Two More Teaching Vignettes

For this month’s blog post, I offer two more vignettes from my classroom experience.  My intention, as in the last column, is to communicate what I think of as the essence of teaching, which is the emotional—not just intellectual—bond between teacher and student.

But first, with the end of the school year 2018-19, we would like to announce several changes in this Blog’s editorial board:

We bid farewell to Art Duval, who has been on the editorial board for five years, and was one of the founding board members for this blog.  His service and guidance have been indispensable to me in guiding the blog, and his voice will be missed.

We welcome two new editorial board members:

Yvonne Lai received her S.B. in Mathematics from MIT and Ph.D. from UC Davis, specializing in geometric group theory and hyperbolic geometry. Following a post-doctoral position at the University of Michigan in mathematics, she took a second post-doctoral position, this time at the University of Michigan School of Education. There, she began doing research in the area of mathematical knowledge for teaching, in the group led by Deborah Ball and Hyman Bass. Lai is now an associate professor in the Department of Mathematics at the University of Nebraska-Lincoln. She is founding chair of the MAA’s Special Interest Group on Mathematical Knowledge for Teaching, a member MAA Committee on the Mathematical Education of Teachers (COMET), and a member of the writing team for the NCTM publication Catalyzing Change.

Ben Blum-Smith received a B.A. in anthropology from Yale University in 2000, an M.A.T. in mathematics teaching from Tufts University in 2001, and a Ph.D. in mathematics from NYU in 2017, with a thesis in representation and invariant theory of finite groups. He worked as a middle and high school teacher in public schools in Cambridge, MA and New York City, and then as a mathematics professional development specialist for high schools and as a faculty member of Bard College’s teacher training program, before beginning his training as a research mathematician in 2011. He is currently a part-time faculty member of Eugene Lang College’s Department of Natural Sciences and Mathematics, and has also taught in the Bard Prison Initiative.

His research interests lie in invariant theory, algebraic combinatorics, their applications to data science, and connections between mathematics and democracy. He was a 2018 TED Resident, developing a TED talk about the relationship of mathematics and democracy, and is a founding organizer of the Mathematics and Democracy Seminar at the NYU Center for Data Science. He remains involved in teacher professional development through Math for America, an organization devoted to the career-long professional growth of teachers. He is also engaged in mathematical outreach. He has led math circles with students and teachers at the School of Mathematics, the New York Math Circle, the Westchester Area Math Circle, the LREI Summer Institute, the Center for Mathematical Talent at NYU, and the MathLeague International Mathematics Tournament, and is regularly a faculty member and faculty mentor at the Bridge to Enter Advanced Mathematics, an organization focused on creating a realistic pathway for underserved students to enter the mathematical sciences.

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The theme of these two vignettes is how the teacher must value the student, must see as much value in his or her ideas—correct or incorrect—as in our own.  Because sometimes they are correct.

I:  Cecelia and the Grapes

This anecdote took place in a high school remedial class.  For many years, I would take the 10 or so students who were not on any grade level at all and teach them together in an ungraded classroom.  These were students who had struggled with mathematics, and had been failing, for many years.  Many had learning disabilities.  Some had significant troubles at home.  All hated or feared mathematics.  It was my job to un-teach them this fear.

Of course, the worst thing you can do in teaching remedial students is the same thing over again, even if you go slower and talk louder, even if make the definitions crystalline and the logic pristine.  It won’t help.  The students will tune out, will continue to hate and fear mathematics—and worst of all, will revert to the defensive learning habits that caused their failure in the first place.

Remember that definition of mental illness?

So in this class I used activities.  Games.  Manipulatives.  Students measured and weighed to learn fractions.  They walked the corridors to learn geometry.  They went up and down stairs to add and subtract signed numbers.  (Many of them would confuse left and right, but almost never up and down.)  They analyzed card tricks—performed with different sized decks—to develop algebraic representation.  My administrators, whose support of my work was vital and unflagging, were kept busy apologizing for my students being in the corridors or stairwells or shuffling playing cards.

I love this kind of teaching.  It requires enormous flexibility, creativity, spontaneity.  It means that you have to be right on top of the students’ cognition, reading their minds as best you can.  And since their minds were full of ideas so different from my own, this was a challenge.

The other kind of teaching I like best is with gifted children—and for much the same reason.  You cannot give them just more of the same, and faster.  Sure, they will appear to succeed.  But eventually they will hit a wall, will find some material that they cannot just read and understand.  They need to experience early that understanding can be the result of struggle, or they will not have the means to surmount that wall.  The teacher needs the same flexibility, creativity, spontaneity that the remedial classes require.

And for either group of outliers, you need the same ability to read minds.  These students will come up with things you haven’t thought of yourself, which are correct or incorrect in the most amazing ways.

Cecilia was a fifteen year old girl in my remedial class.  She was not angry, not stubborn, not resistant to school.  Yet she had trouble with mathematics.  I was trying to get her to pass a first-year algebra exam—without cramming her full of test tricks and meaningless technique.  In this classroom, I had the time to do it.

I had given the class a challenge problem:

Marty ate ten grapes on September 1.  Then he ate twelve grapes on September 2.  Then  he ate 14 grapes on September 3.  He kept eating two more grapes each day than he ate on the previous day.  How many grapes had he eaten at the end of September 10?

Remember: this problem, for these students, had nothing to do with an arithmetic progression.  It was a simple arithmetic problem.  We had been working on how multiplication for natural numbers is repeated addition, how the distributive law could shorten computation, how to recognize ‘complementary numbers’ that added up to 10 or 100 or 1000.  In short, we were busy taking arithmetic off the paper.

But this problem stubbornly remained on the paper.  Students tried this and that.  They got wrong answers, thinking this was yet another multiplication problem.  They tried to fit the problem into a pattern they had seen before.  Nothing worked.

They worked in pairs, but I had an odd number of students that day.  So Cecilia was part of a group of three.  And she was not interested.  She rearranged her books.  She stared out the window.  She powdered her nose.

My job was to keep the wheels turning.  So I came over to her group and asked what they were doing.  One student, Tim, had a good idea, although he didn’t know it.  He drew lines to represent the numbers of grapes for each day:





. . .

. . .

. . .


Tim could have observed that the first and last line, the second and next-to-last line, etc, complemented each other to form a rectangle.  But he didn’t.  I encouraged him and made a mental note to come back to this idea later on.

Other students, of course, were busy adding things up by hand.  Students younger than these are perfectly content to solve a problem using such busywork.  They don’t see it as compulsive or boring.  But teenagers do—and that was lucky for me.  They don’t learn much from tedious computation.

Cecilia saw me come over and wanted to look busy.  So she picked up her calculator.  I always let students use calculators, if they could tell me their plan for using it.  If I sensed abuse, I would forbid them the calculator and discuss what they were going to do with it.  Only after they had a coherent plan would I let them have the machine.

Cecilia’s calculator was pink and heart-shaped.  Each of the keys was a different colored rhinestone, and each was also in the shape of a small heart.  She began pressing the keys.

“What is your plan?” I asked.  She had none, of course, and was busy looking busy.

I said: “I’ll talk to Chris.  When I come back, I want to hear your ideas.”  She looked at me as if the word ‘ideas’ was not in the English language.  I fought hard the temptation to lecture her about paying attention to the task.  That never works.

I went over to Chris, who was more or less randomly multiply numbers together—by hand, to impress me.  I asked why he was doing this or that.  He had the (correct) idea that somehow the small numbers made up for the big numbers.  But couldn’t express this arithmetically.  He had missed the essential circumstance that the numbers were evenly spaced—formed an arithmetic progression.  He knew that 10×10 would be too small, and 10×28 would be too big.  I let him work a bit more.

Then I came back to Cecelia.  She was staring at her paper, but not vacantly.  I could see on her face that something had happened while I was gone from her desk.  On her paper was written

5 x 38 = 190,

and on her face was a look of relief—not quite a smile—but a look that told me that she knew what she was doing.

“How did you do it?”  I asked.

“With my calculator,” she replied.  This was not mere adolescent backtalk.  She really thought I was asking about the arithmetic.

“The small numbers make up for the big ones, so you can shortcut multiply.”  And she started to explain.  As I write this, I cannot recall how she explained it.  Her words made no sense to me, whose mind was full of formulas for general terms and partial sums.  I tried to listen, but quickly got lost in her verbal explanation.

No matter.  She clearly understood the problem.  She had figured out that pairs of numbers added to 38.  I was delighted, but she was merely relieved.  I got her to explain her ideas to the others in her group, who also didn’t quite understand her words.  But I gave them another similar problem, and they could do it—and certainly used Cecelia’s ideas.  So something had been communicated.

I never recovered Cecelia’s words.  To this day I don’t know how she thought of the solution, nor how she managed to communicate it to others in her group.

Years ago, I saw a film called Defending Your Life.  It took place mostly in heaven(!), and some of the characters were angels.  It was explained that angels are not really different from humans, that humans only really use 10% of their brain’s capabilities, but angels use 90%.  Maybe.  My point is that for 10 minutes I had an angel in my classroom.

Then she went back to powdering her nose.

II. Cold Weather: An Unfinished Story

Here is another incident that occurred in a remedial classroom.  The students in this class were studying linear equations, starting with a story and generating mathematical models for the situations. They had worked on stories about cars traveling at constant velocity, about Mary working at the grocery and saving her money, about Bob spending the contents of his piggy bank at a constant rate, and so on. Then I gave them what I knew was a hard example for them, just to see what they would make of it:

At 50 degrees Fahrenheit, 30 people will complain about the temperature of a building. For every drop of 10 degrees in temperature, five more people will complain. How is the number of complaints received related to the temperature in the building?

I had expected a table of values something like this:


D 50 40 30 20 10
C 30 35 40 45 50

… and eventually the equation C = − (1/2) D + 55.

This was not meant to be a realistic situation.  I have found that such things do not trouble students. In this case, they had fun thinking of the occupants of the building shivering at their desks.  It was in fact a cold January day.

The students understood the situation well enough to make a table of values. But they could not write an equation. At first, they could not decide which variable should be independent and which dependent. I described to them how an historian might look at the number of complaints to infer the temperature, but most of us would think the other way around. They had no trouble with this, once it was pointed out.


They were thrown by the fact that the table did not start at 0, although some of them had learned to extrapolate to get the value at 0. They were confused by the fact that the temperature went ‘down’, not ‘up’. And I had not yet talked about what to do when x jumps by more than 1 in a table.

As they worked, I observed. They were still not secure with the concept that the equation must be true for every pair of values they knew. They had somewhere learned to follow the ‘key words’ of the problem, so they had various ideas about how the words themselves generated the equation. And all the equations were wrong. This gave me the opportunity to show them that substituting values, rather than looking back at the words of the story, was what would tell them if their equation is correct.

Work was proceeding as I had expected, until Selma stopped me in my tracks. Selma was a vivacious 13-year old, the kind who seems to want to cling to her childhood. She must have weighed about 75 pounds sopping wet, all sinew and energy. And delighted with life.

Selma, among others, gave the equation C = 30 + 5D. Many students had realized that 30 and 5 play roles in the equation and were simply guessing about where to put them. One reason I selected this problem is that the slope is not an integer, and so it is less likely that they would get the correct answer by guessing. When asked, the class quickly saw that this equation was wrong.

But Selma persisted.

“Do I have to do the equation your way? Can’t I do it another way?”

There is only one answer that a teacher can give to this question, and I gave it. It turned out to be the best question I’d received all week.

“Well, what’s another way to do it?” I asked.

Selma came up to the board, and wrote the following table:


d 0 1 2 3 4 5
C 30 35 40 45 50 55

“See,” she said, “C = 30 + 5d.”

I was about to repeat my tiresome argument about plugging in values, but — just in time — I noticed the top line of her table.

“What is d?” I asked.

“Oh,” said Selma. “My d is different from yours. My ‘d’ stands for ‘drops’. One ‘d’ is one drop of 10 degrees. So when the temperature drops 10 degrees, we have 35 complaints: the 30 we had at first, plus five more. And for every drop, we add 5 complaints.”

I was speechless. But the class wasn’t. “That’s wrong!” “That’s right!” I had no trouble engaging them, but I myself didn’t quite know what to say.

So I played it safe. I told Selma that I understood her reasoning, and her representation. Could she use her ideas to get an equation in terms of the temperature Fahrenheit? She understood what I meant, and she also understood that she was right.

But what should I have done next? I might have exploited this idea of changing variables that Selma stumbled on. What are some fruitful directions? What Selma had discovered, without knowing it, was that a linear change of variables does not affect the degree of a polynomial function, so that a linear relationship remains linear. The trick of choosing your variables wisely is an old one. It lies behind much of the work of the Renaissance algebraists, and was made into an art form even earlier by Diophantos. I am still not sure what the best pedagogical strategy might have been on this occasion, but I feel that there is more here I could have done.

I often structure lessons so that a problem remains open at the end.  This time, life structured the open problem for me.

Some of this Blog post will appear in the forthcoming book, edited by Hector Rosario:  Mathematical Outreach: Explorations in Social Justice. Singapore, World Scientific Publishing, 2019.   Other portions have appeared in “Anecdotes and Assertions about Creativity in the Working Mathematics Classroom” (with Mark Applebaum), in Leikin, R., Berman, A., and Koichu, B., Creativity in Mathematics and the Education of Gifted Students.  Rotterdam: Sense Publications, 2009.




This entry was posted in Active Learning in Mathematics Series 2015, Classroom Practices, Communication, Faculty Experiences, K-12 Education, Mathematics Education Research and tagged , , , , , . Bookmark the permalink.

One Response to Two More Teaching Vignettes

  1. vijaynilatkar says:

    Thanks for sharing such information i really like it

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