Skip to main content

How To Talk To Your Kids About Math (And Why You Need To)

When I started elementary school my mother, like all mothers, began asking me “what did you learn in school today?” I was always eager to share, and did. I talked about the things I was learning in all of my classes. I even remember some of those conversations–like the time I told her about cumulus clouds. After that I remember mom pointing them out to me regularly, and we’d bond over the ensuing conversation…yes, about clouds. Then middle school came. The concepts got more serious–governments and their roles, Greek tragedies, etc.–but she still asked her question, I still answered, and we still bonded over those conversations. It seemed that no matter what the subject was, I could always enjoy a nice discussion about it with my mother. That made me happy; it also made me want to keep learning. But on the horizon was one subject that would eventually drive a wedge between us: math.

Continue reading the article on the Huffington Post here.

Helping All Students Experience the Magic of Mathematics

Mathematics is a beautiful subject, and that’s something that every math teacher can agree on. But that’s exactly the problem. We math teachers can appreciate the subject’s beauty because we all have an interest in it, have adequate training in the subject, and have had positive experiences with it (at the very least, we understand a good chunk of it). The vast majority of students, on the other hand, often lack all of these characteristics (not that this is their fault). This explains why if I’d start talking to a student about how exciting the Poincare-Hopf theorem is, I probably wouldn’t see anywhere near the same excitement as if I were to, say, let them play with the new iPhone. This may seem like a silly hypothetical, but I believe it brings up all sorts of important points. For one, what does it say about our culture (and our future) when young people would rather be playing games on iPhones (or watching Youtube, or being on Facebook, etc.) than studying math or science? What causes our culture to be the way it is? How did companies like Apple and Facebook get students so interested in these activities? What are they doing that we math teachers aren’t?

First, let me admit that there are many, many differences between getting exciting about the new iPhone and getting excited about math,* but I’m interested in one of them in particular: you can see, feel, interact with, and experience the iPhone. Moreover, Apple thinks very carefully about every aspect of the user experience well before they release their next phone (there are, after all, billions of dollars at stake).

Sadly, the way math is taught in many places, students’ experience with mathematics is often confined to a blackboard or piece of paper. They also spend the majority of their time interacting with math in a very different way, e.g., trying hard to get the right answer before the homework is due as opposed to playing around with the content to discover something new, as a first-time iPhone user might do. And what about the Steve Jobs or Jony Ive of the class—the instructor—who is supposed to make it all magical? Oftentimes that person follows the “definition, theorem, proof” style of teaching, which is likely only “magical” to already math-inclined students. My point: we (the math teachers) are the most important drivers of our students’ interest in and excitement about mathematics. Collectively, we are the Apples and Samsungs of the math world. And if we teach math like we discuss it amongst ourselves,** we’re likely to continue losing the vast majority of students to other careers.

So, what should we do? I say we look to Apple, Samsung, and all the other companies that have successfully hooked our students on their ideas and products. Sure, they have hordes of people whose sole job it is to make their products fun, cool, and relevant, but why can’t we do that, too? Why can’t we, for example, give out a survey the first day of class that asks students about their hobbies and interests, and then, at the very least, choose examples and applications for the rest of the course that align with those interests? In fact, why don’t we just structure our courses to make mathematics something that our students candirectly experience and is directly relevant to their lives?*** Let me call this the Everyday Mathematics (EM) approach.

Here’s an example. Instead of reviewing the graph of a sine function by drawing a sine curve, explaining what the frequency, amplitude, or period are, showing examples where these parameters change, and finally discussing a Ferris wheel, picture this instead. You pull up a chart of human sleep cycles, you explain that the average cycle length is 90 minutes, that there are four stages of sleep—with Stage 4 being “deep sleep.” You ask your students to find the formula that best fits the sleep chart. Then you ask them: at what times should you wake up to avoid feeling groggy (which happens when you awake near the bottom of a sleep cycle)? You would then guide them to the revelation that they can now use their formula to predict these times and other interesting things, too. Presto! Sine and cosine have now become relevant; they are now concepts that help explain every student’s sleep cycle and can help them avoid morning grogginess. In other words, this EM approach has made this particular topic at least relevant to your students’ lives. I wouldn’t be surprised if, when you move on to tangent, some of your students would start wondering “Hey, what can tangent do for us?” (By the way, how often have you heard a student ask that?)

In general, the EM approach begins with a topic or phenomenon directly relevant to your students’ lives. Then, you (the instructor) build a lesson that slowly guides students through the math you would have taught anyway, except that now there is context, that context is personal for each student, and there is a point to all of it that students can buy into (in the example, helping them sleep better and explain morning grogginess).

From an instructor’s perspective, the EM approach may seem like a lot more work than a more traditional approach. However, I myself was able to generate enough of these EM-like examples (pertinent to Calculus I topics) to write an entire book about it, mainly by just spending a few days being very observant about everything going on around me and then putting on my mathematician hat to see the math behind it. Granted, this approach might not be appropriate for all courses—it probably wouldn’t work in a course on cohomology—but that’s okay, because by that point that student is probably more interested in how that subject relates to other areas of mathematics.

The EM approach may not be the answer to our national crisis in math, but I think it is a step in the right direction. At the very least it realigns our presentation of the content with our students’ interests. It also attempts to emulate the successful efforts of corporations to get people excited about their products, since the approach puts our students—and their interests—first, and then scaffolds on our content goals (as opposed to the other way around). In my experience using the EM approach, I have received some of the most enthusiastic responses I’ve ever gotten after teaching certain concepts. I would love to hear about your own ideas to make math fun, relevant, and something students can directly experience.


* There are, after all, people who spend weeks in line waiting for the new iPhone; I’ve never heard of a student camping out outside a classroom for weeks waiting for a course to start.
** This would be like Apple unveiling its iPhone by talking mostly in technical jargon—after all, that’s how the designers, engineers, and programmers think. I doubt their press events would be so well attended were this the case.
*** No more talking about the largest area a farmer can enclose with a given amount of fencing, or about a ladder falling down the side of a building, for example.

An Insider’s Perspective on Our National Math Crisis

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):

 


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).

 


 

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.

 

 


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:::

 


 

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.