As a math teacher, it’s easy to get frustrated with struggling students. They miss class. They procrastinate. When you take away their calculators, they moan like children who’ve lost their teddy bears. (Admittedly, a trauma.)

Even worse is what they *don’t* do. Ask questions. Take notes. Correct failing quizzes, even when promised that corrections will raise their scores. Don’t they *care *that they’re failing? Are they *trying* not to pass?

There are plenty of ways to diagnose such behavior. Chalk it up to sloth, disinterest, out-of-school distractions – surely those all play a role. But if you ask me, there’s a more powerful and underlying cause.

Math makes people feel stupid. It hurts to feel stupid.

It’s hard to realize this unless you’ve experienced it firsthand. Luckily, I have (although it didn’t feel so lucky at the time). So here is my tale of mathematical failure. See if it sounds familiar.

***

Thanks to a childhood of absurd privilege, I entered college well-prepared. As a sophomore in the weed-out class for Yale math majors, I earned the high score on the final exam. After that, it seemed plausible to me that I’d never fail at anything mathematical.

But senior spring, I ran into Topology. A little like a bicycle running into a tree.

Topology had a seminar format, which meant that the students taught the class to each other. Twice during the semester, each of us would prepare a lecture, then assign and grade a homework assignment. By reputation, a pretty easy gig.

My failure began as most do: gradually, quietly. I took dutiful notes from my classmates’ lectures, but felt only a hazy half-comprehension. While I could parrot back key phrases, I felt a sense of vagueness, a slight disconnect – I knew I was missing things, but didn’t know quite what, and I clung to the idle hope that one good jolt might shake all the pieces into place.

But I didn’t seek out that jolt. In fact, I never asked for help. (Too scared of looking stupid.) Instead, I just let it all slide by, watching without grasping, feeling those flickers of understanding begin to ebb, until I no longer wondered whether I was lost. Now I *knew* I was lost.

So I did what most students do. I leaned on a friend who understood things better than I did. I bullied my poor girlfriend (also in the class) into explaining the homework problems to me. I never replicated her work outright, but I didn’t really learn it myself, either. I merely absorbed her explanations enough to write them up in my own words, a misty sort of comprehension that soon evaporated in the sun. (It was the Yale equivalent of my high school students’ worst vice, copying homework. If you’re reading this, guys: Don’t do it!)

I blamed others for my ordeal. Why had my girlfriend tricked me into taking this nightmare class? (She hadn’t.) Why did the professor just lurk in the back of the classroom, cackling at our incompetence, instead of *teaching* us? (He wasn’t cackling. Lurking, maybe, but not cackling.) Why did it need to be stupid topology, instead of something fun? (Topology is beautiful, the mathematics of lava lamps and pottery wheels.) And, when other excuses failed, that final line of defense: I hate this class! I hate topology!

Sing it with me: “I hate math!”

My first turn as lecturer went fine, even though my understanding was paper-thin. But as we delved deeper into the material, I could see my second lecture approaching like a distant freight train. I felt like I was tied to the tracks. (Exactly how Algebra 1 students feel when asked to answer those word problems about trains.)

As I procrastinated, spending more time at dinner complaining about topology than in the library *doing* topology, I realized that procrastination isn’t just about laziness. It’s about anxiety. To work on something you don’t understand means facing your doubts and confusions head-on. Procrastination pushes back that painful confrontation.

As the day approached, I began to panic. I called my dad, a warm and gentle soul. It didn’t help. I called my sister, a math educator who always lifts my spirits. It didn’t help. Backed into a corner, I scheduled a meeting with the professor to throw myself at his mercy.

I was sweating in the elevator up to his office. The worst thing was that I admired him. Most world-class mathematicians view teaching undergraduates as a burdensome act of charity, like ladling soup for unbathed children. He was different: perceptive, hardworking, sincere. And here I was, knocking on his office door, striding in to tell him that I had come up short. An unbathed child asking for soup.

Teachers have such power. He could have crushed me if he wanted.

He didn’t, of course. Once he recognized my infantile state, he spoon-fed me just enough ideas so that I could survive the lecture. I begged him not to ask me any tough questions during the presentation – in effect, asking him not to do his job – and with a sigh he agreed.

I made it through the lecture, graduated the next month, and buried the memory as quickly as I could.

***

Looking back, it’s amazing what a perfect specimen I was. I manifested every symptom that I now see in my own students:

- Muddled half-comprehension.
- Fear of asking questions.
- Shyness about getting the teacher’s help.
- Badgering a friend instead.
- Copying homework.
- Excuses; blaming others.
- Procrastination.
- Anxiety about public failure.
- Terror of the teacher’s judgment.
- Feeling incurably stupid.
- Not wanting to admit any of it.

It’s surprisingly hard to write about this, even now. Mathematical failure – much like romantic failure – leaves us raw and vulnerable. It demands excuses.

I tell my story to illustrate that failure isn’t about a lack of “natural intelligence,” whatever that is. Instead, failure is born from a messy combination of bad circumstances: high anxiety, low motivation, gaps in background knowledge. Most of all, we fail because, when the moment comes to confront our shortcomings and open ourselves up to teachers and peers, we panic and deploy our defenses instead. For the same reason that I pushed away Topology, struggling students push me away now.

Not understanding Topology doesn’t make me stupid. It makes me bad at Topology. That’s a difference worth remembering, whether you’re a math prodigy, a struggling student, or a teacher holding your students’ sense of self-worth in the palm of your hand. Failing at math ought to be like any failure, frustrating but ultimately instructive. In the end, I’m grateful for the experience. Just as therapists must undergo therapy as part of their training, no math teacher ought to set foot near human students until they’ve felt the sting of mathematical failure.

This post hit close to home with me as I was sitting in my office, reading blog posts instead of doing work. I don’t think I’d ever thought about procrastination as an expression of anxiety before, but I realized as I read this that that is exactly what I was doing.

indeed, James.

Hi, Ben —

I just read your post on Slate.com and was overwhelmed by memories from over 40 years ago of how I felt in 5th-grade math when I was suddenly moved up to the 6th-grade class where I immediately –and silently — crashed and burned. Of course, because math is cumulative, I never recovered. Throughout high school my transcript was bimodal — Cs in math and As in everything else. When I went off to college in 1977, I was placed in a pilot program for female students with math anxiety, supported by a federal grant, I believe. My wonderful teacher Jeanne Trubek helped me get through a semester of calculus, but I never got past my math anxiety.

I went on to graduate from law school, marry and have children, who turned out to be pretty good math students (my son is a first-year student in college and is considering majoring in math). To my great surprise, I actually enjoyed helping them with math when they were in elementary school. In particular, I loved thinking about all the different ways you can solve any given math problem. Who knew math could be so creative? Well, not I. Gradually, I realized that up until 5th-grade I loved math. This realization has shaped my perspective as a parent that kids shouldn’t be rushed or pushed ahead, but really need to be allowed to develop their understanding of math or any other subject at their own pace.

Hi Alex, thanks for sharing your story.

It’s so true what you say about pushing students ahead. The benefit (getting a year or two ahead of the game) is often dwarfed by the potential cost (throwing a kid into the deep end before she’s ready, which can be a discouraging – or even terrifying – experience).

I’m happy that you had the chance to rediscover math through your children. Math really is creative! And there’s so much to learn in elementary school, from concepts to skills to habits of mind – especially the idea you mention, that there are many ways to approach every problem.

That sounds like a great program you found in college – can I ask where it was? I’d be curious what techniques they used.

And good luck to your son in picking a major!

Best,

Ben

Hi, Ben —

The math anxiety program was developed at Wellesley College in the late ’70s and was funded by a grant from the US Dept of Education, I believe. I’m going to try to find out more about it and how outcomes were tracked. I’ll let you know what I learn.

The critical feature of the program, I think, was simply that the faculty understood that we, the students, had gaps or holes in our math education, but that we

were, nonoetheless, capable and motivated students.They were extremely patient with us, as I recall, but also insisted that we actually understand the material — in other words, they didn’t let us just nod our heads and say “I get it.” I’m so grateful for that even now.

Alex

Thanks for following up, Alex. That sounds like a simple but obviously powerful philosophy – patience and high expectations.

As parents or educators, we have to walk a fine line when it comes to highly-able students. If we push a student too far, too fast, they may crash and burn, as you say you did. On the other hand, when you have children chomping at the bit to learn more math, and being held back because their classmates aren’t ready to move ahead yet, we may kill their enthusiasm for the subject. If we judge poorly, we can hurt a child either way.

That’s well-said. In response to this piece, I’ve heard from several people with Alex’s experience of getting pushed too fast. But a piece titled “What It Feels Like to Be Bored Out of Your Skull In Math Class” would probably have drawn the opposite response. Both are dangers to avoid.

Alex, Thank you so much. Teachers are never sure they’ve actually accomplished anything – its so wonderful to read this.

Jeanne Trubek

I’m delighted that this somehow made its way to you, Jeanne. I struggled mightily with calculus, and, thanks to your patience and excellent instruction, I learned so much — not only about math, of course, but about persevering. Do you recall that the student evaluations asked “Has this course taught you to think in new ways?” Calculus did that for me. Over the years I’ve been so curious about the results of the math anxiety program. Can you tell me anything about that?

Hello Alex, What year did you take calculus ? I was variously an adjunct or a leave-replacement faculty at Wellesley, so I was not really part of the decision-making or evaluation processes. In general, it seemed to me that addressing just the anxiety was not sufficient; people needed to have the holes in their mathematical backgrounds filled, but in a non-threatening, non-competitive manner. If the holes are not filled, when they get into a later course that requires this background, the anxiety returns. Too often math is used as a filter, for no reason except that it is relatively easy to use. I eventually finished my PhD and spent most of my teaching career at Emmanuel College in Boston; we tried hard to build a math department that focused on really teaching (and not sorting). From student feedback, I think we succeeded. I am now retired; I hope the department keeps its teaching focus. This could be a very long discussion (I have about 40 years of thinking about how to teach math in me).

Just eaves dropping on this conversation. My wife attended Emmanuel College in the mid 80s as a Biology major. She also wrked in a lab at Wellesle College for a number of years. Perhaps you struggles to teach her calculus.

Hi, Jeanne — I took calculus in ’78-’79 (fall term, I think, but that might not be right). I was the student who would nod and say “I understand” and you would respond “I’m not sure you do; why don’t you try another problem?” (of course, it’s possile that you had more than one student who did this.:)). I agree completely about the need to fill the holes underlying the anxiety. While the math anxiety workshop did some of this, in my case, at least, many of the holes weren’t plugged until my kids took math in elementary school and middle school. As I helped them — at first =– and then later, as they explained things to me, many gaps in my knowledge and understanding of math were filled in. Like you, I could go on and one about teaching math, though obviously from a very different perspective. It’s particularly interesting to me that my son, who is now in college and who has always been a very strong math student with a real love of math, shares many of my views despite the difference in our ages and in our experiences with the subject. Similarly, I think you and he would agree on many things — just one example, in both HS and college he has seen the math filter at work, particularly in economics classes. His take on it is that the students who lobby hard to get past the filter are often successful nothwithstanding the fact that they don’t have the prerequisites. He has also mentioned that he sees male students as much more likely to push this envelope. Additionally, he has seen more male students continue in math despite getting Bs or even Cs while some female students drop the subject when they get grades lower than an A. Here’s a link to a recent Washington Post column about a study on this subject:http://www.washingtonpost.com/opinions/catherine-rampell-women-should-embrace-the-bs-in-college-to-make-more-later/2014/03/10/1e15113a-a871-11e3-8d62-419db477a0e6_story.html

Enjoyed the article. I would like to use this in my math classes at UOP online. Can I have your permission to copy and post the article. I would just use the URL as reference but these change and I’d like to keep the article for my permanent reference library.

Thanks for your support in teaching math.

Dick

Absolutely! Thanks for reading.

What a great, honest, article. You explained what I’ve felt many times, better than I ever could have (I’m a math Ph. D. student).

Thanks for reading! I’m glad it resonated. It was surprisingly hard to own up to the experience – my first draft had like 500 words of boasting and delaying (“Here, let me explain what topology is…”) before I actually got to the failure part. Being honest about our anxieties seems like a surprisingly big challenge for students at all levels – even grad students, from what I hear!

As a topology Ph.D. student, I feel the same way. I’ve found that when thinking about research, it’s like navigating a huge landscape, but certain hills are higher than others. You can tell the height because it starts making you feel dumb, just like you described… so to learn anything interesting you have to go through this mental process in some way, and it’s a large part of what makes research difficult / painful, I think.

Well said. I found the definitions in topology to be impressive little mountains of their own – I can only imagine what the open research questions are like!

I too admit to feeling very stupid at math on occasion. This was especially true when the topic is geometry, which others seem to get much better than I. (As I have gotten much older, I am less likely to feel stupid when I don’t understand something. Perhaps it’s one of the advantages of aging. ) As a college professor, I’m sure that I often lead others to feel stupid. Thanks for your remarks to remind me of the “other side.”

Thanks, Dad!

Some grad student friends were telling me that their advisors/professors seem impervious to this anxiety. I wonder how true that is. Are they just so brilliant that they’ve never had the experience? (Maybe – one is a Fields medalist.) Do they just have thick skin? (Maybe that’s a crucial skill for becoming a professor; it means you won’t give up when the going gets tough.) Or have they just grown more self-assured with time, so that they can say “I’m confused” without feeling like they’re saying “I’m stupid”?

Your experience suggests the third option – which is the most reassuring!

I think you’re right about the third option. As a grad student, I used to feel stupid. Now as a professor, after having proved a few results of my own, I know I’m not stupid. If I don’t understand something, it’s only because I haven’t thought about it hard enough. From first principles, via simple examples, etc. I often now look to Polya’s maxims: What is the simplest case that I don’t understand? and then try and understand that; Look back at what I think I now understand, and see if it makes sense in the simpler settings, so that I can explain it to someone else.

On that note, lecturing is always useful. It forces you to simplify and break things down to the simplest non-trivial case, from which we can then generalize.

Thanks for the honesty in your article. It’s not only refreshing, but also well articulated.

Thanks for reading, Victor.

Your comment makes me want to go read that Polya book – I got it as a gift a few years ago, and I’ve foolishly left it on the shelf since then. I like that approach of identifying the simplest hard case, and working from there.

That’s very true about lecturing being a good way to improve understanding. Teaching calculus has definitely forced me to unify and simplify my way of thinking about the material.

I had a similar experience with algorithms, and before that discrete math, and before that calc, and before that …. I don’t think (except for brief, fleeting moments) that I have ever experienced mathematics as anything other than a source of fear, anxiety, and self-loathing. I really wish the right people had understood that when there was still time to do something about it.

Speaking of math teachers holding your sense of self-worth in the palm of their hand, I have been crushed by a few. One of them (a calc teacher) asked me point blank “what are you even doing here?”. Not a high point in my life. On the other hand, It was another math teacher who first pointed out to me that I might have a learning disability (some kind of dyslexia, but it was a possibility I came to accept only when it was too late). In the end, I am a successful software engineer. I forced myself to endure just enough math to get through college. I guess I’m a bit of an intellectual masochist. I go through it, but it was nothing but anguish.

I am aware that computer science is a branch of mathematics, but with the kind of work I do, I really feel that I am drawing on different mental faculties. Or perhaps I never approached math with the right attitude. All I know is that it started when I was very young. I got my first arithmetic assignment handed back to me with nothing but red marks on it, and the sense of humiliation and disappointment has been with me almost my entire life.

Think what you do with your red pen.

Thanks for sharing – it’s impressive you pushed through for a career in CS against such a strong (and unfair) headwind.

It’s sad that your teachers took such a narrow view of mathematical ability. One of our big sins as teachers (and I don’t excuse myself) is relying over and over on the same limited forms of instruction and assessment. If lectures and computations aren’t your thing, you’re out of luck in most math classes – which is a waste, because lots of people who struggle with those things are still inventive problem-solvers, with a skill for rational analysis.

I’m glad that the flaws of the system didn’t hold you back professionally (even if they did cause you private anguish). Your story is a good cautionary tale for those of us who dismiss failing students as just “not math people” – a dangerous and misleading label.

Being as how psych was your secondary major, you might find Carol Dweck’s mindset research interesting and informative on this point.

Thanks – vague memories of reading something by Dweck in college, but I’ll check her out again.

“As I procrastinated, spending more time at dinner complaining about topology than in the library doing topology, I realized that procrastination isn’t just about laziness. It’s about anxiety.”

You describe a sense of hopelessness and surrendering.

In light of your own personal experience of “math anxiety”, how might a math teacher — not a college/university math instructor — reverse that anxiety in their students?

Such is an attitudinal difference. I never experienced math anxiety because I know that I shall eventually understand the material albeit difficult and I persevere not procrastinate.

The only measure of “anxiety” is not understanding within the usual time span (one term) compared to the smarter or advanced classmates who understand immediate or fearing not understanding by the time for the midterm or final exam.

“how might a math teacher reverse that anxiety in their students?”

That’s a really good question. I wish I had a surefire answer. (An upcoming post has some other thoughts on the issue.) If other teachers are reading, I’d love to hear their thoughts.

In my experience, the following things help: (1) Giving students space to ask questions one-on-one or in small groups; (2) Peer tutoring (which can give them a chance to voice questions and realize that they’re worth asking); (3) Patience, humor, etc. – a little human sympathy never hurts in any situation; (4) Keeping aware that a trivial interaction for me (“Uh, your classmate just asked that, but I’ll answer it again”) can be emotionally laden for the student.

Did you ever make another go of topology? I bet with another attempt, you could conquer that topology course.

I haven’t yet! But it’s a good idea. I’ve been meaning to take a class in complex analysis, too.

Fear not folks, Mr. Orlin earned his beard!

I think this is quite applicable to the way we teach math in elementary school. We march through a series of lessons, whether or not each one of the students has understood the previous concepts. By the time kids reach 4th or 5th grade, it is difficult to for them to admit that they really don’t understand re-grouping in subtraction or the idea behind multiplication. You can see it in their eyes – they really don’t understand. They will pretend that they get what you say to them, but you can tell they don’t. But it doesn’t matter, tomorrow is a different lesson and you will leave today’s lesson behind — until it comes back in an even less comprehensible form in the future.

That’s really well said. The fundamentals are hard! K-5 math is full of rich, subtle concepts. I’m sure a lot of later struggles in math can be traced back to an incomplete understanding of those underlying ideas.

I found your article fascinating. I teach “struggling” Algebra I students. They come to me twice a day, so as to teach at a slower pace. Coming out of junior high, they are “beaten down”. I try very hard to give the attitude of “I can do math!” I have also taught piano lessons and remember that giving too much information too quickly results in frustration. So I try to give just enough when beginning a new concept. Many times they are frustrated and remark,”When are we ever going to need this?” My reply is, ” I’m your Brain Coach.The exercise it takes to work this will make your brain stronger” But still some cannot grasp it. I’ve heard there is a study about how high an IQ one must have to grasp algebra; is there?

Thanks for reading, Carol. That sounds like really good work you do – knowledge of Algebra 1 opens up so many doors for later coursework (especially in college).

I’d be surprised if there’s a specific IQ needed for algebraic competence – IQ is such a rough and oversimplified construct, and algebra requires such diverse tasks. It’s true that at the lower end of the IQ spectrum (sub-70, roughly the bottom 2% of the population), people have trouble with some basic academic skills. If the IQ is low enough (sub-50), people never learn to read or do arithmetic, much less algebra. But at that point we’re talking about an extremely small fraction of the population (less than 1 in 1000).

Getting students engaged, of course, is much trickier. Have you checked out Mathalicious? http://www.mathalicious.com/

This makes me wonder if perhaps part of teacher prep programs should include prospective teachers taking a subject area course in a field that they have traditionally excelled in but at a level that is just one or two steps ahead of where they are. We don’t want it too many steps ahead where they will have no hope of catching on but also not something that with a tiny bit of extra time they will be able to figure it out.

I wish I had read this post before I became a math teacher myself.

Great post, Ben.

Thanks, Geoff! One of my teacher training seminars did a brief activity like that – “here’s a dense passage of Emerson, now read it quickly” – to show what frustrating texts are like for students.

That’s very clever. I like that idea.

This applies for other topics, too. Genetics, biology, etc. And, it also explains my procrastination. I had never pinpointed the anxiety, but it’s definitely there. Something to keep in mind with my students.

Yeah, I can imagine a similar experience in a genetics or bio class. I procrastinated more as a first-year teacher, actually, than I ever did as a student – the stakes were higher, and I was worse at the job, so my anxiety was greater.

Wow – I had almost the EXACT same experience with topology. As a double major in mathematics and statistics, I LOVED numbers and math throughout school – but topology was like hitting a brick wall. I would love to re-take the class someday, but since I’m a statistician, it’s a “back burner” desire…

Well, we should compare notes sometime if we ever wind up back in topology class!

I was exactly the opposite. I struggled with Statistics and Analysis (especially Complex Analysis) but I loved topology (and algebra)! I eventually decided I was just not cut out to be an Analyst and concentrated on Algebraic Topology. It took me three tries to get through the Analysis part of my Comps.

Interesting. I kind of sprinted through my math major, and never really figured out whether I preferred analysis or algebra. An algebraic approach to topology sounds pretty elegant – part of my trouble with Topology was that every concept had a monstrous analytic definition.

For me it was – I think – something to do with making the jump to somewhere where you lose a framework you relied on and have to completely change your frame of reference – and there is so much you need to get hold of at the same time before it starts to make sense. It was like trying to learn French by first memorizing all the irregular verbs without ever having heard it spoken

When I started grad school (for the 2nd and 3rd times), I was an older returning student and I can vouch for having had the experience of being in classes that I once would have excelled in, but were at the time one or two degrees above where I was. I can remember the feeling of panic more times than I can count, thinking that I would never manage to understand it. I think it does inform the way I teach young children.

Oh my, this sounds just like my experience with real analysis at Harvard. I will always be grateful to that professor for the pitying c- he gave me. All the pedagogical research I have read since has convinced me that I could perhaps someday learn analysis if I had the right practice and coaching…but I would still approach it with a lot of anxiety. I

The experience was very humbling. Like you, I think it really helped me as a teacher to understand how debilitating that kind of anxiety can be. And it hs made me determined to make sure my kids see their failures as a need to work harder and practice more, not as a lack of talent.

Yeah, helping kids contextualize their failures is so important – I’ll look around for a link when I find time, but I know I’ve seen research saying that a student’s view of intelligence (fixed and unchanging, or fluid and influenced by hard work) is one of the best predictors of their success in school.

But it’s not just work. I got some very good advice from an Algorithms professor: “You have to play with it.” [stop sniggering]. What he meant was that solving your problem requires understanding it, and to understand it, you have to look at it from different angles. Sometimes it just takes a few days of chewing on it subconsciously before you even know where to start. It had really never occurred to me that the solution wasn’t supposed to be obvious.

I had always assumed if that if the answer to a problem wasn’t obvious, it’s because I hadn’t paid enough attention. When we teach math, we organize things for clarity. Unfortunately, this creates the illusion that there’s always an obvious, linear, progression of concepts. It hides the messy truth of how the knowledge was actually acquired: through many twists and turns in a process of creative exploration. And, as you progress in mathematics, you have to bring more and more creativity to bear. This has many implications, but the most germane is that it’s very difficult to be creative when you’re in a state of panic.

I’d never considered the drawbacks of clarity before, but I think you’re right. I remember some talented lecturers in college who lulled me into a false sense of confidence; it all seemed intuitive when they talked about it, so how hard could it be? Then I’d sit down to work on the problem set…

Among the other implications: it’s a good idea to teach some exploratory lessons where students have to struggle and collaborate to reach a new level of understanding. (Ideally every day would be like that, but time and energy are sadly finite.)

wow. yep. 😉

Came to your page because leaving comments on Slate is a drag…

You are spot-on with this. I had a similar experience in High School Calc, and then picked a dual major that needed no more math: Int’l Relations, and Philosophy.

Fast forward 20 years, and I am in a PhD program in Educational Measurement and Statisitics…and the Statistics are starting to treat me the same way. Hopefully, with 20 years’ experience, and your article, I’ll be better positioned to soldier through it.

Thanks!

Good luck with the program! I’m sure you’ll rise to the challenge of the stats. The rigor of a philosophy major is a lot like the rigor of a math major, I think.

Definitely true in the sciences, too. A lot of my students are scared of balancing equations and stoichiometry.

I never “got” literary analysis in my English classes in high school. I had a great teacher in 12th grade, who basically told me to make up some fantastic story but to be sure to back it up with quotes from the text… exactly the opposite from my science classes, which start from the data and end with conclusions. But it seems to be okay “to be bad at” all of math and/or science at once, but is okay to not enjoy (only) Chaucer or (only) modern American lit. There seems to a be a tacit societal acceptance of this fear of math.

Hmm. I think you’re right that when people say “I’m just bad at math,” others tend to leave it alone.

I suspect that’s partly because mathematical failure is seen as about intelligence, whereas struggles with the humanities are seen as a matter of effort, impatience, or taste. Maybe if we saw success in math and science the same way – as about a thousand diverse factors, not just raw brainpower – then the “tacit social acceptance” of math phobia in some circles would fade, too.

So how do you encourage students to just keep at it, ask questions, get more resources? Society says it’s okay to hate all of math, so students just drop it. If you don’t get some chunk of history or English class, you have to keep at it because… well, you just have to suck it up and get through the course. But math, oh yeah, you just aren’t a numbers person.

I may have it easy in that sense – the culture at my school is pretty conducive to math learning. For one thing, English is a second language for most of the students, so a lot of them found success in math early on (since the linguistic barrier is relatively lower than in most subjects).

I’m not sure what I’d do if I found myself in a community like you’re describing. I might focus on changing the culture, resorting to the usual teacher tricks – pulling in higher-interest material, colorful applications, games… It strikes me as a tough uphill battle. But I imagine the surest way to change students’ view of math is to put different math in front of them.

Have you hit on any good strategies?

(Strange, I think this is posting in the wrong order.)

I’m mostly thinking of my own experiences struggling through English and history. I cannot remember names or dates to save my life (and I’ve tried a lot of techniques), so history was painful for me. I had to go through the four years of high school social studies courses, and remember very little from them. In college, I took art history, and, being a visual learner, I actually learned a lot of history through art (although I still can’t tell you who painted what or when). It’s sad how much history I don’t know (and know that I don’t know it), but I’ve met few people who are sad for not completing science courses.

In my chemistry classes, I find that I get students to at least try complicated calculations when I dramatize the amount of pain they’re about to go through. It’s pretty fake (“oh gee, yeah, this is kinda annoying!”), but it at least makes some pretend empathy or something, and puts them more at ease. I also try to make it relate-able for them, which increases interest, but not necessarily their desire to actually perform.

Like you said, a little humor/empathy seems to go a long way in making it through those technical steps. I often wax poetic about computation (“And in the midst of the glorious battle, we must summon a new weapon – the Pythagorean identity for sine and cosine…”)

A lot of students probably see long computations the same way I see weightlifting. It’s like, “That looks painful. Why the heck would I ever want to do that?”

This is a great summary of frustration. Can I reproduce this for my community college students? With proper credit of course. If not, I will direct them here. This question has confounded my attempts to deal with it while teaching chemistry!

Absolutely! Thanks for reading, Eileen – I hope your students find it helpful!

This was a huge stumbling block for me when I was tutoring lower-level math. I literally didn’t have the words to explain why things were the way they were a lot of the time because to me (not a math genius, but I got A’s in math all through school) it was just obvious, even though to my students it clearly wasn’t.

Sometimes the most elementary stuff is the hardest! Concepts like place value, regrouping, multiplication, inverse operations… there are so many obstacles there, but they’re so far in my own rearview mirror that it can be hard helping students over them. I admire the heck out of my sister, who’s an elementary math specialist and always knows exactly how to bring those ideas to life.

Teaching “struggling” Algebra I students takes lots of patience, lots of going around the room and teaching one-on-one, thinking of more examples to get their “lights to go on”. I’d like to talk with you elementary math specialist sister. I sometimes tell them, “you have to struggle a little with this”, which they don’t want to take the time to do….

I’ll try to put you in touch!

Thanks for sharing your experience. Coincidentally enough, I had my worst college math experience in topology, although in my case it was more related to other things going on in my life… I failed my midterm, dropped a course for the first time in my life, felt unnecessarily ashamed, and did a great job the second time through when I could truly internalize the material.

In the intervening years, I happen to have become a college math instructor, so I’m curious: What do you think your teacher could have done to help you avoid this situation? You suggest we should bear the student’s perspective and potential fear in mind, and I try to do that already; but how can we, as teachers, not just be aware but actively help?

Thanks for reading!

I was probably a lost cause in that class – it was my last semester in college, and my mind was on anything but topology. Nothing the professor could’ve done.

But it’s a good question you have about actively hepling. I certainly don’t have all the answers – I’d love to hear your thoughts. The best thing I’ve come up with is to try to create one-on-one and small-group situations, where students feel comfortable asking questions, without fear of judgment or time pressure. Easier said than done, but a good goal to strive for.

What are you doing to help move students past the symptoms and on to getting it?

Less than I should be. Less than I wish I were.

I try to connect with them personally. I try to frame the material heavily, with lots of time spent discussing how the pieces fit together, so that if one piece is missing they can still make out the big picture. I try to create low-stress opportunities for them to ask questions. I try to react positively no matter what questions they ask, and fight back the awful defensiveness that I feel when they ask about something I thought they already knew. I try to see the material from their perspective, and anticipate obstacles. Mixed success in all of this.

Do you teach? Any tips?

Excellent article. Effective pedagogy needs to flip its orientation and focus extensively on error. No two people have the same patterns of understanding/misunderstanding — although a teacher experienced in error-based pedagogy in a particular subject will begin to recognize particular “error profiles” which can serve as a foundation for diagnosis.

The fact is that while most teaching is fixated entirely on “the correct answer”, students take a great variety of paths to get to their (often incorrect) responses…and it’s only by respecting those paths that a teacher can move a student away from a set of erroneous responses. Teachers don’t like doing this because it feels like a waste of time; students don’t like it because they are uncomfortable subjecting their own mistakes to careful analysis. But…as a way of building understanding of the material, it works — and once students are no longer afraid to speak up and say, “I don’t get it,” the entire class becomes far more engaged.

If you’d gone to a class called “Topology: twenty common mistakes and misconceptions,” I suspect your experience would have been entirely different.

(BTW, I teach music education)

One of the things I frequently do as a substitute teacher is to thank kids for asking questions and for making mistakes. I will often say something like, “I am SO GLAD you made that mistake, because that give me the chance to highlight this problem that some people frequently make…” or “Thank you for having the courage to guess, even though you weren’t sure.” or even, “What a great question – it doesn’t work quite that way, but you are really thinking about the process!”

Yes! Those little encouragements go a long way.

Thanks, Warren – that’s a nice articulation of the philosophy. I like that term “error profile” – it’s a concept I’ve thought about but never had a word for.

I’ll keep your comments in mind for a unit I’m teaching right now in precalculus. One of my go-to maneuvers in lecture is to make an error, and have students explain to each other what I did wrong. But that’s usually informal and spur-of-the-moment – some prepared materials on common errors might go a long way.

You say you’re bad a topology. So does that make a student who says “I’m bad a math” correct as well? 10,000 hours my friend. Almost everyone can become really good at something if they’re willing to put forth the time and effort. It begins with a willingness to engage in productive struggle.

Other than that one line, I love the story.

Fair point. “Unsuccessful at topology” would’ve been a better phrase than “bad at topology.” I went with the more melodious wording over the more precise one.

Today, I watched a very capable student reduced to tears during my math lesson. She is ordinarily a student that catches onto the material quickly, and even requires an extra challenge. Today, however, she looked confused, yet insisted she did not need any help.

“I hate this game! I hate math!” She continued to protest. “I don’t want to work with my partner! This game is boring! When are we done?”

She is six years old.

I pulled her into the hallway to discuss the content of this very blog post. (How timely!) We talked about the symptoms, and about developing a growth mindset even when all we want to do is pretend we understand. I showed her the math content in a slightly different way — we were working on the inverse relationship between addition and subtraction — and her tense brows started to relax. “Oh, it’s like that?” She agreed to try again another day.

These feelings of failure can start early. I won’t pretend that my one conversation with this girl — a smiling, happy, bright 6-year-old girl — will prevent this from reoccurring. At least, after reading and giving this post some thought, I recognized the symptoms faster the usual. While it’s better to be proactive than reactive, some reactive treatment of math anxiety is better than nothing. Until we figure out a better way…

That’s really beautifully said. Why am I the sibling with the blog, again?

You’re the one with the blog because it’s 1am and all I can think about is how clumsy many of the phrases in my posted comment are. Next phone call: we talk about failure in writing! I had some interesting thoughts this weekend about the role emotion plays in written communication.

Meanwhile, I am thinking about making this blog post required reading for my 8th grade tutoring students… and their classroom teachers. One teacher lamented that the 8th grade girl does not advocate for herself enough when she’s experiencing partial understanding. Perhaps this would arouse some empathy? Once there’s empathy, we can start to confront the issues at hand.

As a student, what’s a better way to try and combat procrastination and anxiety about math? I’m a college freshman taking Calc 2 right now and it’s absolutely killing me. I’m not a math major, genetics actually, but I’ve always considered myself good at math, having taken AP calculus in high school and other advanced courses. After doing well in those classes and our first quizzes, it came as a huge shock when I was handed back our first test with a 48 circled in the corner. As the semester has gone on, I’ve gone to my TA for help, as well as looking up explanations from khanacademy and other online tutors. However I still feel like I only half-understand what we’re doing most of the time, my brain just doesn’t think about things the way my professor teaches it. I appreciate what you say in your article that being bad at topology didn’t make you stupid, it just made you bad at topology.That’s exactly how I’ve been feeling lately, and it’s hard not to carry it over into other classes. I know many people consider Calc 2 the hardest calculus course offered at my university, but that really doesn’t make me feel any better. What are some tips for students in terms of studying, dealing with a professor who doesn’t have a similar learning/teaching style and just making it through the class?

Hey Emma, it sounds like you’re already doing a lot of the right things – asking questions, staying on top of the work, seeking out diverse explanations.

Sometimes when we struggle, it’s because there are missing pieces somewhere behind us. You might find it worthwhile to comb through your Calc 1 (or AP Calc) notes and see if there are any topics there that you feel shaky on. If so, working on those might help you with the Calc 2 material.

Most of all, keep it up – you’ve shown great initiative already – and remember to be patient. Your story reminds me a little of a 9th grader in the first geometry class I ever taught. She was very bright, but geometric proofs (and my approach to them) didn’t agree with her. She wavered all year on the B-/C+ line, and told me often how frustrated she was. That was three years ago – and the same student is in my AP Calculus class this year, earning an A. Couldn’t be more proud of her.

Point being, it might take time for that hard work to pay off, but it usually does.

Good luck! I know you’ll do well. Keep me posted on how things go!

It’d be tough to agree with you more on this. Math is a very peculiar subject in that it builds on itself so linearly and so completely. Therefore there’s a sort of unstable equilibrium when it comes to understanding it. You can understand steps 1-100, but if you don’t get step 101, and you never put in the work to do so, then you won’t get steps 102-10,000 and you’ve effectively “fallen off the cliff”. Then you start spiralling out of control and singing the anthem that you mentioned: “I hate math”.

Lots of other subjects aren’t so delicate. I could not understand biology yet still do fine in my physics class. I could be confused about grammar and past participles but still do pretty well in my literature class. I could know nothing about medieval history but know everything about Chinese history. Thus, if people are confused by something in, say, the history of the modern middle east, you don’t hear them yelling “I hate history” because they’ll probably end up doing fine in next semester’s history class. Putting off mathematical understanding to “next semester” simply doesn’t work.

Sure, this makes math more difficult in this sense, but its beauty and importance should be enough to motivate teachers and students to recognize this added difficulty and work to overcome it. Once one starts to slide down that cliff, they must be pulled up by their teachers and fellow students, because with each passing day they’ll fall further and further down. This is on both students and teachers.

Great post. I stumbled across this blog, but I’ll be here to stay!

That’s really well put. I’ve been thinking a lot about what separates math from other subjects in this regard – I don’t think a post called “What It Feels Like To Be Bad at History” would have had nearly the resonance that this one did. But as you say, the cumulative nature of math seems like a pretty plausible explanation.

Oh Mr. Orlin, how I wish you had been one of my math teachers growing up.

I was labeled “gifted” in second grade and was lucky enough to leave the grind of classes once a week to learn atomic theory, classical art, logic problem solving and other very cool things. I loved it all. Then I had 4 days in regular class. And I started experiencing the downside of the label. I was struggling with long division and decimals, and I still (many more years than I care to admit) remember vividly, the teacher exclaiming in front of the whole class, “You have two professor parents and you’re in the gifted program, and you can’t get this? I don’t understand. Your sister never had these problems”

I was done with math from then on – smote by that triple whammy.

Years later, I had to take an algebra class to get my bachelor’s. I was terrified. More terrified than of spiders. More than a colonscopy. Luckily, my sister (who by then was an 8th grade science teacher) said, “Meh. You can do this. If you can calculate IV rates and measure heart rhythms and respiratory peak flows, you can do this. I’ll tutor you.” I got my first A in a math class. I was stunned. I wasn’t afraid to ask questions in class, because nobody knew my parents, nobody knew I was “gifted” and NOBODY knew my sister.

For all you students who read Mr. Orlin’s blog, please, please, please don’t let others’ labels define you. I’m still not a math “fan” but I know it’s manageable. It’s not about the “A.” It’s about having some sort of understanding. Really.

Hey Carolyn, thanks for sharing that story. Another commentor above (Alex) had a somewhat similar experience – it seems like the gifted label can backfire sometimes, especially if it makes students miss important instructional time.

That comment about your sister must have carried a particular sting – children are so wary of sibling comparisons. My older sister always had a gift for music, and (not coincidentally) I never practiced my interest much. On some level, I figured that she was the musical one, and I’d never stack up, so why try? (And that was without anyone scolding me, as your teacher scolded you.)

I’m glad you came back around to conquer that math class. Keep telling your story to students in need!

Whew! Déjà vu. Point-set topology: D. Grad seminar in logic: Dry mouth, rapid heartbeat, inability to look anyone in the eye. I’m your poster-boy. Took 14 yrs to get my Ph.D. Now emeritus professor of math & computer science.

Advice a teacher gave me that got me through it: “You never understand Course N until you’ve taken Course N+1.”

Advice to self that got me through it: Do I still love math? Then keep trying.

Example: I taught a computer graphics course for the first time in an intensive 3-week term. Every evening I stared at a blank piece of paper and thought, “Tomorrow by noon my students are going to know this material, so it can’t be that hard.”

Check out Parker Palmer’s book The Courage to Teach.

I LOVE that advice…

Thanks, Gene. I’ll check out the book.

It’s so reassuring to hear that professors had the same experience back in their day. That’s good advice about course N+1. I’ve found that even after taking N+2, N+3, and N+4, there are still new discoveries and connections to be made in course N.

I am a college and law school graduate. I took calculus in college and got an A-. To this day I do not understand the essence of calculus. I have read a couple of “math made easy!” books over the years. The clearest on calculus I might have read is “it measures the rate of change in things.” I’m intelligent enough to understand calculus and other mid-level math subjects, but have never encountered an intro/survey that didn’t quickly go down a rabbit hole and repel me. Frustrating – math really interests me, I remember actually liking calc, probs & stats – solving the puzzles. Not many places you can get to the perfect “right” answer and savor it.

Hey Scott, thanks for reading. I think there’s a tight connection, actually, between math and law – both formal, logical systems built on rational arguments, where there’s often a right answer to be had. (At least, I like to think of the law that way!) Our taste in math sounds similar. I love the concepts, and view the technical details and computations mostly as a matter of necessary bookkeeping. I wish I could recommend a Calculus text that explores the beautiful ideas and skips the technical details, but I haven’t read one. There’s a Cartoon Guide to Calculus, which might be worth a shot.

(I’m actually working on a Calculus project like that, but it’s in the early stages.)

Thank you for such a thoughtful article; I originally read it this morning in Slate. As teacher, on item you might want to keep in mind (and I think you touched on it with your topology class example) is that a lack of understanding of math is not necessarily a persistant thing with students. That is, students might genuinely understand the content for several weeks into a course, only to find themselves completely stumped when a new piece of content is introduced.

Such a thing happened to me during a statistics course. I got through the first few chapters just fine, but when we arrived at hypothesis testing, I was completely thrown. In my case, I think the textbook did a poor job of bridging the content between the section we had just finished and the new content on which we were embarking. It took me a couple chapters more before I fully realized poorly I was understanding things. Fortunately I was in graduate school and experienced enough to recognize what was happening and retrace my steps. However, that wouldn’t be so easy for a youngster.

I think this kind of problem happens with math because of its cumulative nature; you have to learn it in sequence. With a subject like art history, while there is clearly a progression of styles and a chronological nature to the historical events behind the art, it is entirely possible to have a meaningful discussion about art of a certain period without possessing a thorough understanding of what came before it.

Well said, Peter. It means that math teachers have to be especially vigilant for the signs of struggle.

Hypothesis testing is a tough cookie, and an especially good example of that cumulativity you’re talking about. Hypothesis testing builds on confidence intervals, which builds on the distribution of sample means, which builds on the normal distribution, which builds on concepts of standard deviation, which builds on deep ideas about data sets themselves…

Wow. Topology.

I am a high school math teacher, and I started graduate school for math part-time last year. You couldn’t get a C in the program in order to stay in and I started with a B and an A, so I thought I was kicking ass and taking names…well…I got a C in topology. That burned.

And as I read this, my students are taking a test in trig…and I resonate with the faces their hatefully sending my way now that I’ve read this. I can’t believe that I sucked at this stuff just six months ago and I’ve already forgotten the sensation.

It’s easy to forget, isn’t it? Good luck with teaching trig – that’s some of my favorite material to teach, and some of my students’ least favorite material to learn!

I loved this post! I’m a word person, NOT a number person. Math and I aren’t really on speaking terms most of the time and when I first read your post, I wondered what topology was…was it the study of tops? Ironically, my husband is a math major. What a difference it makes when a teacher breathes into our lives and compassionately understands. My son has a lovely math teacher and it’s doubtful he’ll ever be a star student in her class, she has cared and praised him for his efforts. I’m now sharing your post with her. Thanks for writing this!

Thanks for reading and sharing! I hope you reconsider that view of yourself as “not a number person,” too – I think anyone who relishes language can find the elegance in mathematical ideas as well!

Great article. I just posted one of my many moments of failure and what liberated me. (www.geniusinchildren.org) Reading your story I’m thinking that your greatest cause of pain was your image as a smart person–in fact a very smart person who was smart in math. Failure would be no ordinary failure; it would be a devastating blow to your self-concept. If true, more more data point for Carol Dweck

Thanks, Rick. I think you’re exactly right – and I suppose I’m honored to be another data point in Dweck’s ever-expanding sample. Just followed your link, by the way – looking forward to checking out your blog.

An English lesson: “disinterest” does not = lack of interest. Look it up in your dictionary.

Touché.

In my feeble defense, “uninterest” is a clunky and unappealing noun. I would’ve had to substitute “lack of interest” or “lack of engagement,” which would fix the meaning but disrupt the rhythm.

But your point is well-taken. Between that and a computational mistake I made in another post, I could retitle the blog “Math with Bad Drawings, Diction, and Arithmetic.” Or perhaps more concisely, “Math with Bad.”

I found this quite interesting, my perspective is rather different so I thought I would leave my two cents here. For me, it is somewhat hard to really put myself in this position, because I’ve never really felt this way. I’ve never been embarrassed about asking questions. When taking General Relativity as a senior, I got frustrated because the classes just did not seem to build on themselves in a clear, logical way. Eventually a third of the way through the semester, it turned into a bit of a mini rebellion with me and several of my friends telling the professor we just didn’t understand what was going on. We spent an entire lecture asking questions, beginning to end. It helped a little (although I never fully understood relativity, it’s a tough topic).

The funny thing is that while this rebellion was going on, there was another group of students that kept saying that things were clear, etc and to just shut up. Me and my group of friends were all top students, and had probably scored 20 percent higher than most members of the other group in all the previous physics classes we’d taken together. I think there is some kind of correlation there. Being aware of your own ignorance is one of the most powerful tools you have. If you combine that with being willing to ask questions, you can do anything.

That same year, I used to play ping pong every week with a computer science professor. He was a very smart, no-nonsense guy. I told him about my tendency to ask a lot of questions, and how it was sometimes a bit much. What he told me has stuck with me till today: some of the smartest people I know ask some of the dumbest questions I’ve heard. Because they want to be absolutely, 100 percent sure that they get it.

That really stuck with me, and reinforced my penchant for asking questions. In fact, where I used to be somewhat concerned to ask a question if it was too simple, now I no longer care. I want to be completely certain that I get it, and I won’t settle for anything less than that. Maybe that is a philosophy you can push towards your students.

Just my two cents!

That’s all so true. Your experience sounds like the flip side of mine – same phenomenon, but you had a much better approach. A lot of the strongest students I’ve taught are just like you – voracious questiion askers.

I wonder how significant it is that you had a peer group in Relativity that took the same view. I imagine knowing that you all felt the same way made it easier to step up and say, “This isn’t just me.”

Hi, your experience of staging a mini-rebellion really resonates with me (just finished my senior year in undergrad physics), mostly because it’s something I wish my peers and I could have done, but something we would never have done…because the social culture here is to keep questions for office hours and email, and not interrupt the flow of the lecture. And that’s quite frustrating.

When you say, “In fact, where I used to be somewhat concerned to ask a question if it was too simple, now I no longer care. I want to be completely certain that I get it, and I won’t settle for anything less than that.” It’s a lesson I’m struggling to learn, and it’s quite heartening to hear it from so many different people (distinguished lecturer, my supervisor, strange academics on the internet…). But it’s hard to let go of the feeling that sometimes the questions some otherwise insightful people ask are a little too simple/nitpicky/whatever, and that combined with everyone’s busy schedules makes prioritizing the ‘interaction’ part of learning quite anxious for me.

At least, that’s my excuse for not asking questions in class, and probably peers like me, and *might* (wild assumption laden guess!) be an alternate explanation for that other peer group in your class.

(Also, congratulations on knowing some relativity. I have some scars from special relativity, myself!)

I had the same experience with real analysis. Having skated through the math curriculum to that point, it was a brick wall. And all the more frustrating because my peers seemed to just keep on skating effortlessly by. The subject seemed like a smooth rock face, I didn’t even know the questions I should be asking. I still have the textbook, unread for all these years, like a totem: I really should try and learn that. For all that I couldn’t do the work, some of the concepts are still my favourite pieces of math: the many variations of infinity, the rational and the irrational.

The ideas are beautiful, aren’t they? Countable vs. uncountable, rational vs. irrational… Honestly, unless you’re in research mathematics, that appreciation is probably the most important thing you could take out of a real analysis course.

(…said the blogger who’s never actually taken a full course in analysis…)

Reblogged this on The Actuarial Pug.

I have spent the last hour emersed in this incredibe conversation. I am a 6th grade science and math teacher who plans to leave the pencils in the binders tomorrow so my students and I can build our own conversation around this blog as we share it together.

My struggle with being a math teacher is working with the tools of traditional math. I love math (something I didn’t discover until I was in highschool) which is the reason I dislike traditional math with its companion textbooks and workbooks. Yes, there is a place and time for such things (not really, I only say that in case…) and for some beings (myself after school) and I do apply them ocassionally in my instruction (just in case).

As someone who is passionate about the wonders of math I am discourged when I see the clock watchers activate, the bodies slumpers slide or the glazers go into zombie mode. I never see these heart wrenching signals of a lesson gone bad when we are “doing” math.

Such as…analyzing food packaging to determine ratios and percents; taking out the food to determine grams, surface area, and volume; having a competition with other classrooms via Skype to see who can stack the food the highest (Oreo cookies) to apply measurement, design and statistical data. Eating the food so we can explore calories to energy. Today we watched Numberphile’s Metal Math song video about Phi and the golden ratio (must and did also watch the companion making of). They are exploring Kepler at home, ViHart is one of our favorite mathematicians.

More succinctly…Relevance. If I don’t connect what they need to learn

to something in their lives their/my attention, retention, and application dip below grade level (whatever that means…grade level).

We work hard in math, our class motto is “Work is synonymous with Opportunity”. And though I strive, an edutopia I do not have. They rebel, they slump, and they watch the clock as it gets closer to lunch…but they also say great things about math and they know it is okay to not be passionate about it like I am (although many are) and most importantly, like so many of you pointed out, they know not getting math doesn’t mean a person is less intelligent, in fact, if we are not hitting walls when engaged in math, walls that require pounding , diligence, sweat and tirades, then we are not doing math. And oh the celebrations when that wall comes tumbling down.

Thank you all for adding relevance to our math conversation tomorrow…

All Good Things

This was so nice to read, Debra! Your class sounds like a wonderful place – that brief catalogue of lessons is totally stacked with great ideas. A lot like the class I imagine Vi Hart would teach.

Let me know how the conversation goes! I bet your kids will have some pretty interesting insights.

And that’s a beautiful image about the walls tumbling down. Gives me the idea to piece together a list of some of my favorite quotes from this conversation…

A nice insight into recognizing challenges.

The last math class I ever took was “Calculus on Manifolds” (I had to drop it because I was listlessly nodding/drooling because I couldn’t keep up).

Nice job getting some shut-eye, at least. Neither of us really learned about manifolds, but at least you caught up on sleep.

Thank you for your wonderful article. As a former urban school math teacher, I’ve faced the same problems from my students, and I also recognize the same patterns in me as I learn new skills. You may appreciate Brene Brown’s TED talk:

Thanks for having the courage to teach!

Thanks for sharing this – it’s a beautiful (and really funny) talk. Strongly recommended to anyone browsing these comments.

I need to have my 12 yr old read this. Math is his thing and the school is not teaching him it. But, if there is something he is unsure of I always say, “Ask questions. That is what your teacher is there for.” Unfortunately he has a teacher who thinks because he is so smart, he shouldn’t need help. They do part of their math online and today when he did ask for help she told him to go watch the online video to figure it out. He still didn’t understand it but didn’t feel he could ask her for help again. Pretty sad. He got some of them wrong and sort of flipped out about it. He’s used to getting 100% on everything. He is however skipping 7th and 8th grade math and jumping to Algebra I next year so maybe that will help. He gets it when it comes to algebra.

I hope he enjoys Algebra 1! Too bad that the teacher didn’t answer his questions – hopefully that won’t turn him off from asking in the future. A tough but irreplaceable lesson for academically advanced kids is that uncertainty and confusion aren’t signs of failure – they’re signs that you’ve finally gotten to the interesting part!

When you took Calculus, did you ever solve the Inverted Pendulum Problem? This is the problem corresponding to balancing a broom upside down on the palm of your hand. Most kids will do this at least once in life. It’s a classic problem in Freshman Calculus.

You first solved this problem (in your head) as a toddler at the age of 12 months. Vertical columns are wobbly, and liable to topple over. A toddler is a vertical column, discovering how to balance upright without toppling over. How can it be that a 1-year old toddler, who doesn’t even yet have language, can solve a problem in Rocket Science — a problem in Freshman Calculus? Isn’t it amazing that the operational part of our brain can implement the solution to a problem in Calculus without being aware of the code or theory that the practice (of standing up without falling over) is based on.

Your blog post is full of personal observations of your emotions, ranging from frustration to chagrin, as you struggled with the challenge of apprehending the subject matter.

Did you know that the interplay of emotions and learning is also a meta-cognitive process that is modeled by the Calculus?

If you didn’t know about that, you can discover more here:

Cognition, Affect, and LearningThanks for the link, Barry. I like learning more about the psychology behind these sorts of learning experiences – nice to revisit and develop some of the ideas I saw in college psych classes.

I read this and my eyes have filled with tears. I can remember the fear I felt in school every time it was time to do math. I can remember the other students teasing me during homework check because the majority of my answers were wrong. I can remember sitting at the table trying so hard to do my math homework, and not understanding any of it! I can also remember the feelings of failure walking around with me on a daily basis, and they still do! The worst part is I’m 41 years old with two undergraduate degrees, and a masters degree. II hated math and my math teachers. The reason is simple. I wanted to understand, and they didn’t do their jobs! I took notes, ask for help and listened attentively in class. Instead of showing me where I was going wrong they would ridicule me in front of all the other students! Once in the schoolyard I had to deal with my peers. Since I wasn’t good in math I didn’t care and I didn’t try anymore. Eventually, I dropped out of high school in the 10th grade. So, you are probably wondering if I’m a high school drop out how is it I have 3 college degrees? Well, the answer is simple. I wanted to learn! I wanted to understand! After I dropped out the desire to go back burned, and my heart ached constantly. Finally, I returned to school. I was whole again. My fear of math remains, but now that I’m older I have the courage to fight my fear and win! Thanks for sharing your story!

Thanks for sharing that, Alisa. I’m glad you came back around to school, even if math class never felt like a comfortable place. Have you read Lockhart’s Lament (http://www.maa.org/devlin/LockhartsLament.pdf)? It’s a scatching (and very funny) take on the problems with math education, and I think it might resonate with your experience.

Dear Mr Orlin,

I have enjoyed reading your article very much, and it inspired me to write a related article for my own blog: http://polytropy.wordpress.com/2013/05/04/limits/ You say student procrastination is not just from laziness. I say, I hope that’s right, because if so, then there is hope for the students: they do have the potential to appreciate learning. But how many students are there that really don’t care?

Hi David, thanks for sharing that. I plan to check out those links on non-standard analysis. I usually do a day or two on the delta-epsilon definition of limits with my students, not with the goal of getting them to write proofs, but with the hope that they’ll gain comfort with those two quantifiers and the tricky logical structure (“for all ___, there exists a ___ such that if __, then ___”).

I believe that procrastination is usually more than just laziness. But as you point out, there are at least two (and certainly more) reasons students don’t invest in math classes: (1) fear of failure, as I discuss above; and (2) lack of interest or curiosity. I’d be lying if I said the latter wasn’t part of my problem as a second-semester senior in that topology seminar. The two obstacles demand different (though related) solutions: for (1), we have to make math classes a comfortable and supportive place to learn, where questions and mistakes are valued; and for (2) we have to find ways to engage our students with the material (a broad and far-reaching challenge). With any given student, though, it’s worth remembering that either of these (or, more likely, some combination of the two) could be the underlying problem.

My weakest showing in math class was an algebraic topology class with the same professor! It was not at all intuitive for me and I too had moments of “I hate topology” or “I suck at topology” instead of “I’m struggling too much with the material to stay up on this class.”

It’s interesting now that I’m both taking and teaching classes. I get very frustrated when I ask if everyone follows or if there are any questions and all I get is two or three nods. Then, come test time, it looks like it went in one ear and out the other for a disturbing portion of the class.

Then a day later in my PDE class (in which I’m a student) our very kind, patient professor will ask if we follow or have any questions and I’ll sit there trying to remain unnoticed (as most people know, that’s an uncommon role for me) just praying he doesn’t ask me. It’s easy to feel like you’re far enough behind that communicating your confusion will just slow everyone else down and that, in turn, is easy to feel ashamed of.

Anyway, this is a very beautiful post.

Seems like grad students have a special insight into this process, being students and teachers simultaneously.

It’s hard, as a teacher, to find quick checks of understanding. I usually tell students to turn to their neighbor and talk about the problem, so that I can eavesdrop on a few conversations and see what snags they’re running into. But that might work better in a high school class than a college one.