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.

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I find it very true, when I was in school this was the way we were taught, if you can’t understand it keep working on it and you’ll figure it out. But over the years in construction I’ve learned that my skills can go higher than I thought. working with fractions and lots of numbers in my head brought my skills up to a new level. I was able to do math in my head without any problems. Now 30 years after high school I’m learning math I never thought I would be able to. I don’t believe in this ceiling of which they’re talking about. if I can do it, anyone can.

Hi Michael,

I am very pleased by the success you have experienced. However I might add a word of caution about :”If I can do it, anyone can do it”. While what you have described is quite possible I believe that it is at least equally possible that your elementary school teachers did not know how to bring you closer to your own ceiling. In other words, it is possible that you were left to “stagnate” mathematically below the level of what you were capable of attaining. In my opinion all that we can say for sure is that in the past 30 years you have become closer to reaching your ceiling, whatever it might be. And in that context, there may be others whose ceiling is lower than the one you have currently achieved. It is just a thought on my part; a thought that is based on over 50 years of being in the classroom. In any event, I congratulate you on your achievement and I hope you manage to climb even higher!!

“It is just a thought on my part; a thought that is based on over 50 years of being in the classroom.” Perhaps the difference in the height of the ceiling of learning in this case lies within the context of math learning. The contrasting conclusions were drawn from 50 years of observation within a classroom and that from the necessities of making a livelihood.

I entirely agree with you, John. My point of view would be that a ceiling still existed but it was higher than the one that was needed to understand K – 12 math. It goes back to my belief that it is isn’t whether the ceiling exists that‘s important but rather the fact that for one reason or another too many of us reach our breaking point before we even come close to reaching our ceiling.

No, I do not believe there is a fixed ceiling in anything we need to do. There is a breaking point, where we get frustrated and give up, but if we were forced to go higher in our skill, we could/would. I believe if we started with math in preschool and were forced to stay in it and progress until college graduation, we would be amazingly equipped mathematically. When I went to school, it wasn’t required to take algebra, geometry or precalc, and I hated math, so I gladly quit as soon as I could. I’ve always regretted that, that no one told me the importance of math and forced me further. I feel if all students, globally, were required to take math and continue progressing for 16 years of school, we would see wonderful advancements in mathematical concepts and we’d learn the sky is the limit when it comes to math. Practice makes perfect!

Hi Neil.

I believe that you say makes a great deal of sense. However the paradox I see in your argument is your concept of a breaking point. Based on my own anecdotal experiences, the breaking point could be the point at which you lose all patience and drop out because people were pushing you beyond what you are able to achieve. If that is true, then it might happen that for most students who were forced to follow a strict 16 year math curriculum, they would sooner or later reach their breaking point (presently it seems to be more on the side of sooner rather than later).

Of course none of us can see what our mind looks like so it is easy to endow it with all sorts of properties that may or may not be true. So let’s look at a more concrete situation involving “practice makes perfect” that might help shed more light on the matter (and then again it might not). Suppose I can run a 100 yards, in say, 13 seconds but I believe that practice make perfect. More specifically, let’s say that I believe that if I practice hard enough, I can always run faster. So I work hard and I soon get my time down to 12 seconds for running 100 yards. And I keep doing this and I keep improving. So far, so good. However if this could happen indefinitely, it would mean that eventually I would be able to run 100 yards in no time at all.

So this leads to a rather interesting situation. Namely I firmly believe that I can always improve; yet I know that there is a certain point that I cannot go beyond. What this means, at least in my mind, is that I will always keep improving in the sense that I get closer and closer to what I will call my breaking point but I will not be able to go beyond that. In mathematics, it’s what we call the limit of a sequence. For example, 0.9 is less than 1. 0.99 is closer to 1 than 0.9 is but it still doesn’t equal 1. If I continue doing this I see that as I annex more and more 9’s the decimal I am creating gets closer and closer in value to 1 but it never actually equals 1. In the real world we can only measure accurately to a certain number of digits. So while, say, 0.9999999999999999999 is not exactly equal to 1, we often say “close enough”. More specifically, the number of digits exceeds the number of decimal places to which we can accurately measure. So in this sense there might be a limit to how much I may be able to learn.

Admittedly it might be dangerous to look at the way we internalize math in the same way as we internalize improving our running speed but I present the analogy only as a way to try to make my point clear. However there comes a time when less is more and I apologize if I seem to have gone over the edge.

I think that analogy is completely wrong. I would say it’s more like travelling a certain distance. Maybe you can only go a few miles one day, then you go a few less the next day, but over time, you build up endurance and you start travelling farther every day. But the point is, no matter what speed you travel, you keep moving at least little bit every day.

I say this as a postdoc in physics. My research involves developing new theoretical, very mathematical concepts in density functional theory. If I had a “ceiling”, I would have to admit that at some point, I would become useless as a researcher when I reach it. The only way to keep going is to keep moving forward.

So, do I come from some special breed of human being who has no “math ceiling”, while everyone else does? I really doubt it. I find this same concept applies to pretty much anything–for example I’ve taken piano lessons my whole life (or at least, between the ages of 2 and 18). I still practice and sutdy music a bit. True I’ll never be Mozart–so I suppose you could look at that as a “ceiling” of sorts, but that doesn’t mean there are certain concepts in music that I just will never be able to learn, or styles of music that I would never be able to compose.

Keep studying and you will get better. You will learn more. Mathematical concepts will come more naturally to you. I can’t imagine what could possibly make your brain shut off and say “I’m incapable of grasping additional mathematical concepts” if you’re actually motivated to learn.

Hi Mark,

I don’t think that we are in disagreement. More specifically (as I think I said in my comment to Neil) you can keep improving but after a while the improvement is imperceptible. Otherwise the conclusion to your analogy is that you will eventually be able to walk an infinite distance. In terms of a mathematical analogy if you look at the sequence 1/2, 2/3, 3/4, …. n/(n+1), …. You see that each term is great than the previous one but no term will ever be as great as 1.

I guess the difference is that we can see a “gimpy” leg but we can’t see a “gimpy” mind consequently we mkaew such statements as “If you study hard enough you can achieve anything you want”; but we don’t say to the person with a deformed leg that if he tries hard enough he will eventually run a sub 4 minute mile. We can tell him that with enough effort and physical therapy he will probably be able to run faster than he currently can.

Again, it’s just my own opinion that is subjectively based on being in the classroom for almost 60 years. Some of those years were at MIT where I produced the video course “Calculus Revisited” that can be viewed on the Internet; but at the other end of the spectrum 40 of those years were spent teaching basic math to mathematically at-risk adult learners at both the community colleges and in prisons. I am currently working on my own website http://www.mathasasecondlanguage.com where I have posted all of my work in arithmetic and algebra for anyone to use free of charge.

“Completely wrong,” Mark? Way to pull your punches!

Herb Gross’ sixty years of experience teaching math at a wide range of levels to countless students vs. what? Your individual experience as a physics post-doc. Yes, you’re a special breed of person: only a very small number of people have what it takes to reach the levels of abstraction in which you operate comfortably.

Herb is talking about a much wider swath and range of people, few of whom are headed to physics or mathematics Doctorateland. I’m better at mathematics than the vast majority of people, and I’m not that great at it relative to virtually anyone who has done doctoral work in the field or any math-intensive discipline like physics.

Michael, you realize that to be a physics postdoc, I had to take 4 years of physics and math in college, 7 years of classes and research in grad school, in addition to teaching for approximately 6 and a half of those 7 years, right? In addition to another year and a half of research in my postdoc so far, collaboration with other people who do research and teaching, and physics education research seminars.

My point, precisely. You have a way above average knowledge of higher mathematics and have associated for years with similar people. Herb isn’t commenting on people like you. Neither am I. What part of that is unclear to you?

Herb, what can I say, but in my experience, if you are motivated, you can learn pretty much anything. I don’t think there will be a level of math that a motivated student can’t learn. Why would your brain suddenly say “enough math, I need to learn something else”?

Another issue is that most of the time, physics and math are not taught in a way that a majority of students can learn easily (and even for those who do learn that way, research shows that they benefit from other teaching methods as well). There is a lot of resistance to the kind of restructured curricula that would significantly improve the way a lot of students learn, largely because it’s a radical departure from the traditional lecturing approach. Not only does it require a lot of effort to make the necessary changes, but we have a long history of doing things one way, which ingrains the attitude “if it was good enough for me, it should be good enough for the next generation of students”.

I hadn’t seen the last comment you made to Herb, but now that I have, I think you are arguing about typical (average) people by holding out your atypical experience and that of people with similar interests and abilities. I have no idea whether you have tried to teach or otherwise help people who are not operating in mathematics at the level of a physics doctoral student, but I’m certain that if you had, you wouldn’t be taking this rather odd position. No one here is denying YOUR experience. But to suggest that because you have had such encounters with difficult and abstract mathematics and its applications to physics that it follows that ANYONE can do so or that no one is going to have the sorts of “my brain needs to do something else now – and maybe for a very long time” experiences that you seem not to understand. Consider yourself fortunate, gifted, or just really great at what you love. But not everyone comes to the table of mathematics – applied or pure – with that same set of feelings. Ask any mathematics teacher at the K-12 level.

You’re obviously intelligent. I’m puzzled that you’re still arguing a position that simply doesn’t hold true for literally millions upon millions of people.

Michael, what part of I had to teach students for more than 6 of the 7 years I was in grad school, or the fact that I attended (weekly) physics education research seminars didn’t you understand? The classes I taught were largely for 1st or 2nd year physics/engineering students (many of whom still end up switching majors) and occasionally for life science majors, who have no interest in physics whatsoever (algebra based, rather than calculus based).

And since it doesn’t seem possible to edit posts here (I think), let me assure you that physics education research is not concerned with “how do we teach quantum field theory to graduate students”, but rather, “how do we build a solid foundation in students taking their first physics class”.

I’m arguing the position that I am because I think you have fundamentally the wrong idea about what it means to learn physics and math. I think millions of people do, and this is part of the problem. When people see it as something impossible for certain groups–or for themselves–to learn, that, in reality, becomes the biggest obstacle to learning.

Teachers share some of the blame, as I pointed out before. It’s difficult to work with students who don’t get something that you think should be obvious. But more often than not, you can learn to see it from their perspective, and fix the underlying misconceptions, although it doesn’t happen immediately, and almost never through lecture.

My mistake: how could I possibly think you’re overstating your case and missing the point Herb Gross made very, very clearly? I bow in the face of your superior experience and insight. Any personal struggles I’ve gone through as a LEARNER of mathematics are almost certainly due to my own shortcomings, my lack of grit, and so forth. The fact that I never used or implied the word “impossible” doesn’t matter. And that notwithstanding, mucking about in with high school kids (in Detroit and many similar communities) with high school kids who still can’t subtract has given me the odd sense that the issues are a little more complex than just the attitudes and beliefs of the students or their teachers. It’s a big world; I think you’re looking at a limited part of it and claiming that this suffices to speak meaningfully about what’s true for everyone. It isn’t, and your take simply doesn’t suffice to address the non-elite, deeply impoverished in a host of ways areas that I’ve been working in for a long time. There’s not a teacher I’ve encountered in that journey who couldn’t be doing a better job (myself included). Yet if you spent your days dealing with what they deal with (and what their students deal with) on a routine basis, you’d thank the gods of physics that you don’t walk in their shoes.

On Sun, Jul 10, 2016 at 7:17 PM, Math with Bad Drawings wrote:

> mcpalenik commented: “And since it doesn’t seem possible to edit posts > here (I think), let me assure you that physics education research is not > concerned with “how do we teach quantum field theory to graduate students”, > but rather, “how do we build a solid foundation in studen” >

Well, this is essentially what the research shows, not just my own personal experience, but whatever. I guess it gives you security just to feel that some people are incapable of learning. Can you explain where my experience teaching (between lecturing, being a lab and recitation TA, and in the help center–essentially one on one tutoring) is invalidated, or where the mounds of evidence pointing toward physics (likely math, too) being taught in an extremely inefficient way that puts a lot of people off and hinders learning is wrong. Or even where the research into neuroplasticity hinted at in the article above is invalidated?

The difference between me, by the way, and other people who don’t do the work I do, is very unlikely to be some innate genetic or epigenetic difference. It’s mostly in how I’ve directed my interest and effort. And I will say it is often very disappointing to see the techniques for tackling problems with which people enter college. It’s abundantly clear that a lot of people start with totally the wrong approach to even thinking about problem solving–search for the right equation/formula, plug numbers in, or whatever. But you can teach people to think differently–to try to catagorize problems. To match principles, rather than specific equation. To sort through a tree of thought processes from top to bottom. Whether it’s physics or advanced mathematics, it’s kind of the same either way. That is a learned skill. That is a skill that can be taught. And that’s really the only skill you need to avoid getting caught at some arbitrary ceiling–although, it’s much more difficult to acquire or teach than the bare minimum skill set required to pass a class. And that’s why so many students don’t pick it up.

Traditional lectures (specifically at the college level, according to research, but also in high school, from what I remember) don’t usually convey learning in this manner. For physics, the research indicates that the best way to solve this is with integrated lab/lecture/recitation, with students working in teams and a teacher that guides the learning and manages the teams, but rarely lectures in front of a board. Of course, this requires well structured problems, and some trimming of the number of things that you try to teach in a semester as well. For pure mathematics, I’m not familiar with the research, but I would imagine that similar concepts apply.

Nell, over the narrow spectrum of K-12 math, with or without AP calculus tacked onto the end, you would likely find that most people can learn more math than they do. But even there, some people have been so scarred by bad math teaching that the damage may be close to irreparable. It’s been my experience, too, that some people really have cognitive problems with math that may be organic and that really hamstring them severely, regardless of effort on their and their instructors’ parts. It’s not a huge percentage, but it’s not nonexistent and must be acknowledged in any conversation about ceilings.

That said, as you go past undergraduate, lower-level calculus into the more abstract levels of mathematics, it’s a rare talent indeed who doesn’t bump into barriers. We can think of them as temporary obstacles, detours, blocks, delays, or more serious ceilings. They may be plateaus or they may be long-term barriers that are only bypassed through long, dedicated effort over time, if at all. But they do exist for nearly everyone, regardless of enthusiasm, and the sky proves not to be the limit nor practice adequate for much growth, let alone perfection. We’re not talking about calculation-driven math that revolves around memorizing and drilling on algorithms, but deep abstract concepts. It’s not easy to will or practice oneself through such challenges, at least not in my experience, and I’m not someone who gives up readily or isn’t willing to wrestle over decades with things that resist my understanding.

Yes, we can, most of us, do much more with math, and those who teach could teach more math more effectively to more students than is usually the case. But let’s not pretend that it’s simply lack of conviction or effort that stands in the way.

Perfection in maths means that all your proofs are OK and can be published in a peer-reviewed journal.

Mathematics is still the topic that needs live instruction. Presentations at a blackboard are a necessity. Even the tone of the professor, or the way she breathes and marks the finer points, can help you with understanding. The explaining half-sentences are extremely important, because they keep up the context, weave together the statements, and keep up-to-date your working memory.

And this is the main barrier to teaching more math more effectively to more students. It needs personal insruction, best delivered to 2-15 students per group, or one by one. If they are in a group, students have to have the same basic knowledge. It’s not History, that can be recited and regurgitated without context. The mathematical context always has to be in your head.

“It’s not History, that can be recited and regurgitated without context.” Wow, now there’s an analogy that proves how education – in all subjects – has lost its way.

I don’t know much about math (other than I suck at it), and less about teaching math, but I know something about history, and NOTHING is more context-dependent than history.

Just as math isn’t about knowing which number to move where and memorizing multiplication tables, history isn’t about names and dates, presidents and kings, wars and treaties. Just as math should be about understanding WHY the numbers move and the relationships between them, history should be about evaluation of what is recorded and investigation of why some things happened and some things didn’t. It’s an exact parallel.

And don’t even get me started about what’s been done to literature in schools.

Hi Guys!

I didn’t intend to start a fire storm. I only wanted to point out that in my opinion people can keep achieving up to a certain level (which might not be the same for everyone), after which they would become saturated. Otherwise everyone could be a poet laureate or a Nobel Prize winning scientist just by trying sufficiently hard.

I think that where there is no controversy lies in the fact that for a multitude of reasons (including, but not limited to, bad teaching and poor student habits, etc.) too many students become saturated well below the level of their true capability.

Because of the comparison: good, better, best, people tend to believe that best outranks good. Yet the truth of the matter is that one of three very bad things can be the best of the three.

So from my point of view, all I am saying is I don’t believe that everyone can become good at math but at the same time I believe everyone can become better at math.

In closing I want to thank Michael for the support he has given to both me and my work, Developing my video courses has been a labor of love and I feel blessed by the positive comments my work is eliciting from viewers around the world. I guess a lot of it is due to the name recognition that MIT enjoys.

Anytime, Herb: it’s 100% sincere as I think you know.

I don’t hear you saying, and I know I’m not saying, that we can know where someone else’s ceiling is. I’ll never say to a student, “I think you’re now done growing in mathematics.” And that’s easy enough for me to avoid: I don’t know enough higher mathematics well enough to rule out the feasibility that the weakest students I encounter could never climb higher up or further out along any of the branches of the tree of mathematics. If I wake up tomorrow and suddenly discover that overnight someone has downloaded Andreas Blass’ mathematical breadth and depth into my brain, perhaps I’ll be better positioned to pass such judgments.

On the other hand, I say quite frequently on Quora when folks post questions there about why they’re not making progress learning mathematics at higher levels that they may have hit a point where they need a break; where they need a different point of view; that maybe there’s something they’ve missed along the way that’s necessary to understand before they can move forward.

I’ve also said unhesitatingly that some mathematics is notoriously difficult according to people who do serious mathematics. A friend in NYC who had finished a master’s in mathematics at MIT told me that there was one class he hit when he started doctoral work that gave him fits (it may have been algebraic topology, but it’s about 30 years since I heard this and am no longer in touch with him). Now frankly, he knew more math in his pinky than I’m likely to ever know. If he said that that class was really hard, I’m reasonably sure that it was, and that I would not find it easy because of my grit and go-getter attitude. Should I ever get to the point where I’ve got the prerequisites under my belt, perhaps I’ll find out. But I know that there are areas of mathematics so esoteric that few people even within the world of professional mathematicians know anything about it. And every year, the totality of mathematical knowledge grows apace. No one does or possibly can know more than a tiny fraction of what’s currently “in the literature”; it’s likely been a very long time since anyone on the planet could be said to be competent in the full range of what comprised human mathematical knowledge and achievement. The problem goes well beyond a matter of willpower, grit, or any other such trait.

I’m a very patient person when it comes to mathematics. It’s not my goal to be a research mathematician or PhD. in the field. Whatever I learn now is gratifying to have understood. Sometimes it’s deeply exciting, aesthetically pleasing, and downright fun. I try to pass along my sense of excitement, appreciation, and pleasure to others, including teachers I coach, students I teach, and various people I encounter in other contexts. Sometimes I’m even successful at doing so, though it can be a tough sell, particularly to those who resist even hearing the first word after “mathematics.”

I don’t particularly need the concept of a “mathematical ceiling,” but neither do I need to kid myself as far as my own vaguely defined limitations. I’m always open to surprising myself, even after days when I’m forced to grapple with how dense I can be. I wish everyone would be willing to be patient when trying to solve a (potentially) interesting mathematical problem, but the fact is that most people are not. If I get my foot in the door, I can usually find something that will pique someone’s interest, but it’s not always possible to get a lot of folks to even answer the doorbell.

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“I stopped at multivariable calculus” means “Hey, I didn’t win, but I’m proud of making it to the final four.”Multivariable calculus? I thought that was preseason.

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I feel this article is disingenuous at some point. A lot of these examples are indeed those that can simply be grokked with enough time, a textbook, and maybe a MOOC. That said, as someone who has a master’s in statistics (and did well with it), at some point, math goes from “here’s a line that satisfies this equation” to “here’s a little bit of notation, now all aboard the derivation express train to the punchline!”–AKA the way all of the cool new ideas are presented in mathematical, peer-reviewed journals.

I am pretty certain that there are a great deal of people that completed a major in a STEM topic in undergraduate studies, or even have a master’s, and are not able to grok those journal papers written by professors for professors, or people with the equivalent academic pedigree.

Or, put another way:

There’s a reason that there is a very small number of people who are at super-successful firms (Pfizer, Google, Renaissance Technologies, etc.) in highly quantitative roles and a bunch of other people who aren’t them. The difference is far more than a little bit of elbow grease, perseverance, and some online coursework. If getting the equivalent skills of a PhD (not the degree, but the ability to just digest the kinds of papers found in journals) were that easy, I think it’d be something that a lot more people would have.

Empirical evidence shows it’s not that simple.

Yeah, I tried to be careful not to say that “everyone is capable of everything,” which is a statement that’s half false and half meaningless.

My point is that the ceilings we hit are rarely about cognitive ability. In secondary/undergraduate mathematics education, I find the ceilings are usually caused by the long-term toll of not understanding basic concepts. Elsewhere, ceilings can come from lack of effort, lack of access, poor strategy, bad timing, the demands of family life, or – yes – cognitive ability. But I tend to think (from personal experience, more than any sophisticated reading of the literature) that “IQ” is a weird-ass construct and is almost certainly overrated as a determinant of life outcomes.

There is a lot of reason to believe that “IQ” is an absurd notion that has come to carry a ton of not-so-wonderful baggage, not the least of which is its historical use for highly racist purposes. Required reading in this regard is Stephen Jay Gould’s MISMEASURE OF MAN. Though there have been subsequent critiques of parts of this book that might hold water, Gould raises so many important points that can’t be dismissed readily. Further, there are documented historical facts about how IQ was used/abused in the US in the first half of the 20th century that even if he got some other things wrong in his zeal to expose the sordid history of American intelligence testing, the book would be invaluable nonetheless.

As for mathematical ceilings, the question isn’t whether most of us have them. Einstein famously said, “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.” Take that however you like: my guess is that he was both encouraging the average person to never give up grappling with mathematics and exposing the fact that even “geniuses” hit ceilings. Indeed, the more mathematics you study, the more likely it is that you’re going to hit ceilings. Whether a given one is temporary or permanent for a given individual is just about impossible to say. Our problem as a nation is that we don’t seem to know in general how to help and encourage students to keep pushing against their math ceilings and to successfully break through them as much as possible. Instead, we have developed a nation of people who are prone to quit mathematics as soon as they hit a barrier and who are surrounded by a culture that encourages and reinforces giving up when it comes to math. The irony is that in virtually any other arena but math and science, people who given up early in school are considered to be embarrassingly stupid in addition to simply being ignorant. The latter can generally be remedied. The former is a stigma not so easily gotten rid of. But ignorance AND stupidity in mathematics seem to be badges of honor among the vast majority of Americans. And given that gaining a basic understanding of a wide range of mathematical areas is not only useful but doesn’t have to be nearly as daunting as most of us think, the widespread innumeracy in this country should be a national disgrace.

I think the concept of a “math ceiling” masks a more serious problem. Let me see if I can make a valid analogy. We know that there has to be a ceiling in how fast a person can run. More specifically, if we assume that hard work and a lot of practice can help us increase our speed, we could eventually run a 100 meters in “no time at all”. That is, once we got down to 9.5 seconds, by hard work we should be able to improve the time to 9.4 seconds, then 9.3 seconds, etc. So here we can see there is a ceiling. Thus it is a given that each of us has a ceiling when it comes to how much time it takes us to run 100 meters. So the question isn’t whether we have a ceiling but rather how close we can come to reaching our ceiling.

For example suppose there is a person who currently runs 100 meters in 14 seconds. This might be the person’s ceiling but it is quite possible that if the person is willing to work a bit harder s/he could improve his or her time to 13.5 seconds. However, for whatever the reason is, the person might believe that 14 seconds is their ceiling. And if the person has a “gimpy” leg, we might even believe that 13.5 seconds is very close to that person’s ceiling. The difference is that we can see a “gimpy” leg but we cannot usually see a “gimpy” mind; and it is therefore easier to fantasize what our math ceiling is.

So in my opinion (based on nearly 60 years of teaching mathematics) the problem isn’t as much about a math ceiling as it is about people who, for one reason or another, is willing to settle for less than what the person is capable of attaining. And what we, as teachers, can do to encourage students not be become discouraged in their quest to earn more be it math or anything else.

If I had to come up with a one sentence creed it might be “Not everyone can become good at math but everyone can become better!”

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As a final year maths undergrad, with good grades but very bad understanding of the subject, I can confirm I’ll never get any better at maths. I spent last year cramming and learning by rote just to get a good grade, and it definitely worked. But it also made me lose all patience with the subject, so I’ll never take the time to go back and address the small misunderstandings that prevent me from progressing.

And now, in final year, I’m expected to approach questions with the understanding of a final year. But I have the understanding of a first year at best, so everything becomes even more stressful, and reinforces the fact that the only way I’ll pass my exams is by memorisation once again, because there’s simply not enough time to fix the problem properly.

😦

I’m curious: what do you expect to do when you graduate? Please don’t teach. Please.

I feel for you, having gotten similarly stuck but at a much lower level. If you’re anything like me, you’ll spend some years running away from all things math, until you finally find a way – or someone who can help you find a way – to grapple with it on your terms. If you’re lucky, it won’t be too late for you to get back to what appealed to you about math in the first place.

Good luck! But please, please, don’t teach.

Karen Carlson,

I’m sure you mean well, but wouldn’t it depend upon what level of teaching someone proposed to do before that sort of advice is given? If we all have a ceiling (or an amorphous upper bound) of understanding in mathematics, then no one should ever teach any math following your reasoning. I think it’s possible that someone who recognizes that s/he’s hit that ceiling might still be an effective teacher at lower levels, particularly if they have gained insight into the sorts of struggles students at that level and below are likely going to face and how to possibly overcome them. That’s been my own experience as a math teacher, at any rate.

Hi Paul – ok, let me clarify: I wasn’t referring to “math ceiling” or any level of mathematical competence, but instead to the “I’ve lost all patience with the subject”.

As a student, I’d love to learn someone who’s still learning, or re-learning, who knows s/he has some gaps; but I don’t think learning from someone who’s feeling so defeated by the subject , any subject, is going to benefit anyone.

Then again, I know a math prof who lost interest in academic math, and found his way back via teaching his kids. So maybe that’s a valuable path.

In any case, I’ve never taught anything, so I realize my pov isn’t necessarily accurate. I just saw someone really sad, and I puttered around for a couple of days hoping to see a response; when I didn’t, I jumped in. And now you’re here, so things are looking up 😉

Hi, Karen,

Good point. I hadn’t reread the original comment and wasn’t thinking about the overall feeling of defeat. His(?) experiences are different from mine. I stopped caring about math in the middle of high school, then became interested again in my 30s. And from that point, I mostly did quite well. Eventually, I started informally studying math (and am still doing so) that stretches me beyond my comfort zone, but since it’s just on my own, I don’t get a feeling of total defeat, just temporary realizations that what I’m up against is either way past my current knowledge (and might always be), or simply something I need to come back to after a break of some length. When the stakes are low (just trying for my own satisfaction to stretch my understanding), there’s less likelihood of feeling “beaten” by having to stop.

Whether FP will come back with a fresh attitude is impossible for me to know, but I hope it happens. And as you suggest, outside mentor(s) can make a big positive difference in that regard.

Michael (Paul’s my middle name)

I hope FP recovers, too. And your low-stakes idea might be a good way to start – play for the fun of the game, without worrying about player standing. Fortunately, lots of options out there on this set of tubes we call the internet, though i’m not sure how many are at FPs level. I’m flaming out of a no-stakes linear algebra mooc as we speak – which has me feeling pretty beat-up, to be honest, but I’ve got other things that keep my ego on life support.

(I have no excuse for mistaking your name – my apologies!)

Where are you in the linear algebra course? Is the pace just too quick for your schedule? That tends to happen to me in MOOCs: I get ambitious, but then I have trouble staying with it for various reasons. Depending on what you’re dealing with, feel free to ask me (mikegold@umich.edu) and I’ll see if I can help.

It’s more a matter of everything being pretty fuzzy, to the point where I’m not even sure what questions to ask. I’m actually a couple of weeks ahead (they opened 3 weeks early for “brush up before school starts so I got a head start), and I’m in a “let’s sit here and see if I feel like continuing in a couple of weeks” holding pattern, but right now, I’m pretty sure I’m done. Everything’s floating around vaguely with no connective tissue.

I think part of it is that it’s run by the computer science department and they’ve included a significant amount of material on coding matlab algorithms for various operations; I get the importance, but it took me a while to find out that “axpy” and “flops” and “memops” weren’t part of linear algebra per se.

But there’s some good stuff too, and I’m enchanted with 3Blue1Brown’s vids, so I’m just going to have to poke along, see if I can create a structure to pin it all to.

but thanks so much for your email, that’s very kind of you.

I don’t know Matlab at all (other than having heard of it), so I couldn’t help there. The 3Blue1Brown videos are remarkable; they helped me realize some big picture things I’d never realized before. I didn’t grasp everything he was getting at on first viewing, but a lot of it made sense. Of course, he’s not pausing much to let you contemplate new information or interspersing problems for you to work out to test your understanding, so I think to squeeze more juice out, it will take me time to sit down with a textbook to pull examples from for self-testing. He’s not trying to teach computational algorithms, but rather some sense-making of a number of important ideas. And despite having gotten an A in linear algebra at University of Michigan, that was in 1992 and it’s not like I retained a ton of that knowledge since I rarely have an opportunity to use it.

You might like the free video lectures on linear algebra from MIT with Gilbert Strang. I started watching the series but found his style a bit distracting. You might fare better, and since there’s zero time pressure, you can take forever and watch videos as many times as you want. Perhaps doing that with the 3Blue1Brown “essence” material in mind will make the straight-ahead work from Strang easier. And no MatLab needed. 🙂 Feel free to touch base. I’m no expert; I do like the idea of trying to work with someone else on linear algebra as a way to refresh, solidify, and deepen my own understanding. You should know that I was a literature grad student in the ’70s and hated math in high school for the most part. But I wish I had a better visual imagination for spatial aspects of math.

I had an interesting experience when I was teaching a secondary school algebra course. I had one boy who was failing all his algebra tests and he asked if I could give him some 1 to 1 support for a bit as he needed to improve his grade. We sat down for a while and I asked him to explain how to simplify expressions, solve equations and so on and he could tell me the methods and explain the ideas, he clearly understood all the concepts of algebra that he had been working on so we did some examples. All was going well, I couldn’t see what the problem was. This kid was good at algebra. Then, I gave him an example with a few negative numbers in it and he started guessing wildly. Turned out that he had never properly got to grips with multiplying, dividing or subtracting negative numbers. So we spent the next few hours going over some very basic number work. We went back to number lines and multiplication tables and so on. Next test he passed with flying colours. In that case we found the missing bit of futon and patched it up before too much damage was done. But I can imagine most of the time this doesn’t happen, teachers simply don’t have the time to spend with all the students figuring out exactly which bits of knowledge are missing. And the boy could easily have decided that he was rubbish at maths and stopped trying.

Maybe we have an inate ceiling, maybe we don’t. But even if we do I believe that most people in their life don’t even get close to theirs.

Hi Rebecca,

I thought I would comment on why I think that students have so much trouble with subtracting negative numbers is because they have been taught to think of subtraction in terms of “take away”. That definition works fine for positive numbers but when students are shown the “add the opposite” rule and rewrite 8 – (-3) as 8 + (+3) = 11, they can’t understand how it is possible that we subtracted something from 8 but the answer was greater than 8.

What I think should have been taught is a less formal version of the mathematical definition. In other words, mathematician define b – c to be the number that must be added to c to yield bas the sum. The paraphrase that I use is that subtraction tells us the gap between the two numbers. In that vein, 8 – (-3) means the gap between 8 and (-3); or, in other words, what must we add to (-3) to obtain 8 as the sum. To make the solution more visual, we might use the profit-and-loss model, in which case the problem might be what transaction is needed to convert a $3 loss into an $8 profit? I believe that without even thinking about “add the opposite” rule, almost everyone would realize that to you first needed a $3 profit just to “break even” and then you need an additional $8 profit; so all in all, it takes an $11 profit to convert a $3 loss into an $8 profit. Other model work as well. For example, if the temperature and hour go the temperature was 3 degrees below 0 and now the temperature is 8 degrees above 0, it means that the temperature increased by 11 degrees. My belief is that once students internalize the definition in this way, their ability to compute with signed numbers will improve, and in some cases, dramatically.

An analogous thing happens when we divide a number by a fraction. For example I have had students use the “invert and multiply” rule correctly to obtain 12 ÷ 1/4 = 48 and then ask me why they got the wrong answer. When I tell them it is the correct answer, their response is “It can’t be because when you divide the answer has to less than what you started with”. This problem would not have existed if the student had bene taught that the quotient of two numbers gives the relative size of one number with respect to the other. In that vein 12 ÷ 1/4 means the number by which we have to multiply 14 to obtain 12 as the product. To make this more visual for the students to internalize, we could ask them “How many 1/4 pounds of candy does it take to equal 12 pounds of candy?” It is not difficult for the students to see that it takes 4 quarter of a pound to equal 1 pound; and therefore it takes 12 X 4 pounds to equal 12 pounds.

My point is that too often we teach only the “how” while neglecting the “why”. If you want see more about my approach you can go to my website (www.mathasasecondlanguage.org. All of my work, including my videos, are available for anyone to use free of charge.

Depending on the student, I recommend supplementing/complementing what Herb is saying with a visual and/or tactile model. One of the ones I really like is the chip model for integer addition and subtraction: http://www.learner.org/courses/learningmath/number/session4/part_c/

Thanks, Michael. I would add that when I conduct a workshop for elementary school teachers, I show them the chip model but only AFTER I have shown them the logic that underlies subtraction. More specifically, using the chips is sort of a fun game for the students and they enjoy manipulating the chips but it doesn’t explain the logic behind the method. In fact I use the chip model as a concrete illustration for “personifying” b + (-b) = 0.

Hi, Herb,

I usually start with the arithmetic, too, but I inevitably go to other models even if all the students in a class claim to “get it.” Trust, but verify. 🙂 And if a student’s grasp of something is so tenuous that learning or hearing about another viewpoint or model will shake that grip loose forever, well, let’s just say that my skepticism about that “grasp” is elevated accordingly.

That said, in one-to-one teaching or in cases where I know that particularly students may suffer paralyzing anxiety at numerical/symbolic approaches initially, I may look at the number line or chip models before “mathy” symbols, terminology, etc. Like many things, too many variables for me to reduce things to formulas when it comes to teaching a topic.

I hear what you’re saying but I think there are “easy” ways to paraphrase the mathematical symbols in a way that makes it easier for students to grasp the concept. For example, the way I would proceed to demonstrate that 8 – (-3) = 11 by first asking the question “By how much did the temperature change if it went from 3 degrees below 0 to 8 degrees above 0?” Once they understood that the temperature increased by 11 degrees, I would show them how it translated into 8 – (-3) = 11. Our experiences in the classroom might be different but in my case, I never had a student who couldn’t grasp this approach.

And I would motivate the chip example by using it as a way to visualize the axiom that (+b) + (-b) = 0. I might have them make their own tiles, labeling one set P (to denote positive) and another set N(to denote negative) In that vein they saw that a P tile and an N tile cancel one another.

Within the limits of our educational structures there is an imposed limit on the duration of the energy that both students and teachers can commit to any subject, or to any student. Coexisting with this are the real world demands to get on with courses, or put a roof over our heads and food on the table. Coexisting with these are the distractions of interest and other talents that are more easily accessed and for the world at large more beneficial. What a soup! Certainly imagination and commitment can overcome some of these constraints. For myself it is the belief that until someone is shoveling dirt on my coffin, there’s hope that the chain rule is knowable and observable in the real world, never mind real numbers! Let’s discover our own limits, and derive our function. We can allow others the same pleasure.

I think you have expressed the situation very well. In fact, I had the same thoughts in mind while I was developing my own website (www.mathasasecondlanguage.org) . However, in my earlier responses to this blog, what I was trying to show is that even if none of these “distractions” were present and we devoted more time to learning math, there would still have to be a ceiling somewhere. In other words we could keep improving yet not get beyond a certain “limit” It is analogous to a runner improving his speed every day, yet still not being able to get beyond a certain limiting speed. While I hope that viewers would become good at math just by using my material on the website, my primary goal is to help mathematically at-risk people overcome their fear of math.