In a previous blog post I wrote about the myriad problems that math education faces. So what about solutions? Where can we find I model of how what good math education could look like?
I say we look to language education. If you want to know how solve a problem in math education, look at how the analogous problem is solved in language education. Of course language education has problems too — many kids, especially boys, stop reading as they grow older — but language education is fundamentally in better shape than math education.
Here is a list of six best practices from language education that suggest revolutionary changes in math education.
1. Universal literacy
It all starts with societal attitude — we as a society expect that all kids will achieve a basic level of language literacy, including both reading and writing, but we do not expect the same of math. Far from it. It is almost a badge of honor in the United States to say you were never good at math. Even worse, parents and elementary school teachers are often the ones who pass on the message that it is okay to be no good at math, especially if you are female or a minority. If we are to improve the state of math education in the US, this attitude must change.
The statistics tell a similar story. Far more kids in the US achieve basic proficiency in reading than in math — 37% vs. 25% for US 12th graders, according to the Nation's Report Card. Science is worse, at 12%, writing is on par with math at 27%, and none of these statistics is where it should be.
2. Support for catching up
My sister in law, a teacher, recently told me that her former principal is now tutoring 3rd grade math. Why? Because she knows that if kids fall behind in understanding concepts like fractions, they are very likely to be stuck in lower-track math, severely limiting their college and job options — a terrifying situation, especially for students who can't afford tutoring. The problem here is that we label kids very young ("fixed mindset"), instead of assuming that all kids can achieve math literacy ("growth mindset"), and provide support for all kids to get there.
We can do better. Remember that widespread literacy is historically unusual. In the middle ages, few people received advanced education, and if you needed something written, you hired a scribe. Countries like the United States made a decision to provide free education to all citizens, with the belief that literacy was essential for democracy. And so we created an expectation of literacy, with support systems outside of school like reading programs in libraries and encouragement for parents to read to their kids at bed time. What would that look like for math? Look no further than bedtimemath.org.
3. Shift from mechanics to meaning
Somewhere around third grade language education shifts from mechanics (recognizing letters, sounding out words, spelling, grammar) to meaning (reading stories, comprehending essays, writing compelling arguments. The emphasis shifts from "learning to read" to "reading to learn". The assumption here is that once kids are fluent with a certain level of mechanics, they have achieved a basic level of competency. Kids continue to learn mechanics, but education broadens to include reading and writing.
That shift never happens in math class. Kids learn the mechanics of arithmetic, then move on to the mechanics of algebra, functions, geometry, trigonometry, etc., ("learning to compute") without ever using math as a tool to do other things ("computing to learn"). Math remains an abstract exercise in getting the right answer, without understanding what it means. No wonder the most common complaint from math students is "when are we ever going to use this?"
Those fortunate to take physics (calculus), cooking (fractions) or engineering (algebraic thinking) do find that the math they are learning has real meaningful applications. My wife Amy Jo learned to value math (logarithms) when she learned in graduate school that it was the key to understanding human perception (the human visual and auditory systems perceive ratios, not absolute values).
In order to change math education so it too shifts from mechanics to meaning, we would have to do two (radical) things: 1. teach the underlying universal concepts of mathematical thinking that apply to all areas of math, and 2. define what concepts need to be mastered in order to achieve a basic level of math literacy. (I will drill down on these big ideas in a future blog post.) I expect that this shift would occur a bit later in math than in language, perhaps around 6th grade, instead of 3rd grade, and it would occur at different ages for different kids.
4. Read what you're interested in
Once kids are able to read, the entire library becomes available. Kids are encouraged to read what they are interested in.
In contrast, conventional math education is structured as a linear sequence of one year specializations: algebra, geometry, algebra 2, trigonometry, calculus, etc. It's like climbing a narrow path up the side of a mountain. Fail any one class, and you fall off the path, ending your math studies for life. Only the few who climb beyond calculus reach the meadow at the top of the path, where they can choose where they want to go next.
While it is true that some areas of math should be studied in a strict sequence, the road to mathematics is far less linear than school makes it seem. There is no reason students in grade school shouldn't get a taste of geometry, algebra, trigonometry, statistics and calculus. And there are good reasons mathematics should be studied as an integrated subject, with all the strands integrated. In fact, most countries in the world teach "integrated math" in high school. In this scheme, the course sequence is called simply math I, math II, and math III, and every year includes bits of algebra, trigonometry, statistics, and geometry.
Linearizing the study of mathematics creates the false and demoralizing impression that math is an endless meaningless march toward a goal that few will ever reach. If language education were structured in such a linear fashion, then you might read biographies one year, plays the next, then drama, with the belief that you must read books in this sequence, and once you fall of the path, the door to the rest of literature is forever shut.
5. A rich children's literature
Imagine a world in which kids learned to recognize letters, read words, and decipher sentences, but never read a book, or even knew that books existed. Kids might get good at spelling, and enjoy conjugating verbs, but few would fall in love with the power of words to transport us to imaginary worlds.
Now realize that is exactly how mathematics is taught. Math too has a rich, dramatic, and imaginative literature, but kids in school never see it, or even know it exists. The literature of math includes stories about problem solving, games and puzzles, biographies and histories, and great theorems and discoveries. Math textbooks give lip service to math literature through word problems, but make no mistake, solving a word problem is to real world mathematics as analyzing an isolated sentence is to reading a captivating novel.
The solution? Expose kids to the literature of mathematics. Have kids play math games and do math art projects. Create a rich children's literature of math stories that teach kids problem solving skills,. Tell the stories of the people, places, and events from the history of math. Put emotion and humanity back into math, and make it a social experience. Wondering where to start? Try Mathical, an annual prize for the best kid's math story books "from tots to teens."
The next step after reading is writing. We accept that all kids learn writing skills, not because all kids will become published authors, but because all people need basic writing skills to function in their lives, and because writing deepens the experience of reading.
For math, the next step after problem solving is problem posing — asking your own mathematical questions without necessarily knowing what the answers will be. All kids should learn this skill, not because they will become research mathematicians, but because all people need basic problem posing skills to function in their lives, and because problem posing deepens the experience of problem solving.
Problem posing is almost completely missing from mathematics education in the US. Yet in my own work I have seen that when kids are given the chance to ask their own questions, they respond with tremendous enthusiasm, far beyond anything else I have seen in math class. It is as if they have been waiting for an opportunity to inject their own creativity into mathematics.
Here are a few thoughts on how to teach problem posing:
Have kids invent their own test questions for whatever topic they are currently studying. Kids typically invent much harder problems than teachers would, and are more motivated to solve problems if they know they were invented by classmates. Furthermore, inventing test questions makes students reflect on what skill is being taught, and what a challenging question would be.
Have kids play games and puzzles, and then invent their own. Start by having kids alter the rules of familiar games, before going on to do more original work. When I give students the assignment to invent their own puzzles, most students immediately have an idea of what they want to do. It is as if they had been waiting for someone to give them permission to invent. I do light coaching. Mostly I let kids critique each other. And I publish a book of the finished puzzles for them to show to friends and family — publication is a strong motivation to do a good job.
Create a classroom culture that encourages kids to ask questions. Kids are naturally curious about the world around them, and that includes math. What is the largest number? What is rounder than circle. As a teacher, you don't have to know the answers. Give kids a chance to voice their questions, and spend time wondering. You'll create a culture of self-motivated researchers.