PCMI #3: Talking and Listening

We were introduced to a protocol called “Talking Points” by Elizabeth Statmore (@cheesemonkeysf). Participants take turns agreeing, disagreeing, or expressing uncertainty about a list of statements. Each response must have a reason, and no commenting is allowed. The process occurs a second time with the same statements, and final opinions are tallied during the third round.

The protocol got me thinking, ‘What does it take for students to talk and listen to one another?’ No single reflection can do this question justice, so I’ll just go through two thoughts, one about talking and one about listening.

1: Students won’t speak if they’re rarely given the chance, and even if given the chance, they’ll rarely take it unless they think their thoughts are worthwhile. That’s why Elizabeth Statmore’s Talking Points and Ilana Horn’s ideas about competence have been so valuable in helping us think critically about our practice. Student talk requires space and intention. Protocols like Talking Points give students the opportunity to speak and a framework for structuring their thoughts, while surfacing our students’ competencies (like being willing to take risks or highlighting the strengths of others) make it more likely that our students will exercise them through dialogue. Recent posts by Dylan Kane (who makes a compelling distinction between getting students to feel competence and getting to students to recognize their competence) and Deb Barnum (who makes a great point that such protocols can “keep our conversations grounded in evidence”) resonate with me in these respects.

2: Listening to someone else’s ideas about math is different from listening to someone else’s ideas about other subjects. Because of the high stakes people feel about their own math abilities, it’s hard to accept an idea that’s different from your own because it may mean your idea was incorrect or incomplete. To paraphrase one teachers’ comment: you’re not going to listen to someone else’s thought if that would mean yours is wrong.

I wonder if this makes listening to mathematical ideas less about hearing what another person says and more about being willing to be influenced by them. How someone sees mathematics is closely tied to this issue. If math is mainly about being right and wrong, a person’s own thoughts are more likely to be a barrier from those of others. As teachers who want our students to become not only better talkers but also better listeners, we have an obligation to normalize for our students the idea that doing math requires making mistakes and revising our thinking. This is a necessary step toward ensuring that receiving feedback doesn’t get in the way of recognizing ones own competence. But this is also multi-dimensional work, and so helping our students form a healthy understanding of math is merely one aspect of designing a classroom that takes into account how students position one another and themselves. But it’s a step in the right direction.

PCMI Reflection #2

Today’s reflecting on practice focused on using student-generated questions to make math more meaningful for students. In addition to great conversations and an interesting activity on analyzing data sets, two concrete resources were shared by members of our group:

Screen Shot 2018-07-05 at 3.49.36 PM


“Meaningful” is a really interesting word, especially when it comes to math education. I think for a lot of teachers, the term means “real world applications”. But I don’t think that has to be the case.

I like the idea of asking students to create their own questions because their questions can still be meaningful to them without the questions having to do with real world situations. I think having students engage with questions that they’ve created centers the students while they experience math. Other group members also mentioned how asking students to ask their own questions creates ownership, improves metacognitive skills, gives students voice, and creates new opportunities for students to show competence in math class.

To me, the opposite end of the spectrum is the case (far too common in classrooms, probably even my own from time to time) where adults impose “real-world/fake work” activities on children which they think children will be interested in.

In The Having of Wonderful Ideas, Eleanor Duckworth has this wonderful* quote:

First, the right question at the right time can move children to peaks in their thinking that result in significant steps forward and real intellectual excitement; and, second, although it is almost impossible for an adult to know exactly the right time to ask a specific question of a specific child – especially for a teacher who is concerned with 30 or more children – children can raise the right question for themselves if the setting is right. Once the right question is raised, they are moved to tax themselves to the fullest to find an answer.

I’m particularly interested by Duckworth’s use of the word “setting”. That is, it may not be enough to simply ask students to ask their own questions. Rather, the setting must be structured for students to ask good questions. My students know when they’ve made something they’re proud of and when they’ve made something just because I told them to. So I wonder if the motivation and meaningfulness from asking a question comes not from the question itself, but rather from how much they’ve found their own question to be worthwhile (and also truly theirs).

Unfortunately, we didn’t have enough time to discuss how we might approach the work of asking students to create their own questions. But instead of seeing “create your own question” as a simple pedagogical trick, I do believe it should be done as part of a larger project to change the way students view mathematics and their mathematical selves in ones classroom.

*did you see what I did there?!

PCMI Reflection #1

I have the privilege to be currently attending the 2018 Teacher Leadership Program (TLP) at the Park City Mathematics Institute (PCMI). The TLP is a 3-week conference of math teachers doing math together, reflecting on their teaching practice, and working in small groups to achieve certain professional development goals.

The theme of this year’s reflecting on practice is “Mathematics and Motivation”. Using Ilana Horn’s book Motivated, the facilitators will be helping us develop a framework for thinking about motivating and engaging our students.

On the first day of reflecting on practice, we were asked to contribute one word (ONE word, c’mon!) that described a “good math student.” Here’s the wordle of the results:

(do you see the “NotAJerk”?)

Having noticed that curious was the most common term, we launched into discussions and activities centered around the potential noticings and wonderings that our students might have about mathematical concepts such as linear equations.

By the end of the session, we were asked to reflect on the following question:


This image has stuck with me since and is the main reason I wrote today’s post. For me, it raises the question:

How have I been constructing competence in my classroom?

For a better sense of this question, I should mention the work of Melissa Gresalfi. In this article, Gresalfi argues that competence, rather than being a set of skills particular to an individual student, should be seen as the relationship between what a student does and the opportunities for learning that he or she was given based on the classroom context. In this sense, competence is constructed by the tasks and participation structures designed by the teacher and the way they are realized by the students. This is important because how competence is defined in a particular classroom determines what kind of learning can take place.

For example, competence could be constructed in a classroom as correctly carrying out procedures established by the teacher. But this is not the only form of competence that could be constructed. It could also mean experimenting, making mistakes, and learning from those mistakes. In both cases, the classroom sets forth an image for success, and that image determines the path that students must take to get there.

Looking back at my own classroom from this year, I believe that competence was defined for my students as following procedures, reasoning about them, creating original mathematical methods, justifying them, and applying known methods to new situations. Today’s closing question raises the idea that curiosity can (and probably should) be added as a component of competence in my classroom. I praise students for solving problems with originality and justifying their reasoning. However, I should also (but don’t yet) consider whether my students do so out of intellectual urgency or whether they are simply complying with classroom norms – i.e. “I used a different method because I was supposed to.”

I would also like to start holding my students accountable for being curious about other people’s reasoning and sense-making. In discussing accountability, Gresalfi points out that teachers should not only think about for what students are accountable (what students must do) but also to whom students are accountable (who students must convince). In many classrooms, including my own, I am the only one to whom students must ultimately justify their work. Sure, I tell them to discuss their reasoning in pairs or in groups, but in the end I am the only one to whom my students feel accountable. In this situation, students in the lower-right-hand corner of the graph above continue to be rewarded while students in the upper-left-hand corner fail to have their competence of seeking and sharing ideas acknowledged.

Here’s my final thought for now: How does one go about construcing an open view of competence that includes curiosity? Probably, the first step is to critically examine the practices and rituals of ones own classroom. In Engaging Children: Igniting a Drive for Deeper Learning, Ellin Oliver Keene asks the reader to consider when your students speak up in class. Is it only to answer a question posed by the teacher? Or do they also speak up to ask questions, challenge one another’s thinking, and pose areas for further exploration? The nature of students’ unprompted discourse may be an excellent starting point for examining the ways that competence has been constructed in the classroom.

A Few Reflections on the Regents

Two days ago, my 8th grade students had their last day of middle school. They also received their report cards and Regents scores. In my school, every 8th grader takes a math Regents exam. This year, I decided not to look at my students’ scores. I would find out at the same time they did: at the end of the school day during PM homeroom.

The last day of school was marked with lots of tears, jokes, and goodbyes. For me, the highlight was sharing the Instagram account for my cats. Throughout the day, the students were their silly selves, and it was great to connect with them one last time before summer.

This made it all the more disappointing to see the impact that receiving their Regents scores had during the final moments of school. Even though I still didn’t know their scores, it was terrifyingly easy to tell how they did simply based on their demeanors walking out of PM homeroom. Some students were clearly elated, some were disappointed, some were ashamed, some were questioning their worth and abilities, and some were simply coming to terms with their scores, good or bad.

It happens every year. Regents scores reconfigure my students’ relationships with themselves and others. Despite my constant messages not to let a single number define them, it’s hard for them not to allow their scores to shape how they see themselves as learners and doers of math. I’ve had students who were once confident in their abilities cry because they received an 89 (out of 100) instead of the 97 that their close friend received. I’ve had other students receive perfect scores, believing that such a score grants them authority and expertise over others. Even the teachers are affected. Once the scores came out, I saw an Algebra teacher talking about how she had “a low rate of students who achieved above mastery this year” and how she was going to spend the summer restructuring the curriculum. I, too, couldn’t help but attach scores to students once I found out.

In the end, it seems like no one wins. Students who receive lower scores (where students’ conceptions of “low” have been unduly influenced by our school’s and parents’ obsession with certification success) have their agency and self-worth chipped away. Meanwhile, the higher scores serve to entrench the beliefs held by higher-achieving students that their mathematical abilities boil down to their performance on standardized exams.

I must confess that in some ways I allow the prospect of the Regents exam to affect how I teach throughout the year. As one of my students pointed out in his mathography:

Mr. Peralta’s class was really fun in 6th grade because we played a lot of games during our double period. This year was less fun, but I understand because we had to do Regents prep.

However, I don’t consider getting high scores on the Regents exam a priority in my teaching practice and instead like to think I have different overarching goals for my students, three of which include:

1. Engaging with sophisticated mathematics.
2. Developing and nurturing their mathematical identities.
3. Becoming aware of and attending to issues of equity, access, and power when learning and doing mathematics with others.

I have conflicting feelings about my responsibility to ensure that students are prepared for the Regents. Inevitably, I’m left with questions about the compatibility of my goals as a math educator and my obligations with respect to standardized exams.

Must I choose one over the other? Is it actually true, as some would say, that if I focus on authentic learning, the scores will follow? Do I take that chance and disregard the Regents? If so, who bears the risks? Is it fair for those students who need the scores for scholarships and high school admissions? What do I owe the students? In what way is my thinking asset- or deficit-oriented?

In theory, it’s easy to say that I shouldn’t let standardized exams interfere with my teaching practice. But the reality of standardized exams is so deeply embedded in the culture of my schools and in the mindsets of students, parents, and staff that it’s gravitational pull is impossible to avoid. To anyone who says that Regents exams are simply neutral measuring devices, I simply can’t see how that can be true.



I recently gave my students a Mathography assignment. The directions were to “write about yourself and your relationship to mathematics. Start off by telling a little about yourself – your interests, hobbies, family, and any other aspects of your background that you’d like to share. Talk about your experiences with mathematics since as early as you can remember. Discuss any favorite lessons or aspects of math which you enjoy (or don’t enjoy). Talk about any math that you do or that you see outside of school. Talk about how you see yourself as a learner and doer of mathematics. Most importantly, talk about why you feel the way you do about mathematics.”

Beyond this, I did not give them any structure or constraints other than the fact that it had to be at least one page long.

The day I assigned it, I shared my own Mathography. I stressed that their Mathographies did not have to resemble my own, and that it was okay to share positive and negative experiences or views about math. It was important, however, that they try to discuss the root causes or sources of their feelings and views.

In Why (Urban) Mathematics Teachers Need Political Knowledge, Rochelle Gutierrez said, “All mathematics teachers are identity workers, regardless of whether they consider themselves as such or not.” It follows then that a Mathography may be a way to assess at least one aspect of my teaching practice this year. If mathematical learning is not just the factual repertoire and techniques students develop, and if it includes their attitudes about math, their disposition and engagement with math, and their sense of how math is defined, what doing mathematics looks like, and who can do mathematics, then a Mathography that asks students to reflect on their mathematical identities may be a way for me to know how well (or poorly) I’ve supported (or undermined) these aspects of their mathematical learning.

In the list below, I’ve shared snippets from my students’ Mathographies. A couple of comments. First, by way of background, I teach at a public midle school in eastern Queens. Technically, it’s part of the NYC DOE, but it’s also in one of the wealthiest districts in NYC. The school is racially diverse and there are some students from lower socioeconomic backgrounds, but the school mostly draws on students from the surrounding middle class and upper middle class communities. The snippets come from 6th graders in my school’s gifted and talented program. This year, they took Prealgebra, and next year they’ll be taking the 9th grade algebra Regents exam.

I chose passages that stood out in some way. Maybe I thought it was funny. Maybe what a student said cohered with my philosophies on teaching math. Maybe it didn’t. I realize my particular selection likely reflects my biases about math and math education and maybe also some desire to “look good”. But I did try to choose a wide range of thoughts and experiences, some reflecting well on my teaching and some perhaps reflecting poorly. I also acknowledge that there’s something fundamentally lacking by leaving out the context surrounding each passage. Then again, I was careful to include only brief passages so as not to include anything that could be revealing or identifying about my students.

Mathography Snippets

“I see myself as a limited mathematician who can do greater things. I see myself as a person who wants to do more; wants to learn more…People who think math is hard; they quit and give up easily. But to others, math can be a hobby, a passion, a love for a subject. And that’s me: a person who doesn’t give up, a person who challenges himself to do his best.”
“My classmates seemed like everything was easy, like they’ve done this a million times so I felt alone…Knowing that I was not smart when I came to math it put me down not that people were better than me but it was just that I felt…dumb. Then we had our first test which I didn’t do really well on which made me even more sad and everyone was doing really well and again I felt alone. Not only that but it put a lot of stress on me.”
“I remember in third grade, when I first was taught to do long division, I just sat for over an hour just dividing any numbers that popped into my head.”
“I can’t wait to learn the math skills that ninth graders learn. I can impress my friends with my exceeded knowledge in the field of math, and even use it to help overcome the challenges of daily life.”
“I wish that I tried harder in class. I used to like math because it was easy so I’m guessing that the reason that I don’t like math anymore is because I’m bad at it. But I’ll slowly try and improve in math and maybe one day I’ll like math again.”
“I thought that math was (excuse me) dumb, because I didn’t understand why math was so important in life and why a person should be learning math. Everything came back down to counting, I said to myself, and I knew how to count…It was my fatal flaw that led me to like math. I think it was because that I wanted to be just like my sister, with her walls covered with paintings and pastel drawings, plus all the awards, medals, and certificates that cluttered her bookcases.”
“Math is not only about the patterns and numbers and symbols and signs, but about the way you interpret the numbers. Math is like a ladder. If you are missing one step, you can’t go higher. So, if you miss something, you can’t go on.”
“Ms. […], one of them, was so interested in both math and art, and brought math to life with her art. Art was my talent, and I was exhilarated to be  applying art to math.”
“What I don’t like to do are word problems and decimals. I don’t hate math, but I don’t love math either, so I just like math. Math is a part of my life because my family is conflicted in it…If there wasn’t math in my life, then who would I be?”
“Mr. Peralta, our new math teacher not only taught, but encouraged us to use creative thinking and create our own formulas…Instead of sticking to the curriculum, he encourages us to make our own. There isn’t just one way to do math, but multiple ways, and none of them are incorrect. In math class, mistakes don’t matter, but but rather benefit you, for they allow you to learn and grow.”
“I remember the summer before second grade my grandfather would give my sister and I about ten problems everyday to solve. I never really understood what I was doing…I recall crying because he had to explain the same thing over and over, but I still couldn’t understand.”
“My mom was an after school teacher and wanted to teach me about numbers. However, I threw a big tantrum that I didn’t want to study and ended up hurting myself from all the running away and kicking. That was just the beginning though.”
“Math can be really helpful. It could get you the job of your dreams. Being good in math looks good on your resume.”
“Most people have a strong opinion about math but I don’t. Either they like it or not. For me, it depends on the teacher.”
“At my Sunday school for math, the teachers don’t speak English well and have a Russian accent so it’s hard to understand and they teach the ‘Russian way’.”
“I remember that when I was testing to get into the G&T program, the tester asked me which shape was the circle and I chose the triangle. A TRIANGLE…Another time in pre-K, the teacher mad me spell mop and I spelled it wrong. Then she made everyone spell it slowly and loudly RIGHT IN MY FACE! After that experience, I never felt very confident about my spelling skills.”
“One of my favorite lessons of math is when you get to learn to think critically and solve certain problems outside the box…Also, it is very satisfying when you solve a hard problem on your own. I like math because it is sort of an open free world that you can use any method or idea you want, as long as you can solve the problem and get the right answer…Why I feel the way I do about mathematics, is probably because of my mom, who influences me a lot.”
“A math topic that I really enjoy is our current topic – multiplying binomials (forwards and backwards problems)…The only topics that I don’t like are State Test Prep topics. I don’t find it fun because we have already learned everything and I find it boring to do the same thing over and over.”
“It is the one subject where there is a strict right and wrong. No ‘good’ answers, only right and wrong answers.”
“I started to gain momentum and by the time I made it to 100 by the end of 5 minutes, my mom hugged me and said, ‘Oh my gosh! You are correct!’ and that’s when I knew math was my thing.”
“How I see myself as a learner and doer of math is not just memorizing formulas and facts, but as in finding patterns, solving puzzles, and doing something that will benefit me in the future.”
“Math, so far, was just something that I didn’t want to be beaten at. Not because I was amazingly good at it, but because of my pride…Because of my realization of my identity as a math student, the burning flame inside me has (temporarily) died down, and I can appreciate mathematics for what it really is, not as a field of study where I am compelled to compete with others concerning my skill at it.”
I began middle school at […]. I’d never before in my life been taught such material in such a way. I actually began enjoying math. I began to see and be awed by the complexity and simplicity that occurred at the same time in mathematics.”
“I got really bored at the idea of ‘learning’ math, and I say ‘learning’ because I didn’t actually learn much. I didn’t get the idea or the concept of doing and ‘learning’ something I had learned by myself. I’d also had some more common sense and kind of figured out everything on my own. I didn’t understand why I had to do something I already knew and just keep doing examples. According to my previous teachers, it was to make sure we understood wheat we were learning, to me we weren’t learning anything.”
“The way I view math is…complicated to say the least of it. It’s kind of a love hate relationship. I can really love the challenge, but I also partially have anger issues so I get frustrated kind of…A LOT.”
When I was in third grade a thing I had trouble with was the times tables. Whenever we were waiting on line, the teacher would always ask us times tables. She expected us to answer in a snap. However, I was always too slow.”
“I actually have to work now, and think about whether or not the answer I got is right or makes sense. Basically, the reason I don’t love math is because I’m a very lazy person and I don’t like to work.”
“One other detail that I do not like about math is the amount of test prep that is created by the NYS exam.”
“If I learned one thing from mathematics in my entire math life, it is probably the fact that mathematics is easier to people who enjoy it, not to the people who are forced to do it.”
“I’ve always disliked ‘school math.” My earliest memories in math were about my mom who taught me a simple, but challenging view of numbers. Those were my happiest memories of math. I was challenged, but happy to spend time with my family. I felt that that was when I progressed the fastest.”
“I feel the way I do about mathematics because doing math required you to think. If you do not think, there is no point of working in the first place.”
“I learn Korean math from a Korean teacher by Skype. Korean math is basically math but just a bit more challenging.”
“Math has always been my favorite subject. I don’t really know why, but it was. My sister excelled in it, so I guess I wanted to be like her…My parents always compared me to her…Well, when she was my age, nothing was this competitive, the work wasn’t as hard, nor did she do 13 extracurricular activities.”
“This was probably why I loved the cubes so much. I would fiddle and fidget with them when I was done with the classwork involving them. It made me think about how that one tiny thing the size of the tip of a thumb could turn into something the size of my head and then even bigger by putting multiple of them together.”
“The thing I like about math is how it’s so flexible and that it can be used to solve some of your everyday problems. What I don’t like about math is the way math is presented, because it takes away the creativity and fun of math, and turns math into a thing that most kids dislike.”
“I think I hate it because I’m bad at it, but my math average went from an 87% to a 93%, so I say I’m good.”
“Growing up as an Asian-American in an Asian family and household, I was always involved with math and learning outside of schools such as academies, my whole life.”
“As an (very bad) analogy, math would be a daily skills, like getting ready for school in the morning, while “hands-on” learning would be like taking a field trip outside of school.”
“Sometimes I wonder how people know how to solve questions so quickly while I’m struggling with them. It sometimes makes me feel annoyed and jealous that I’m so slow…I have realized that to solve my problems involving math, I should just keep trying and take my time if it takes me a lot of time to figure things out.”
“To this day, I still picture the Abacus in my head when doing mental math like multiplying two-digit numbers by one-digit numbers or dividing two or 3-digit numbers by one-digit numbers.”
“In certain topics of math, I am often fascinated and amazed at how everything fits together and makes sense. I usually learn well in math when I see someone else solve a problem and use it as a model that I can use to solve the same sorts of problems.”
“Math is sort of fun outside of the curriculum, which sometimes a student must do in order to get the most out of his or her educational life. Math is not the same when you view it from a different perspective, which is why math is sort of enjoyable to certain students and not to others, who see it as set-in-stone numbers and formulas, and nothing more.”
“Some subjects require you to memorize things, then forget about them after a test–but, math is different. If you understand a topic, you can remember it for the rest of your life.”
“I didn’t really understand most strategies used for visualization. Of course, it’s helpful for students to visualize problems. But, when forced onto students, it can confuse more than contribute.”
“Extremely long word problems that are used to trick you don’t help comprehension. By conning a student into not understanding a word problem doesn’t make them better at the subject. I don’t find it difficult, just tedious.”

Joyous Math

Michael Pershan recently made this post in response to criticism he received on Twitter. The criticism:

Pershan’s original post talked about microskills, and his follow up made a distinction between “abstract joy” and joy situated in a classroom context.  Pershan puts forth the idea that joy can come from feeling success and confidence in the math classroom, including feeling success over mastering a microskill.

For me, this criticism and his response raise questions I’ve had with the concept of “joyous math”.  Pershan quoted Francis Su, and so I will too:

So if you asked me: why do mathematics?  I would say: mathematics helps people flourish.  Mathematics is for human flourishing.

The idea of bringing joy and a sense of flourishing to children through mathematics is not new.  James Tanton strongly advocates for “joyous” approaches to math education.  His homepage has this stated goal:

The goal of this site is to demonstrate the beauty of mathematics, its wonder and its intellectual playfulness, and to work towards bringing true joy into mathematics learning and mathematics doing for one and all.

The concept of “joy” (better yet, the lack thereof) is sometimes used to criticize the current state of math education.  Possibly the best known criticism of math education comes from Lockhart’s Lament:

In place of meaningful problems, which might lead to a synthesis of diverse ideas, to uncharted territories of discussion and debate, and to a feeling of thematic unity and harmony in mathematics, we have instead joyless and redundant exercises, specific to the technique under discussion, and so disconnected from each other and from mathematics as a whole that neither the students nor their teacher have the foggiest idea how or why such a thing might have come up in the first place.

But he’s not alone, to varying degrees.  Here’s a recent book by Alfred Posamentier, who was a teacher in the Bronx and later the dean of math ed at City College in NYC:

Here’s a snippet from (one of) Sunil Singh’s pieces against the current state of math education:

My own personal lament is that the holy trinity of theories — number, graph, and game — are nowhere to be seen in most K to 12 math curricula. Prime numbers are merely a definition. Mathematical mavericks like Martin Gardner, John Conway, Ivan Moscovich, etc. are unknowns. Algebra pops out of a can in high school — with a sequel to boot. There is no organic and seamless bridge between arithmetic and algebra. Teaching algebra as an appendage to teenagers as opposed to teaching it as a circulatory system earlier on is one of the clearest indicators of the mismanagement of mathematics by education. It’s like going to hardware store. Aisle 3, top shelf: nails, washers, and Algebra I. 

I could go on.  But the main issue, as I see it, is an utter lack of joy in the math classroom.  The question then is “What does joyous math look like?”  To many of the mathematicians who advocate for joyous math, it usually takes the form of an elegant or unexpected idea, connection, or result.  Singh cites videos such as these as representing a form of “joyous” math:

I don’t disagree that mathematical ideas like the square-sum problem are mind blowing.  Every time I watch a Numberphile video, I think to myself, “Wow, that’s amazing. I want to show this to my students”.  I’ve gone through James Tanton’s exploding dots videos and read his book Mathematics Galore!  I have the fortune of attending amazing professional development seminars through Math for America (shameless plug!).  I try as much as possible to learn new math, new connections, and new ways of thinking about concepts I thought I already knew.

But here’s my question: will math that makes me feel joy also necessarily make my students feel the same way?  I’ve got 20+ years on my students.  I (pretty much) already know the material that they have yet to learn.  So when I attend a PD where we do familiar math in a cool new way (e.g. clothesline math), I’m stoked because I find myself saying “it makes so much sense now” and “that’s an amazing connection that I had never seen before!”  Moreso, when we’re learning math that is new to me, such as Patrick Honner’s favorite theorem (Varignon’s theorem), I am equally stoked because “this is so elegant compared to what I’ve seen” and “this is so refreshing from the stuff I’m so familiar with”.

But will my 6th grade students, who are fresh to a lot of the curriculum I teach, feel the same kind of joy as I do when I introduce them to clothesline math, exploding dots, visual patterns, and Varignon’s theorem?

Frankly, I’m not sure.  I’ve done some math with them which I consider beautiful, and they don’t seem to be into it.  I thought creating personalized polynomials in Mathematica using Lagrange’s interpolation formula was going to be a hit – I thought wrong.  Or how adding consecutive odd integers starting from 1 always gave a square number because…

img(no excitement whatsoever)

On the other hand, we’ve done math which I don’t consider particularly exciting but which they found really interesting, like using the ladder method to find the gcf of two numbers.

Politicians get a lot of slack for thinking they know about education simply because they’ve been in school.  I worry that teachers fall into the same trap of thinking they know about children simply because they were kids once.  There has been a lot of work done on how children think.  I would be interested in any work that has been done to systematically understand what makes children feel fulfilled.

If making children feel “joy” and “fulfillment” is part of our end game, it can’t come merely from what we adults consider joyous and fulfilling.  I sometimes find myself rolling my eyes at what I, as a 13-year old, once thought made my life complete.  But I don’t regret what once fulfilled me, and I know it’s made me who I am today.  The mathematics we have children do must honor their sense of joy and fulfillment and not merely ours.

That brings me to how I started this post – Pershan’s response to criticism that his way of teaching microskills lacks joy.  I agree with his idea that success can equal joy.  But I also wonder if the question of joy is so context-dependent that it really boils down to his students.  If they feel fulfilled in working through and conquering the micro-challenges he presents to them,  and if they are in fact growing in their mathematical maturity, then perhaps he’s creating joyous math for them and that’s enough.


Sequence Diagrams

I introduced “sequence diagrams” as a visual tool to help students think about sequences (alongside graphs, tables, and visual patterns).  Instead of modeling the use of this tool, I had the students explore it through a series of questions.

The sequence diagram below shows an arithmetic sequence where the second term is 5 and the thirteenth term is 49.

How much is each little arrow worth?  // Most students realized that counting the number of little arrows and “spreading out” the +44 equally to each one would do the trick. Formally we introduced this concept as slope. Though unconventional, I prefer to introduce slope in terms of a unit ratio, i.e. the amount that y changes when x increases by 1. Usually, slope is simply introduced as “the ratio between the change in y and the change in y”. But ratio has many different interpretations, and I believe introducing slope as a ratio is not enough – one should make a choice about which ratio interpretation to use when introducing slope.



What’s the y-intercept of this function? // The nice thing of sequence diagrams is that boxes can be added and removed as suits its user. Students were quick to add boxes to the left. 


What’s the value of box X?  // I gave my students hints in the form of two more questions: How many +4 arrows are there from box zero to box X?  How large is the big arrow?



What happens if we change all the little arrows from +4 to x4?  // The shift to geometric sequences and exponential functions was very natural with a sequence diagram since many of the same patterns of reasoning carried over. Students were able to figure out a sensible expression for the value of the large arrow, thereby finding the explicit formula of an exponential function without the need for direct instruction.



Reflection: Sequence diagrams aren’t all that different from tables. But for some reason the boxes and arrows helped the students reason about sequences better than cohorts of students in the past.

The inspiration for these sequence diagrams came from my recent experiences reading a book called “Visual Group Theory”.  In college, I never really understood abstract algebra. I could get through the tests OK but it never really stuck.  Visual Group Theory framed the foundations of abstract algebra in terms of Cayley graphs and everything made so much sense after that.

Like Cayley graphs, sequence diagrams encapsulate many concepts (sequence progressions, y-intercepts, slope, explicit formulas, recursive formulas) into a single, modular, visual package that can be shrunk, extended, and edited to suit different purposes. I hate forcing students to memorize formulas, like A_n = A_1 + (n-1)d or f(x) = ab^x. This is my attempt to avoid that.