Or, Math Class is Too Full of Spoilers
In grad school, my wife took a class that assigned no homework. The topic was an advanced, hyper-specific area of research—the only plausible problems to give for homework had literally never been solved. Any answer to such a question would have constituted novel research, advancing the field and meriting a publication in a professional journal. The professor assigned no homework for the simple reason that there was no practical homework to assign.
This tickled me. I’d never thought of good questions like a fossil fuel. A nonrenewable resource. Built up over eons and consumed in minutes.
But the thought kept popping back up: Good questions are a resource. And in this new light, something started to make sense, an uncomfortable little fact that had nagged at me since my first year teaching.
I lurched sideways into the profession, like a bowling ball that hops into the next lane and knocks over the wrong pins. I knew nothing about pedagogy, other than whatever naïve intuitions I’d gathered as a student. So that year, whenever we encountered new types of problems, I simply told my geometry students exactly how to answer them.
It felt natural; it felt like my job. Need to prove these triangles are congruent? Do this. Need to prove that they’re similar? Do that. Need to prove X? Do Y and Z. I laid it all out for them, as clean and foolproof as a recipe book. With practice, they slowly learned to answer every sort of standard question that the textbook had to offer.
Months passed this way. But something wasn’t clicking. I kept seeing flashes and glimpses of severe misunderstandings—in their nonsensical phrasings, in their explanations (or lack thereof), in their bizarre one-time mistakes. Despite my best intentions, something was definitely wrong. But I didn’t know what.
And, more worryingly, I didn’t know how to find out.
I’d already coached them how to answer every question in the book. How, then, could I diagnose what was missing? How could I check for understanding? For every challenge I might give them, every task that might demand actual thinking, I’d equipped them with a shortcut, a mnemonic, a workaround. The questions were like bombs defused: instead of blasting my students’ thoughts open, they now fizzled harmlessly.
Good questions are a resource, and I’d squandered mine.
I couldn’t articulate it then, but I began seeing questions differently. They were not just obstacles to overcome, or boxes to check. Instead, they were my fuel. My kindling. My only real chance to ignite reflection, curiosity, and the impulse to seek out deeper truths.
Questions were not just things to answer; they were things to think about. Things to learn from. Giving the answer too quickly cut short the thinking and undermined the learning.
Good questions, in short, are a resource.
Solving a math problem means unfolding a mystery, enjoying the pleasure of discovery. But in every geometry lesson that year, I blundered along and blurted out the secret. With a few sentences, I’d manage to ruin the puzzle, ending the feast before it began, as definitively as if I’d spat in my students’ soup.
Math is a story, and I was giving my kids spoilers.
Now, I don’t believe in withholding the truth from kids indefinitely, or forcing them to discover everything on their own. But I’m aware now of a danger: if I tell them too early how the film will end, they may turn it off halfway through, and I’ll have a hard time convincing them of everything that they’re missing.
Questions aren’t merely targets that need to be hit. They’re also our arrows for hitting bigger, more elusive targets: concepts, connections, ways of thinking.
Questions aren’t the enemy. They’re the ammo.
As I move through my career, I find myself increasingly overcome with the opposite difficulty: too many good questions. Too much cool stuff. A sprawling wealth of worthwhile problems, such that my students and I couldn’t possibly get around to half of them.
But sometimes, I still slip up. I think my students are learning deep concepts and strategies, but they’re only parroting phrases or mimicking procedures. Too late, I realize what’s happening… but by then the suitable questions are used up. They can answer without understanding. And it’s back to the drawing board for me.
Reblogged this on thebloggeraboutnothing and commented:
very true, students will lack the capability to read more to advance their knowledge.
Ben, as you know, I am thinking of moving materials in my optimization course into a MOOC (Massive open online course). Your post about questions raises a lot of interesting questions for me to ponder. Should I make questions relatively straightforward and easy to grade? This might be appropriate if the students studying the materials want to develop certain mathematical or software skills. But how do I ask questions about modeling in practice, problems that involve a lot of context, and where answers require subtle tradeoffs and good judgment? And for the very mathematical students, how can I provide questions that permits them to explore and learn for themselves?
I don’t have answers. But I thank you for helping me to think about the questions.
Hi Dad, some tricky questions indeed!
Dan Meyer’s written some very smart stuff about issues like these. One of his points: that not everything needs to be marked by computer – or marked at all. Simply asking good, open-ended questions, and then storing student responses (for later feedback from a teacher, or peers, or even from no one), is an improvement on not asking those questions at all.
For those open-ended questions, I could imagine you providing an exemplary answer, along with some guidance on how to reflect on your own answer.
I’m a MOOC enthusiast.
Theoretical questions are welcome, but please don’t score it too hard, because that is frightening, instead of inspiring.
Hard, multi-part questions are very instructive, in which the questions go into the facets and interrelations of a more complex problem.
For learning, it is best if students have unlimited (or 40) tries, because that inspires them to solve it, whatever the cost. If the time is over, or if the solution is correct, do give a full-fledged, interesting solution automatically. Or even two different solutions.
I don’t think it will be quick or easy to develop such a MOOC program, but it will be worth it. 🙂
Please check that your instructions are unambiguous even if they are read by an Indian or Australian reader. Standard expressions, naming the same thing with the same word is a feature here, not a style mistake. Please define new words, or tell explicitly, that they are synonyms in your specific science context. It is language issue, but can severely mix up non native speakers and children. There may be up to 1000 questions about one such glitch in your discussion board, ahem.
“M. Night Shyamalan had no idea what he was doing all along!” Legit lost it at this one.
Do you have a network of concept-driven teachers you can bounce ideas off of? When I was in community college, my Calculus teacher was extremely concept-driven. He used to give us assignments like, “Find the volume of a Krispy Kreme doughnut and answer the following questions about the concepts behind integration.” We as students totally hated the assignments, but we learned a lot.
My Calc teacher didn’t work alone, though. He found other like-minded professors who understood that the key to succeeding in math is understanding the practical application of mathematical concepts.
Switching gears slightly: what about having the students create their own questions? One student may be interested how geometry relates to how many textbooks she can fit in her backpack, while another student may want to learn how geometry affects a race car’s performance on the track. By creating and solving their own questions, they have to start by exploring the concepts.
Sorry for the long reply. Clearly, you wrote an intriguing article today.
I feel like I’ve got two networks to bounce ideas off of: my colleagues at school (who are great) and the people I follow on Twitter (where you’ll find a shockingly good professional network for math teachers).
As for having students create their own questions, I definitely agree that math goes better when kids get to bring their own ideas and enthusiasm to the table. Sometimes it’s important to scaffold that very carefully, with narrowly tailored assignments (“Describe a situation that can be modeled by the binomial distribution”). Sometimes you can just kind of sit back and let them run wild (“Describe a situation where probability provides some insight.”)
In the tunnel-vision of a teaching week, I’m not always good at doing this myself, but your comment is a good reminder!
Students can be a resource for questions. Have them investigate topics they are interested in–ask and research questions. They may need some guidance. You could propose ideas to explore related to what you are teaching and topics you know they find interesting, such as, How are transformations used in animation? Why would they contribute to reduced costs?
Loved this post. I still remember the first time that I realized I was spoiling math for students by helping too much and therefore taking away the natural excitement and curiosity that people have towards figuring things out for themselves. The hard part for me continues to be in how to foster and get students better at persevering at questions. Depending on their background, by the time that many students get to high school, they are not used to being asked questions that are hard or know how to grapple with questions without being rescued by a teacher so it becomes not just an issue of the teacher asking good questions and not giving away the answer, but also teaching students to appreciate this process.
Yeah, well said. The times when I’m tempted to “spoil” the story is usually as a quick fix for classroom management: kids are unsure how to start + I haven’t successfully taught them how to dive into a tough new problem = a recipe for 20 hands in the air and everybody quickly abandoning the task to talk about last night’s football game. Telling them how to do it is the easiest way to keep them on task, and I suspect this is part of what keeps teachers in “tell” mode. It’s harder to run a classroom where the kids are on unsure footing.
It’s ironic because we have thousands of problems that some think are good, some think stink, but none of which if we provide solutions for would be information sufficient to solve (“spoil”) any other problem. Nonetheless, it is true that “America” may run out of novel K-12 questions because that is one of the limitations of narrowly drawn standards like Common Core. We’re already seeing a manifestation of that limitation when PARCC, by threatening students, attempts keeps its finite selection of problems a secret.
Standardized test makers have always had finite sets of problems that they kept secret. They would add to the pool and retire a few every year, just as PARCC will do. So, nothing new.
I write math problems for a living. I’ve written problems for 25 years and I don’t expect my well to ever run dry. There will always be interesting connections to make to math.
Some of this sounds familiar from the recent NY Times Magazine article “Why do American’s Stink at Math.” It’s an excerpt from Elizabeth Green’s new book Building a Better Teacher.
Reblogged this on Snowbird of Paradise.
I am reblogging this to Snowbird of Paradise. Great post!
Reblogged this on One of Thirty Voices and commented:
Questions are the lifeblood of my teaching. I hate to tell kids how to do something, and yet, like this author, I slip up, or time constraints cause me to give away too much, too soon. For those of us who truly see good questions as “kindling” to ignite our students learning, I gotta share the following blog!
(from an interview with Fields medalist Cédric Villani at http://www.huffingtonpost.com/2015/05/07/cedric-villani-mathematic_n_7223966.html)
Finish this sentence: In the year 2025, we will…?
…will have even more problems than we do now — and we will address them with courage and pride!
This is a great post. Sometimes I feel like I am the only one who thinks this way and then someone will blog or tweet about it and let me know that I am not alone. There are a lot of issues that cause teachers to tell but I find it is usually because they have never experienced teaching a different way (I know I was always told in my math classes) and so they don’t know how to do it in their classroom. The one thing that I wanted to add was that I am constantly hearing that we are preparing students for jobs that don’t exist yet. I ask myself how are we supposed to prepare students for something we don’t even know about? I realized that it is through assessing situations, making up and answering their own questions, and problem solving that we can prepare them for thier future life. Again thank you for this post.
Knowing the answer to a Math question is never the end of the question. If a student can answer to the question “why is it so?” then you exhausted the question.