There's a national crisis in math; the U.S. ranks 25th in math out of 27 countries studied by (OECD 2012), and 51 out of 144 in a recent World Economic Forum report on the quality of math education (WEF 2014). Though there are many potential drivers, there is one fundamental and often hidden driver of the crisis that is often not discussed: the way mathematics is taught.
Nowadays most math lessons follow a "TTA" (short for Theory Then Application) approach. Here's an example (which I hope won't stop you from reading the rest of this article):
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Definition: We say that two variables y and x are directly proportional if there is some non-zero constant k such that y=kx, and inversely proportional if there is some non-zero constant k such that y=k/x.
Example of Direct Proportionality: The linear function y=2x; here k=2.
Example of Inverse Proportionality: The rational function y=4/x; here k=4.
Theorem: If y is directly proportional to x, then x is directly proportional to y.
Proof: By assumption, there is some non-zero constant k such that y=kx. Since k is non-zero, divide both sides by k, so that x=(1/k)y. Defining l=(1/k) this becomes x=ly. And since l is non-zero, by definition this equation says that x is directly proportional to y.
Another Example: The circumference C of a circle is directly proportional to its diameter d, since C=pd (where p is Pi); here k=p.
One Last Example: The miles y driven in a car is directly proportional to the number of gallons x of fuel the car uses: y=ex, where e is the car's fuel economy (in miles/gallon).
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Okay, welcome back (if you're still there). If you're into math, this TTA teaching style worked well, and you probably felt like this was a great way to present the concepts discussed. But since the vast majority of us aren't into math, the TTA approach only serves to exacerbate the crisis. Here's why.
Most of us actually learn math best through direct real-world connections. This is a fact that every elementary school teacher knows all too well. After all, they help kids learn to multiply and divide by first using manipulatives and then abstracting the concept. (Imagine how unsuccessful the reverse process would be.) But at some point, math instruction becomes more TTA-like---typically this happens in middle school Algebra classes---and the real-world connections are no longer the first thing students see when they get to a new concept, the math is.
From them on, the TTA approach becomes the standard teaching approach for all higher-level math courses. And unless you learn well from this type of approach---in which case you're likely destined for a career in science anyway---you quickly start to think that maybe math isn't for you. Problem is, that often rules out science and engineering majors too (since they depend heavily on courses like calculus).
A more effective approach---both in my opinion and experience---is the complete reversal of this process: the ATT approach (Application Then Theory). Unlike the TTA approach, ATT keeps the real-world problem the central focus. It gets almost all students (granted, depending on your ability to choose appropriate applications) interested at the beginning of the lesson---not the end---by immediately answering the "why should we care?" question. Taught this way, the math discussed has a natural context and now becomes relevant.
Let me return to the example and illustrate the ATT presentation of the same concepts.
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Let's say you're planning a small road trip with a few friends. There aren't any gas stations along your route and you're wondering if your 1/4 tank will get you to your destination. Google says it's a 60-mile trip; your car gets 30 miles to the gallon, and it has a 12-gallon tank. Echoing Keanu Reeves: "Pop quiz hotshot. What do you do?"
Solution: Let y be the distance traveled (in miles), and x the number of gallons of fuel used by your car. Then from the fuel economy information, y=30x. When y=60, x=60/30=2, meaning you'll use 2 gallons of fuel to get to your destination. You currently have 1/4 of 12 = 3 gallons. So, go for it, enjoy the trip and don't worry about those gas stations!
The relationship between the distance traveled and the number of gallons used is part of a broader concept:
Definition: We say that two variables y and x are directly proportional if there is some non-zero constant k such that y=kx, and inversely proportional if there is some non-zero constant k such that y=k/x.
In our road trip example, the variables y and x were directly proportional, with k=30.
:::Insert the theorem and remaining content of the original presentation:::
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Notice how the ATT approach, among other things, anchors the abstract concepts of direct and inverse proportionality in a context that is interesting and directly applicable to students' lives. Hopefully you'll also agree that the ATT approach is more likely keep a student interested in the lesson; it keeps the reader engaged, and helps make the concept of proportionality relevant.
Taught the ATT way, math becomes something intimately connected to the real world, and with this perspective one even begins to appreciate how interconnected everything is (who knew that there were linear functions hiding in your car?). But perhaps most importantly, this approach helps dispel the myths that math is abstract, too difficult to learn, and inapplicable (or only applicable if you want to be a scientist). So, the next time you hear someone say any one of those things, help the cause and point them to an interesting example that follows the ATT rubric. And if you can't identify one yourself, take that as a challenge, because math can be found all around you.