the first post in a series
(see also parts 2, 3, and 4)
“I’m planning a series of posts,” I told my dad the other day as he drove me home from the airport. “The title is How to Avoid Thinking in Math Class.”
Before I could get any further, he rubber-stamped the idea. “That sounds great! I always tell people, the point of school is to help you not to think.”
It’s a good thing he was the one behind the wheel, because if it were me, I’d have slammed the brakes and spat my latte all over the windshield.
“Really?” I sputtered. “I meant the exact opposite. I was going to catalogue all the silly ways that kids avoid doing real math. You know: blindly carrying out operations; hating word problems; worshipping right answers; focusing on procedures instead of ideas. I was going to take inventory of all those cognitive shortcuts and how they impede learning.”
“Well, sure,” he said. “But they can’t think all the time.”
“Well, isn’t thinking the whole point of math education?” I countered. “Math class is like a gymnasium for thought. You work out daily with a trainer who knows how to stretch your limits. You watch your mental muscles build. You see the payoff of intellectual risk. You learn that just as people can grow faster and more agile, they can grow smarter, too.”
“Yeah,” he mused, “that’s part of it. But that’s not the whole thing.”
“Why not?”
“Because thinking is hard.”
I had to admit, I knew what he meant. Just a few hours before, on the plane, I’d been reading a book when I realized that my eyes were just gliding over the words in a mindless imitation of reading. Meanwhile, my attention roamed and drifted. I wondered which sandwiches can and cannot be improved by bacon.
“Sure, sometimes we focus deeply,” my dad continued as he absentmindedly accelerated to make a yellow light. “But our natural state is autopilot.
“It’s like he says in that book,” my dad added. This is my dad’s usual level of specificity; he once described his favorite television character as, and I quote: “Oh, you know. Him. The guy.” Luckily, this time I knew which book he meant.
“Thinking, Fast and Slow,” I said.
“Right. By Khina… Khena…”
“Daniel Kahneman,” I said. My dad nodded.
Kahneman (a psychologist and Nobel laureate) describes human thought as governed by two systems. The first is fast and automatic. It employs shortcuts, heuristics, and dogmas—quick-and-dirty rules that save our mental strength. The second system is slow and deliberate. It musters focus, frames scenarios, tests assumptions, and reconciles contradictions—all those cognitive activities that exhaust students and excite teachers.
“We can only do that deep thinking on special occasions,” my dad said. “It’s just too tiring to focus intensely all the time. It’d be like sprinting everywhere instead of walking.
“That’s why the goal of school has to be automaticity,” my dad concluded. The Sunday morning roads were empty, and we’d nearly made it home. “Take learning your times tables. You’ve got to know them cold so that you can go on to finding common denominators, or reasoning about algebraic functions, or whatever. You need each task to become automatic before you can move onto the next intellectual step.”
“I guess you’re right,” I said. “But how do you reach automaticity? Isn’t the only way to get there by thinking deeply? Don’t you need to wrestle with the ideas before you can make them automatic?”
He drove quietly for a moment. “Yeah.”
“So then,” I said, “you’re saying that the purpose of school is to make you think hard now, so you won’t have to think later.”
(All this, of course, to allow you to think even harder.)
“Yeah,” he said, signaling a turn and pulling into the garage. “That’s learning, pretty much.”
And that, in a nutshell, is what I hope to write about over the next month.
In teaching math, I’ve come across a whole taxonomy of insidious strategies for avoiding thinking. Albeit for understandable reasons, kids employ an arsenal of time-tested ways to short-circuit the learning process, to jump to right answers and good test scores without putting in the cognitive heavy lifting. I hope to classify and illustrate these academic maladies: their symptoms, their root causes, and (with any luck) their cures.
But, as my dad points out, it’s not always bad to avoid thinking. It’s often healthy, even necessary. So I also hope to highlight the good heuristics, the sensible shortcuts, and the wisdom of seeing math class as an effort to program your autopilot.
I hope to make posts on the following topics (subject to abrupt change, regional blackout, second thoughts, and personal whim):
- Number Smoothies. Throw the numbers in a blender, and press “puree.”
- Dish-Washing Robots. When thinking less means working more.
- Word Quarantine. Do math with no word problems! Then, paint with no pigments, and breathe with no air.
- Garden of Doubt. Watering the seeds of uncertainty.
- Sharpening the Axe. The wisdom of Abe Lincoln, math teacher.
- Little Waldens. As Thoreau advised: “Simplify, simplify, simplify!”
- Church of the Right Answer. Where do my students worship?
Yes, surely, maths, in a way, is thinking hard so that you no longer need to think hard. Except new things come along about which you have to think hard, and the things you previously thought hard about help you to think hard about the new things. Or something.
Yeah, I think that’s true. With practice, thinking hard becomes less… well, hard.
This is a really interesting post, so thank you for it.
The way I see learning like this is a layering of understanding. It takes hard thinking to grasp a concept, and to employ it against problems at first. Later it is ‘obvious’. But that doesn’t mean you stop thinking, just that your context stack no longer needs the loading of all that hard thinking to use that learning.
The car example is that when you first learn to drive, it is terrifying, thrilling, difficult and tiring, even to drive at 20 in a residential area. Later, you stop thinking about the driving, not so you can fall asleep but so that you can focus on the next layer up; the route you want to take, whether there are likely to be children or other dangers at this time of day, weather conditions and their effect on safety. Sure, you could switch off, stop thinking, or you can level up.
Perhaps it is a bit like a wonderful chip which has a lot of programmable hardware available, and a built-in programmer. So as time passes, it takes things it does a lot, and that are hard, and slowly builds up bits of hardware circuits that take the load off, thus freeing it to take on bigger and harder problems that would not otherwise fit in its capacity to calculate stuff.
Remember once upon a time, algebra was hard. All those rules you had to keep in your head, all those little tricks you had to try to remember when you were simplifying. These days that is all second nature, a second nature built from years of effort, and you look at an equation and see its structure, the elemental blocks that form it, and see how you might map it onto a problem you already have a handle or a proof for.
Layer on layer.
Mmmh, pancakes.
Thinking about this, with that picture it should be possible to forewarn students about how learning will *feel*; this will seem hard, and confusing, and like there’s a lot to remember, but that is just how it feels to be learning something.
It could mean that they discover the difference between understanding what the teacher is saying, and fully grasping the concept; you haven’t fully grasped it if you still have that awkward, slow thinking when you’re trying to use it.
The upside is that it is only temporary, and you can expect the slow awkwardness to turn into fluent immediacy with time and effort spent grappling with it.
First, on your first comment: Yes, exactly. Beautifully said. Driving is a good analogy for this basic developmental trajectory, though you could use any task. In basketball, you first learn to dribble, pass, and shoot; when that’s automatic, you can run simple offensive maneuvers (picks, screens, whatever); after that, you can learn whole schemes that employ these maneuvers in complex and flexible combinations.
Second: I find myself doing this fairly often, especially at the start of a big unit. “This feels weird and impossible now, but I promise you, in a year you’ll be doing this in your sleep.”
Quite interesting. Being a teen, I hope I can use one of your future ideas!
I hope so too!
Something tangentially related but still relevant: students’ overreliance upon memorization and how their choice to memorize rather than understand overwhelms them. Over a couple of decades I have witnessed students listing “formulas” and reciting procedural steps without trying to understand the underlying connections, and then they get angry that they have to remember so much. Time and time again I tell them that I remember very little; mostly I figure it out as needed. The majority of students conveniently forget that so they can later accuse me of making them remember too much. I think that this underlying difficulty is why students hate word problems; without the step of abstracting the words to math symbols, the problems don’t seem to conform to lower cognitive thinking. Two problems that look nearly identical to me can look totally different to the student who doesn’t want to learn.
Yeah, that’s all really true (and concisely said). I’m planning two separate posts on those exact issues: one on rigid procedural thinking, and one on word problems (and the particular challenges they pose for rigid, procedural thinkers).
Part of the issue here is the standardized tests students take. My daughter, who is now a math major in college, also remembers very little and derives it as needed. This makes her deliberate and good at solving math problems, but she routinely tests lower on mathematical ability because the standardized tests emphasize speed. So finding the right balance is really important – they need to memorize enough, but not too much.
since am a maths student myself, i can relate. Its quite an interesting post. Thanks for the innovative thought.
Thanks for reading – glad you can relate!
I’m just waiting for Microsoft Word to add automaticity and multisensory to its dictionary 😉
Problems that look nearly identical to me can look totally different to a student for lots of reasons. I’ve found that assuming the student wants to learn is important (especially for the student who doesn’t think s/he wants to), tho’ sometimes it is a desperate cause…
The connection of the language to the math concepts is usually horribly missing, and unfortunately too often attempts to make the connection are things like “write down why you did that,” and the student memorizes that part, too, and that’s accepted, and the gap between comprehension and formula copying widens.
When I read authors with assorted perspectives about math — David Berg, Robert Moses, Jo BOaler and Marilyn Burns are names that come to mind but I know there are more– generally tucked into their discussion about “what works” is something about spending time describing math in “learner terms” and then working with that to understand the (often at least a little deeper and more abstract) mathematical meaning and how it’s expressed. David Berg *really* works at building the automaticity of the connection in true “structured, systematic” direct instruction style; others are much less formal. I think the importance of that is highly underrated; I wonder whether “automaticity” is, in many brains, very nested in language. (I know it is for me; I learn even motor skills through using words to tell the body parts to do stuff… but my wiring is a tad atypical.)
1. Yeah, when “automaticity” gets the red squiggly, I’m like, “C’mon, MS Word, get with the nouns.”
2. It’s a discomforting truth that verbally intensive questions (“define,” “explain,” “discuss”) are often just as gameable as strictly numerical ones. Since I sometimes treat those questions as an antidote to rote/superficial thinking, your point is well-taken.
3. Funny – for me, slow and deliberate thought is VERY verbal and language-dependent, but quick, automatic thought is much less so. Sounds almost like the opposite of your inclinations. I think you’re right that the concept of “automaticity,” as I’ve introduced it here, needs fleshing out.
When I spy a student using a formula to answer the “explain” question (which is usually “find the sentence in the book that has the key words in the text and rephrase it”), I *try* to make the connection between what they copied and the math they’re doing. Sometimes, though, the gap is huge.
Once I’ve got students in the works again, I’ll be able to come up with examples of scaffolding for that process — “which concept is this solution an example of, and why?” … “why is that the right answer?” “Why couldn’t you just…?”
(When I think ‘automaticity,’ I hear “AUTOMATION!!” (which I*think* is from the Limeliters version of John Henry… and wish it were on YouTube to confirm…)
What sandwiches can’t be improved by bacon?
Pulled pork.
Also, arguably, grilled cheese. But if you throw tomato and jalapeno (or onion) in, then you’ve got a big net improvement.
I’m looking forward to this.
Thanks – the remaining post are all in various states of half-written gibberish, so we’ll see how it comes out!
This is interesting for me as a student, because I simultaneously employ tricks to get the right answer without thinking too much and hate them. Part of the problem, of course, is that school is so test/results-centric — nobody (other than the teacher and myself) cares if I learned Topic X as long as I scored well on the test about Topic X and get a good grade in the class that colleges can see. When you combine that kind of thinking with lots of tough classes, shortcuts become the only way you really manage to keep doing well.
Yeah, I think you’re exactly right. I know some of my students feel the same way: all things being equal, they’d like to learn cool things and think deeply, but their top priority is securing their academic and professional futures, and that sometimes stands opposed to genuine learning. So on behalf of the giant test-driven educational machine for which I work, and of which you are an (admittedly participating) victim: sorry about that.
I’d love to say, “Just focus on the learning! The rest will take care of itself!” But I know that’s only partly true. Colleges look at grades, grades depend on tests, and shortcuts help your test scores. I can’t fight that fact.
But I will say this: someday you’ll be done with this semi-dysfunctional system. You’ll be out in the world. You’ll want to do cool things – in academia, business, the arts, government, whatever. For that, what will matter is not your grades (no one will care by the time you’re 25) but your abilities.
Creativity. Grit. Ingenuity. Problem-solving. Your wealth of knowledge. Your curiosity, independence, and ability to learn new things.
Those skills can be developed. It just takes time and effort. So even though school doesn’t always reward those skills as much as it should, for your own sake, don’t wait until you’re 22 to start working on those. Start now!
College admissions is a zero-sum game. Your dream school has a limited number of spots, and for you to get in, someone else must get left out.
But the real, post-college world ISN’T zero-sum. Smart, curious, thoughtful, talented people go on to work on all kinds of problems–sometimes together, sometimes alone, but rarely in competition. The more of them there are, the cooler the world becomes, and the more we all benefit. There’s always space for more!
I never comment, but I will here: I’m really looking forward to reading these upcoming posts. I can see them already in my minds eye, writing themselves from some of my least favorite moments from my own class. It’s good to hear that you, too, face this particular challenge
Oh man – I sometimes feel like I face nothing BUT this challenge! I hope the posts live up to expectations, but please comment with your experiences when I slip up or leave gaps (as I inevitably will).
5th year teacher at a Waldorf high school where we have a lot of out-of-the-box, creative thinkers. I’ve noticed many of the brightest mathematical minds can be slow, make a lot of mistakes, and take more time than their peers, and so they aren’t always the traditional “A” students but they really own their math, and I try to praise their original work and encourage them to develop the wisdom and discernment to know when to rederive everything from first principles and when to make life easy on yourself and follow the procedure!
Wonderful Idea. Looking forward to the rest.
But – fulfilling my position as a pedant I must point that it is Kahneman and not Khaneman!
Thanks – fixed!
I’ve always been a fan of Knuth’s statement:
“Science is knowledge which we understand so well that we can teach it to a computer; and if we don’t fully understand something, it is an art to deal with it. … We should continually be striving to transform every art into a science: in the process, we advance the art. ”
“Things you don’t think about” seem to play a similar role to “science”.
Yeah, I like that, especially the last bit. Seems true that for every “art” (cooking, teaching, whatever), there are projects it can’t yet tackle, until the current tasks are boiled down to science.
I also like the connotations of “art” (inspiration, focus, directed creativity) in contrast to the more laborious-sounding “Kahneman’s System 2.”
I remembered running across this quote. Sounds like your father might approve.
“It is a profoundly erroneous truism repeated by all copybooks, and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of operations that we can perform without thinking about them. Operations of thought are like cavalry charges in a battle–they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
A.N. Whitehead
Ah, that last line is just gorgeous. ‘Fresh horses’ is exactly the resource I feel I’m often lacking.
I had really admired the posts on your thoughts on memorization and on your teaching philosophy and simply can’t wait for this new series 😀
Thanks for reading! We’ll see how it turns out…
Many years ago when I first started teaching math, so around 1980, I saw the “The Student’s Guide to Problem Solving.” Don’t know where it came from or who the author is. Often true then and (sigh) still often true now.
Evan Romer, Susquehanna Valley HS, Conklin NY (retired)
——————————————–
The Student’s Guide to Problem Solving
Rule 1. If at all possible, avoid reading the problem. Reading the problem only consumes time and causes confusion.
Rule 2. Extract the numbers from the problem in the order in which they appear. Be on the watch for numbers written in words.
Rule 3. If rule 2 yields three or more numbers, the best bet for getting the answer is to add them together.
Rule 4. If there are only two numbers which are approximately the same size, then subtraction should give the best results.
Rule 5. If there are only two numbers in the problem and one is much smaller than the other, then divide if it goes exactly, otherwise multiply.
Rule 6. If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers in the problem.
Rule 7. If the rules 1 – 6 don’t seems to work, make one last desperate attempt. Take the set of numbers found by rule 2 and perform about 2 pages of random operations using these numbers. You should circle about five or six answers on each page just in case one of them happens to be the answer. You might get some partial credit for trying hard.
Rule 8. Never, never spend too much time solving problems. This set of rules will get you through even the longest assignments in no more than 10 minutes with very little thinking.
So funny xD how did you even come up with this idea of blog? really cool 🙂
I think your idea is incredible. I am really curios to see how the next posts are going to be. 🙂
I found you by accident but now realize it was serendipity. I love, love, love how you explain learning. I’ll be sharing this site with my friends and hopefully send you more traffic. That means, keep writing. You’re good at it.