Imagine that you are a sailor on a leaky boat that is on fire, and sailing in the wrong direction. Which problem would you fix first?
Well, that depends. If the leak is slow and the fire is raging, then you put out the fire first. If the leak is gushing and the fire is small and contained, you would fix the leak first. It makes sense to fix the most urgent problem first.
What you would NOT do is fix one problem and declare victory. If your goal is to get to your destination safely, then you must fix ALL the problems, no matter how difficult. Anything less will not get you where you want to go.
Such is the situation with math education. The problems are so deep and so numerous that it is tempting to fix one problem, and give up on the rest. And certainly we have to prioritize if we are to make progress. But if we are to get the ship of math education back on course, we, collectively, must fix ALL the problems of math education. Nothing less will get us where we want to go.
Fixing all of math education may sound impossible or impractical. And indeed it is a formidable challenge. Well-meaning entrepreneurs who have launched successful businesses frequently grind to a halt when they try to start their own innovative schools. Resistance comes from all sides — standardized testing, textbook publishers, parents, admiinistrators, government officials, and the students themselves trying to get into college.
But change is in the wind. America is losing its competitive edge, math and STEM education are struggling all over the world, colleges are becoming impractically expensive, and the internet makes us dream of free education right now for everyone. I say we face the problem with eyes wide open, assess the full range of challenges we face, and look for the smartest moves that get us where we want to go.
With that in mind, here is my survey of the problems plaguing math education, and steps we can take to fix them. I've grouped the challenges into three problem sets that range from tactical to strategic: Mechanics, Meaning, and the Math itself.
Problem Set 1. Faulty MECHANICS (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.
1a. 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.
Solution: the "flipped classroom.”
1b. Visual learning. I love the Kahn Academy. My high school aged son hates it, because he, like many 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. Other kids are primarily audio or kinesthetic learners. Education needs to address all learners, not just kids who learn in words.
Solution: teach every lesson three different ways.
1c. 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, now entering schools across the nation, replaces standardized multiple choice tests with richer tests that include essay questions graded by human beings — a better way to assess mathematical understanding.
Solution: better assessment.
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.
Problem Set 2. Lack of 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 situations. Unfortunately most adults, including many teachers and administrators, think this is how it must be. But teaching only the rote procedures of math is like teaching only the grammar and spelling of English, without explaining what words mean, and 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.
2a. 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.”
Solution: use problems that kids care about, and excite student interest.
2b. 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.
Solution: read good stories about math in use.
2c. Ask 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. One 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.
Solution: Give kids freedom to ask their own mathematical questions, and pursue their natural curiosity.
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?
Problem Set 3. The wrong MATH (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.
3a. 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.
Solution: give kids an overview of mathematical topics and what they are for, long before they have to study them formally.
3b. 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.
Solution: give kids an explicit process model for problem solving.
3c. 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.
Solution: build and use better computer tools for doing math. Revamp the curriculum to assume the presence of such tools. Emphasize solving interesting problems, de-emphasize or delay learning about the mathematical mechanics for carrying out the computations. In other words, teach mechanics on a need-to-know basis.
So there you have it, my assessment of the problems in math education. Certainly I've left out many important practical problems, like teacher training and funding. I've chosen to focus on the problems that I think matter most.
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 together on all fronts.