I like to keep an eye on the Google search terms that bring people to this blog. Some warm the cockles of my heart. Some chill the cockles of my soul. Some bewilder the cockles of my mind, forcing me to Google things like “why do people search for such strange terms” and “what exactly are cockles.”
And the occasional search term will tap into a matter of real depth, like this one: “my students are failing my math class.”
It’s bleak. It’s discouraging. And if you’ve taught math, it’s an experience you know.
We’ve all endured days when it felt the whole class was falling short. You demonstrate how to bake brownies; the kids burn theirs into coal. You model a skateboard move; they gasp for air, having somehow placed the elbow pads over their nose and mouth. You say, “Don’t divide by zero.” They light their hair on fire and then, while you sprint for the extinguisher, they divide by zero.
On such days, my psychic antibodies kick in. “This isn’t my fault,” I say. “Their prior teachers didn’t teach them anything. Or maybe they’re lazy. Or wait—why didn’t I think of it before?—it’s all the administration’s fault.” School administration, federal administration, National Aeronautics and Space Administration; it doesn’t really matter, as long as the blame weighs upon shoulders other than mine.
Of course, “blame” has no good place in the classroom. The question “How did we get here?” matters only insofar as it informs the real question: “What do we do next?”
When a whole class fails, nobody wins. The kids suffer twice—first on their transcripts, and second in their blessed little hearts, where they’ll lose faith in themselves, or in the fairness of schooling, or (most likely) in both. The teacher suffers, too—facing worried administrators, outraged parents, and inner doubts about being “cut out” for the profession.
So let’s take it as a given that you can’t fail ‘em all, and explore some possible causes.
- Pacing.
My first year teaching geometry, I sampled the whole platter of teacher mistakes. But I really gorged myself on one in particular: bad pacing. I marched through one topic per day, with no breaks for water or rest. Who needs synthesis? Who needs review? Isn’t this stuff obvious?
Well, math you know is always obvious. Math you’re learning never is. It’s healthier to learn one thing well than to “see” five.
- Assessment.
When I write a test, I’m often tempted to overload it with interesting, challenging, novel questions. To make it a true test of wits and wills. But a test should assess basic skills, too. It needs a good mix of easy, medium, and hard, if it’s going to hold an honest mirror to students’ abilities (rather than a horror-movie mirror in which they can’t see their own reflection).
A bad grading scheme can create problems, too. It’s easy to turn a test into an all-or-nothing affair—by asking lots of similar questions, or many short questions with no partial credit, or questions where 1c is impossible unless you nailed 1a and 1b. That’s a recipe for whole-class failures.
Grades aim to reflect the quality of a student’s work, and “quality” is one of life’s most flexible concepts. Mathematics—despite its reputation as objective—leaves teachers a lot of latitude.
- Background
There is no “first rule of math teaching,” except perhaps “talk about math teaching.” (This is math education’s primary difference from Fight Club, which it resembles in all other respects.) However, there is a truth that I find to be almost universal: Whatever you expect students to know upon arrival, they probably know less.
That’s okay. Learning is hard. Teaching is hard. Summer wipes a giant eraser across all of our mental whiteboards.
Every student has gaps: concepts they’ve never learned, techniques they’ve never nailed, anxieties they’ve never addressed. I can’t pretend those gaps reside in the past, that they’re debits on the ledger of some prior teacher, irrelevant to my work. Those struggles are here, in the present, weighing down my students, derailing their learning. I’ve got to survey the landscape of their thinking—to see what’s there, not what I want to see.
When you ask Google something you should already know—“how much do library books cost” or “who is Tom Hanks” or “what is pasta made of”—the search engine doesn’t snark or seethe or pass quiet judgment. It just shows you the answers: “they’re free”; “the mayor of Hollywood”; “pasta molecules.”
A good teacher is like a sentient Google. No finger-pointing. No recriminations. Just a benevolent omniscience, helping everyone to take the next step.
I just love your conclusion though it is next to impossible to be that way, every day. I teach design and fashion but there are times when I am tearing my hair apart (at times literally) trying to figure out how to get my students to do their assignments on time and in a way that can get them better grades. So I laughed out aloud when I read the topic of this post
post vagene
It would be really interesting to see things from the perspective of a math teacher/professor. As a student, I never really think about how many years of practice that you professionals have, and how difficult it may be to relate to students who have had much less experience with these topics. Great post as always, and one that has given me a greater level of respect for math teachers. Especially that analogy to Google which is profound if you think about it.
Reading this post is encouraging after having just graded a calculus test that half of the class failed.
I realize that it was overloaded with very hard questions, but nevertheless I will give them a second chance when they return from vacation.
Do you teach high school or college? I’m a senior in high school, and I am really enjoying calculus right now. So I think that hopefully there are people in your class that greatly enjoy learning from you. I just wanted to tell you, as a student, I think all teachers (especially math teachers) are great people who deserve more recognition for the hard work they put in!
I am a college professor.
I taught trig for a while at the high school level. I entered each year with the idea that my students had NO IDEA what geometry was and so wrote up a “everything you need to know about Geometry but failed to remember” cheat sheet that i allowed them to use. Even during tests. Every single test was “open book” and even though they had my permission to “look stuff up” I only had one student ever do so. None of them, ever, remembered the basics that were needed to learn trig. I always assigned the even numbered problems in their text. Those had “the answers” in the back of the text. That way, I hoped, they would check their work. The students that actually learned the most though never skipped the odd problems as well. I loved teaching. I HATED the administrators though.
I think you have to start “at the begining.” i.e. no prior knowledge.
First there is the time gap — not just one summer for a trig course. Since Algebra II comes in between Geometry and Trig, it has been over a year since your students have had any Geometry discussion.
Even so, I am looking at a college level book on analysis for students with 2 years of calculus under their belts. The first chapter discusses the definitions or rational irrational numbers, real numbers, complex numbers, the distributive law, equality and inequality, absolute value, and the triangle inequality, all of which “should have been covered” in Algebra I.
Looking stuff up — I had one final, where I came across a question that meant absolutely nothing to me. I finished the rest of the test, pulled out my book read the unit relevant to the question and answered the question. It was the only time I can remember where the open-book saved my butt.
Pacing and background make a nice composite problem, too. I’m teaching a college pre-calculus course right now and since most of my students are freshmen in their first term, they all have exceptionally different backgrounds in terms of the math they know. Some of them have already taken calculus. Others struggle to multiply fractions. So trying to pace the class to line up with their different backgrounds and engage the more knowledgeable students while not losing the students without as much background has been a huge challenge.
The most frustrating situation is being told “you know know this”, especially in mathematics. I didn’t know anything about factorials until calc 2 in university, when I was admonished in class by my instructor. My calc book didn’t include a remedial instruction, but luckily the internet existed, so I was able to find a video on their use. The realization that a previous instructor hadn’t completed the prescribed lessons that I would need to continue my education was a harsh one. :\
After a 30-year career in engineering, I *just* started the process to become a STEM teacher.
I am now very afraid.
I heard an analogy once that really struck me. When you go to the dentist with a cavity, the dentist doesn’t say, “Why do you have that cavity? Didn’t your parents teach you to brush? What brand of toothpaste do you use? Do you not brush enough minutes? Are you too lazy?” Okay, the dentist might THINK that, but she doesn’t say it. And she sees it as her job to fill the cavity. Likewise, as teachers, when students come to us with holes in their education, it’s not productive to ask them why they have that hole. The most helpful thing we can do is help them fill it. For whatever reason, this analogy has stayed with me and made it a lot easier for me to be patient with people who arrive “unprepared” for some specific task.
Great post, and I always enjoy your artwork. 🙂
First, there’s always partial credit on my tests. If you make a mistake in 1a or 1b and use that mistake to get your answer for 1c, I give half credit for 1c if the math is correct. It’s only fair since I believe the process it super important. Test corrections is also a separate gradd in the grade book. Mh students need to see the things they did right and understand how errors early on impact the final answer.
Second… What about those kids who turn in a blank test but when you sit down to talk to them about the test, they can tell you how to solve both problems 1 and 2? Why am I holding a test with a grade of 0% if you know how to do this?!?
Reblogged this on K. S. Senthil Raani and commented:
Such an interesting blog that anyone who teach math can relate to.. “Learning is hard.. Teaching is hard..” True! Learning to teach is hard..
I go all prepared spending days to motivate the students for a particular topic, just to face a very few audience. Back in days, we were evaluated based on our attendance. Trying to pull myself to the ‘new generation’, keeping up the spirit, I try to explain in an articulated way, so that the students do not sleep. Being an average, not-so-competitive student, I try my best to sympathize with the students. However, as an instructor, sometimes teaching is more difficult than learning.
Starting to prepare a course-time table before the course starts, assessing a fairly average point of ‘what they are supposed to know, by now’, answering overly smart students, trying to reach those who feel inferior, solving as many as problems I can to describe the beauty of the course, balancing between teaching-preaching-coaching-lecturing, setting-up questions that train the students and at the same time not to disappoint/deject them…
While I try to avoid comparing the way I was taught in my university-based Bachelors and Masters, I try to empathize with the students remembering the sluggish-not so competitive-average student me; I still have a long way to learn teaching and be the teacher who I have pictured myself to be.
I think you left off inappropriate expectations as a reason for “failing”.
I’ve seen classes that are “failing” because the teacher didn’t realise that kids can’t learn all of Trigonometry in a month the very first time they see it.
I’ve also seen teachers very pleased with how their classes are doing, but only because the work set is so easy that they aren’t actually learning very much. Those teachers then can’t work out why, after doing so well, their students fail the independent exam.
It’s tricky, because while we need to have realistic expectations, we can’t be bound by them. We need to recognise some students should be working well above, and sadly some well below, the expected level. I find it best to teach to the individuals, but to judge the overall pacing and level by the class.
Why weren’t you my math teacher? Yes, yes yes the blame is all on you!
You know I’m kidding, right?
I’m not a math teacher. I don’t even play one on TV. I do tutor a first-year, home-schooled girl of middle school age as a favor to a friend who was afraid to “teach math” to her own child. She’ll go back to public school for high school next year, so I’m aiming to unlock her own access to the skills she has to make algebra a useful tool in her box, not a dark mystery that belongs to the loud-mouthed boys in her class.
I often find myself thinking, sheesh, I should know more useful skills for this difficult job! I wish I knew “how to teach” math. I’m winging it.
On the other hand, I’m also constantly telling this smart young teen that she just needs to TRY to do something on every problem, because most of what she needs is already there in her brain. She’s afraid to “be wrong” and so does nothing due to anxiety. I hope, if I do nothing else, I break her of that habit. Anxiety is probably a much bigger issue nowadays in our test obsessed culture.
I don’t assign anything. I recommend stuff. If she doesn’t do it, I get creative during our sessions and that’s way more stressful for her. She’s learning to do my recommended stuff to avoid my frightening creativity. I’m a real loose cannon, it turns out. 🙂
Can we as teachers just let go of what students “should” know and meet them where they are? As a kid I switched schools frequently, and somehow the new school was always ahead of the old one. Teachers never wanted to help me catch up because I “should” have learned the prerequisite skills last year (or whenever) and so the fault was always mine. I grew up thinking I was just stupid at math. It didn’t occur to me until college that maybe I just had gaps in my education. At university I had an amazing math teacher who never, ever made me feel stupid no matter how, well, stupid my questions were, and I decided I wanted to be her when I grew up.
Fast forward, one bachelors degree later: Now, as an 8th grade math teacher, my students’ prior knowledge ranges from high-school-ready to must-take-shoes-off-to-count-past-10. Whatever students come in lacking is NOT their fault. Gaps happen for all kinds of reasons. Some of my students are re-entering public school after being homeschooled (or unschooled); some had sub-par teachers more than one year in a row; some have learning disabilities, health issues, or family drama; a significant number don’t speak English and/or are new to this country; some by their own admission just goofed off last year. What good is blame going to do? This is now; you’re there to help; show them how to fix it. Telling a kid they “should” already know something doesn’t help them learn; it just makes them feel like you’re against them and reinforces their feeling that math sucks! Let’s stop “shoulding” on our students and support their growth as much as we can.
(And by the way, I HAVE been yelled at by the dentist for not flossing! Your dentist is nicer than mine for sure!)
Honestly even as a pre-service teacher this is encouraging to read. I really appreciate that you are giving practical tips to watch for when teaching. Recognizing that there may be times (especially in my upcoming first year) where I am pacing poorly, over extending their background knowledge, or just writing bad assessments, but that it can be recovered from and it is normal. I will be considering these things as I enter my student teaching semester and finish up my final field placement observation semester.
I think it is great to be thinking about us as knowledge givers and not to shame them for not knowing certain facts. I want my students to feel comfortable in my class.
Pacing, assessment and background: PAB. seems a good rule. I try to follow it.
The main problem is how to fit the whole syllabus in one semester taking care of the P and the B. I have not been able to solve this yet.
For the A, I think that a good habit is to have some easy questions that go to the core of the subject (together with those difficult questions we teachers love above all).
For instance: on of the main concepts in a 1st year Calculus course is that of derivative. I am sure that all of us teachers explain the definition slowly and carefully, trying our best to make good pictures in the blackboard of the slopes of the secant lines and all that stuff. And even spend some time providing examples of different physical interpretations. That is P and many times B.
But nobody will ask that in an exam.
Instead we will look for some problem to be solved with a subtle argument where the students need a “happy idea”(*) , together with all the knowledge of Calculus, Trigonometry and Algebra they are supposed to have gathered since they started to count to 3 many years ago.
If we think that the concept of derivative and its interpretation is important, at some time we should check clearly and straightforward if the students know it.
Sure all of us can think of different examples of concepts so easy and essential that it is amazing how students can have arrived to college without knowing them. In my opinion, the reason is not that nobody has explained them before. Maybe, it is that nobody has asked them before.
(*) “idea feliz” in Spanish; is there an English equivalent expression for this?
I’m super on board with the kind of questions you describe.
One pattern I find is that education drifts towards “scalable” questions – question styles that could be asked year after year, in classroom after classroom, on quiz and test alike, and still feel like they’re serving some purpose. Computational questions fit this well.
Quick conceptual checks (“Give an example of a real-world derivative with respect to something other than time, and explain its meaning”) aren’t always scalable but can be really useful in the classroom.
When we design a system that puts all the weight on exams, and then our exams don’t include those sorts of questions… well, it’s understandable that students think they don’t matter.
(Also, I don’t know of a good English equivalent for “idea feliz”! Maybe something like “brainstorm” or “stroke of insight”? I think those work as translations in context but they’re not perfect substitutes.)