Math education is a leaky boat that is on fire 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 get us where we want to go — we have to address all the problems simultaneously. Broadly speaking, there are three levels of problems: assembly-line teaching, the lack of meaningful projects, and the outdated choice of topics and tools.
Problem 1. Assembly-line teaching
The most urgent problem with math ed is how teaching is delivered. Math is taught as a death march through an endless series of details that must be learned in order. This one-size-fits-all conveyor belt approach to education virtually guarantees that students gradually accumulate holes in their knowledge. Khan Academy founder Sal Khan calls this Swiss cheese knowledge. And little holes in math knowledge cause big problems later on — problems in calculus are caused by problems in algebra, which are caused by problems with fractions and place value.
Here are three ways to fight the fire of poor pacing.
Self-paced learning. The Khan Academy addresses the problem of fixed pacing by providing a complete library of short video lectures. While the lectures themselves are rather traditional, the online delivery allows students to view lectures when and where they want, and to pause and rewatch sections as much as they need. An online dashboard shows teachers how far each child has progressed, allowing teachers to assign lectures as homework, and use class time to tutor kids one on one on exactly what they need — a format called the “flipped classroom.”
Multiple modalities. Teaching math by lecturing at a whiteboard makes as much sense as teaching music through notation. Most people can’t look at notation and hear music. For students to hear the music of mathematics, they need to experience it through all their senses. K-1 math educators do use stories, movement, and manipulatives to teach math concepts; this needs to continue through all grades.
Testing for understanding. Nothing can change in education unless testing changes. Traditional standardized tests assess only rote memorization of math facts and procedures. The drive to teach to these tests is what compels teachers to race through the material without taking time to bring it alive. A good alternative is portfolio assessment — students demonstrate understanding by showing a collection of projects they’ve done.
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 2. Lack of meaning
Traditional mathematics education focuses on teaching rote computational procedures — adding, dividing, solving quadratic equations, graphing formulas, and so on — without tying procedures to meaningful applications. 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, integrated with other subject areas.
Warren Robinett: 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. 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 using math, so we can put math in a personal emotional context, and get excited about participating in this human activity. But wouldn’t reading about people doing math be deadly boring? Not if you are a good storyteller. Math 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 young daughter as to my adult friends.
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. Identifying the right question often takes more effort than answering it. So we need to encourage kids to ask their own questions. Mathematical free-spirit Marion Walter wrote a book on this called Problem Posing. For instance, some teachers have students make up test questions. Students invariably invent much harder questions than the teacher would, 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 be creative with how they present puzzles. Some focus on art. Some focus 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 topics?
Problem 3. Outdated choice of topics
The mathematics we teach in school is embarrassingly out of date. Our curriculum is essentially the same as what was taught during the industrial revolution, when calculations were carried out by hand. 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 the topics we teach as mathematics.
Re-evaluate topics. Do kids really need to learn how to factor polynomials? Not really. That’s why math standards are starting to change what topics they include, devoting more time to topics like data collection and statistics. Kids still need to be fluent in mental arithmetic, but we need to teach skills appropriate for a world in which digital tools exist.
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 to teach a similar program for the process of doing mathematics. The full process starts with asking questions. Math teacher Dan Meyer argues in his famous 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.
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 spend most of their time teaching calculating by hand — the one thing that computers do 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 math. And students can work on interesting real-world problems that are impractical to tackle by hand. Here is a diagram from Conrad Wolfram’s initiative computerbasedmath.org comparing traditional math ed with a more balanced approach.
Those are the three biggest problems of math education. Of course there are other problems, like lack of training and funding. My point is that there are many problems to fix in math education, and that solving just one will not get us where we want to go.