lim x → 5 = ???

February 2, 2007 at 3:30 PM

Browsing around this morning, I stumbled across the following picture:

Ah, Limits!

Math bored me throughout elementary school. Frankly, it bored me until I started calculus, and I think I've pinpointed why.

Assuming for a minute that the story behind this rather humorous picture is true, with what (purely speculative) methods or misconceptions does the student approach problem?

Glyphs instead of Numerics

Rather than having a conceptual understanding of the numbers behind a process, the student is attempting to perform a purely symbolic manipulation. This sort of approach often works with simple algebra, fractions, and other elementary school level maths; it breaks down badly in this scenario.

No Graphical Understanding

A simple plot of 1/(x-5) would reveal the asymptote at x = 5, making explanation of the limit much sounder. Indeed, in the math of nonlinear dynamics, a simple equation such as

d?/dt = 1 + 2cos?

can be best described with a graphical approach. Even with respect to easier calculus, it's much simpler to find the intersecting volume of two rotated curves if you can plot (and rotate) the curves correctly--a visual grounding in mathematics is key!

Lack of Prerequisites

Of course, there's the chance that this student doesn't know the concept of infinity, whether symbolically or theoretically. This, combined with (1) and (2) would produce the given answer--turn the number 5 counter-clockwise by 90°!

So What?

So, what does this have to do with math boring me during elementary school? Actually, a lot. The tendency is for teachers and students to focus on formulaic knowledge rather than graphical. Some students are able to learn with this emphasis--but it's my opinion that such learning is attained in spite of formulae.

Algebra, "Advanced" Algebra, and Geometry were--to me--the trifecta of bordem. The sum total of their material could (should?) be taught in the space of 3 semesters, rather than their sum-total of 6. Each class spent the first 1/3 reviewing fomulae that teachers assumed (often correctly) students no longer remembered. I, unfortunately, didn't benefit from the review--but that's completely beside the point. This isn't a "look how amazing my math skills are" post, but rather one that asks "what can be done to make math more appealing and learnable?".

My suggestion? Do away with the formulaic routinue, and emphasize graphs more. Rote memorization is never a good way to learn things; however, it is a consistent way to get good grades. Rote memorization directly undermines the principle that grades are to be reflective and indicative of learning. Regurtitating the same formula without understanding its implications sets students up for forgetfulness; requiring them to show graphically what the formula means forces them to either understand or evince their lack of understanding.