I had a surreal moment this year. I’d almost finished a lesson when one boy, usually a hyperkinetic little bundle of enthusiasm, raised his hand.
“So, like, I don’t really understand anything you’re saying,” he informed me, “But I can still get the right answer.”
He smiled, waiting.
“Which part is giving you trouble?” I asked.
“Oh, you were talking about this extra stuff,” he said, “like the ideas behind it and everything. I don’t… you know… do that.”
I blinked. He blinked. We stood in silence.
“So is that okay?” he concluded. “I mean, as long as I can get the right answer?”
Here it was, out in the open: the subtext of practically every class I’ve ever taught. I’ve grown accustomed to yanking my side of the rope in an unspoken tug-of-war. The teacher emphasizes conceptual understanding. The students conspire to find shortcuts around it. So it always goes.
But I’d never heard a student break the fourth wall quite like this. It was as if Peter Jackson popped up on camera saying, “I know you want a good story, but what about a bloated trilogy full of mind-numbing battle scenes instead? You’ll still buy a ticket, right?”
“Is that okay?” my student repeated. “I mean, I can get the right answer!”
He had a point. What else is there?
There’s a powerful ideology at work here, one my student has perhaps internalized without realizing: the unshakeable belief that math is all about right answers, and nothing more.
The Church of the Right Answer.
Millions of students worship there, and it’s not because they hate ideas. They do it because schools exist, in part, to sort and label them. “Honors.” “Remedial.” “Vocational.” “College-ready.” And the primary mechanism of this sorting—society’s basic equipment for doling out futures to young people—is grades.
Make no mistake. Grades are a bona fide currency. When I write a score on a test, I might as well be signing a paycheck.
And as for our tests—no matter how well-intentioned, no matter how clever and fair, there will always be a back-road to the right answer. There will be something to memorize—a procedure, a set of buzzwords, whatever—that will function as a fake ID, a convincing charade of understanding.
Some tests are better than others, but all of them can be gamed.
It’s tempting to blame the kids: these grade-grubbers, these incurious mercenaries, shuffling through our schools, forever demanding to know whether an idea will be “on the exam” before they deign to learn it. But let’s be honest. From right answers come good grades, and from good grades come all worldly prizes and pleasures.
Students worship right answers because that god delivers.
I’m not fond of this reality. But most of the time, I simply work within the confines of the grade-driven system.
I write the best tests I can. I mix in posters, projects, and presentations. I dabble in standards-based grading. I give second and third chances to succeed. I sing the praises of learning, in the best tenor I can muster. And at day’s end, when my job demands that I sort and label my students, I try to do so with sympathy and transparency.
In short, I’m constantly nudging students to think more deeply, but I never really challenge the dogmas of the Church of the Right Answer. I’m a good, rule-abiding cop, in a city where the rules are sometimes grossly unfair.
That doesn’t always satisfy me. Some days I don’t want to nibble at the edges. I crave a more radical assault: a reformation, a new religion.
Some days, I want a Church of Learning.
Now, I don’t have 95 theses ready to nail on the wall. It’s easy to critique a system, and damn hard to build a better one. You want a magic plan for education, a foolproof blueprint for helping the next generation to flourish? Me too! Send me the link when you find it.
In the meantime, I’ll tell you the four theses I’ve got, and let you supply the other 91.
Think of all the people you know who are curious and thoughtful. People who ask real questions and listen to the answers. People who read weird, well-written stuff, and then tell you about it. They’re not so scarce, are they? In many places, they’re numerous and thriving. The soil is not as barren as it seems.
I’ve seen it often. I’ll meet a student who knows everything about quadratics: how to factorize them, how to graph them, how to complete the square. And, like most students burdened with such knowledge, they view quadratics with pure contempt.
Then, with a little coaching, they’ll suddenly see what a quadratic expression is. They glimpse the idea beneath it all. And all of those useless fragments of procedural knowledge finally snap together, swift and tight, into a whole and gorgeous understanding.
The idea pops like a firework, and the kid lights up like a sky.
Real learning touches something deep inside people. It endows a sense of power and purpose and wonder that you’ll never get from regurgitating facts and executing shortcuts. Once you start thinking, you want to keep thinking.
This is a theology that doesn’t require a steeple. We don’t have to march on Washington, or take a scalpel to the face of education. We’re talking about the simple conviction that ideas are worth celebrating. Such as love of ideas can take hold as… well…
As an idea.
The Church of Learning doesn’t require that I reject the economic structures around me. I don’t need to drop out of school, eat from dumpsters, and refuse to clip my nails. Sure, the school-to-workforce pipeline doesn’t always honor the things I care about. But that’s okay. My material life need not dictate the terms of my intellectual life.
I can be a capitalist, and still be curious.
(In fact, I haven’t subtitled my memoirs yet, but dibs on “The Curious Capitalist.”)
In a zero-sum game, every winner needs a loser. I can’t gain a dollar unless you forfeit one. I can’t triumph unless you fail.
It’s pretty much how we’ve built our educational system. Students compete for finite resources: admissions spots at elite schools, A’s in a tough class, titles like “valedictorian” and “cum laude.” If no one is failing, we sometimes worry that we’re doing something wrong—perhaps setting our standards too low.
Somebody has to fail, right?
To some degree, this continues after school. Adults compete for jobs, for customers, for parking spots. But—and I can’t scream this loudly or often enough—
Life isn’t just competition!
Wherever my students go—business, academia, government, Mars—they’ll reckon with problems. They’ll create things. Their hands will get good and dirty. And if they topple the challenges, fix what’s broken, and imagine wonderful new things that everybody needed and that nobody ever realized, then the world will get better.
Really.
A better world is not like a fancy college. The spots aren’t limited. The more we populate the planet with curious, thoughtful, talented people, the better it becomes.
There’s a word for this idea: “education.”
When my students leave, it won’t particularly matter what grades I bestowed, or what tests they aced, or what shortcuts to the right answers they could and couldn’t find. What will matter are their abilities.
Creativity.
Grit.
Ingenuity.
Their wealth of knowledge.
Their curiosity, independence, and capacity to learn new things.
Those skills can be developed. It just takes time and effort. And that’s what teachers are for: to help make it happen.
Like my students, I have good days and bad. They and I are jostled by the same storms, pulled by the same tides. Sometimes I scold them for worshipping at the Church of the Right Answer, and sometimes I play back the tape and discover that I’ve been preaching its sermons myself. I try to track the harm and the good that I do in the classroom, measure the one against the other. It isn’t easy. There are no right answers.
But of course, life’s not about right answers.
***
Thanks for reading! This is the sixth (and final!) in a series of posts about How to Avoid Thinking in Math Class. See also parts 1, 2, 3, 4, and 5.
Love it thank you! Also do you have an tips on ways to help the new to school kids into loving math? Currently my 4 almost 5 year old says Math is her favorite subject. I’m super excited for a couple reasons, I really liked math, but fell off the wagon as girls don’t do that and two she is a girl who likes math and math is the world these days. Even computing has so much math in it these days, so I really want to encourage it as much as I am encouraging the reading and the spelling words. Maybe you know of books or places to look? Thanks in advance and no worries if you don’t. Loved this article.
I know this is science but the principle is the same really I think
Teresa H: Your daughter is probably too young for this now, but I really liked the Math For Smarty Pants book. In general, I would encourage playing puzzle games which teach the concepts without the numbers. I loved games like that when I was little and now I’m finishing my PhD in engineering. Also, I thought the Set card game was good for abstract thinking and looking for patterns.
Ben: This is such a great blog! I look forward to each new post. Keep up the good work! 🙂
I would look into mensa puzzle books, suduku books, and Ken Ken books (once she can multiply and divide). Also I would see if your local college has a camp for little nerds, I went to one as a kid and was hooked on math. NPR recently had a great article too on math games to play with your kids at coffee shops. If you want I can try and find it too. Also matchstick puzzles are a great way to con your children into doing math.
Hi Teresa, I think the three commenters above have given better advice than anything you’d get from me!
Its a nice one! yesicanbd
Now we’re down to brass tacks. To me I’ve felt at this point like a Buddhist or someone in the Matrix (same thing, I know) whose realized the deceptive nature and now wants others to wake up. So I think the church metaphor is a good one. Go forth, Ben!
Where did the ‘just right’ student ever get to on this idea?
We are great in number, but not it power, GoldenOJ.
Thanks for reading, John! In the spirit of the matrix, I’ll now go write two over-budgeted, underwritten follow-ups to this piece.
How about “Collaborating is not Cheating” for your list of tenants? Most students, up until AT LEAST the junior year of high school have an inherent fear of working together because they think they’ll be accused of cheating (well, that or they cheat and don’t care). In true learning, peers should help and teach one another as much as possible. If one student has an understanding of the concepts, but can phrase it in a better way (a good teaching tool, rephrasing concepts), then everyone might benefit form hearing this new viewpoint. My biggest peeve is teachers who don’t allow for wiggle room, especially in math. There’s more than one way to approach a problem, provided you show your work!
Agreed! Being able to do your own work is really important (it’s easy to think you get something when your friend is the one holding the pen). But the way you reach independence is usually through lots of collaboration!
Sometimes in class, I try to make the difference between “things you need to remember for the next couple of weeks” and “things I want you to remember five years from now.” The second list is much more important to me.
Yeah, that’s a worthwhile distinction. Can’t overuse the second category, but the fact is that most classes have a few deep ideas, and those are the biggies.
Here are a few practical suggestions from my own teaching to engineering students in HE (They have exactly the same attitude).
1. Don’t tell them the answers.
2. Make them tell you how they know it’s right.(“I just know” is not an answer!)
3, Give them the occasional problem that spews out several answers, not all of which are any good (spurious roots is a start).
4. Give them problems where there isn’t a right answer (What’s the best shape for a cereal box?)
5. Get them to raise their eyes from the desk and look outside, Let them formulate the questions, from real situations. “How do they figure out the timings for traffic lights” is a good one.
6. Give them answers to some problems, but not always the “right” ones.
Of course, the responses you will get are “It’s not fair” and “This isn’t maths” (like they know!)
You could even take some of these a step further and implement direct math into them, making the students back up their answer with mathematical reasoning, if they absolutely need a direct relation, then go more general and have them explain further without the math, or just in concepts. (This may work best for younger students, who will have a harder time understanding more abstract math).
Yeah, those are all good suggestions!
To some extent, I think you get coevolution of teacher tactics and student responses. Teachers come up with a fresh idea to encourage real thinking. At first, it has a big impact, and students think more. Over time, though, the fresh idea calcifies into something rote and predictable. It gets swallowed by the system. And so teachers go back to the drawing board.
I think this definitely plays out in an individual classroom; not sure to what extent it plays out that way on a societal level.
Reblogged this on Paul Karam Kassab.
I like this post. One tiny question though: when you say “they’ll suddenly see what a quadratic expression is”, what exactly is the “is” that it is?
To be sure, I understand the sentiment you’re conveying. For example, when someone realizes that a logarithm is an exponent, or that a derivative is a linear operator (and I don’t mean, “Differentiation is linear”, but rather, “When you take the derivative of a function at a point, the result you get is a linear operator), it’s the same idea.
I’m just having trouble filling in anything poignant for X in the sentence, “A quadratic expression is X.” A locus of coordinates satisfying a specified relationship? Certainly, but that applies to any function. A planar slice of a bi-infinite cone? Without question, yet every textbook I’ve seen on the subject has stated and illustrated that fact numerous times.
Or did you only mean a quadratic in one variable, ie. a parabola? If so…I still don’t see what there is to “click” for a student. Now completing the square, yes that has a wonderfully-clarifying visual illustration which most treatises inexplicably omit, but the quadratic itself?
Yeah, I mean “set of points in the coordinate plane satisfying a relationship.”
(BTW, since you mention it, conic sections are my single least favorite topic in the American math curriculum. As taught/presented/tested, they’re completely alienated from their most interesting feature, which is that they can be described as loci. They’re offered instead as impenetrable Cartesian equations, whose relationships to their graphs students must memorize cold because they don’t have the time or the preparation to understand WHY such a locus would correspond to such an equation. It’s the distilled and weaponized form of every misconception kids have about graphs.)
You’re certainly right that quadratics aren’t anything special in this regard. The revelation kids are having is really “what a Cartesian graph is,” not “what a quadratic is.”
I only got into your set cus I only learnt how to get the answers, didn’t learn how or why it happens.
Yeah, I suspect you’re not the only one who feels that way! This year is probably a change of direction for a lot of guys in your class. Doesn’t make you bad mathematicians (you’re certainly not!); it just means there’s more to learn. And everybody’s always got more to learn!
Great article. I’m also a HS math teacher. I made a video of my philosophy a year ago and we have a similar view. Take a look if you have a few minutes.
Lovely video, Rick!
Yeah, that’s a gorgeous video!
I sent a link to this post to Grant Wiggins. I am sending you the link to his post. It is a fascinating account of the confusion about what readers actually get from the teaching and the feedback. there is a reference to math grades as well, but his analysis has a lot of bearing on math as well.
https://grantwiggins.wordpress.com/2015/02/11/maybe-we-dont-understand-what-readers-really-do-and-why-it-matters/
It’s a great post! I always love reading Grant’s stuff. At once soothing and humbling to read somewhere who knows 700 times more about education than I do.
Amen! Preach it brother! I’ve seen the Promised Land…
My few, not-very-original dictums to nail up with the Ninety-Five Theses on the Power and Efficacy of Right Answers:
● Whenever possible, design tasks that have a divergent element.
(so, for example, http://y4ist.blogspot.fr/2015/01/number-patterns-with-cuisenaire-rods.html )
● Whenever possible encourage divergent thinking.
(Number Talk, “What’s your strategy?”)
● Whenever possible don’t do tests.
(this is more possible in primary / elementary)
● Whenever possible de-emphasise tests.
(Don’t use fear-of-the-test as some kind of stick. Don’t treat them like some kind of Eucharistic moment at its altar in its apse.)
● Whenever possible, don’t give numerical assessments.
(Instead, “This was good”, “This is how you could improve”.)
● Whenever students initiate something, big it up.
(Maybe this doesn’t happen a lot in Secondary, but it can in Primary.)
(for example: http://y4ist.blogspot.fr/search/label/Puzzle )
Excellent bullet points. And I think in the right secondary environment, students initiate a lot of really cool stuff. (I see it at my current school often; not so much at my old school, where the only full-time math teacher was this stodgy uninspiring guy named… uh, me.)
Great post! I am guilty of this on many occasions particularly when preparing students for national exams, so this is a timely reminder.
Food for thought: Answers are important. However, they are not for the questions that we set for students in the classroom because these are solved questions. Try solving an unsolved math problem and you become a mathematician. Start solving a real problem…What is important is not the answer, but the ideas, skills and techniques that lead to it. This is what we change the world with, not answers to some math problem set by my math teacher.
That’s a really good point. Right answers DO matter; bridges and software are built out of ’em. But the species of “right answer” we emphasize in school is a sickly, miniature version of the real thing.
Can I play Devil’s Advocate for the moment?
I was that kid in class. I was, “I get what you’re saying, but while you were talking, I solved the problem, so why are you still explaining the concept to me?” kid. I annoyed my teachers with that attitude to the point that one or two of them probably assume I was after the grades, not after the learning.
There’s a very different kind of math student from Group A, which wants to soak in the concepts, and Group B, which wants to earn the grade. Group C contains the students who just get it. They don’t know why they get it, but they do. These are the students to whom logic and critical thinking are second nature (note that I did not say “common sense”). I was/am the type of student who sees a procedure and understands without an explanation why it works. I was always the first to hand in a correctly solved logic puzzle, and I always scored higher on word problems than on straightforward formula work (back to the common sense thing–I regularly made operational errors, swapping a negative for a positive, etc.)
I graduated high school with a B-minus GPA and I flunked several classes in college. However, I learned a programming language with a week and a half of sub-par training and am now considered one of the most knowledgeable employees in my department. The logic of the computer language is as natural as breathing to me, where as many of my colleagues cannot seem to grasp its rhythm.
All of that being said, I’m curious if your student is really one of the members of Group B, as you seem to think, or if he’s a member of Group C. Maybe his brain is just wired completely different from the students around him, and maybe what he doesn’t understand is how other people don’t see the logic in the operation.
Sorry for the babble. I’m not entirely sure that even made sense near the end…
I love a good devil’s advocate!
You’re right that an unchallenged student can exhibit a lot of symptoms that teachers misunderstand. I definitely can’t pretend to have the perfect read on all my students; I’m sure I’ve underestimated many of them in a variety of ways.
There are some settings, though, where a knack for problem-solving isn’t enough. I see this particularly with algebra. Many of my students are so quick and clever that they can solve early algebra problems by sheer intuition. But that’s a dead end. Eventually, the setting gets tough enough that their intuition fails. And then, they’ve got nothing to fall back on, because they haven’t learnt or practiced the techniques that actually DO generalize. This describes many of the 15-year-olds I currently teach: really clever kids who aced the last few years of mathematics and are now, paradoxically, struggling a bit, because they didn’t acquire the underlying algebraic concepts. Their number sense was so good that they never needed to state patterns externally and explicitly – which means they’re lacking practice in the essence of algebra.
Not sure if I’m now devil’s advocate to your devil’s advocate, but there we are.
I would argue that you don’t really “get it” until you can effectively explain it to someone else. So when I have a student who claims that they don’t need the “explanation”, I challenge them to justify their answer and method to another student, preferably one who doesn’t get it as easily. Knowledge is useless without effective communication, and often the effective communication teaches us facets of the concepts that lay a better foundation for future learning. When you don’t thouroughly care about the how and the why, you limit your future prospects for learning. You can only get so far on intuition and innate ability.
please forgive my spelling errors in the above comment
I love this! I just made a comment for others to view your blog on FaceBook! Having been a high school math teacher (retired now), I loved teaching, and loved the students. They were my “life-support” each day. Thank you for being one who teaches/taught/ and loved it.
I needed this blog to cheer me up so that I can come up with ideas of my on at jmmurphey.wordpress.com
YOU HAVE THE BEST ONE I HAVE OBSERVED AND IT IS PRECIOUS!
Thanks for reading!
First of all, let me just say that I loved your post and the ideas it conveyed. It was a fitting end to a great (and hilarious and thought provoking) series.
Also, i have to agree with cverhuls that SET is a great card game for abstract thinking pattern visualization, and fun.
Unfortunately, bashing on the lord of the rings series may have moved you from my favorite blog to my second favorite, or maybe even third favorite. I would have expected better from you. Tisk tisk.
I thought he was bashing the latest Hobbit films which pale in comparison the the story quality of the preceding trilogy of films.
Yeah, i’d be a little bit more okay with that because the hobbit series wasn’t as great but it did not have tons of mind-numbing battle scenes, except the last movie, which made me think he was referencing the LOTR.
Simply, wow.
Yet another well written post.
Thank you!
Reblogged this on Pathological Handwaving and commented:
“Some tests are better than others,
“So, like, I don’t really understand anything you’re saying,” he informed me, “But I can still get the right answer.”
“Super. How would you say to YOUR student, who gets a wrong answer, that he did it wrong? How do you know you’re right, and how do you explain it? Obviously I failed to say the right things to get the idea across to you. And you’re lucky to have the idea without needing to hear what I said. But what are you going to say, later?”
A student who has failed to “get” the lesson is at least half-likely to have encountered a teacher to failed to “send” it, right?
Sure, agreed in principle. My description leaves out lots of details, so I see where it might sound like a failure of teaching. (God knows I commit enough of those every day.)
In this case, though, I’m talking about a 6th grader able to solve simple linear equations by mechanical procedures. He’d already been taught this before I met him. But he lacked the recognition that he was “doing the same thing” to both sides. (This describes 80% of my sixth and seventh graders.)
I like the question “How would you explain it to someone?” as a way of eliciting and verbalizing intuition. But it wouldn’t do much good here, because I’d already heard him explain it: “First you do 37 minus 7, then divide by 3, because there’s a 3 in front of the x. But what does that have to do with ‘doing the same thing to both sides’? I’m taking a 7 from this side and putting it on that side.” Pure mechanics, no concept.
Reblogged this on sheldonjoseph.
Really interesting post, thank you. Sometimes I think we are in danger of institutionalising Microwave Maths – just recognise the question’s pattern, insert the numbers into the memorised method and ‘ding’ a few seconds later out pops the Right Answer. And this isn’t true only for maths – I spoke with an experienced English teacher recently who lamented that she no longer shared texts with her students and extracted the characters’ motivation and nuances but was focussed solely on grades. There are even books listing mark-worthy answers for English Literature, thus “Romeo’s balcony speech was idiomatic of the recently introduced chrysanthemum flower” (made up, but by this I mean some insight that only a very able 16-year old student would arrive at independently, now available to anyone bothered to ensure the contents is piping hot) and the exam markers must be fed up of paper after paper stuffed with these cardboard answers that meet only the marking scheme.
Tom Lehrer, “New Math” — c. 1967 — http://en.wikipedia.org/wiki/New_Math#Popular_culture
“In the new approach … the important thing is to understand what you’re doing rather than to get the right answer.”
Great thoughts! My only suggestion is that we need more lessons/activities that involve discovery, that involve situations in which students find patterns, that mimic real life where things are messy and there’s more than one right answer or alternate answers that may appear “right” at first glance. Class discussions can allow students to share what they discover/realize/see and decide for themselves what makes the most sense.
what a great post from start to finish – thanks! loved the idea of “breaking the fourth wall” 🙂
>And as for our tests—no matter how well-intentioned, no matter how clever and fair, there will always be a back-road to the right answer. There will be something to memorize—a procedure, a set of buzzwords, whatever—that will function as a fake ID, a convincing charade of understanding.
From Sheldon Ross, A First Course in Probablity: give a combinatorial explanation of the identity nCr = nC(n-r).
I defy you to produce a student who can produce a satisfactory explanation sans understanding.
Reblogged this on Raja Oktovin's Blog.
I could say a lot of things about this post, but I’ll just say: Loved it!!
And I don’t worry about writing a too simple response, since there are no right answers, hehe