Reply to Struggling in Hon Precalculus

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[quote=Anonymous][quote=Anonymous]PP, you seem to have skipped the part where they [i]learn and remember [/i] the stuff they "already know and learned". Most humans aren't computers who memorize everything at first sight.

When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.

https://www.joelonsoftware.com/2006/10/25/the-guerrilla-guide-to-interviewing-version-30/

Skip to the part about Jared the bond trader and then Serge Lange the world-class mathematician and professor.

"Serge Lang used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and [b]Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t. [/b]
The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.

You see, if you can’t whiz through the easy stuff at 100 m.p.h., you’re never gonna get the advanced stuff."
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My curiosity about this origin of this story was piqued  :D , so here's the pdf of the actual article where Serge Lang discussed the test, December 1969 of the Columbia Daily Spectator (page 6 in the pdf):

[url]https://spectatorarchive.library.columbia.edu/?a=is&oid=cs19691208-02&type=staticpdf&e=-------en-20--1--txt-txIN-------
[/url]
Nowhere did Lang say that students who can solve 5 simple problems in a few minutes will pass his calculus class with flying colors. Here's what he actually said in the article:

[i]After receiving Mr. Wyer's letter, I decided to give a short test to check Mr. Wyer's opinions. The test consisted of five problems, and was given to the 1A sections. One can draw some conclusions: a) The test is very easy and students unable to do reasonably well on such a test [b]should not be taking a calculus course...[/b].[/i]

So your original quote is highly inaccurate. While it's more provocative and makes for a good story, it doesn't change the fact that being fast at doing very easy problems does NOT automatically predict success in math. All we can conclude here is that this is an absolute minimum of what the students are expected to know before taking the class (btw, if you're curious, in the article you can find the actual 5 questions that were asked). So no, Serge Lang did not believe that students who can do easy problems with speed will get As in his class (and why would anyone believe that? Correlation is not causation). The test was simply a low bar filter to identify students who could be at risk of failing because they don't understand fundamentals. Actually doing well requires much more than that, namely working hard and thoroughly learning the material.

Even more interesting in the article, is Lang's take on memorization:

[i]Although a couple of students raised an objection to the time limit on the test, most students felt the time was reasonable, and in any case, part of the test amounts to verifying that students have reasonably fast responses to the type of question involved. Another point raised by some is that the question on sines of angles involves "memory." Which memory? Part of being properly acquainted with the basic facts of trigonometry is to be able to draw the proper triangle for and angle of 30 degrees, or 45 degrees, and determining the sine from that triangle, possibly using the pythagoras theorem. [b]The type of memory involved is that of understanding, not the brute force memory... [/b][/i]

It's not about memorization and regurgitation at high speed, it's about understanding. Speed develops naturally as a consequence of understanding. Doing homework (i.e working on stimulating problems) is by far the most important part of learning mathematics. Initially it's a slow painful process, but over time it solidifies concepts in students minds. If done well, it can lead to mastery of the material by the end of the semester/year.

I also take issue with the work analogy you make:

[i]When someone at work has a quick question, they want a highly paid subject matter expert to recognize the core issue and then provide a quick correct answer, not think for a while to work it out.[/i]

You can't actually compare kids education with adults performing tasks at work. Under this analogy you are suggesting kids should just ask for an answer and not try to think about it first, which defeats the whole purpose of learning.

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