Sunday, February 28, 2010

Innovative Ideas That Make Sense for Those Hungry for Math Instruction

It is a mistake to suppose that requiring the nonmathematical
to take more advanced math courses will enhance their understanding
and not merely exacerbate their sense of inadequacy.
-- William Raspberry

I have a dream. A recurring dream and I don’t think I’m alone. In fact, I just had this dream AGAIN last night. My dream is that I am contacted by someone from my undergrad school who tells me that they discovered there was a mistake in their records and I didn’t actually pass algebra (sometimes this is geometry, but usually algebra) so I will need to go back to school for a semester and take the class or all my degrees will be canceled. I’m shocked to learn this but confirm it is true. So I enroll in the class and once there I struggle with the class and hate it as much as I did when I was actually in high school and college.

What does this mean? When will I ever use this?

That was my math mantra and to my frustration these questions were left unanswered by my instructors and met with rolled eyes from other students who wanted to move on because they had accepted that math had a god-like quality and was not meant to be questioned.

Recently, the PBS News Hour featured a series of YouTube math videos that cover everything from basic addition to calculus. In them, the narrator poses a problem, and walks through the steps to solve it. After just four years, these videos have attracted tens of thousands of students a day, and are used by schools and students around the world.

Yet as popular as the videos are, and, as powerful as they certainly have been for those having difficulty with a concept, it still doesn’t answer those two questions that resulted in my distaste for the subject.

What does this mean? When will I ever use this?

The classes are ideal for the student who has accepted math should not be questioned, but for a student like me where the real learning lies beyond the “how” and into the “why,” such videos are of little value without the answers. Academic studies and anecdotal evidence alike support a simple (and perhaps obvious) fact: students learn best when the instruction is meaningful and relevant. This is particularly true in mathematics where, starting in the middle grades, content becomes increasingly abstract. During my high school and math classes, while I was often the only one to actually speak up, I don’t think I was alone. In hindsight I probably had silent supporters in the shadows too embarrassed to share their frustration.

To take this to a real-world example let’s take the algebraic concept of “slope” which is defined as “rise over run,” “∆y over ∆x,” or “y2 – y1 over x2 – x1.”


What does any of this mean to me? Does any of this make sense? Isn’t this just another procedure/algorithm to memorize? And, “Why should I care?”

My answer to the equation: Nothing; No/Yes; I don’t.

While the aforementioned videos are a valuable tool for differentiating instruction, for me reformatting traditional content for YouTube and the iPhone helps students learn the algorithm better, but the fundamental questions are left unanswered and my annoying mantra still exists.

What does this mean? When will I use it?

These are not just good questions, but critical ones. Like many students, I was led to believe Algebra was an isolated subjected created for the sole purpose of teaching critical and higher order thinking skills out of context. But the reality of what math actually is, that they never taught me in class is that math is:

1) A set of logical tools that we [humans] created to
2) Explore the world around us.

Math skills such as slope were not, as many students might assume, codified in the Big Bang. Divide thy riseth by thy runeth was not the Eleventh Commandment. Instead, at some point in our human development we had a question, we needed a tool, and this is what we came up with. To illustrate this point, here’s another real-world example.

Question: what are percents, and why did we invent them?
Answer: because they allow us to compare things that are otherwise difficult to compare.

In one store, we save $4 for every $10 that you spend. In another, we save $9 for every $25. Where should you shop?

There are any number of ways to approach this. One method would be to compare how much we’d save if we spent the same amount: we could spend $50 in both stores, and save either $20 or $18, respectively.

But what if the numbers weren’t so clean? What if, instead of $10 and $25, the amounts were $12 and $25? Here, the “multiples” approach is a bit more cumbersome, yet the underlying question remains: Where should we shop?

At some point in our history, mankind faced a situation like this and said, Lets just pick some number and compare everything to that. For whatever reason, we picked 100. So now instead of finding a common multiple, we simply ask, How much would we save if we spent $100 at each store.

So when could we use this?

According to the Wheel of Fortune wheel, bankrupt should come up once out of every 24 spins. If in an actual episode it comes up three times in 60 spins, can we conclude that the show is rigged? For every 100 spins…

Babe Ruth got 2,873 hits in 8,398 at-bats, while Ty Cobb got 4,189 hits in 11,434 at-bats. Who did better? For every 100 at-bats…

That’s the logic of the percent. It’s not magic. It’s not ordained. It’s simply a useful tool.

What about slope? A traditional source might ask us to calculate the slope between (16 , 629) & (32 , 729). But if a student is only taught the procedure, what does he actually know, and how long will he remember it?

But what if we instead approached the problem as:
The 16GB iPad costs $629. The 32GB costs $729. How much is each additional gigabyte of hard drive space?

With this simple question, a student might reason:
  • Slope
    If an additional 16GB costs an additional $100, then Apple is charging $6.25/GB.
  • Y-Intercept
    If Apple charges $6.25/GB, then 32GB would cost $200.
    But since the actual cost of the 32GB iPad is $729, the base-cost must be $529.
  • Equation
    The cost, C = 529 + 6.25g
  • Evaluation
    Based on the equation, the 64GB model should cost 529 + 6.25(64), or $929, but it actually costs $829.
    Therefore, iPad pricing isn’t linear.

Of course, this emphasis on context does not mean that math classes should revolve entirely around real-world problems such as Wheel of Fortune, batting averages and the iPad. At its extreme, this would be just as limiting as rote-procedure, albeit in the opposite direction.

Instead, effective math instruction involves a three-step process:

1) contextualize a problem to explore a skill ($/gig)
2) generalize the skill (change in y due when x increases by 1)
3) apply the skill to a wide-range of real-world topics (effect of music tempo on running time, marginal benefit of another piece of Halloween candy).

Unfortunately, teaching too often addresses only the second step. It’s understandable, then, why so many students construe mathematics as an arbitrary collection of meaningless steps; why so many ask why they have to learn it; and why so many absolutely hate it.

Fine. This all sounds good. But don’t teachers already feel overwhelmed by the demands of teaching? Won’t this approach take three times as long? Doesn’t the author get that I have to cover this material before the end-of-year test?

These are legitimate questions. Fortunately, addressing the meaning behind and applications of mathematics has a strange effect: it actually saves time, and allows teachers to cover more material in more depth, and with better results.

The earlier question about saving money at a store? In the “spend $50” approach, we implicitly addressed months of instruction: common multiples; the lowest common multiple; equivalent fractions; simplifying fractions; and ratios & proportions. We then extended this to percents with the “out of 100” step, and could have easily included decimals by asking, How much do we save for every one dollar that we spend?

Likewise, the iPad example addressed most of the topics surrounding linear functions. Yet were any of the steps arbitrary? Was there any place where students would have asked, What does this mean?, or When will I use this?

Of course, this is not to say that rote practice does not have its place in math education. But for students like me, the practice comes after the fundamental questions are answered. The practice is not a substitute for learning procedure or a replacement for understanding. But, practice, after-school tutoring programs and drills-based YouTube videos play a more effective role after, “What does this mean? When will I use it?”has been answered.

In the end, true innovation and lasting progress in math education will come not by repackaging or rebranding methodologies, but by emphasizing meaningful and intentional instruction. And this requires math teach-ers, not simply math do-ers. Once a context is set for the videos in the Frontline special their on-demand nature has a more valuable place.

At its heart math is simple. We would do well to pull back the curtain and remember that.

Math is a tool. It’s a tool that we created—that we continue to create—to make sense of the world. And in our efforts to guide students through math, we can’t ignore the world. We can’t ignore the sense. To do so is to ignore mathematics itself.

I imagine some math teachers, many who grew up just accepting the idea that it was okay to teach math without answering these questions, may agree with this philosophy but feel it would be unrealistic to expect them to be able to answer these questions for all math concepts. Furthermore, they already have a curriculum to follow, standards to meet, and a textbook they use. How could one begin to teach this way???

There is help in a site featured in the New York Times’ Freakonomics blog this month in a post called, “Making Math More Appetizing” The blog explains the site as follows: Mathalicious provides free math lessons, including supporting materials, for teachers and parents. The organization hopes to “transform the way math is taught and learned by focusing not only on skills but on the real-world applications of math, from sports to politics to video games to exercise.” So far, they’ve used the Pythagorean Theorem to determine how big a 42-inch TV really is; used percentages to examine environmental issues; and asked whether music can kill you.

The site is broken down in two ways:


Middle School Math
Algebra II


Fractions, Decimals, Percents
Functions & Equations
Graphing & Plotting
Number Sense & Operations
Probability & Statistics
Ratios & Proportions
Shapes & Measurement

The lessons are written in an ease-to-use, teacher-friendly format which makes sense since the site founder was a public school math teacher and later a math coach who worked with teachers to improve instruction by teaching for conceptual understanding and relevance. This site helps provide the answer for students like me who were hungry for the answer to the questions “What does this mean? When will I use it?” before being able to consume an out-of-context lesson. With a foundation like this, kids may find they no longer need tutors, and the question of "when will I use this?" will be a thing of the past.

This post was written jointly by Lisa Nielsen, The Innovative Educator and Karim Kai Logue, the founder and CEO of Mathalicious, which creates meaningful and real-world math content for parents, students and teachers.


  1. What is an example of a textbook that teaches math using a traditional approach? Every textbook that I have seen gives explanations and applications for concepts such as slope and percent, not just procedures.

  2. I love the idea of connecting math to real world applications, in fact, I find that merely discussing how and why certain concepts were studied is sufficient to get students interested. I recently related the development of the trigonometric functions to my students, and how the mathematician who named them was probably merely comparing 1000s of triangles and noticed a pattern. They were engaged and fascinated. I poked around the Pythagorean TV lesson and thought it was problem though, its estimated to take 4 45 minute periods - I have one period of time to teach the theorem and its converse. Its tough to be this creative in that amount of time.