Math education is a leaky boat that is engulfed in flames and sailing in the wrong direction. There is so much wrong that it is hard to know where to start. And fixing just one problem won't save the ship — we have to address all the problems simultaneously.
Now that I've got your attention, let's unpack the nautical analogy and see what we need to do to save the ship of math education. My goal is to lay out the full range of problems in math education, so we can decide where to act.
Broadly speaking, there are three problems with math education, which I equate with fire, leaks, and sailing the wrong direction. Here are the problems, and ways to fix them.
1. Poor pacing (fire). The most obvious and urgent problem is that the mechanics of math are taught as a series of blink and you'll miss it lessons, with little opportunity to catch up. This one-size-fits-all conveyor belt approach to education guarantees that virtually everyone gradually accumulates holes in their knowledge — what Khan Academy founder Sal Khan calls Swiss cheese knowledge. And little holes in math knowledge cause big problems later on — problems in calculus are often caused by problems in algebra, which in turn are caused by even earlier problems with concepts like fractions and place value.
Here are three ways to fight the fire of poor pacing.
> Self-paced learning. The Khan Academy addresses the urgent problem of pacing by providing short video lectures that cover all of K-12 math. While the lectures themselves are rather traditional, the online delivery mechanism allows students to work at their own pace — to view lectures when and where they want, and to pause and rewatch sections as much as they need. All lectures are available at all times, so kids can review earlier concepts, or zoom ahead to more advanced concepts. Short online quizzes make sure that kids understand what they are watching. And with an online dashboard that shows exactly how far each child has progressed, teachers can assign lectures as homework, and use class time to tutor kids one on one on exactly what they need — a reversal now known as the "flipped classroom.”
> Visual learning. I love the Kahn Academy. My high school aged son hates it, because he, like most students, is a visual learner, and Sal Kahn's lecture stick largely to traditional symbolic math notation. Mind Research is a nonprofit co-founded by a dyslexic learner, Matthew Peterson, who sought to teach math without words. Mind Research now produces a full K-12 math curriculum that communicates range of math concepts through wordless games (yes algebra can be done without traditional notation). Not only do the games reach visual learners, the very lack of words causes students to want to talk about their experiences with each other, thus deepening their understanding. Math education needs to address all learners, not just kids who learn in words.
> Testing for understanding. Nothing can change in education unless testing changes. Traditional standardized tests born of the No Child Left Behind era use multiple choice tests that assess only rote memorization of routine math facts and procedures. The new Common Core State Standards for mathematics, just now entering schools across the nation, boldly replaces standardized multiple choice tests with richer tests that include essay questions graded by human beings — a better way to assess mathematical understanding.
If we douse the fire of poor pacing in math education, we will increase test scores and student confidence. But there is more to mathematics than teaching the mechanics well.
2. No meaning (leaks). Traditional mathematics education focuses on teaching rote computational procedures — adding, dividing, solving quadratic equations, graphing formulas, and so on — without tying procedures to meaningful problems. Teaching math this way is like teaching the grammar and spelling of English without bothering to teach the meanings of words, or letting kids read books. No wonder the most common complaint in math class is “when are we ever going to use this?”
Here are three ways to plug the leaks of meaningless math.
> Use math. In our increasingly digital society, kids spend less and less time playing with actual physical stuff. All the more reason to get students out of their desks and into the world, where they can encounter math in its natural habitat, preferably integrated with other subject areas. My friend Warren Robinett told me “a middle-school teacher I knew would, after teaching the Pythagorean Theorem, take the kids out to the gym, and measure the length and width of the basketball court with a tape measure. Then they would go back to the classroom and predict the length of the diagonal. Then they would go back to the gym, and measure the actual diagonal length. She said some of the kids would look at her, open-mouthed, like she was a sorceress.”
> Read about math. Before we learn to speak, we listen to people speak. Before we learn to write, we read books. Before we play sports, we see athletes play sports. The same should apply to math. Before we do math ourselves, we should watch and read about other people doing math, so we can put math in a personal emotional context, and know what the experience of doing math is like. But wouldn't reading about people doing math be deadly boring? Not if you are a good story teller. After all mathematics has a mythic power that weaves itself into ancient tales like Theseus and the Minotaur. My favorite recent math movie is a retelling of the classic math fable Flatland, which appeals as much to my 7 year old daughter as to my adult friends. Here's a list of good children's books that involve math.
> Create your own questions. In math class (and much of school) we answer questions that someone else made up. In real life questions aren't handed to us. We often need to spend much time identifying the right question. The simplest way to have students ask their own questions is to have them make up their own test questions for each other. Students invariably invent much harder questions than the teacher would dare pose, and are far more motivated to answer questions invented by classmates than questions written by anonymous textbook committees. Mathfair.com goes further to propose that kids build and present their own physical puzzles in a science-fair-like setting. Kids can apply whatever level of creativity they want. Some focus on art. Some on story. Others add new variations to the puzzles or invent their own.
If we plug the leaks of meaningless math, we will grow a generation of resourceful mathematicians who understand how to solve problems. But are we teaching the right mathematics?
3. Mathematics itself (sailing in the wrong direction). The mathematics we teach in school is embarrassingly out of date. The geometry we teach is still closely based on Euclid's Elements, which is over 2000 years old. We continue to teach calculus even though in practice calculus problems are solved by computer programs. Don't get me wrong: geometry and calculus are wonderful subjects, and it is important to understand the principles of both. But we need to re-evaluate what is important to teach in light of today's priorities and technologies.
Here are three ways to update what we teach as mathematics.
> Re-evaluate topics. The Common Core State Standards take small but important steps toward rebalancing what topics are taught in math. Gone are arcane topics like factoring polynomials. Instead, real world mathematics like data collection and statistics are given more attention. As Arthur Benjamin argues in a brief TED talk, statistics is more important than calculus as a practical skill.
> Teach process. The widely used Writer's Workshop program teaches the full process of writing to students as young as kindergarten. The process accurately mirrors what real writers do, including searching for a topic, and revising a story based on critique. We need a similar program for the process of doing mathematics. The full process of doing math starts with asking questions. Math teacher Dan Meyer argues passionately in his TED talk that we do students a terrible disservice when we hand them problems with ready-made templates for solution procedures, instead of letting them wrestle with the questions themselves. Here is my diagram for the four steps of doing math. Conrad Wolfram created a similar diagram for his Computer-Based Math initiative.
> Use computers. In an era where everyone has access 24/7 to digital devices, it is insane to teach math as if those devices didn't exist. In his TED talk, Conrad Wolfram points out that traditional math teachers spends most of their time teaching calculating by hand — the one thing that computers do really well. By letting students use mathematical power tools like Mathematica and Wolfram Alpha, teachers can spend more time teaching kids how to ask good questions, build mathematical models, verify their answers, and debug their analysis — the real work of doing mathematics. And students can work on interesting real-world problems, like analyzing trends in census data, that are impractical to tackle by hand.
So there you have it, my assessment of the problems in math education. I've left out many important practical problems, like teacher training and funding. My point is that there are many problems to fix in math education, and that solving just one of these problems will not get us where we want to go. Let's be aware of all the problems, and move forward on all fronts together.