“Schools are government-run and their employees are unionized; how do you expect to get anything to change?”
-Evan Miller
Concerns about math games
Explorations in Math is an interesting Seattle organization that I have been learning about recently. I went to their volunteer training last week and have been discussing my thoughts with family and friends ever since. Those conversations have hugely helped to clarify my concerns. The conclusions deserve to be written down.
This is the draft of a letter to Explorations in Math:
What I Like
That’s easy — I fully share your mission to help all elementary students succeed in math. I think this is one of the most important goals there is. You are clearly making a difference and changing attitudes and that is awesome.
Concerns
These concerns are largely spurred by books I have read about education psychology. They make me nervous that some aspects of Explorations in Math’s approach could be counterproductive.
1. Losing games may perpetuate students’ assumptions that they are bad at math.
Most of the math games have winners and losers. If a student consistently loses the games, doesn’t that reinforce the notion that they are bad at math? Weaker students can be paired with other weaker students so that they do not always lose. But kids know what is going on. They know they are in the “dumb group”. Even if they still enjoy playing the games, do they gain any confidence in their math abilities? One of the success stories that Explorations in Math gives is the observation that kids start playing math games of their own accord while waiting in lines. Are all of the kids playing? Or only the ones who consistently win?
2. Sugar-coating may perpetuate the idea that real math is hard and boring.
I don’t doubt that games are a good way to teach and learn some math concepts. But I worry that it sends the underlying message that math needs to be sugar-coated in order to be fun. After all, we don’t need to resort to games to teach reading. Instead, we give students reading materials of increasing difficulty level, with content that is interesting to them, and help them progress. We can do the same in math with well-designed curricula, teaching students how to solve math problems on topics that interest them, at gradually increasing levels of difficulty as they master each concept.
3. Arithmetic is one of the least interesting parts of math.
Most of the math games center around mental math and memorizing multiplication tables. These are important building blocks. But the really interesting aspect of math is the ability to answer a wide range of questions with just a few simple tools (such as fractions and algebra). Mental math helps students understand these tools by allowing them to solve problems quickly, without a calculator. But in the real world, we use calculators and computers to do the calculations, and the important skill is being able to translate everyday problems into math problems. The further you progress in math, the more uncommon it is to see actual numbers. Instead, you work to understand abstractions and learn how to model the world and make precise predictions.
4. I don’t see any hard evidence of results.
The Explorations in Math website includes great testimonials and anecdotes. But have math test scores improved in the schools you have worked with? In particular, have scores gone up among disadvantaged and historically weaker students? I’m as disillusioned with “teaching to the test” as anyone, but the tests do measure students’ math ability. If you are improving attitudes but having no effect on test scores, what has really been accomplished? The recent documentary Waiting for Superman points out that American students score among the lowest of developed countries in math scores, but score higher than anyone in confidence. Confidence is only useful if it is backed by reality.
It may be that the real problem is that all of our hands are tied by the school board’s mandated, poor choices of math curricula. Is Explorations in Math designed to be a side-run around that very stubborn obstacle? Is it designed as the best approach given the constraints?
What motivates a child?
That seems like a simple enough question. If you asked a selection of teachers, parents, and psychologists to answer it, I wonder how their answers would differ from each other. Is it a solved problem? Or a complex mystery on the frontier of science? Or somewhere in between?
The myth of ability
“Historically, societies have always been divided by myths of difference: between peasants and nobility, slaves and slave owners, or minorities and majorities. Today, the most pervasive and enduring of those myths — the myth of ability — is being challenged.”
-John Mighton (The Myth of Ability)
There are no excuses for letting children fail
John Mighton writes in The Myth of Ability:
As far as I am aware, no program in mathematics was ever developed with the expectation that every child in the program would excel. To most educators, the idea of an entire class doing well in any subject seems absurd.
That was published in 2003, but today it is still widely assumed that the “curve” in test scores is a “natural” result of innate differences in human intelligence.
Through a tutoring program that he later extended to classroom teaching, Mighton showed that in fact every child could excel. And he found that such a result requires only two essential ingredients (which match my definition of profound: obvious only in retrospect). They are:
1. The teacher must actually believe that every student can excel.
2. The curriculum and teaching methods must be designed, tested, and refined in a way that treats any student’s failure to learn as being a failure of the curriculum and teaching methods.
This is entirely analogous to creating usable software. Instead of blaming problems on “user error,” you blame the software — and, critically, use that knowledge as an opportunity to improve the design. Creating great software is not easy, but it’s also not rocket science.
Mighton has experimented with various best practices in teaching, many of which have been well documented elsewhere in the psychology literature (and popular literature by authors such as the Heath brothers, Malcom Gladwell, Carol Dweck, Martin Seligman, etc.). For example, he neatly sums up the research on the importance of flow:
Nothing focuses the attention of children more sharply than the feeling that they are meeting a series of challenges and succeeding brilliantly.
But the crucial thing that John Mighton has shown is not in the particulars of his curriculum or philosophy. Rather, it is the simple, incontrovertible fact that he made every single student succeed in math — without requiring extra money or super-human energy or even teachers who previously knew anything about math.
It is an existence proof.
And it means that all of the excuses are bogus. “They can’t focus.” “They don’t care.” “There’s not enough time or money to get through to them.” Every student brought to him as “unteachable” was in fact taught to excel. And it was not even particularly difficult. It was certainly not rocket science.
But if we know how to teach in such a way that every child succeeds, why are we not doing it? Mighton says,
I believe the answer lies in the profound inertia of human thought: when an entire society believes something is impossible, it suppresses, by its very way of life, the evidence that would contradict that belief.
I think it’s harder than that, and a good analogy is racism or sexism. When injustice is ingrained — when “that’s the way it has always been” — elaborate excuses and rationales must be crafted to avoid the conclusion that well-meaning people are perpetuating discrimination (in this case, discrimination against the very students they purport to help). It’s a terrible conclusion to come to. To accept it means admitting that for decades we have been undermining the potential of millions of eager young students.
After seeing how children flourish with even a modest amount of attention, I have come to believe that when a child fails a test it should be regarded as a failure of our system of education. And when millions of children, year after year, fail tests they could easily pass, it should be regarded as the failure of an entire society to care for its young.
Supporting change implies acceptance of this terrible conclusion: that for generations we have been letting students fail, letting poverty persist, and letting the economy correspondingly sink — and that there is no excuse for it. How can we live with that guilt? Especially as a student or teacher in the system that is perpetuating the injustice — someone who has the power to make change?
I think that Dweck’s “growth mindset” is a good place to start. It helps us accept the truth as a learning opportunity to do better.
Because the fact is, we already know how to do better. And putting it off is only making the situation worse. As Matt Wilka and I concluded, “just do it!”
When students realize who school is for
“When teachers are judging them, students will sabotage the teacher by not trying. But when students understand that school is for them—a way for them to grow their minds—they do not insist on sabotaging themselves.
“In my work, I have seen tough guys shed tears when they realize they can become smarter. It’s common for students to… adopt an air of indifference, but we make a mistake if we think any student stops caring.”
-Carol Dweck (Mindset, p.201)
Math interfaces
“Mathematics, as currently practiced, is a command line. We need a better interface.”
-Bret Victor [link]
Deciding that a department is for everyone
Why do so many Williams students major in mathematics? The Williams Alumni Review magazine gives this answer:
The mission of the math department had long been to identify and educate the most talented students, which meant the College graduated about a dozen math majors each year. But new department chair Frank Morgan and some of his colleagues contemplated a more inclusive view of the discipline…. “Everybody deserves a chance to do this,” Morgan says. “It’s like music—people should have a chance to enjoy math.”
Today the reconstituted math department graduates five times as many majors… a third of them women. More than half of all Williams undergraduates complete multivariable calculus [and introductory statistics]. Most impressive of all, 12 percent of the College’s graduates major in mathematics at a time when… the national average hovers around 1 percent.
Has [this] led to a dumbing down of the discipline? There’s much evidence to the contrary. [Professors from elsewhere call the department] “unquestionably the best teacher-scholar math department in the country.”
In other words, the department did not become popular by chance or by working harder at it than other departments. Rather, it made a decision to become a popular department rather than a selective department. The whole design of the curriculum and staffing is different when popularity rather than selectivity is your goal.
Is this related to Dweck’s growth mindset? Does the belief that all students can enjoy and pursue math make it more likely that they will?
The next question is: why have so many departments not made this decision, choosing instead to continue to prioritize selectivity? Shouldn’t everyone also have a chance to enjoy physics and anthropology and comparative literature?
Are academicians too focused on being “serious”? Do they take the fixed mindset, believing that only people with the right “talent” and “drive” can succeed in their field of study?
Ability is learned, not fixed
“After forty years of intensive research on school learning in the United States as well as abroad, my major conclusion is: What any person in the world can learn, almost all persons can learn, if provided with the appropriate prior and current conditions of learning.”
-Benjamin Bloom (via Mindset by Carol Dweck)
The pieces for educational software
Who do you need in order to make outstanding educational software?
- Artistic and child psychology experts from children’s TV
- Curriculum designers and subject experts from textbooks
- Interaction designers and programmers from computer games and apps
- Teachers who know the kids and can test prototypes with them
What pieces do you need to bring together?
- Psychological: “flow”, storyline, context
- Aesthetic: beautiful, interesting, simple
- Emotional: for a purpose
- Reliable: solid programming
- Convenient: internet, app store
- Low-cost: software product that runs on widely-available devices
- Effective: learning goals are met
Once the disruption in education takes hold, software with all of these pieces will prove very popular.