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.

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I would also like to give this to my students at the beginning of the fall term. Please let me know if I may do so. My wall was differential geometry (all I remember is that brackets change superscripts to subscripts in tensor notation, and that’s all), but this is a much better explanation for a freshman class. Thank you very much for articulating this so well.

Hi Joshua, feel free to share it – I hope your students find it helpful.

Ben, wonderful article, and thanks to Slate for leading me to your blog! (Wasn’t sure if my last post took, so I apologize if this looks like a repeat!) Twenty-plus years ago, I’d talked myself out of engineering, and changed my major to computer science after getting a C in multi-variable calculus. It wasn’t until after I’d graduated from college and started working as a technician in the petroleum industry (where I found myself revisiting those “boring” concepts as I write and troubleshoot database code), that I found a deeper appreciation for math, that it is about more than memorizing “magic” formulas or doing long multiplication in your head (though it can’t hurt!); it’s also about finding patterns and relationships, and arriving at creative solutions to everyday problems (“there’s more than one way to skin a cat”). Like another commenter on this thread, I now enjoy helping my preteen daughter with her math homework, and I can’t believe that I’m now the one telling her that dealing with improper fractions is a piece of cake! I’m saving a copy of your write-up for my daughter when she’s older; she’s expressed an interest in the sciences, and I’d like her to know that she should not be discouraged at the first sign of difficulty or a bad grade (whether in math, or any other subject)!

Hey Daisy, thanks for reading! I’m glad that math eventually reared its less-ugly head to you. (Multivariable calc is often taught in a very technical, computational way – not surprisingly, it turns a lot of students off.) Good luck to your daughter with her education – when she hits her first big hurdle in a math or science class, I hope she knows to keep on pushing.

Dyscalculia may be the problem with some people. I wish my math teachers would have been aware of it. It is the numerical equivalent of dyslexia and there is a list of symptoms. http://en.wikipedia.org/wiki/Dyscalculia I wish someone would have recognized the symptoms and had taken me seriously when I told them that I thought I had number dyslexia. I was made aware of it when I was in my late 30’s and once i was diagnosed it was like a huge weight was listed off me and when we started reading some of the symptoms a lot of things started to make sense.

Keep up the good work.

I get math but hate doing it :)

I was at a reunion of my old students – possibly your year? – and your name came up. And then another old student pointed me to your blog a week later. I am very pleased to find you engaged in teaching. So hello!

Nice post. I’ve had the same experience. Multiple times, unfortunately. Differential geometry was one where I checked out, as above. Randomized algorithms was one where I didn’t check out and still came up short. I’m not sure which of the two felt worse.

Someone once gave me the observation that for almost everyone, their mathematical career ended in failure. In particular, we keep taking courses until we hit the wall and that is our last taste of it.

Your story above suggests that topology is difficult, but it doesn’t demonstrate that you are bad at it. One reason for avoidance might be to avoid that self-knowledge being too clearly demonstrated. But then you never really know. It’s definitely a risk and reward calculation.

Hey, it’s good to hear from you! I remember hearing a few years ago that you were out here at Cal – are you stil in the Bay Area?

That’s a good point about math careers almost all ending in failure. It’s different than, say, organized sports. If you’re a good HS athlete who isn’t good enough for college, then your career usually ends with 12th grade (where you’ll still have success). Same for a college athlete who isn’t good enough to go pro. But in math, most of us get a bitter final send-off.

I’d say that I’m “bad” at topology in the same sense that I’m bad at algebraic geometry, skiing, and (for that matter) drawing. I currently lack skills. Maybe I’ll never reach the level of “really good.” But with effort I could certainly improve.

I think you’re right, though, that one of the big barriers to improvement (especially for people who already feel successful at math) is the fear of hitting your ceiling. Way easier for me to check out and blame my bad habits than to risk finding that I really CAN’T handle topology.

Hi Ben. I am a professional mathematician who advocates for better public school math instruction (see the WISE Math blog to which I link) up here in Canada. A fellow advocate from the U.S. pointed me to this post. A very good message, thanks for it. There’s something in there for teachers at any level. I’m sure some of my students feel chewed up and spat out (I KNOW some do) although I really have a lot more compassion and interest in their situation than they seem to believe. I must have a threatening exterior or something. I do, however, seem to reach some, even rescue a few strugglers, and it is gratifying to see them blossom.

On a personal note: many of the specifics of your tale here are personally familiar to me. I also was successful in Mathematics, completing an Honours Degree in the subject at UBC. In my final year I took Topology for the first time. There were two students, and classes were held over coffee in the professor’s office.

The class was run exactly as you describe except for a set of cardinality 2. As you can imagine, this scaled up the anxiety factor around one’s turn to lecture. Worse, if the other student destroyed a subject too badly, it was your job to pick up the chalk and fix or finish it properly, on the spot. Fortunately (sort of) this particular task fell more heavily on the other student than on me … if you get my drift.

I sailed through the first topics, rather enjoying the point-set introductory material, cardinal and ordinals, and so on. About two lectures past the introduction of Hausdorff spaces I started to freeze up and panic a bit. Oh, I didn’t mention: No textbook. Our resource consisted of handwritten notes by the professor. Definitions and theorems, with no proofs. That was largely our homework: Prove the theorems, and present your proofs as lectures.

Though I struggled mightily at points, and experienced most of the inner struggles you mention, I felt at the time, as now, that it was a hugely valuable learning experience, and do not resent it in any way.

It was actually my second experience of this type. I had, for some reason, real difficulties with the way the second course in PDEs was taught, and really thought I was going to fail the course. The second half of the course lectures didn’t “connect” for me, and the text was impenetrable. Anxiety, I think, creates a huge mental block. It opens up an inescapable vortex of self-propagating ignorance.

A couple of days before the exam, sweating through my notes for the nth time, I slammed my textbook shut. “I’m smarter than this! I am good at math! Why can’t I do this? Maybe it’s just not presented the way my mind thinks!” So I did something that usually is not the right thing for students. But at that time it was the right thing for me: I decided I was smarter than the text, and I should be able to solve all the problems in my own way. So I spent a couple of days “inventing” a completely different way around the major questions of the course, at least the ones that were problematic for me. Knowing how, I worked it out with full mathematical rigour. It was actually remarkably easy (math really isn’t as complicated as it appears — the hard part is fitting into someone else’s way of formulating abstract ideas). For the first half-hour of the 3-hour final exam I wrote a 2 or 3 page summary of how I intend to approach these particular problems, with a sketch of how I justify it. I declared that I would be using the approach shown in all the problems of that type in the exam … and I did. The prof must have liked this, because I got a great mark on the exam and passed with a good grade. I never did discuss it with him. But much later I revisited the material and learned that the approach I “developed” was simply another standard way of organizing the material.

Now I don’t recommend that students always try to invent individualized ways of doing math. In fact this can be a disaster when teachers try this as a fundamental paradigm of math instruction — as at least one contemporary school of pedagogy advocates. Cognitive science tells us that this form of instruction can be very effective for experts (perhaps this is why it worked in my case), but it is a very poor way to teach novices (certainly very few students are equipped to provide rigorous justifications for original methods). That said, I don’t think there’s harm in struggling students stepping back and saying, “Now, if I was to do this from scratch, how would I approach it?” instead of trying to blindly apply a method that they don’t understand or can’t seem to use properly. But I would advocate that they always use this mental experimentation to help leverage their understanding of the mainstream method. The objective should be to master that, or one remains unprepared for later topics that reinvest that knowledge later. And they may not have teachers/professors who offer the kind of grace I received in that course.

Hey, thanks for reading – it’s great to hear your story.

Liike you say, it can be tricky to persuade students that you’re actually on their side. One-on-one conversations work better than reassurances to the class as a whole – but beyond that, I guess, it depends on the kid.

That topology class sounds like the stuff of my nightmares! But that’s a very cool experience you had with PDEs. You’re absolutely right that blazing your own trail only works for the most advanced students. My dad (also a mathematician) always struggled to keep his focus during lectures, so he developed the habit of just listening for the statements of theorems, and trying to prove them himself. He usually couldn’t, but he credits that habit with building a lot of his intuition about proof, and how to identify the deep ideas behind a theorem. It’s a little like your adventure of inventing PDEs from the ground up.

It’s reassuring to hear your take on using mental experimentation to “leverage understanding of the mainstream method.” That’s generally how I run my math classes – I shoot for a constructivist, first-principles approach to the material, and do guided explorations of a few deep ideas (like the limit definition of a derivative), but mostly adhere to convention. Nice to hear an expert vouching for that approach!

Anyway, it’s great to hear from you, and best of luck up north with WISE!

Hi Ben. Thx for the reply. Just to clarify, I didn’t “intent PDEs” from the ground up — I got stuck on some of the methods in the course (I don’t remember now but it had to do, I think, with separating solutions and identifying stable and unstable points). Since these made the bulk of the latter half of the course I knew I was sunk if I didn’t master them somehow. So I did what your dad said, except I had the quiet of my dorm room and tons of scrap paper, and lots of time, on my hands.

Your dad’s approach to lectures is like mine. In fact, I’ve been a prof now for over 20 years and I still can’t maintain presence of mind through an entire lecture by someone else. So I do things like I did as a student: I focus on a key result or example, stop watching what’s on the board, and start working it on my own. If/when I get stuck I look up and see what the speaker is doing with it. But I usually try to beat them to the punch. Also, something I read somewhere works wonders for attention: adversarial listening. Try to disagree with everything the speaker says (to yourself). Have a little mental debate and see if he wins. He usually does.

I dislike the term “constructivist” in the context you mention. The term comes from cognitive science, and is a perfectly good theory of learning, but it is not prescriptive at all. Education theorists have picked up on the term and use it for a prescriptive method of teaching, and in the end it has little to do with the theory of learning — though proponents will talk as if they are essentially one and the same. To a constructivist in cognitive science, a student who sits in a lecture taking notes and digesting the content is “constructing their own understanding” — this has little to do with how information is received; it has everything to do with how the learner assembles it into “knowledge”, internally.

Among cognitive scientists this habit some have of conflating the theory of learning with the method of teaching is called the “Constructivist Teaching Fallacy”.

To distinguish I like to use the term “pedagogical constructivism” for the teaching method. Cognitive scientists use the term “minimal guidance instruction”. And here I note that you do “guided explorations”; I do the same. In a sense this splits the difference between transference of knowledge in a direct lecture and having students figure things out for themselves.

It has always been my view that as students mature they should be encouraged more and more to take the initiative in understanding how mathematical facts arise. Ultimately the instructor should guide the student through the material but the student should be doing all but the hardest analytical proofs themselves. That ideal is hard to attain by the best of students. I’ll say beyond realistic, for the average. However, all benefit from a certain amount of exploration. The teacher must not, in any case, abdicate the responsbiility to teach.

One other thing cognitive science has demonstrated — minimal guidance instruction is effective only for those who already have expertise. It is a disastrous way to teach novices. For, to effectively explore one must have models of what appropriate exploration looks like, and one must have a rich context of knowledge within which to understand what the exploration means. This is also a good reason to have “guided” exploration rather than unguided.

In my mind this does not preclude the use of discovery even in primary years — it only means that the teacher must step in more at that age to help the student structure the knowledge and to understand how to explore. I gather you teach high school. By this time students who have mastered things are likely already independent explorers, at least until they hit a new level of abstraction and need much more guidance. I imagine your student appreciate both being given chances to explore and your presence assisting them to explore effectively.

I like how the Quebec curriculum specifies this idea:

http://www.mels.gouv.qc.ca/progression/mathematique/index_en.asp?page=arithmetic_01

If you mouse over the information button for first category, “Student constructs knowledge with teacher guidance” you get this instruction, “An

arrowindicates that the teacher must plan for the student to begin this learning during the school year and continue or complete it the following year, always requiring systematic intervention from the teacher”. The key difference between this approach and pedagogical constructivism being “always requiring systematic intervention from the teacher”.Thanks, that’s a good breakdown of the constructivism issue. I was actually shooting for the cognitive science definition when I used the word, but you’re clearly right that the theory’s name has been adopted by a family of basically unrelated teaching practices. A teacher should be an expert, and it seems silly to deny students that expertise when they could really USE an expert model to go off of.

I should also compliment you on your Blog title, Math with Bad Drawings. An excellent idea, and I gather a subtle reference to the mathematical saying that a proof should never depend on the accuracy of a diagram. Some people even draw diagrams out of scale deliberately, to make this point. I like to draw the 4-cube with wobbly spaghetti for edges to underscore that the abstract content of the picture is the essential part and that ANY way it is drawn tends to deceive our minds, which don’t think well in 4D, and in any case there doesn’t exist a truly faithful 3D representation of this object, especially not when we use 2D drawings to represent the 3D representation…

Thanks – hadn’t thought about that implication of the title before, but it fits. That reminds me of a story I have about the limitations of using 2D models for 3D systems… it may appear in a blog post soon.

No offense, but it seems to me that this article is trivializing the truly traumatic scope of failing at, say, high school mathematics. Since math has been made synonymous with (general) intelligence, any larger failure will leave you with a destroyed self-esteem, inferiority complex, self-hatred, shame, and social stigma – and all of these for life. I got very suicidal thoughts in high school because I now “had to” internalize the realization that I am an idiot, a worthless idiot. To compare passing struggles in topology to wholesome experiences that will leave you a depressed, near-suicidal husk of your former self feels insulting.

First of all, I’m sorry that the system failed you so terribly – high school’s a vulnerable time, and one narrow-minded teacher can inflict real damage.

Second: I’m sorry if I made it seem like my experience was comparable to one like yours. Mine was a paper cut; yours was a deep wound. And I acknowledge the craziness of someone with a paper cut claiming, “Hey, now I know what all physical pain is like!”

But this pain – the one you felt so deeply – is different than physical pain. It’s hidden. It’s stigmatized. It’s rarely talked about. And – as you did – people often blame themselves for it. My primary hope with this piece was to bring that pain out into the open. To show the people suffering from it that they’re not alone, that they’re not stupid, and that it’s not their fault.

That’s why I used the universal language I did. To show that such experiences are made of the same substance. Not to claim that my puddle of pain can compare to others’ oceans.

Thank you for an honest and encouraging reply. Mental trauma does develop one’s thinking (to an extent), but at the price of almost everything else. I honestly think there should be more thorough “vetting” of one’s “true” mathematical competence before high school actually starts, so that one isn’t unwittingly setting oneself up for a fall. Going naively in (as I did) is a potential recipe for disaster, a scarring for life.

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Man, I remember the first time I failed a Math Test… My second year of Algebra in High School and I just didn’t get logs. I cried in class and had to go sit in the hall. My teacher came out and got me through it, but it wasn’t for a few more years that it finally clicked that everyone sucks at math before they learn math.

…Actually, that should be the motto of math class; “We all suck at math, let’s get together and get better.”

Very true. It holds any time you learn anything, really. I’ve started rock-climbing, and it’s important to remind myself, “I’m already better than I was, and still not as good as I will be.”

To anyone who isn’t good at math, you need to just do what I do to learn complex math ON MY OWN.

I slept almost every day in my AP Calculus, barely touched homework (just enough to not bring my entire grade down to a C), didn’t pay attention at all, didn’t study a single time, and the teacher disliked me for all of the above. The only things I would actually do were graded classwork, quizzes, and test. I got a 100% on my midterm and a 95% on my final.

Why am I telling you this? It’s because you don’t have to practice math nonstop in order to understand it. You need to simply understand how to look at it the right way and learn how to get from point A to point B. Use the Trial and Error method.

All you need to do is look at the problem, find the correct answer to the problem, write them down next to each other, and find a method of transforming the problem into the answer. As long as you understand basic rules of mathematics, such as PEMDAS, you will eventually see a solution. Once you find this solution, find a similar problem and try to solve it using the method you just found. It should work, but if it doesn’t, don’t be discouraged. You could have made a slight error somewhere a few steps back that made it so you couldn’t get the answer, but your methodology may have been correct.

Think of a problem like a puzzle. It can seem confusing with so many pieces looking the same, but you just need to find the puzzle pieces that slide in together. You can’t really find that out until you observe and narrow your results using trial and error.

Reblogged this on Bock Math and commented:

Every math teacher should read this article, to understand why our students behave the way they behave. Every math student should read this, to understand that the way he/she is feeling is felt by others, it’s not his/her fault, and it’s often fixable.

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Possibly the best mathematics education post I have ever read! My course was Abstract Algebra. I have never taken Topology. You just nailed what all math teachers must and should read. We have all been in this situation. We just don’t admit it publicly. And if a teacher hasn’t been in this situation . . . . . then I am not sure that Math education is a good place for them.

Thanks for reading–I’m glad the post resonated. I’m hoping to take another stab at Topology someday; I hope you get the chance to do the same with Abstract Algebra! I found my experience of struggle/failure useful to draw on in my teaching; and I bet the experience of overcoming such a block would be powerfully useful, too.

Nice post. My demise came in Analysis, when I couldn’t even see the point of open and closed balls. However I was confident (arrogant?) enough to blame the teacher and changed my major to Operations Research. Some great ideas.

Ah, my dad does OR! A cool field.

This was the first time I’ve been to your blog, but I’ll be back! I tell my ninth grade geometery students such a similar story about myself, though it was first semester of junior year of college and I had both Topology and Advanced Calculus and ended up getting Ds in both of them. I re-took them (separately!), passed, and finished the rest of my math major just so I could be a math teacher, the career I chose in tenth grade when I was an “amazing” math student. The experience of thinking I was working hard, not getting help when I knew I was floundering, and tackling it all over again after failing made me the teacher I am today!

Thanks for reading, and for sharing your story! There’s a good moral there about providing enough challenge for our students, but not drowning them. (One extremely difficult class at a time is enough!)

Thank you for sharing. I agree with your assessment on several levels. My understanding of failure came as I took the Praxis exam, more than once. It made me feel stupid and inadequate. My goal was so clear (teaching math to others) that I persevered. I saw those very emotions in my students last year, but did not recognize them for what they were. So I get to say that I also failed at being the teacher my students needed. In spite of that failure, I have spent the summer trying to become the teacher I should have been last year. I have learned a lot, and am still learning. I think I will carry this blog around with me.

Thanks for reading. Teaching is definitely a job we grow into. I’m still embarassed by almost everything I did in my first year in the classroom. Teaching math is a little like learning math, in that they both take perseverence!

Hello Ben,

Thank you for articulating this truth about Math studies. We all run headlong into that same wall at some point. We secretly think ourselves stupid and most of us walk away. But some of us accidentally discover the secret that learning math is repeated exposure to the material until it becomes familiar. This was a truth that my high school math teacher tried to instill in us. Somehow that message saved me from total despair during my struggles first with Calculus and then with the Introduction to Topology course. It was my Turkish roommate who talked me into memorizing all of the theorems in the Topology course. What could be more iterative than the memorization process which slowly peeled away the obscurity cloaking the beauty of Topology.

Best of all, the total terror that incomprehension fuels magnifies the sense of joy when that same incomprehension transforms into understanding. It is that success in the face of extreme adversity that binds us to Mathematics. Can one truly invest oneself in Math without the experience of those two poles?

This is a great article. I’m a Computer Science undergrad teaching a Calc I lab to freshmen this semester, and it’s more difficult than I expected.

I thought “Hey, the lecturer will do all the hard work — all I need to do is give them a worksheet and then help along! I’m basically a glorified tutor…” — I thought. But the problem is that they don’t want to ask questions, and it just makes it a little bit worse. I try to get them into groups, so that they talk to each other instead of me (it’s easier to ask for help from your peers than from the teacher, plus the “strong” ones in the groups aren’t afraid to ask questions — so I put the students that aren’t good with those that are good, one for each group, and I hope that it works out). But the problem is that in one of my labs, there is not one really “strong” Math person that I could put with each group – I’m basically putting all the bad students together. Any advice?

Yeah, that sounds tough!

I think you’ve got the right instinct–kids are often more willing to ask peers than teachers for help.

Sometimes I’ll divide students into small groups (usually pairs), and then then circulate around the room, looking over their shoulders, asking questions, hearing how they justify their work. If I hear the same misconceptiosn repeated (or even a single mistake that I think is illustrative), I’ll call the whole class’s attention back to the front, and say, “Here’s a good question that came up; I’d like to show this to everyone.”

Past that, I try to make sure to honor their thinking, no matter how “off” it ma be. (“I see–it looks like you’re using the fact that both those terms contain x to group them. But it turns out that x^2 and x^3 can’t be condensed in quite that way…”).

Anyway, good luck! Your first teaching experience is often a learning experience as much as anything.

I could’ve written this article, substituting “Graduate Level Abstract Algebra” for “Topology”. In a way you are lucky that you had this experience in your undergrad years. I’ve been teaching for seven years before I got my asymptote handed to me. It was really painful, twice-a-week therapy, until I realized that it was professional development. I was never evil, but I am much more empathetic with the plight of the struggling student that I could ever have been before this humbling experience. Thank you, again, for making such a clear statement of the symptoms. I needed that.

Thanks for reading. “Professional development” is exactly the right phrase. Teachers get to spend all day talking about stuff they know, so it’s easy to forget what it’s like to be struggling to learn.

Yes! Yes! Yes! I agree on so many levels. Like your professor, I never, ever want to crush. Thank you for sharing your story. I have teachers I am going to share this with!!!

I’m glad you found it resonant! I find it easy to forget what power we hold as teachers.

Ben,

I appreciate how inclusive your message is. Thinking about how to make all my geometry students feel included is something I think about a lot. As you move up in mathematics, the group of people with whom you can have a significant conversation is ever-shrinking, but math is much more fun when everyone gets involved.

Definitely. For all the press mathematicians get about being antisocial, it’s really a very social profession, and collaboration is huge at every level from kindergarten through professor.

I’m going through this right now and it’s difficult because I’m homeschooled and my mom is my teacher and in my household my mom has always been harsher in words than my dad. The real problem is that I’ve had a simple pre-algebra calculator for two years now. So when facing problems that required a graphing calculator my current calculator didn’t help and made me feel stupid because I never knew I needed a graphing calculator. So I started watching tv till 2:00 in the morning and falling asleep during math later. I was barely getting any work done and cheated because I thought I was stupid not knowing how to do the math that clearly no average human mind could do alone, and to top it all off my sucky calculator can’t even do negative fractions. So about a week ago I found out I needed a graphing calculator and that I was not as stupid as I thought. But I had lied to my mom about doing good in math and which lesson I was on because I was afraid of how she would react. I cried (not the most manly thing to do) and thought I was a failure. Every movie we went to see that had anything to do with college or school scared me because I was afraid I wouldn’t even make it that far. I am so scared now. I don’t want my mom to yell at me even though its not my fault but the fact that I lacked the correct equipment for the subject. Do you think that she will understand if I explain it to her?

Please give me some advice

Also I have had really bad anxiety ever since this stuff has happened.

Hey Cory, thanks for reaching out. I’m not sure I’m qualified to give advice–I’m just a teacher, not an expert or specialist or therapist, but I’ll give it a shot.

The first issue is making sure you understand the math. For that, you need a teacher who knows what you know, and what you don’t know, and can help you along the path. I’m assuming your mother knows the mathematics that you’re learning. But if you’re mostly just working independently, or the two of you are figuring things out together as you go along, then I’d recommend trying to find an additional teacher to help you specifically with math. Depending where you live, there may be resources for homeschool students, or groups where homeschool families pool their resources to bring in tutors or teachers, to cover topics they’re not experts in themselves.

Graphing calculators are very powerful tools, and it’s great to be able to use one. But almost every topic in high school math should be possible to tackle and understand without one. So if the graphing calculator really made such a huge difference, there’s a chance that the topic is worth revisiting.

Second of all, there’s the issue of your relationship with your mother. For that, you need to be honest. Maybe writing a letter would be easier than confronting her in person. Try to express everything you told me–that you’re feeling worried about the math, that you felt the lack of technology was holding you back, and that you’re feeling anxious and guilty about it now.

I can’t speak as a parent, but as a teacher, it always disappointed me when my students cheated. It’s my job to help students understand math, and I can’t do that when students lie to me about what they know. It made it hard for me to trust them, which is crucial for any healthy relationship. That’s especially true in your case, where your teacher is your parent.

But how the students dealt with the cheating afterwards could go a long way towards rebuilding trust. If they were honest with me about their reasons, admitted that they’d done something wrong, and showed a willingness to work to relearn the material and to rebuild the trust, then I was much more satisfied with the experience. We could both see their cheating as a mistake that led to a growth experience.

Im in 7th grade and im doing so bad in Math in my previous years i was in Advanced classes. Had excellent grades.Then in 6th grade i guess my work enthic just got poor.I got a low grade on my Nys Math Test. So they switched me into regular common core classes. Heres the thing im in honors but not in advanced.But my goal is to get back into Advanced so when i got my report card i showed my assistant principle and she i needed to get my MATH grade up. i FEEL so overworked im trying believe me but there is times where id get lazy. Can someone please help me.

Hi Lilly,

Coming from a 7th grade math teacher, I can tell you that your school is truly trying to act in your best interests for a class placement right now! There is a HUGE jump in required logical skills starting in about 6th to 7th grade math and sometimes I find that my students’ brains just need to get older before things really start to click for them with what the common core says you should be able to do. Being in an honors class still means that you are most likely getting almost the same material as the advanced class, just at a slightly slower pace (at least that’s how I teach my classes) so you have more time to learn the material better. I’d encourage you to keep working at the class you are in right now and try not to worry too much about what the name of your class is right now. Effort makes a HUGE difference in achievement, but unfortunately, a lot of math for you right now is dependent on how long your brain has actually been in existence. Give yourself time for your brain to make more connections between the parts, so things will start to come more easily. You can do it!

-Abby

Great article! I’ve always been an English and language whiz, but I’ve never “gotten” math – I think it started in third grade, with long division. My teacher would do her lesson at the front of the class, then have us do problems on our own. I didn’t get long division. I asked her to help me. Her response? “I know nothing.” At the time, I wasn’t enough of a wise@ss to say “really? If you know nothing, how did you get a teaching job?” Instead, a generally insecure kid terrified of failure and of looking stupid, I curled in on myself and doodled. Algebra was nightmarish; geometry made me cry. When I finally got up the courage to ask when I didn’t get something, the teacher – the teacher! – shamed and made fun of me for asking questions that “everyone” should have already known the answer to, to the point that I fled the room in tears.

The only time I ever really got math – and liked it! – was in college, in an astronomy class. I’m sitting at my tiny lecture desk looking at the equation for determining the absolute magnitude of a star, and math MEANT something, suddenly. And on the few times I’ve heard conceptual, wacky math explained to me, I think it’s fascinating, but it’s like another world where my mind refuses to walk. I still can’t do fractions worth a damn. Or long division.

I’m in tenth grade and I currently have a 67, if not less now, in geometry. I didn’t ask for help at first because I thought I’d get better. Now there’s barely anytime left in the class and I’m terrified the final exam coming up is going to do me in. My teacher goes over the material so fast I get lost and it’s like I’m the only one just not getting it half the time. There’s no hope to pass with higher than a c now I just do NOT want to fail. I really need help.

And just how does your story help?

Reblogged this on karjna and commented:

A CS professor posted this on the student conference; great read.

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Great Article, I just got out of bed after reading this on my smartphone, i’m going to stop moping around now. I just got a quiz back from my teacher and I received an F, first quiz of the semester. The bad news is I completely understood all the material, I’m trying to think of a way to end this comment, but I think the better thing to do is to say thanks for whatever insight I just received from this article and study for the quiz that we are having tomorrow,

Thanks for reading – I hope the course goes well!

Im an Algebra 1 student and I cried when my teacher called me to answer a math problem on the board,I was sweating alot and my heart was beating fast,worst feeling ever,I gotta get a tutor

Your Topology is, in my case, something roughly equivalent to a Calculus 3 course, which would count as the last Math course (Industrial Chemistry, equivalent of a B.A.) of my life.

Studied theory for about 3 months straight and then only exercises and problems for another month. The teacher approved my choice of books. Had trouble with solving nearly all the exercises i did, but seemed to overcome them with the help of a local math forum and WolframAlpha.

I failed the exam (couldn’t do any of the exercises b/c i didn’t figure out any of the required substitutions that were the only thing that made the exercises artificially challenging and suffered a nervous breakdown).

What should i do? What should one do when insufficient math skills may keep him/her away from the goal (a science B.A. followed by a nice job) forever? This has gone on since i was a child. I’ve always been the last one in my classes to grasp Math, yet the first one to grasp Science and especially Chemistry.

Fortunately, in my country (try to guess what) any student can fail as many exams and years as he/she wants without no consequences other than a waste of time and money, so, apparently, time is on my side.

By the way, i love your article. It makes me feel less lonely.

I failed 3 times at entering univ. exam, all I had to do was a math exam. But because I knew I’m not stupid I just kept trying… today I’m a 2nd year undergraduate student of pure mathematics… when my family and friends knew I failed so many times and yet I was now entering a math course they respected my resiliance… today I still strugle a lot and when I dont feel ready I don’t go to the exams, but I will never give up… my goal now is to go as further as I can reach and later tell the world about my experience and how everything is possible if you never stop trying… Because of how I suffered a lot in the past, I know how it feels… it still does from time to time in present days. But in the end I hope my story will encourage future generations… that’s one of my life’s goals… your post just made me realize even more how my mission is important. You’ll ear about me in the near future, just wait for it :p . And thank you for this post… is really important pedagogical material.

You mention that when you don’t feel ready you don’t “go to the exams”. Are you Italian too? ;-) In most countries it’s either sit your exam, ready or not, or pack your stuff and go away. The major exception is Italy, where it’s only a matter of money if you take too much time (“fuori corso”), even if Italian uni exams are somewhat harder. Either way, it sucks when you want to graduate but you’re facing your own limits. Most of the time it happens with Math and, to a smaller degree, with Physics, especially heavily calculus-based Physics such as electromagnetism.

I didn’t know that… I tought my countrie’s system was equal everywhere else. I prefer not to share that info but isn’t Italy ;D, altough is in Europe. In my univ. if you are not ready you can skip the exam and your name doesn’t even appear… in then end you don’t even get a zero… you only get a description like “not evaluated” and that’s it!

i am complaing about my maths because i failed the whole term except fourth term .it is my maths i amm struggling so much please help me to go to grade8 pleaseT_T

My math process basically involves reading, taking notes, trying the homework, crying, trying to figure out what I don’t understand so I can ask a good question (typically, I just don’t get it, but what “it” is is the key part, and I need more information before I can figure that out. I’ve been crying all morning about my statistics homework. I keep trying to ask the question in different ways, but it’s just not leading me to any answers. I have a mean of 107 and a standard distribution of 15, and an x that is equal to or greater than 120, and I have tried normalcdf, I’ve tried InvNorm, I’ve watched an hour of Khan Academy videos, and I’m no closer to an answer and there are bunch of problems I have to do before 11 tonight that I also don’t understand.

In my experience (and I’m an adult student going back for a bachelor’s) – people who are good at math are not always good teachers of math. My professor is just such a person. He doesn’t even understand my questions because he doesn’t comprehend that someone isn’t getting it!

Perhaps a really good math teacher should begin by teaching students to ask questions – good, bad or otherwise! I am sorry I did not see this last night, but if you still need help, tell me what the question was that involved the mean, standard deviation and x equals to 120. I suspect you were supposed to give the probability of that number 120 being “under the curve”. Or perhaps you needed a z-score? I’d love to help.

A (hopefully good) math teacher.