But I've been thinking lately that the expectation and need for these most fundamental skills is often not communicated to our students; in the era that frowns on structure and drills for automatic knowledge, many of our students have never seen such a requirement assessed directly anywhere. Of course, I'm thinking of the times-tables drills that people my age did in the 2nd or 3rd grade, and nowadays may possibly be done in the 8th grade or high school by the more exceptional and dedicated teachers (so I hear).

Might it be the case that in any class, there's at least one specific skill that is expected to become automatic, even if many of us overlook communicating and drilling on that? For example, it's occurred to me that we might expect the following regular speed drills to take place:

- In early grammar school -- Times tables.
- In late grammar school/college remedial arithmetic -- Negative number operations.
- In junior high school/college remedial algebra -- Matching a slope-intercept equation to the graph of its line.
- In statistics -- Estimating the area under part of a normal curve, or interpreting confidence intervals and P-values. (?)

What do you think? Have you used timed drills to communicate the expectation of automatic skills? And for anything other than times tables?

In beginning algebra (community college), I give a timed multiplications test on day one. I have it on a spreadsheet, so I can randomize. It has one instance of every one-digit problem (7x8 or 8x7, not both). They can retake until they get it all. I don't have timed tests in any other classes, perhaps I should. Calc II students should be able to do derivatives quickly.

ReplyDeleteSue, that's really good feedback, I'm glad to know about that. For first-day algebra, I was leaning towards a timed sheet on integer operations, which would include a lot of multiplications and divides along the way.

DeleteActually yesterday at the end of my algebra classes I did a timed worksheet on matching linear equations to graphs -- it went over pretty well, and the students who were present said it was helpful.

I've been assuming automaticity with simple algebra in my engineering classes, but I suspect that it is not there. Perhaps I should do some timed drills in the Applied Circuits class on some very simple examples.

ReplyDeleteMy students at the end of the algebra course seemed thankful for the speed-graphing drills. I'll also now try doing stuff like that for prerequisites on the 1st-day, maybe it will motivate some people to get to the math workshop early to fix gaps. I'd be interested in hearing how this works for anyone else.

Deletegswp, I'd like to see what you come up with.

ReplyDeleteI don't have any quickie tests to check pre-requisites in any of the classes I'm teaching now. It's worth thinking about.

ReplyDeleteOn the other hand, I like a different emphasis on the first day. I want them to see that math is about thinking, not about memorizing procedures. Hmm... How to address both issues?

Definitely a prioritization issue, as there's about 100 "would be nice" things on the first day. For algebra, among the top things for me are a correct order of operations (the only arbitrary thing; almost everything later can be explained by it), and seeing variables in translated identities (thinking abstractly, not about specific numbers, and incidentally reviewing some basic properties).

DeleteOrder of operations isn't as arbitrary as some people think it is. Strongest operation comes first. And I'm guessing it was developed so that 2x^2-3x+7 (and things like it) would have the right meaning without parentheses. Hmm, I wonder if we could get students to figure out what order of operations they'd choose, if they were the queens and kings of math.

ReplyDeleteAgreed, of course, that the existing order of operations boils down to "strongest first". But it seems like the brevity in one direction is offset in another. Say we ignore juxtaposition, and denote [oop] for normal ordering, [oop*] for hypothetical reverse ordering. While [oop] 2*x+5 is shorter than [oop*] (2*x)+5, the translation of a common statement like "three times the sum of a number and eight" is [oop] 3*(x+8), longer than [oop*] 3*x+8.

DeleteJust a thought -- to date I haven't seen a claim that [oop] is inevitable that convinced me. If I found one I'd be excited. :-)

Maybe I've been convinced too easily? Maybe polynomials just feel like natural sorts of things to want to describe. When translating from the particulars of a situation (three times as much as eight more than x is a fine example), there is probably no one oop (I'd call it ooo) that is better. But once we're hanging out in the world of mathematical stuff, I want to talk about how many of each power of x (polynomials) easily, and that requires the current oop.

ReplyDeleteYeah, here I'm thinking of the fast way to convert a string to an integer. Like: [oop] 3x^2+5x+2 = (3x+5)x+2, which in [oop*] would be 3x+5x+2. Which also has the advantage of being more computationally efficient (4 multiplies for [oop] vs. 2 multiplies for [oop*]). You could then define degree as the number of times "x" appears in the expression, etc.

DeleteThis is so cool! I sometimes feel like I don't have enough imagination. This is definitely the way to play with it. If students could build their own, and discuss advantages and disadvantages, I think they'd have a deeper understanding, even if they ended up thinking the convention wasn't the best chioce.

ReplyDeleteNice, I think that's mostly my software engineering/numerical analysis bleeding through. :-)

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