One afternoon, the head of my department caught me in the staff room and posed a musing question.

(He later confessed that he was just curious if he could play puppet-master with this blog. The answer is a resounding yes: I dance like the puppet I am.)

So, do we have ceilings?

The traditional orthodoxy says, “Absolutely yes.” There’s high IQ and low IQ. There are “math people” and “not math people.” Some kids just “get it”; others don’t.

Try asking adults about their math education: They refer to it like some sort of NCAA tournament. Everybody gets eliminated, and it’s only a question of how long you can stay in the game. “I couldn’t handle algebra” signifies a first-round knockout. “I stopped at multivariable calculus” means “Hey, I didn’t win, but I’m proud of making it to the final four.”

But there’s a new orthodoxy among teachers, an accepted wisdom which says, “Absolutely not.”

You’ve got to love the optimism, the populism. (Look under your chairs—*everybody *gets a category theory textbook!) But I think you’ve got to share my pal Karen’s skepticism, too.

Do we have a ceiling, Karen?

Karen works hard. Karen asks questions. Karen believes in herself. And Karen still feels that certain mathematics lies beyond her abilities, above her ceiling.

The chasm between students (“everybody’s got a limit”) and teachers (“anyone can do anything!”) seems unbridgeable. A teacher might say “You can do it!” as encouragement, but a frustrated student might hear those words as an indictment of their effort (or as a delusional falsehood). Is there any way to reconcile these contradictions?

I believe there is: the **Law of the Broken Futon**.

In college, my roommates and I bought a used futon (just a few months old) off of some friends. They lived on the first floor; we were on the fourth. Kindly, they carried it up the stairs for us.

As they crested the third-floor landing, they heard a crack. A little metallic bar had snapped off of the futon. We all checked it out, but couldn’t even figure out where the piece had come from. Since the futon seemed fine, we simply shrugged it off.

After a week in our room, the futon had begun to sag. “Did it always look like this?” we asked each other.

A month later, it was embarrassingly droopy. Sit at the end, and the curvature of the couch would dump you (and everyone else) into one central pig-pile.

And by the end of the semester, it had collapsed in a heap on the dusty dorm-room floor, the broken skeleton of a once-thriving futon.

Now, Ikea furniture is the fruit-fly of the living room: notoriously short-lived. There was undoubtedly a ceiling on our futon’s lifespan, perhaps three or four years. But this one survived barely eight months.

In hindsight, it’s obvious that the broken piece was absolutely crucial. The futon *seemed* fine without it. But day by day, with every new butt, weight pressed down on parts of the structure never meant to bear the load alone. The framework grew warped. Pressure mounted unsustainably. The futon’s internal clock was silently ticking down to the moment when the lack of support proved overwhelming, and the whole thing came crashing down.

And, sadly, so it is in math class.

Say you’re acing eighth grade. You can graph linear equations with perfect fluidity and precision. You can compute their slopes, identify points, and generate parallel and perpendicular lines.

But if you’re missing one simple understanding—that these graphs are simply the x-y pairs satisfying the equation—then you’re a broken futon. You’re missing a piece upon which future learning will crucially depend. Quadratics will haunt you; the sine curve will never make sense; and you’ll probably bail after calculus, consoling yourself, “Well, at least my ceiling was higher than some.”

You may ask, “Since I’m fine now, can’t I add that missing piece later, when it’s actually needed?” Sometimes, yes. But it’s much harder. You’ve now spent years without that crucial piece. You’ve developed shortcuts and piecemeal approaches to get by. These worked for a while, but they warped the frame, and now you’re coming up short. In order to move forward, you’ve got to *unlearn* your workarounds – effectively bending the futon back into its original shape – before you can proceed. But it’s well nigh impossible to abandon the very strategies that have gotten you this far.

Adding the missing piece later means waiting until the damage is already underway, and hellishly difficult to undo.

This, I believe, is the ceiling so many students experience. It’s not some inherent limitation of their neurology. It’s something we create. We create it by saying, in word or in deed, “It’s okay that you don’t understand. Just follow these steps and check your answer in the back.” We create it by saying, “Only the clever ones will *get* it; for the rest, I just want to make sure they can *do* it.” We create it by saying, “Well, they don’t understand it now, but they’ll figure it out on their own eventually.”

In doing this, we may succeed in getting the futon up the stairs. But something is lost in the process. Sending our students forward without key understandings is like marching them into battle without replacement ammo. Sure, they’ll fire off a few rounds, but by the time they realize something is missing, it’ll be too late to recover.

A student who can answer questions without understanding them is a student with an expiration date.

EDIT, 4/15/2015: What a response! The comments section below is infinity and beyond. It’s like eavesdropping in the coffee shop of my dreams. I wish I had time to reply individually; please know that I read and enjoyed your thoughtful replies and discussion.

We had this conversation at home the other day: about the way I was taught simultaneous equations, of all things, vs the way they tried to teach my son’s class. Our conclusion would have been, had we known about this post at the time, that you are absolutely right.

You have a thoughtful and reasoned perspective on this. I think I’ve felt similarly for a long time and not been able to realize it as eloquently. I am teaching rational functions in algebra 2 right now and the biggest problem is that my student (most of them) never understood fractions very well, or at all. Ideally I want to reteach them fractions but they are resistant so I end up having to give techniques with no hope that they’ll “get” rational functions ever. It’s so disappointing.

As I help my son with his math homework, I find myself pushing against my own idea of a ceiling – one I wouldn’t admit I had. It’s cool to start fresh with the very basic idea that he can learn this, and indeed can a learn anything, if we just play and puzzle it through. I love your perspective on this.

This is really wonderful, especially the closing line. Unfortunately, many students probably believe that answering a question correctly is the very definition of understanding it.

That’s why one has to give challenging problems. It will turn out early what is wrong with the student’s understanding, and it can be corrected in due time.

Josh Foer referenced an ‘OK Plateau’. When we’re young, we’re super-focused when learning to drive, but eventually we get comfortable with our skills, most of us not pushing on to be professional racers or experts in coasting. We all learn to cook, but there’s a point where most alter the trajectory that points toward chef or culinary expert.

Some biology or background can mean short leaps in elevation, but there’s no getting to the top without intent and perseverance. If you’re looking up, you can always go higher. The problem is that when we stop aiming upwards, even a helpful push just takes us sideways.

This is a really interesting connection! Intent and perseverance are essential to learning, and I am wondering how much of this ‘OK Plateau’ is determined by individual students choosing to try hard and ‘look up’, and how much is dictated to them by their access to education.

It seems as though this idea of an OK Plateau is an external, culturally determined ideal. Thus, it likely looks very different for a student depending on their resources in school and the expectations that teachers, schools, family, friends, and communities have for them.

This concept of the OK Plateau is very real, in math and otherwise, and I think that it is extremely important to be attentive and critical of what students learn to be the acceptable stopping point for their drive for learning.

Harry Connick Jr., once said if you practice something long enough, you’ll become an expert at it. A pro basketball player once said, if you practice long enough, be consistent and persistant, no matter what you’ll become good enough to make the pro’s. I’ve always held to that mantra. If I practice long enough, am persistant and consistent, there will be no math ceiling for me. The ceiling is not a true thing, it’s just an excuse when we grow tired and frustrated with math. If we push forward, we’ll surprise ourselves and be amazed how far we can go with math.

Thank you, Ben, for hearing me – when I read the title of this in my twitter feed, I thought, oh boy, it’s more of the “anyone can learn math” thing again, and I just get ignored because, by that view, it’s my fault – I don’t work hard enough, or haven’t tried long enough.

I still don’t think you believe me – but I’m so grateful someone heard.

I’ll be back later. This is one math topic where I have expertise – from the other side.

Unfortunately, many of us were taught the “one way to do things” mathematics and I found my brain needs a broader, conceptual vision of ideas. Since I didn’t get that teaching early on, I could find answers without ever truly understanding what I was doing. This made higher level mathematics both frustrating and inexplicable to me.

That’s a great analogy and I love your last line. I’ll share this with all my colleagues.

This analogy reminds me of the nature/ nurture conversations that have emerged about elite athletic performances. Consider “The Sports Gene” by David Epstein vs. “Outliers” my Malcolm Gladwell. No matter how strong the “raw” material, it takes a village of knowledge, concern, and organized effort to develop “elite performance” in mathematics students. And, of course, it takes effort and time on part of the students.

Two things that, I think, thought, that something deserves to be explored more deeply:

I want to challenge myself and the other readers’ understanding and definition of “conceptual understanding.” Too often, we label “new, non-traditional techniques” as “conceptual,” and old-school techniques as “procedural.” I think furthermore, how is procedural understanding less important than conceptual understanding? I want us to re-think the possibility that we want “both-and,” and not “either-or.”

Secondly – I wonder if there’s some confirmation bias about “how we were taught.” I suspect that we (and our students) have remembered mostly procedural understanding of techniques because they were practiced, easy-to-understand, and connected to mnemonic tricks that stay lodged in the brain. I have witnessed similar recollections from my students, and I know that I did a whole bunch of “conceptual” work with them. Having a deeper, more complete understanding of how to modify those procedures, or explain why they work, is harder to explain. I suspect that most people, if asked, wouldn’t really know if they have a “conceptual understanding” of, say, rational functions. But every new conversation I have with a teacher about teaching rational functions helps paint a more complete, nuanced, thorough picture of rational functions. Every new, challenging problem on the AIME exam will challenge my understanding of logarithms, or geometry, or counting rules.

I think both things are true: there are ceilings for everyone that we’re bound to come to sooner or later. But I think that they may be (mostly) penetrable for people under certain sets of circumstances. And the variables are much more complicated than something as undemonstrable and doubtful as “IQ.” I hated math in high school, ignored it until my 30s, and now have a near-Ph.D in math education from U of Michigan with the equivalent of a BS in math from CUNY. I’ve gone on to study a variety of topics well past calculus (including graphy theory, category theory, lattice theory, surreal numbers, etc.) on my own with the help of textbooks, video lectures, etc. I’m very persistent and take the long view on everything: given enough time, I think I can get difficult/abstract stuff, and I return to topics periodically despite running into ceilings in the past on them. I’m nearly 65. I don’t have close friends who know more math than I do, so there’s really no one to play things off of that I’m grappling with. I just have a lot of curiosity and a desire to know more than I already have mastered or need to know for my work. Of course, not everyone feels similarly about math, and some folks don’t feel that way about anything. But I don’t think it’s “written” that any given person is irrevocably doomed to live below a particular ceiling. Probable, perhaps, but not certain.

I think this may be one of your best posts ever, as you’ve eloquently identified and explained some of my own difficulties with math. I can do arithmetic, do and understand algebra and scientific calculations, but as I approach calculus, my ceiling starts to develop alarming cracks with chunks of plaster falling around me. Like Karen, I’m a superannuated student with an interest in mathematics. I’ve tried MOOCs and teaching myself, but haven’t gained any further understanding from either approach. It could certainly be a case of not finding the right teacher or a setting where I can learn, or maybe I’m overvaluing my own “grit/desire/effort/work/” but I’m beginning to think that part of it may be that my little metallic mathematics bar snapped off around my junior year of college. I’m not sure where or how I can get a replacement part, but I will keep trying for some true understanding. Thanks for writing about math in a way I can understand and relate to, and the drawings certainly help!

I am a tutor – usually handling kids with behavioral issues who are in limbo until the powers that be, administration, advocates, doctors and lawyers, decide what to do with them. As a student I moved between countries so you can bet I missed lots of concepts. Interestingly when it came time to teach my student fractions (and having to buy my own materials but that’s another story) I had to learn them (again or maybe for the first time) in order to teach them. Math became my favorite subject to tutor. I took my student from mid-year fifth grade all the way to seventh grade in nine months.

This is an excellent post. Great job.

I think MPG is correct: students have a wall (or ceiling) and to some extent we can penetrate it.

One thing to consider–why do some students intuitively grasp the reality and other students just go through the motions to find the correct answer, even though they have the same teacher? Isn’t that evidence of the wall, or a cognitive limit? So yes, we can teach students properly and conceptually and delay the point at which they stall.

I went through calculus in high school, but my understanding stopped at algebra. Like another commenter, I came back to math when my son was struggling with geometry, where I first hit a wall. But I’d had 30 years of learning how to learn, and was better at challenging myself and understanding my stall points. And as a result, here I am a math teacher.

However, I’m well aware that my wall still exists. I’ve pushed it out, but the original hold up was the fact that my visual-spatial ability is nowhere near my verbal reasoning and logical abilities. I am compensating for this lack with logic–I redefine problems in ways I can understand. That won’t work out in the land of MVC and vectors and whatever else is out there.

This post, these vids, pretty much sum up my math ability:

http://robertkaplinsky.com/why-depth-of-knowledge-is-critical-to-implement/

Even now, at 60, I am those kids. And those kids are going to be me in 50 years, which is a terrifying thought.

From 7th to 12th grade (1966-1972), I did great in school. Mostly As. Never got below a B, even in math. In 10th grade, “algebra II” it was called, I was very upset after the midterm, in tears. I got a B+ (the things we remember… I remember that +). “See,” said the teacher, “you did really well.” But I still didn’t understand anything. How I did so well on that test – on all the tests – I’ll never know. Luck? The “halo effect” a well-behaved white girl enjoys when show-your-work ends up most of the score? Don’t know.

what I do know, is that I knew I didn’t understand math at all. But I couldn’t get anyone to listen to me, because I kept batting out B+’s.

I tried several times, over the years. When I finally spent a couple of years polishing off my Bachelor’s degree, I started with a summer intensive course algebra/trig – another B+ – and, guess what, calculus, where I got an A. Still didn’t understand a thing. I’m not sure, but I don’t think I got a single calculus problem right the whole time, but I got enough partial credit, somehow, for an A. At one point, next to a long, complicated calculation I’d done, the teacher wrote, “Wouldn’t it be easier to factor?” I wanted to tell him: “Yes, I bet it would, but I don’t know how to factor, and I didn’t knowit was an option, and why would it be easier? How can I be sure I don’t leave something out?” That isn’t the sort of question an A student would ask.

I more or less gave up after that, until about 3 or 4 years ago when I saw Sal Khan (relax, it’s ok) on something like 60 minutes or Sunday Morning, talking about Khan Academy. I ran through the curriculum from number lines to the power rule of derivatives (all he had, at the time – mid-2012). I thought I “knew” trig when I got my Journeyman Trigonometer badge. I was wrong. It was just more of the same: learning how to answer questions, not learning math.

Then I heard about MOOCs. I signed up for Calculus of all things, and met Jim Fowler from OSU; he started a paradigm shift in my view of math class. He was funny. He made jokes. He drew pictures. I passed his MOOC with a near-perfect score. Did I learn anything? Well… I learned it was possible to have fun in math class, which was something new for me, and something important. It’s what I needed to learn at that time.

And I decided to enroll in another math course: Keith Devlin’s Mathematical Thinking. This is the one that changed everything. It’s the first time I’d ever heard someone say, “It isn’t about getting the right answer; it’s about understanding the mathematics that’s happening.” First time I heard, “It isn’t about doing calculations quickly; slow down and understand why you’re doing every step.” He gave us a link to Ben’s “What it feels like to be bad at math” post. That’s when I started following Ben, and when I got into the habit of looking over the shoulders of a lot of math people.

I took Keith’s MOOC twice. I still can’t tell you what it was “about” (logic? proofs? number theory? real analysis? All of the above?), but ever since, a little tiny Keith Devlin has sat on my shoulder (I hope this doesn’t sound creepy; it isn’t, honest) saying things like, “Are you trying to get the right answer, or are you trying to learn the math?” and “Where, exactly, are you losing the thread of reasoning in this?” The former, shames me out of opening Wolfram Alpha at the first sign of trouble; the second, forces me to formulate a clear question, at which point, I either discover I know or can find the answer, or I can at least ask something beyond whining “I don’t get this.”

I see small victories: just recently, I conquered scientific notation for the first time ever (yes, I really am that bad at math) and then, sigfigs. But I have these huge gaps. Proportions, for instance. I don’t even know what that means, it seems to mean differen things to different people. I seem to be able to do problems that claim to be proportions, but if it’s worded, “A is proportional to B” I’m lost. Like in that video with the perimeter. Then there’s parabolas. Inverse functions. And silly little things you math people take for granted: In a fraction, increasing the denominator makes the fraction… um…. smaller? I had to use an example to do that just now (1/27 is smaller than ½), because it isn’t something I automatically “see” the way math people “see” it. I can “see” an even number; which fraction is larger, not so much. I can muscle through it; does that count?

My small victories, while valuable, are random; things I happen to run into, while at the same time, there’s this huge expanse I don’t understand at a very basic level. I’m not going to claim small victories are the same as “getting better at math” – they’re getting better at certain calculations. But I still suck at math. And I don’t know what else to do, besides what I’m doing. Maybe it’s just too late for me; that’s fair. But don’t let this happen to anyone else. I am the cautionary tale.

I remember you from Robert Ghrist’s MOOC. While I really don’t have much commentary on your math ability, I wanted to let you know that your perseverance and kindness is inspiring. Your “Kiss and Cry” thread got me through half of the course, and after that I simply quit because it drove me nuts. But I will be back for another iteration of the course at the end of May. Perhaps I’ll see you there?

Lee – I’m touched you remember me – and what a nice thing to say! I loved that class; it ran back to back, so I took it a second time right away, but I got stuck at the same place, week 4, related rates, and then when ODEs came into the picture I was completely lost. I would like to take it again, but I have more “catch up” work to do – I have to be a lot stronger in basic related rates before I can follow the kind of tricks he’s doing. He’s another of “my Coursera math profs” – I have this whole collection I follow on Twitter, amazing people; I so want to be a success story for them, but I can’t seem to get to success.

Nice to see you! Funny, how I keep running into MOOCers…

Karen, if you’re reviewing calculus from the basics to multivariable, ODEs, and some advanced calculus, let me recommend one of the jewels of the ‘Net, a free lecture series on YouTube by Herb Gross, originally done in 1970 at MIT for engineers: Calculus Revisited. Herb is still around, going strong in his mid-80s, and is one of the best mathematics teachers I’ve had the pleasure of seeing (and exchanging e-mails with, too). He’s a marvel.

You can start here, subscribe to the channel, find support materials online at MIT, and also find Herb’s recent lectures that go back to basic math and continue through at least algebra.

Michael – thanks for the recommendation! I’m in a “AP review” session on Coursera right now, though I’ve been moving very slowly through it (more of a long-term project than a course I’m going to complete). And I have the MITx calculus sections starting in the fall on edX, not sure if it’s the same thing or related.

I’ve been taking calculus for two years, nearly continuously now. At some point, I should learn something. But again, I’m not fooling myself that I’m “understanding math” – I’m just learning how to do stuff. Which is ok, knowing how to do stuff is good. I just wish I understood it at the level all the teachers here seem to think is so understandable.

I still say – some people are tone-deaf to math. Like me. I can learn to play the notes if I practice long enough, but I’ll never know what the music sounds like.

Karen, Herb’s stuff doesn’t require any formal signing up, so if you’re doing a calculus class online via MIT, it’s not Herb’s, I suspect.

That said, you sound like someone who has had a lot of negative messages repeatedly put into her head about mathematics and her ability to really do it. Of course, far be it from me to claim I know you better than you know yourself, but I’m generally skeptical about those sorts of negative messages, even when they’ve fairly become a way of life for someone. Given my own history with mathematics, I tend to believe that given world enough and time, most of us can learn far more math than we’ve been led to expect. Oh, and it helps to have friends and mentors who can help you find the most useful metaphors, analogies, and lore to make sense of what you’re trying to understand.

You already have insight if you know that crunching numbers to get results is the booby prize. I was a star chem lab student who didn’t understand diddly about chemistry back in fall of ’71. When I withdrew from the lab course, my instructor was shocked. I explained that I was failing the lecture course and saw no way at that point in my life to pass, so I was dropping the lab class, too. What he never saw was that I could follow cookbook instructions really well, had a lab partner who DID understand the chemistry, and that I was such a strong writer that the two of us made one hell of a chem lab student. But I knew that I knew nothing and understood less.

Oddly, my son, nearly 20, is aces at chemistry, and started explaining carbon bonding to me the other night. And in very short order, I saw the underlying math, the basic logic, and was nailing the basics, to our mutual delight and my very great surprise. What a difference 44 years makes, apparently. Who knows what I might glean from a serious study of chemistry now. I just wasn’t ready before now.

I’ve taught mathematics for long enough now officially since the late ’80s when I was approaching 40) to have a pretty good handle on a lot of things that will arise for students in the things I teach and throw them completely off-track. And I have a lot of ideas in my bag to help them avoid crashing and burning, if they’re willing to work with me on finding one that works for them. I think that’s the case for most of us, at least up to some point considerably further down the road than calculus. Keep in mind, too, that I’ve had multiple mathematics professors at U of Michigan tell me that they felt they didn’t really understand calculus until they had to teach it. Most of us who learn it never have to teach it, so can we really be too sure that we understand? I am squeezing juice out of Herb Gross’ lectures that I never suspected existed. He’s really marvelous. And it’s not about number crunching, but about understanding what’s really going on. So do give him a try.

Michael – ok, it’s not as scary as I thought (anything with “MIT” in it scares me to death). And he’s a little goofy, which always helps. What did come up a couple of times was “remembering” like the equation of a circle and binomial expansion (I just happen to remember the binomial theorem because I spent a couple of hours going over the AOPS tapes on it about a week ago, but two months from now will be a different story). And again, the tiny Keith Devlin on my shoulder whispers in my ear: “If you say ‘I don’t remember’ that means you don’t understan something, because math isn’t about remembering.” But at some point, you do have to remember something, or you have to reinvent the wheel with every problem. My memory isn’t what it used to be.

As for negative messages, most of those come from me. A lot of it was just being slow – not in the sense of stupid, but in the sense of needing time to think and figure it out, and in the meantime the fast kids (who I perceived as “smart”) would all have the answer, and the class would move ahead, so I never got the chance to figure anything out, I just wrote down the answer, which may be a big part of why I never learned anything. Another good thing about youtube and moocs: the pause button! I can take all the time I need. But I still fear it may be too late for me. I only have so many years left!

I’ll see how Herb’s lectures go. I still suspect they’re going to go over my head pretty quick. Thanks for pointing them out to me! You’ve been very kind.

I think the only sane answer to your department chair’s question is, “Who cares? And why is this anybody else’s ‘ceiling’ any of *OUR* business?”

We’re not asking kids to solve the Riemann hypothesis, in the same way that we are not asking kids to be able to write the Gettysburg Address or Shakespeare’s sonnets in English class. We are asking them to master the fundamentals of our subject at an advanced level so that they have the tools and the skills they will need to make informed decisions for themselves about how far and how high they want to go with mathematics in their own lifetimes.

Unless you are a skilled mind reader (and I know that *I* certainly am not), there’s no way to know in advance who is going to suddenly have that match-strike moment that causes all of their previous confusions and struggle to burn away and blaze brightly and who is going to burn brightly now but fizzle out later.

That’s why we need to keep an open mind about every student. We’re not here to “sort” the students.

I tell my kids all the time to keep in mind that, “The only one who limits me is me.” That, I believe, is the most important part of having a growth mindset.

If students focus on what is in front of them, they can make the entire journey. The destination they choose for themselves is up to them.

– Elizabeth (@cheesemonkeysf)

I agree with the spirit and essence of what you wrote, Elizabeth, but please let me beg to differ in one particular: I do ask my students to prove the Riemann Hypothesis. Oh, not literally, but I want them to know and consider all the great and little unsolved problems that are out there and that they can invent themselves, along with the idea that the young ones might some day grow up to solve one of them. Andrew Wiles had a boyhood dream of proving FLT. If we don’t help such dreams grow by informing students of some of the landscape, including the existence of “grand challenge” problems like the RH and Goldbach Conjecture, etc., how are they going to give birth to those sorts of dreams?

Michael – I don’t disagree with you about hopes and aspirations! But in implementation, I only hope that they’ll understand the “big idea” of the RH—not necessarily that they’ll be able to prove the whole damn thing and move on.

When I was in high school, my giftedness was in piano, not in math. I could play a lot of Bach’s *very* difficult pieces at that stage of my development, but I was not able to play the virtuoso pieces at that stage — at least, in my own mind, not YET.

My goal was to play whatever piece was in front of me right then — not to be distracted or tossed away because I wasn’t at the highest levels of my craft.

That’s how I see my students’ development. They are developing and growing on a continuum. Some of them are at the “Anna Magdalena Notebook” stage. Others are at the “Toccata and Fugue” stage. Technically speaking, the former is easier than the latter, though musically speaking, I find them both equally rich and deep.

The same is true in the mathematics we are doing. Some of what we do in class is mechanical, to be sure, but some of it is so simple and profound it takes my breath away.

THAT’S what I want all my students to take away. I did not mean to inadvertently suggest a new sorting mechanism. To me, it’s all on a continuum.

– Elizabeth (@cheesemonkeysf)

I actually have never discussed the RH in my classroom, but I do throw out lots of other problems well beyond the level of what we’re discussing, as long as I can help make the digression relevant in one way or another, and I’ve found that younger kids are the most open to unsolved problems (I have Devlin’s book on those million dollar problems and the grade 4-8 kids I taught a couple of years ago were very interested in reading/hearing about some of them). The Goldbach Conjecture is far easier to talk about (for me at least), and there are “lesser” unsolved problems the statements of which are accessible without a lot of higher math (FLT was the classic and is still fascinating to many students despite having been solved: I love pointing out that mathematicians hold out the hope/belief that a more accessible proof is “out there” waiting to be discovered/created. I also have some personal anecdotes with which to flavor any conversation about Wile’s proof, including a great little anecdote about Wiles and the late French number theorist Andre Weil.

Getting back to the bigger question on the table: I think it’s important to have a binary vision on the ceiling business. If you don’t know any or much math past calculus (and more to the point, the sort of mathematics one just begins to get a taste of in Keith Devlin’s “Mathematical Thinking” MOOC, it’s hard to see just how much and how fast the price of intellectual poker rises when you leave the realm of (mostly) computational mathematics and start exploring abstract realms. It doesn’t take long to get into the enormously esoteric. Pretty much any excursion via Wikipedia into some upper undergraduate or graduate topic in mathematics will quickly leave one’s head spinning: so many definitions and symbols to deal with, so many truly new concepts, and so much of it difficult or impossible to connect easily with other things you know in math or with real-world applications (at least hard to do that quickly). So if you’ve never played in that realm, it’s a little naive to think that many people are going to spend time playing around above the trunk of the tree of math. Even people who use a lot of math in their work may not stray beyond some of the lowest main branches. And of course as you go out towards thinner branches, you find yourself in places where only a handful of people have enough background and wherewithal to play meaningfully. That’s because the amount of math that’s been developed in the last two centuries or so is vast and almost entirely lightyears beyond school mathematics and calculus.

And on the other hand, if you let that intimidate you, you’ll quickly find lots of reasons to give up. I find that many kids already have found plenty of reasons to sell out their mathematical future by about third grade. It’s frightening. And I blame our entire approach to teaching the subject starting in early elementary grades. We do it almost entirely wrong, focusing on speed and accuracy of arithmetic computations to a point that borders on the insane. We convince kids that math is for the elite, and the criteria we use to sort them (and help them sort themselves out) are often not valid.

Since I’m not alone in realizing that, it’s not surprising that a lot of folks teaching math want to err on the other end of things: they would cut off favorite body parts before stating that there’s a limit to how much or what mathematics any kid can learn. I empathize and to a certain extent I agree. And I see no percentage in telling kids what I think is difficult or where their limits might be. But people can be very strange.

My first high school math teaching job was at an alternative high school. Since humor is a big part of who I am and how I teach, I put up that poster of Einstein where he is quoted as stating: “Do not worry about your difficulties in mathematics. I can assure you that mine are still greater.” I thought it was inspirational. They took it as a put-down. I never saw it coming and was simply gobsmacked at their reading of that quotation. On the one hand, they hated math and resisted just about everything I asked them to do as if I were trying to suck their very life’s blood out of them. On the other, they were enraged at the thought that some crazy-haired dead guy was (they thought) putting down their mathematical abilities. Go figure.

Dr. Gordon Hamilton over at Math Pickle thinks unsolved problems are good for math classes. http://mathpickle.com/K-12/$1,000,000_Problems.html

Do you have your students actually work on unsolved problems or invent their own?

I like that MathPickle site a lot. I have in fact had students work on unsolved problems. Depends on the students and the courses I’m teaching. Unsolved problems with eager younger students is a load of fun, as long as the problems are understandable. And of course, any problem can be thought of as unsolved if you haven’t solved it and don’t instantly know how to do so, or if you don’t know whether it’s been solved.

I get uncomfortable with conversations like this because the terms are so loosely defined. The question of whether or not there is a natural ceiling for a person seems to be based on whether or not they’re physically capable of understanding certain math levels “even if they work hard.” What’s the measure of hard work in this? Karen is offended, above, because to say that one can learn any level of math if only one works hard enough implies that she isn’t a hard worker. She doesn’t appreciate the implication that it’s “her fault” that she didn’t understand.

Without knowing her, I assume she means she worked the hardest amount within what was possible in the framework of her life. I assume she didn’t quit her job, or quit traditional schooling and spend the rest of her life traveling around the planet to visit thousands of different mathematicians, continually training every day with them to the exclusion of all else. She means she just applied the most effort in the most time she had and didn’t slack off in any way.

So what we’re really talking about is understanding math levels within an amount of effort available to each person, which is probably a different rate for each person. To create a unit of math efficiency, we’d need a unit of understanding per unit of effort. The unit of “effort” could perhaps be a unit of focus per unit of time.

We could assume that certain people do have naturally different levels of math efficiency. It could take more time and effort for some people than others. In all of the anecdotal evidence presented in this conversation, it seems that really what’s happening is the math understanding isn’t worth the life-effort at whichever point in their lives they dropped off, not that understanding beyond that point is honestly biologically impossible.

So, you seem to be trying to address something fundamental in the math efficiency we’re teaching kids, to keep the efficiency value at a rate that will still create understanding within the amount of effort people will be able to put in as they age. That seems more realistic than assuming that people will be able to apply infinite effort to math understanding, certainly. But still, it doesn’t seem right to assume that there is a concrete, natural limit to math understanding regardless of effort applied.

I agree there are definitely differing abilities when it comes to grasping math concepts. But I believe the question is “is there a formal stopping point” a ceiling, where absolutely no one can penetrate? Obviously, we will never know, as math is infinite-and actually, math is mostly silly, possibly made up stuff, because who would need to know half of it and why? but math is a skill and a concept and I believe anyone could be a rocket scientist or mathematical genius if the desire, motivation and appropriate teacher were assisting them.

I see it a bit differently–it’s not an absolute wall, but it’s a point where further progress comes only with great and increasing effort–exponential rather than linear. Pushing into the exponential range of a curve is usually not a productive use of resources and is usually only done when there is no other option.

I agree with both Karen Carlson and Michael Paul Goldenberg, specifically MPG’s statement “there are ceilings for everyone that we’re bound to come to sooner or later. But I think that they may be (mostly) penetrable for people under certain sets of circumstances.”

Some of your earlier respondents on Twitter said there’s no subject they (or, presumably, anyone else) couldn’t learn, given enough time, persistence, interest, and the ideal upbringing. But think about what their statements imply: they are basically implying that every human being who has ever lived is born equally good at math, barring a pathological brain defect or genetic anomaly.

They might object, “No, not equally good; if a genius understands something immediately and I understand it after years of study, then the genius is smarter, but I still understood it.” But we can’t just keep extending the time horizon forever, because eventually we would die without having understood lots that we were trying to understand, while mathematical geniuses grasped all of it before finishing grad school. Saying you can understand something after (practically) infinite time means you can’t understand it. That is a practical ceiling.

If it takes you a lifetime to understand a deep, difficult branch of mathematics, then maybe there was no theoretical ceiling preventing you from understanding it, but we can say that time is the relevant barrier, then. There is a “time ceiling” that you could not break — or maybe a “time floor” that you could not get under.

And the opportunity cost is relevant: What might you fail to learn by focusing on a narrow subject for half your life out of an obsession to understand it, or maybe out of a desire to prove me and Karen wrong? If a genius mathematician can understand that and much more and have a normal life and career, but you can only understand a fraction of it through years of narrow focus, then maybe you have a “breadth ceiling” (barrier, whatever) — your mathematical abilities were such that you could only understand a certain number of difficult concepts in a lifetime, while others could understand twice as many, and more easily. Again we see a practical ceiling: your abilities did not allow you understand certain subjects that others could, despite intensive effort.

“But I could have understood any one of them, if I had put enough time and effort into it” is not a rebuttal to my point. If our abilities and our mortality only allow us to understand a fraction of what a better mathematician understands, then we have a lower mathematical ceiling than that person.

Either way, it is simply not true in any meaningful sense that anyone can understand any and all mathematical concepts. This is not only because there isn’t enough time, but because people’s brains are different enough that some people will be better and worse at different things. To suggest that if every human were exposed to the ideal learning environment — or merely the same environment — then they would all understand every subject eventually is very wrong, as far as I can tell.

Fawn, in it to Nguyen it baby!

Very nice post. I hate it. I don’t want there to be a ceiling.

Your argument seems to be that unless teaching is really really good (perfect?), children can be stamped with their math expiration dates. And I would say it that if the potential for learning maths is a function of each teachers perfection level, the decaying exponential function seems to agree.

But there is my rub. I believe this post should be rewritten to say School Math in place of math, every time. School Math is something we want other people to know. It refers to some idealized, dehumanized knowing. If School Math is the goal of Maths Ed, no one will ever reach that ceiling. Or if they did, who would know.

I contend that the drive to teach “School Math” is the crisis of our profession. (But it remains so solidly in place, why? Because it’s priests kill off the blasphemous.)

If maths were reconstituted to be the activity of the human mind, none of us would have a mathematical ceiling. In fact all of us would invent new maths; we would not only be generative mathematical thinkers, but we would be mathematical authors. And in authoring, we are inventing. And as inventors, we have crashed through anybody’s ceiling they, as some sort of math priest, has determined we strive to expiate.

My most sincere apologies to the blog author, I mistakenly thought this was written by @fawnpnguyen. So please ignore the first sentence!

No apology needed – having my writing mistaken for Fawn’s is one of the most flattering things that’s happened to me!

My heart wants to say no ceiling. But I do think there is talent, too. I don’t think I could hit the top end where my advisor worked. But I was able to get to where I could see where he’s at and appreciate what he can do. It’s amazing how guilty I feel saying this, because I am signed up completely on the growth mindset. But if there’s a barrier between Nigel and I, are there barriers for others?

As much as I loved basketball, I was never going to play in the NBA. Could I have gotten good enough to play in college? A small college? I don’t know. Those guys that are my height were a lot faster than I was, and could physically do things I could not.

I would love someone to convince me otherwise on this.

However, I do totally believe that everyone can do meaningful mathematics, and everyone can enrich their lives through the math that they can do. I have never had a student that I thought was incapable of achieving an A in a class I taught. (And I’m old & at a university.)

I had a great uncle (he died in 2013, well into his 90s) who played for St. John’s under the legendary Joe Lapchick and was an original member of the Boston Celtics, retiring right before the merger that created the NBA. He’s in the Jewish Athletic Hall of Fame. And he was all of 6’0″. Pays to be in the right place at the right time: late ’30s to late ’40s.

So absent that good fortune, no, you couldn’t have been in the NBA.

Oh, you want convincing on something about math? 😉

John, I really believe that at some point, whenever you really want to do a thing (including mathematics), you have to just sit down, shut up, and do it. I believe that process is more important than progress. I mean, if I don’t actually glue my butt to the chair and just do it, the mathematics (or anything else) will not get done by themselves. We really get ourselves tangled up in monkey mind if we allow ourselves to get too caught up in measuring, comparing, and sorting ourselves or others. Mathematics is a human activity. If you’re a human being, you can do it. The same is true for writing or music or sculpture or basketball or finger-painting or woodworking. I think we need to get ourselves psychologically out of the business of trying to manufacture the next Fields Medal winner and get back into the business of nurturing creative, deep, passionate mathematical activity.

I believe that if we do this, everything else (including the Fields Medal winners) will all unfold in a much healthier, saner, and more sustainable way. And we’ll have a more numerate, mathematically engaged citizenry.

I believe this is a powerful and important goal for us to aim for.

– Elizabeth (@cheesemonkeysf)

Really interesting to read your comments, all three of you – wish I had more time to reply, but please trust that I’m taking it all to heart.

Elizabeth (so excited I know your name now!), I think I essentially agree with you: the question of “do you have a ceiling?” is not really productive to ask or answer. Maybe you do, maybe you don’t, maybe they’ll bring back those light brown M&M’s someday – in any case, you’re best off just doing the math and seeing what happens. Nothing is foretold.

Michael and John, I think I agree with you guys too. There are six million factors that will eventually halt our progress in mathematics, among them dwindling motivation, lack of time, end of financial incentive, fading curiosity, and conceptual challenges that don’t feel surmountable (or worth surmounting). No reason to lie to students about those things – especially since, with proper teaching, the factors that will halt most of us (time, motivation, etc.) have nothing to do with “how smart you are.”

Another answer to the question, “Do we have mathematical ceilings?” would be, “Well, none of us talking here will ever read the Fermat-Wiles proof. But we can all learn some difficult, exciting, powerful stuff. And only time will tell how far you want to take it – maybe further than you think! Next question.”

It’s probably worth noting in this regard that math textbooks of the advanced kind can be misleading or simply mute on what prerequisites you need to bring to the dance before twirling around the floor with them. I just realized that a little book I have on topology that’s supposed to be introductory is way beyond what I expected and starts nowhere near where it should.

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It’s very cute.. asking people a mathematical question about their limits of mathematics.. “Given these sets of theorems and beliefs about people, what is the ceiling level of mathematics for person P? Show that the ceiling level is either convergent or divergent.”

Go further with one of the analogies.. a student can be eliminated from the math NCAA if they don’t constantly practice and train.. math really is another language.. without practice in speaking or writing in the language of math.. even if a student is at the “final four” level.. after a few years.. the same student if playing a rematch tournament may be eliminated earlier on.. perhaps this math NCAA analogy isn’t appropriate.. i get it.. it’s cute and useful.. but.. perhaps a different metaphor can properly capture all the things that happen when someone learns math and reaches a limit after a while..

If you consider that mathematics pretty quickly enters the realm of metaphysics insofar as it requires a proverbial leap of faith, then I would venture that, as with the notion of the existence of a deity, there are probably quite a lot of people with the potential for increased capability but whose intellectual integrity may never permit it. That is, my ability may be constrained by my reluctance to suspend disbelief at some point along the curve that requires acceptance of “givens”.

Futons apparently have this amazing capability to analyze human behavior and offer insight. I agree with your points, and I, too, tend to be more realistic in my expectations. It remains to be seen how we can successfully ‘raise the ceiling’ (or ‘raise the roof’), but I believe that most young students can do it. Great teachers (preferably those who do not insist that the students are the cause of their own failure) are real catalysts for monumental change, adjusting student attitudes over time in what can only be described as fractal chaos.

It was a little different for me. I never hit a ceiling where I “couldn’t” do the math. I hit a ceiling where I didn’t care any more. My degree required multivariate calculus and differential equations. I aced them and never took another math class.

Ah, the crucial missing concept. When a kid is out of school for a day no one thinks it’s important. But if that is the day the teacher spent the entire class on a crucial concept, that may be the missing piece that dooms the child’s progress in math.

That’s probably what happened to me.

I don’t want to deny your experience or speculations about it, but I doubt that holds true for most people in the big picture of whether they see themselves as capable in mathematics. Few people’s personalities or intellects stand on ground quite that precarious.

On the other hand, whenever students leave class early, I always tell them that I’m about to teach the single most crucial concept in the course and that if they miss it, they’ll be hopelessly lost for the duration. Oddly, few believe me.

Well, there may be other contributing factors to my limited mathematical abilities.

Maybe I didn’t get it because I didn’t study enough or it was beyond me.

I actually don’t think there’s a ceiling- isn’t it about the curriculum focusing also on what the concept is about and examples of real life application instead?This would allow students better conceptualisation and relation to what we are learning. This is a very simplistic example (and i could stand corrected in more abstract higher level math scenarios by more complex ones) but if we were given more time to explore and understand that liner equations help identify relationships between two variables, that when we differentiate we are identifying increasing/ decreasing functions etc etc, then students would understand? How can there be a limit to conceptualisation in the brain- surely it’s about different speeds of understanding rather than a limit?

My own experience with math: i could never push myself to fully comprehend math concepts. I aced tests only by starting off with practicing and rote memorisation, before gradually (and partially) understanding concepts, which got me by. i was never comfortable with extending beyond the syllabus and exploring implications. while the author of this article may say it is a ceiling, i just think the testing and teaching methods don’t support a stronger foundation in conceptualisation for slower students. Which may be just as well, since only those who have talent would be great and care enough to go on in the field- but this does not mean there is a ceiling.

We never dealt with concepts or theories in class. Just how to work the equations.

We didn’t get to word problems until Middle or High school. I think that is way to late.

As a long-ago student I think there is a ceiling.

Up through the lower levels of calculus math made sense. While I had to learn it it was then obvious why everything worked. Past that point, though, I could “learn” the material in the classroom sense–pass the test. It was simply applying formulas, though, it never made sense why one did things and it was a real struggle rather than nearly effortless.

In the decades since I have pretty much retained all the parts that made sense back then–I very well might have to look up a formula but I understand how and why to use it–and I have totally lost everything that didn’t make sense back then.

Us students see the point where it becomes cryptic and say there’s a ceiling, the teachers see us progressing (I was able to pass the classes) and think there is no ceiling–because they’ve never hit one.

This gets down even deeper..

Into how novel problems are solved by our brain. You cant ‘force’ your brain to solve a problem. It just happens. Ten people can have the same information and only one might ‘see’ the solution.

We dont understand how it works.

Its not just math either.. songwriting, business, the first caveman that used a stick as a spear.

Its the same as intuition that Magnus Carlson has at a chessboard or hendrix on a guitar. There’s many people who have berkely phds and have done as much math as Einstein. Theres people on every Friday night that have technical skills as good as Hendrix.

But that mysterious intuition.. their brain ‘connects’ the information in a different way. No one understands how that works.

Thats the wall youre talking about. And im sure it exists as many levels. And is probably some unique confluence of environment and genetics.

Intetesting post.

Great, great post. And the comments section deserves its own post…this really is connecting to our “lizard brains” as both students and teachers.

My two cents:

1–I hit my “ceiling” because of curricular requirements and grading. Altering the sequence or timing of what we ask of students, and also how we respond to the attempts could really change the game. As a student, I struggled to get it “right”, and it took longer and longer with each attempt, leaving me further and further behind the rest of the class. After a few years of this, even sitting together every day, and taking the same tests, I still wasn’t “with” them.

2–Please remember that these conversation about no limits are talking of neuro-typical brains. Those of us that work with neuro-diverse students run into ceilings all the time….but they are largely the ceilings that the teacher puts in place, with a specific order and way that a problem has to be done to show understanding. Again, coming back to the idea of looking at math teaching and assessment differently could remove the artificial ceiling barriers.

Thank you for this, and commenters for your stories. 🙂 I am an Algebra 2 expiration date from 25 years ago, now relearning and teaching it in my pull-out class. Imaginary numbers no longer make me cry, and I have successfully used the quadratic equation…just took a lot of time!

I really like several of the points that have been made. Notably:

1. A mechanical understanding of math instead of a conceptual understanding of math will place a cap on a person’s ability to exceed a certain level of math.

2. It takes different people different amounts of time to learn math conceptually

3. From Karen (I apologize if this paraphrase is inaccurate), mental habits affect our ability to learn math conceptually

4. Interest in a subject increases one’s ability to focus and thus learn new concepts

I think that third point really needs more weight, but possibly in a broader sense than Karen intended. To explain, here is a bit about how my math ability and what I perceive as its origin. I am easily top 1% or better in my ability to do Math. As a concrete example: when taking Algebra I in school, I started being able to intuit solutions to problems which are only addressed in Algebra II. Now the bit relevant to the ongoing discussion: Its my perception that this is because of mental habits I’ve developed over the course of my life. Not mental habits which apply only to math, but mental habits which influence how I approach every aspect of my life. Everything I encounter is a constant part of my mental habit of breaking concepts down into their component parts, and then reconstructing them such that I have an inside-out understanding of it. This is not to say that I’m particularly smart, but rather that the continuous (and I mean that very nearly literally) habit of decomposing ideas into their constituent parts, has greatly developed by ability to do that in any domain I encounter. This is not to say that this mental habit I have is 100% a good thing. It makes me very, very good at conceptualizing things, and very VERY bad at processing some things (like emotions) in real-time.

So the point I’m trying to make is, each person is the result of the mental habits they have developed over the course of their lives. These habits either increase or decrease our ability to do many different things. Attempting to attain (or reverse) the affects of 20+ years of a certain mental habit very well may take 20+ years of a different mental habit, and that assumes that one can simply turn on the needed habit(s). I’ve spent a great deal of time trying to improve my ability to emotionally interact with others, and its taken me 15+ years to get to the point where I can do that in near real-time (I can usually process relevant emotions in under 10 min, but not always).

I don’t know if this is discouraging or encouraging from a “ceiling” standpoint, but I think it does give a means of understanding ability as something other than being either hard work or innate talent (although I’m sure both of those have some effect as well) while at the same time addressing the question of why some people appear to have a higher ceiling than others. 31+ years (in my case) of mental habits is a high barrier to cross.

@mydogisbox

Dear Mr. Orlin,

A friend of mine sent a link to your article. I am copying below the comment I made to him. I thought to trouble you with it because your article indicates that you care very deeply about education and have thought about it in many ways (certainly more than I have). I hoped therefore that you might appreciate an alternative view.

Sincerely,

Minhyong Kim

_____________________________________

Dear Deane,

The article has some good points, but I have to say it agrees only marginally with my own experience of learning mathematics. An alternative view, I suppose, is expressed by the von Neumann quote: ‘In mathematics you don’t understand things. You just get used to them.’

If we pursue the ‘construction analogy’ used by the author, my understanding of mathematics has always been more like a rickety house than the author’s futon. I start out with just one or two comfortable rooms. Some others are adequate, many are draughty, some have broken windows. For a long time, many rooms have no functioning lights, but still are OK part of the time. Of course it’s important that the whole structure has enough structural integrity to be more or less safe for dwelling in, that there is a kitchen and a bathroom that are serviceable, etc.

Some people may prefer to build one room at a time, making sure each one is really solid. But I quite like my large faulty house: the different parts still complement each other in some organic way and I hope to get to working on most of them by and by. Maybe some of them will always be a mess, and I’ll have to wait for someone else to take over (past my own expiration date).

In any case, I’d rather not have an architect come by, look at the bad plumbing, then tell me I have to tear everything down and start over.

But Prof. Kim, with all due respect, I don’t know how much this article is meant for mathematicians. Surely there is big difference between a high school student struggling with the calculus and someone struggling with spectral sequences? I feel, after a particular stage of mathematics education, one’s ability to grasp known math becomes potentially unbounded and the difficulty, instead, comes not from manipulating available furniture, but fashioning new ones.

(btw, thank you for your wonderful answer on Galois groups/Fundamental groups on mathoverflow)

Dear Santaraxita,

I think my own education was haphazard in this way right from the beginning, long before I was a mathematician. But of course, memory is short. I’ll grant that determining what kind of learning strategy is useful for which groups is very difficult.

In order to make clear what my main objection was, perhaps it will be helpful to reproduce the discussion I’ve had with Deane Yang so far. (I’ve also edited a bit the initial comment.)

Best,

Minhyong

—————————————

MK: The article has some good points, but I have to say it agrees only marginally with my own experience of learning mathematics. An alternative view, I suppose, is expressed by the von Neumann quote: ‘In mathematics you don’t understand things. You just get used to them.’

If we pursue the ‘construction analogy’ used by the author, my understanding of mathematics has always been more like a rickety house that I keep extending than the author’s futon. I start out with just one or two comfortable rooms. Some others are adequate, many are draughty, some have broken windows. For a long time, many rooms have no functioning lights, but still are OK part of the time. Of course it’s important that the whole structure has enough structural integrity to be more or less safe for dwelling in, that there is a kitchen and a bathroom that are serviceable, etc.

Some people may prefer to build one room at a time, making sure each one is really solid. But I quite like my large faulty house: the different parts still complement each other in some organic way and I hope to get to working on most of them by and by. Maybe some of them will always be a mess, and I’ll have to wait for someone else to take over (past my own expiration date). Presumably, one of these days, the rate of expansion will slow down enough for there to be some hope of order.

In any case, I’d rather not have an architect come by, look at the bad plumbing, then tell me I have to tear everything down and start over.

DY: I don’t think the article has much relevance to the way you learn mathematics. Or even the way I do. At some point a long time ago, we learned about logical rigor and how to detect and fix flaws in it. So you and I can afford to have large faulty houses, because we can fix up small parts of it as needed.

MK: I agree, to some extent. (I’m still not so good at fixing my own flaws.) But this underscores the difficulty: I certainly don’t regard my own learning experience as something special. The key difficulty for the teacher is to figure out which students are like my house, and which like the bad futon. Maybe most people are a bit of both. Then what?

Possibly more precisely, many people are like the faulty house with a few bad futons in them. I’m still afraid of someone tearing down the house, or even a room, because of a futon.

DY: I agree that neither you nor I are able to fix all of our flaws, but we can certainly find parts of the house which we can fix. Students are simply not trained to even detect the flaws, never mind repairing them.

All I can say is that although the analogy is imperfect, the description matches my experience working with at least some precalculus students during what we call “all day workshops”. If you have the luxury of time, you don’t have to tell a student what to do. Instead, you can probe and figure out where exactly the student gets stuck. It’s often a very basic skill or concept that they should have learned properly a long time ago. And surprisingly often, if you fill in that one gap, the student suddenly knows how to do the rest of the problem on his or her own. I had this happen many times, and it was quite satisfying to see.

MK: My experience agrees almost exactly with yours. This is why we should both disagree with this critical paragraph:

‘You may ask, “Since I’m fine now, can’t I add that missing piece later, when it’s actually needed?” Sometimes, yes. But it’s much harder. You’ve now spent years without that crucial piece. You’ve developed shortcuts and piecemeal approaches to get by. These worked for a while, but they warped the frame, and now you’re coming up short. In order to move forward, you’ve got to unlearn your workarounds – effectively bending the futon back into its original shape – before you can proceed. But it’s well nigh impossible to abandon the very strategies that have gotten you this far.

Adding the missing piece later means waiting until the damage is already underway, and hellishly difficult to undo.”

I’m afraid believing this too strongly will lead to a rigid and anxious belief that everything needs to be done right in some narrowly defined way at the very outset. Many foundational misconceptions in technically proficient students (for example) are often easily fixed, and in a pretty enjoyable way for both teacher and student. Not having studied the issue at all, I have only my experience to go on. And as far as I can remember, it doesn’t agree with Mr. Orlin’s description.

Dear Prof. Kim,

Thank you for posting the interesting exchange. As someone who studies by hopping around inside texts and has built up his ‘house’ of math sometimes one randomly placed brick after other (some have even hung/are still hanging in mid-air). I agree wholeheartedly with your last paragraph.

However, to extend my previous point, I think such a way of construction is free of (serious) hazard only after a certain scaffolding has already been erected (even if only of the ground floor) in the good old-fashioned, linear brick-laying way. And with respect to that scaffolding, I feel Mr. Orlin has hit the nail on the head.

Hi Prof. Kim, thanks for writing. I’ve found this exchange very interesting to read. I think your imagery of the house perpetually under construction is powerfully evocative, and constitutes quite a fair and thoughtful critique of my argument.

I have colleagues who feel similarly. They view my construction scheme (finish and furnish every room before adding a new one) as likely to frustrate students and waste time. They favor a construction scheme more like yours: pin down the basics and move on, to revisit and elaborate later with the wisdom gained from further context.

Ultimately, as is often the case with education, I think both schemes can work. The choice comes down to a complex interaction between the teacher’s inclinations, the student’s experience, and the nature of the material at hand. I do have a few general thoughts, though:

1. As others have mentioned, your approach seems better suited once a certain foundation is in place. This foundation is less about specific content than about meta-cognitive abilities: specifically, the ability to recognize that your understanding is incomplete (even if you’re not sure HOW incomplete), and nevertheless to move forward with both confidence and caution.

This is a capacity many students, in my experience, lack. I’m not sure if acquiring it is a matter of age (I find it typically not present until age 17 or 18), experience (perhaps you first need to know what a fully furnished room looks like?), or explicit instruction (being directly taught the difference between a fully furnished room and a half-finished one). But I do think this sort of meta-cognition is a necessary precondition; otherwise students will be unwilling (or unable) to revisit their partially finished rooms.

2. You describe a wonderful type of experience: helping a student fill in a gap, such that their understanding suddenly snaps into place, like a machine coming to life. I have had these experiences too, particularly when I tutored a small group of my students 2-3 times per week for an hour after school. But I do have a worry: that this sort of hole-filling is only possible if a teacher devotes a fair amount of time, energy, and personal attention to a student.

I believe this gives two reasons for a teacher to take pains to avoid leaving holes in the first place. (A) It is much harder to find holes later than to recognize them as they are forming, and (B) As students advance through school, there is generally a greater presumption of independence and a diminished amount of contact with teachers, making such hole-filling experiences less likely to occur.

3. I suspect that my approach applies most directly at ages 12-18, during which time many previously successful math students begin to encounter obstacles stemming from gaps in background (which they perceive as ceilings on their ability).

For more advanced students, my advice probably doesn’t hold much water. An undergraduate following my approach might benefit in certain ways, while suffering in others. A graduate student will probably feel frustrated and slowed down by the compulsion to master every detail – hurt more than helped. And a research mathematician adhering to my scheme would be quite paralyzed indeed.

4. As always, moderation is an essential virtue. Teachers must use their own judgment to discriminate between critical damage to the futon, and mere scuff marks or scratches. For example, when teaching basic trig functions, I push hard: a student who cannot grasp the definition of sine and cosine on the unit circle probably has a deficient understanding of functional relationships, and needs immediate help. But when teaching logarithms, I’m a little softer: conceptualizing inverse processes is hard for students, and if they can sketch graphs as well as solve several basic classes of questions, I’m happy to move on.

Dear Mr. Orlin,

Many thanks for your kind and detailed reply. I can see that most thoughts that I might have about education must be familiar to you! I hope very much that you will continue to share your valuable insights with the community.

Sincerely,

Minhyong Kim

I for one was perfectly happy with math right on through multivariable calculus and partial differential equations. (Linear algebra was kind of messy, but the teacher I had for that course was generally acknowledged to be a bit loopy.) My proverbial “ceiling” was complex analysis.

I propose that as one advances through a university education, one reaches a point where the audience has dwindled from the swollen ranks of fresh entrants and the subject matter has advanced into the obscure and high-level, where there is less opportunity or incentive to refine the presentation material and the only people left to teach might be those professors who would prefer not to be lecturing. And then you end up with Arfken & Weber as a textbook (sample Amazon review: “may God have mercy on your soul”).

I long hoped to one day go back and attempt to master that material (residues and integrals over the complex plane and so on), but incentive has dwindled over the years. Perhaps someone could point me in the direction of a suitable course? (I found it curious in Feyman’s memoirs that he seems to advocate differentiation under the integral sign over complex integrals, but we never got into that either.)

Reblogged this on Cloudier Than Thou.

So, for all of the optimists who see the ceiling as a delusion: what do we, those past our expiration date, DO to find the missing futon part? Or parts; I suspect I myself may have quite a few screws loose. And to focus the question: I’m talking about basic math, through, say, calculus at the college freshman level, AB stuff. Nothing fancy. I’d settle for really solid algebra. And geometry. I like geometry.

Do we go back to the beginning? Where is that, exactly? And what’s the path forward from there?

Do I continue to do what I’ve been doing, taking every math course I can find, doing everything I can to learn instead of rack up points or badges or certificates, and hope I find the missing pieces? I’ve been following Mike Lawler’s kids, as well as a variety of teacher blogs (since they’re written for teachers, they tend to assume a significant level of expertise, but sometimes I can use the recommended techniques), taking every mooc that seems within reason and pestering CTAs with explain-it-to-me-like-I’m-six questions, finding whatever’s available on AOPS when I run into something fuzzy, and now I have Herb’s lectures; I’ve never had a lot of success with learning math from a book, though I’ve tried that, too (Michael Starbird’s “Effective Thinking” was a nice introduction to a variety of branches of math, but not a whole lot of very basic stuff)?

Is there some other path, or is it just “tough luck, kid, but at least you know how to handle scientific notation and sigfigs, and just what is it you need math for at your age anyway”?

This might be helpful.

http://projectlearnet.org/tutorials/concrete_vs_abstract_thinking.html

The fact that its associated with recovering from Brain injury is particularly helpful since those people are probably having to relearn everything and thus the suggestions assume you are starting with no background experience/knowledge.

I have master’s and PhD degrees in engineering and studied and used graduate level mathematics in both my master’s thesis and PhD dissertation research (abstract linear algebra, stochastic processes, convex programming, a few other areas).

I suspect that anyone who thinks or claims that there’s no “math ceiling” simply hasn’t hit their own ceiling. I think the ceiling occurs even for the highest and most abstract levels of pure mathematical research, where, for example, other pure math theoreticians are struggling to understand Mochizuki’s claimed proof of the abc conjecture. Maybe one could claim that Mochizuki has no ceiling!

I hit my own math ceiling pursuing a thesis master’s degree in feedback control theory, when I realized that the PhD students had the ability to think through function spaces in a way I couldn’t get my head wrapped around, if you will excuse the partial pun. Realizing that I wasn’t going to be able to do theoretical research in mathematical areas of engineering was actually quite freeing, and I moved into a more applied area.

When I tutor or teach mathematics, I find that students appreciate the notion that everyone tops out at some point–even their teacher–and the goal is to see how far you can get. It’s less intimidating than being told that given sufficient time and effort, anyone can understand any mathematical concept. And then each time the student experiences a conceptual breakthrough they realize that maybe they can climb just a little higher.

We can’t all be olympic marathoners, but we can all work to run further and faster.

Love the hypothesis, Roger.

I wonder if this speaks to confirmation bias – or just the human desire to believe in dreams.

Dreams? What is she talking about now, this goofball who isn’t even a teacher so what does she know?

At the end of those singing competition shows (oh, come on, like you’ve never watched them) the winner says, “See, dreams do come true if you keep trying and work hard.” Ok. Except there are 12 other people who were eliminated over the past 12 weeks. And there were 100 people featured in the auditions who didn’t get onto the competition part of the show. And 1000 people who auditioned but never got a minute of screen time. And 100,000 people who never made it into the audition room at all. Out of 100,000 people, one person’s dream came true. Were the others lazy slackers?

Yes, the numbers are fictitous, but it doesn’t matter if it goes from 1000 to 1 or 10,000 to 1. The fact is, one out of 100, or 1000, or 10 dreams coming true does NOT prove that dreams come true. But it makes for a great soundbite, so we all believe it, and we take that narrative and weave it into less competitive (but still competitive, don’t kid yourself) arenas, and wonder what’s wrong with us. And we start yet another algebra course. Because we want to believe. And if the price is “it’s my fault if I didn’t get it”, we’ll pay it.

“We can’t all be olympic marathoners, but we can all work to run further and faster.”

That’s perhaps the most useful point I’ve read here so far. Worrying about where the ceiling for any given person “really” is strikes me as a waste of time for the vast, vast majority of people, few of whom will ever be taking courses on abstract function spaces, hyperreal numbers, transfinite arithmetic, etc. There’s so much mathematics out there that the speculation is that the last human being who was reasonably conversant with all the areas that were known was Leibniz back in the 1600/1700s. But the number of people who bail on mathematics well below their likely ceiling (which is probably not completely fixed anyway) is legion.

Maybe a good litmus test for all of us is to react to/interpret that Einstein quotation: “Do not worry about your problems in mathematics: I assure you mine are more difficult.” I find that inspiring, but as I reported previously, I’ve had students who took it as an insulting put-down. How do you read it?

I don’t think it’s valuable to try to assess “blame” for people hitting what they perceive as an unconquerable barrier to their mathematical progress and growth. Everyone is different and comes to the mathematical table with a unique set of experiences, some of which make learning math easier and some of which make it more challenging. And not everyone is going to be interested in going far down the mathematical road, into obscure or even unexplored territories. So let’s not waste time with tales about grit and genius other than as interesting anecdotes that might apply to us and might not. Instead, we need to do better at getting people to have a realistic understanding of what it means to do math and to be “good” at doing math. It ain’t about speed-computation and robot-like accuracy. It’s about an entire way of thinking (actually WAYS of thinking) that is different from many other ways of thinking (e.g., literary thinking, musical thinking, etc.) that takes enormous amounts of time, effort, interest, and commitment. Not everyone chooses to put that sort of energy into mathematics. Why should they? Because I like it doesn’t mean everyone should. And because I’m good at doing arithmetic and algebra in my head doesn’t mean that that is THE measure of being good at math (it isn’t, though it’s fun and sometimes pretty useful). We’ve done ourselves and our children an enormous disservice by swallowing and promulgating a huge body of lies and myths about mathematics. The first step to undoing that is to stop buying into the bull. No one is going to learn all the math there is to learn. We’re past the point in human history where that’s practicable. Instead of worrying what you “can’t” do, start having some fun. Math can be playful, like most human activities and disciplines. Time to start playing and stop worrying so much.

Michael – I do play. As much as I can. Following Ben’s blog, trying to figure out the more mathematical references he makes (what is it that would prevent someone from understanding parabolas?) is playing. And I play on my own – I’d show you one of my playgrounds, but it’s… embarrassing to show real math people, or even regular people, so I keep it hidden. But I play, trust me. Most of my frustration comes from not being able to play more.

Have you ever wanted to play a game you just couldn’t play? What if you tried to play baseball, but you couldn’t hit, couldn’t pitch, couldn’t throw, couldn’t catch? Is that fun? Community leagues try to make it fun, but we all know it isn’t fun at all, to constantly drop the fly, and strike out, to have teammates groan when you come up to bat. Not fun. And what if the coaches and your parents have tried to help, and you just can’t do anything? You’d take up soccer. And when you find you can’t do that either, you try roller skating, and by golly, you discover you can roller skate up a storm in your driveway – but you can’t really go anywhere because the road is much rougher than the driveway and takes ssoooooo much time and effort to go just a single block so you never get where you’re going before sundown, and you find out you have the wrong kind of skates for the rink, so you rent the right kind of skates but you can’t seem to get the hang of it even after taking the beginner classes – you still fall down, and get run over by the “real” skaters. So you try to make the best ouf of your driveway and the wrong kind of skates, but it gets old after a while; you can’t go anywhere else with it, so it’s a little prison. Not fun.

That’s me, where baseball = algebra/calculus and soccer = geometry and roller skating = Khan academy and all the other courses – including courses leaning heavily on problem solving rather than skills – I’ve taken where I may be able to “pass” (whatever that means) but I can’t access whatever it is I’ve learned in a useful way anywhere else.

I’m stuck roller skating in my driveway, watching the baseball players and soccer kids and even my friends on their way to the roller rink. I can “play” all I want, but I just keep going around and around the driveway, over and over. I haven’t found a way to get outside the 50 square yards of asphalt. And I want to, very much.

I’m willing – eager – to see your POV. I may be the person who most wants to be proved wrong in this entire thread. So how do I expand my playground? Because I’ve been trying for a couple of years now, and I don’t seem to be getting very far.

Have you ever looked at Kalid Azad’s Math Better Explained (http://betterexplained.com)? And if so what do you think about his way of looking at things? I particularly liked his Understanding Calculus With A Bank Account Metaphor article and some of his discussions on imaginary numbers.

Karen, you probably need to show your playground to someone trustworthy who can give you some ideas about what to try to expand it. Sometimes, we’re so close to breakthroughs and don’t realize it. The image that haunts me is the person who gives up just inches from cutting through the mine wall to freedom. I don’t want to be that person. But I’m not going to be the one who knocks himself bloody and unconscious against the wall, either. I take breaks from things that are stymieing me and come back to them. I try something else. Or step away from math entirely for a bit. No one’s looking but me, so why should I worry?

I’d love to have buddies to play with, but as I mentioned way earlier, I don’t. It’s a little dangerous to look for help in some quarters, so I’m picky about playgrounds. If the venue doesn’t feel safe to me, fuhgeddaboudit.

When I was doing graduate work in math education, I did have friends in math classes with whom I could work on homework, study, etc., and some of the experiences I had with that were very rewarding. Finding situations like that again would be a plus, and I suspect you would benefit from having people you trust to do some exploring with. It can be productive to work with people who are at various levels in relationship to you: some who can use your help, some who can help you, some with whom you can share and work together. Each has a lot of potential for growth. But regardless, trust is vital. Can’t work without it, in my experience.

Thanks, Brian, I’ll check it out!

Dear Karen,

I resonated with what you wrote. I have often said that if play has to contain an element of joy, most kids do not play Little League baseball. Instead they “work” it. That is, they worry about making an error or their parents and/or teammates being disappointed etc. In a similar way too many students “work” school rather than “play” (that is, enjoy) school. In sports we recognize that great players seldom become great coaches. Yet academia usually hires its coaches (that is, their instructional staff) by who well they played (that is, by academic degrees, publications, etc.). One of my favorite teaching adages is “Never think so much of your subject that you neglect your subjects!” and this is closely followed by “People don’t care how much you know until they know how much you care”.

I hope you have time to look at my “Calculus Revisited” videos and/or the videos and power point slide shows that appear on my own website http://www.mathasasecondlanguage.com and that you will let me know if my approach was of any help to you. As an aside I feel that in many ways, video lectures can be more helpful and/or user friendly than live lectures because they can be viewed at a pace that is comfortable to the viewer. In a live class the pace of a lecture is too slow for some and too fast for others. But thanks to the “pause”, “rewind” and “”fast forward” keys viewers can Adjust the pace of each lecture to their own individual comfort level. If you would care to, please feel free to write to me at hgross3@comcast.net.

Herb – How nice to “meet” you after hearing about you from Michael. As a dedicated MOOC-er, I’ve been very lucky to run into several math profs who’ve shown me it’s ok to “play” with math. And then there’s Ben, our blogger-host here, who always manages to get a smile out of me. I took a look at several of your vids – the intro and lesson 1 from calculus (I think I need more background, particularly in trig, and even more fundamental areas) and the intro to arithmetic. I’m going to visit again soon.

It’s always great to meet someone else who is so generous with time and talent – thank you so much!

GREAT article – but I was surprised not to see the term “threshold concepts” in the article or subsequent comments. I think that is the “ceiling” you are referring to. I’ve found with my own children it’s pointless to keep on trying to explain something without going back to understanding what concept they’ve not “got” or misinterpreted along the way. Once you discover that, you can address it and everyone can move forward (http://mathforum.org/blogs/max/threshold-math-concepts/)

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http://www.p21.org/news-events/p21blog/1639-readying-students-for-todays-mathematical-challenges

I recall a study I read some 30 years ago or so, which posited that most people have two “math plateaus”. Up until the first plateau, math is almost obvious. At most, you just have to be shown once, and then it’s trivial. From that plateau to the next, learning new math requires work, but it’s no harder than any other discipline. Hard work may be necessary, but it’s just hard work. But starting at the second plateau… any math beyond that causes actual, physiological signs of panic. It’s next to impossible to learn. And everyone’s plateaus are at different levels. I think the people who claimed that there’s no “ceiling” simply haven’t reached their second plateau yet.

nature vs. nurture. Their isn’t really an easy answer to this question.

Reblogged this on Expecto Patronum!.

Everyone has a ceiling, but it is usually much higher than they think and partially self-imposed to boot. In addition, our one-size-fits-all approach to teaching mathematics combined with social promotion sends students at all levels forward without adequate understanding of what they need to proceed. In my experience, starting from mathematics in grade 8 up through my 32 years as a university mathematics professor I have seen this at work, and it only seems to stop at the level of graduate school mathematics. My department has endless debates about what grade in course X is acceptable if course X is deemed pre-requisite for course X+1, and this is not just at the level of pre-college and freshman/sophomore level courses. Even my colleagues don’t grasp the idea that 75% of 75% is 56.25%, so that a C in one course will likely mean a D or F in the next one. Just yesterday I had a student in trigonometry who knew the relevant sub-identities but failed to establish the one at hand because he believed that (a/b) + (c/d) = (a+c)/(b+d).

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I feel a bit like a voyeur in the sense that I often surf the internet to look for comments about my “Calculus Revisited” video course as well as for comments on work that appears on my own website (www.mathasasecondlanguage.com). In my surfing I came across this blog and even though I have never thought about the concept of a math ceiling per se, I think that many of my own subjective thoughts address the idea. For example, a basketball coach will never say to a 5 foot 6 inch tall player “If you were serious about playing basketball you would have grown to be 7 feet tall” However he might say something along the lines of “I can’t help you become 7 feet tall but if you work with me I can make you a better 5 foot 6 inch basketball player than you are now”. In a similar ay it is possible that when it comes to mathematics a person kite have a 5 foot 6 inch “mathematical height”. Maybe that height is his ceiling or maybe it isn’t. In my mind it is an oxymoron of sorts to strive to help the student exceed his or her ceiling but it’s not an oxymoron to try to ensure that students do not “stagnate” at a height that is lower than their actual ceiling. As I often tell audiences, I don’t believe that everybody can become “good at math” but I do believe that with the proper effort on the part of the student and with the help of the “right” teachers every student can become “better at math”.

I guess the fundamental difference is that we can see the physical height of person but we cannot see the height of a person’s mind. So while we would never say to a very sort person that if he had been more serious and worked harder he could have grown to be 7 feet tall; we will often say to people that they can be anything they want of they just work hard enough to attain it. In that expect while I am best known for my “Calculus Revisited” course, 40 of my 50 years in the classroom were spent teaching developmental mathematics to adult mathematically at-risk students; and my goal was not to make them good at math (although that would have been a wonderful outcome had it occurred) but rather to help them get over their fear of the subject and not avoid seeking positions that might require using some math once in a while. I have been bothered by the number of people who are so “afraid” of mathematics that they avoid taking advantage of any such opportunities that might increase their upward mobility.

I apologize for my rambling (maybe 86 year old guys should shut down their computers after dark) but since my name did come up in conversation I did want to acknowledge that I saw it and rad the blog. Should anyone wish to correspond with me “in person” my email address is hgross3@comcast.net.

PS

Thanks for the very kind words Michael!!

Dear Karen,

I’m pleased that we “met”. I’m wondering whether you were watching my videos on YouTube (or ITunes) or whether you were watching on the MIT OpenCourseWare (OCW) website. The reason is that the OCW site contains my supplementary notes and extensive study guides to complement the videos. The course was called “Calculus Revisited” because it was designed to be used by scientists/engineers who had previously studied calculus but were now a bit “rusty’ in their attempts to recall it. My approach was based on the fact that if they had forgotten a lot of what they had previously learned, it might be a good idea for me to review it and that’s why the supplementary notes exist.

The other thing I was wondering is whether you were watching the videos sequentially, starting with the very first one. The reason I ask is that I try to make all of lectures flow from one another in the form of seamless transitions and as a result it can be confusing if a person watches a video without having seen what occurred previously. I’m wondering if it would be helpful to you if you watched my arithmetic and/or algebra materials because my approach, at least in my opinion, is different from most. You might enjoy looking at a few of the arithmetic videos I made for elementary school teachers last summer. I was 86 years old and I was proud of myself for not falling asleep even once during the recording sessions. If you would like to see the videos they can be accessed on the link below.

Thanks for responding to my first message. I look forward to the possible continuation of our correspondence.

Best wishes and warmest regards,

Herb

Herb – I have emailed you in reply so as to reduce stress on Ben’s blog 😉

i absolutely agree with the concept of neuroplacticity. Therefore it is said that teachers can make a difference.. Positive or negative … It depends on the right attitude towards teaching and learning 🙂

Your comments resonate with one of my favorite sayings, “The music is not in the guitar” (or as it is often paraphrased “It’s the singer, not the song”). In this vein, the textbook is the song and the instructor is the singer. As the various math “reform” movements seem to indicate, we are quite good at writing songs (that is, the standards) but fall short in our attempts to produce good singers (that is, the teachers). What complicates the problem even further is that it is possible for one listener to like a singer and for another listener to dislike the same singer. In effect, it is not true that “one size fits all”.

However, because of the Internet and MOOC courses, there are a variety of different singers for the viewer to choose from. In my own case, the MIT “Calculus Revisited” video series I produced has elicited a great number of positive comments and while there have been no negative comments, some viewers have checked the “dislike” option on YouTube. However I feel confident that the viewers who may not have liked the way I sing have found a singer that they do like.

As an aside I should mention that we see this situation outside of academia as well. For example, in sports it is not uncommon for an underperforming player on one team to be traded to another team and then become a top-notch player .

hi Im really really bad at maths but you know what you have actually helped me so thx a lot 🙂

I still believe there is a ceiling. I ｈａｖｅ ａ Ｐｈ．Ｄ ｉｎ Ａｐｐｌｉｅｄ Ｐｈｙｓｉｃｓ． Ｔｈｅｒｅ ｉｓ ａ ｂｉｇ ｄｉｆｅｒｅｎｔ ｂｅｔｗｅｅｎ ｕｎｄｅｒｇｒａｕｄａｔｅ ｍａｔｈｅｍａｔｉｃｓ ａｎｄ ｇｒａｄｕａｔｅ ｍａｔｈ． Ｉ ｆｅｅｌ ｔｈｅｒｅ ｓ ａ ｂｉｇ ｇａｐ ｔｏ ｌｅａｐ ｏｖｅｒ． Ｕｐ ｔｏ ｓｏｍｅ ｐｏｉｎｔ， Ｉ ｋｎｏｗ I reached my limit. Using the futon example, when a load much bigger than the designed capacity, it just breaks. The question is how much is too much. Secondary school and undergraduate math are in the normal usage capacity, i.e. a normal weighed person sitting on it will not break it. However, if a super obese person sitting on it, it breaks no matter what. Then the question is : Is it possible that some futon are poorly built and not even able to take up an normal adult weight? Do we ask too much for our kids?

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