork nature of the worksheets also seemed weird to me. I guess the standard purpose of homework these days is to occupy time rather than practice what you learned or apply those skills in reasonable useful example situations. I am reminded of Feynman's remarks on science textbooks:

Finally I come to a book that says, "Mathematics is used in science in many ways. We will give you an example from astronomy, which is the science of stars." I turn the page, and it says, "Red stars have a temperature of four thousand degrees, yellow stars have a temperature of five thousand degrees . . ." -- so far, so good. It continues: "Green stars have a temperature of seven thousand degrees, blue stars have a temperature of ten thousand degrees, and violet stars have a temperature of . . . (some big number)." There are no green or violet stars, but the figures for the others are roughly correct. It's vaguely right -- but already, trouble! That's the way everything was: Everything was written by somebody who didn't know what the hell he was talking about, so it was a little bit wrong, always! And how we are going to teach well by using books written by people who don't quite understand what they're talking about, I cannot understand. I don't know why, but the books are lousy; UNIVERSALLY LOUSY!

Anyway, I'm happy with this book, because it's the first example of applying arithmetic to science. I'm a bit unhappy when I read about the stars' temperatures, but I'm not very unhappy because it's more or less right -- it's just an example of error. Then comes the list of problems. It says, "John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?" -- and I would explode in horror.

My wife would talk about the volcano downstairs. That's only an example: it was perpetually like that. Perpetual absurdity! There's no purpose whatsoever in adding the temperature of two stars. Nobody ever does that except, maybe, to then take the average temperature of the stars, but not to find out the total temperature of all the stars! It was awful! All it was was a game to get you to add, and they didn't understand what they were talking about. It was like reading sentences with a few typographical errors, and then suddenly a whole sentence is written backwards. The mathematics was like that. Just hopeless!