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.”

Social Justice Pedagogy

In the past month or so, I’ve become increasingly aware of the importance of infusing social justice pedagogy into my math teaching practice. This is not something that has come out of the blue. Years ago, I was quite interested in critical legal theory and immigration reform, having been involved in the world of immigration law before becoming a teacher. For some reason, I never brought my past experiences into my classroom. Maybe it was because I was just getting used to teaching. Or maybe because I considered my transition into math education a “clean break” from my lawyerly past.

But something’s changed recently and I’m not sure what. Regardless, I’m someone who’s just starting to think about how math is not a “neutral” subject as is commonly believed, and how the ways we teach math can help reproduce or interrupt social inequalities.

From my position as a beginning thinker on this subject, I’ve started to change my pedagogy in small ways. Mostly, I’ve been integrating issues of social and economic inequality into the examples I use to promote mathematical understanding. I’ve been asking students to engage with these issues on a mathematical level but also to use mathematical ideas and concepts to get a better sense of the issues themselves. Two resources have been profoundly helpful: Rethinking Mathematics: Teaching Social Justice by the Numbers and Radical Math.

I still think about Nepantla, a term Rochelle Gutierrez uses to describe the tensions inherent in teaching. Incorporating social justice pedagogy comes with a lot of tensions to think about. How do you balance a standardized curriculum with a vision of social justice? How do you tow the line between teaching your students to critique institutions and propagandizing your students? How do you find time to carefully and thoughtfully incorporate social justice pedagogy into your classroom without getting fired or forgoing self-care?

I should also note I teach in a relatively privileged school. With that in mind, two big questions come to mind — why engage in social justice pedagogy at all (1), and how might it backfire (2)? In Educating Activist Allies, Katy Smallwell offers insight into both questions:

Why Social Justice Pedagogy

(1a) Poverty is not just about poor people but about the relationship between people of all social classes. If you want to interrupt the reproduction of unequal opportunities and outcomes, there’s value in understanding the perspectives of those with privilege

(1b) Orienting children from privileged communities toward justice can be an important strategy in the larger project of addressing injustice

Potential Problems

(2a) Injustice is seen by students as “over there” and “in the past”. Injustice is romanticized or deficit stereotypes are reinforced

(2b) Injustice is seen as the result of “bad” people and not of structural forces that systematically oppress those without privilege

(2c) Students who confront facts and figures of injustice incorporate them into heightened feelings of exceptional progressiveness and worldly ease.  At worst, knowledge of social and economic inequalities is seen by them as something marketable to help them get into elite colleges

I’m not sure how I’ll address these concerns as I move forward with efforts to make social and economic concerns more visible to my students. Like all things in teaching, I’ll have to try it out, see what happens, and reflect. Here’s to the beginnings of a journey.

Tension, Grading, Nintendo, Strawberries

The concept of conocimineto leads Anzaldua to construct Nepantla or the space that represents “el lugar no lugar” (neither here nor there), what has been thought of as the “third space”, “between worlds, between realities, between systems of knowledge”…When one lives with this constant tension, there tends to be a greater awareness and conocimiento con (familiarity with) uncertainty. Knowing that everything is conditional, that we may need to pull out another hat to wear at any moment, we are tentative with our ways of viewing the world.

This is the concept of tension, which Rochelle Gutierrez describes in her paper Embracing Nepantla: Rethinking “Knowledge” and its Use in Mathematics Teaching. Recently, I’ve been able to make use of the concept to think more clearly (or better yet, less definitively and more intimately) about my students and my teaching practice. I’ve learned to embrace being in a “messy place”, and this has led to some positive changes in my classroom. It’s also reinvigorated my feelings toward this profession by highlighting the ever-changing, dynamical nature of teaching.

One tension that I’ve recently thought about is the extent to which we give grades. Students view grades as a form of accountability. I want them to feel that their work is valued and recognized, and grades are a way to give them credit for their efforts. But students also attach their identities and worth to grades. Grades can undermine students’ efforts to become intrinsically motivated learners and distort the relationships among students and teachers.

The question is how holding this tension plays out in the classroom. One aspect of tension is the notion that it is never resolved. Having this particular tension means that the extent to which I give grades should remain unsettled and subject to revision. Tension does not create an excuse to be frozen in a state of inaction and stalemate myself. It does require that I search for sources of inspiration to build new ways of thinking as I continue to exist in Nepantla.

One such source of inspiration has been video game design. One game in particular has had a positive influence on my teaching practice. The game is called Celeste, which I’ve been playing on the Nintendo Switch. It’s an incredible game about a girl who wants to climb a mountain, but along the way she must overcome issues of anxiety, depression, and self-doubt. Mechanically, it’s a side-scroller like the original Mario game. The goal is to get to the end of the level without dying from spikes, melting in red goo, or falling off a cliff.

Image result for celeste game

One of the best elements of the game is the strawberries. The game places strawberries throughout each level that the player can collect, if he or she wants to. Some of the strawberries are grueling to get to without dying. They’re not required to beat the game and you can get as many or as few as you like. Unlike how many other games might have handled it, strawberries don’t strengthen your character or unlock extra levels. There’s no actual point in the game to getting the strawberries. But, damn, have I spend a lot of time getting them.

Image result for celeste strawberries

Since playing Celeste, I have changed the structure of my task design. In every task or question I assign (actually, to the extent that I can manage), I now include one or more “strawberries” for students to solve. I still grade the core questions or at least set out a clear expectation that students must engage with them. However, the strawberries are optional. They are not worth extra credit and have no impact on their grade whatsoever. In that sense, there are assignments for which there is no accountability, and yet there is accountability everywhere else throughout. Nepantla.

Source for Optional:


The strawberries are usually tied to our current topic, but not always. Some of them are easier than others, and some are grueling. I hesitate to call them “Extension Questions” for the reason that enrichment shouldn’t be reserved for the strongest students or fastest problem solvers (an idea I developed from Craig Barton’s book “How I’d Wish I’d Taught Maths”).

I don’t believe that this new design resolves my tension with grading. I’m not even settled in how I’ll implement this design next year. Moving forward, here are possible ways I’ll revise my use of strawberries:

  • Call them “strawberries”. I’ll be transparent with my students about where I got the idea. They’ll also find out sooner how much I like my Switch.
  • Give them a chart where they can record how many strawberries they’ve collected. Again, this won’t be for a grade. It’ll just be for them.
  • Admittedly, most of the strawberries were more challenging than the core problems. I should widen the range of questions that I pose as strawberries.


Intersecting Thoughts

I recently worked through two books on education: How I’d Wish I Taught Maths, by Craig Barton and For White Folks Who Teach in the Hood and the Rest of Y’all Too, by Christopher Emdin.

Barton’s book is part personal narrative, part research compendium about how Barton radically changed how he taught math despite being lauded for his high test scores and use of group work, open-ended problems, class discussions, and productive struggle. Barton draws heavily on Cognitive Load Theory, which says that teaching should be structured to avoid overtaxing people’s limited working memories. He highlights differences between novices and experts, defends explicit instruction as useful when students are first learning a topic, and lays out pretty good strategies for dealing with students who are not novices but are not yet experts. His book mentions lots of other theories from cognitive science, including variation theory, self-explanations, and interleaving.

Emdin’s book is also deeply personal and well-researched, which makes sense considering Emdin is a former science teacher and now professor at Columbia Teachers College. Emdin’s work is an outgrowth of his experiences teaching in Harlem, where he had a rough first few years because of the disconnect between him and his students. Emdin’s central thesis is Reality Pedagogy, which aims to meet students on their turf. His proposed strategies for doing so, especially among urban youth, include borrowing practices of engagement from pentacostal churches and barber shops, forming cogenerative dialogues with students, positioning students as teachers in a co-created classroom, and entering students’ cultural contexts.

I want to be cautious against painting overly broad strokes here, but I’d dare to say that these are two different types of books about education. Sure, they’re both about how we as teachers can teach our students better. But there’s something fundamentally different about them. And before anyone gets the wrong idea, it’s not because I think Emdin’s book is about culture, politics, and race whereas Barton’s book is about more “neutral” topics. I definitely don’t think that. The cognitive science in Barton’s book can just as much be viewed under the lens of critical theory as anything else, raising questions such as “Who benefits from the research of cognitive science? Where is cognitive science silent and how do its results help perpetuate the status quo?” Both Emdin and Barton talk about politics, it’s just that the cognitive science in Barton’s book is way more subtle about it.

I think what makes them different is that they touch on separate (yet overlapping and interconnected) aspects of who our students are. Barton touches on the side of our students that can be informed by cognitive science research. Students are not “calculating machines”, but there are general patterns that can be gleaned about how people think and what amplifies thinking or inhibits it. Emdin on the other hand is concerned about our students’ membership and participation in cultural, social, and political spaces. Of course, these two aspects of our students are not mutually exclusive. Working memory fails if students are stressed about bleak living conditions and the realities of urban youth. Racially relevant pedagogy can increase engagement and capture students’ attentions, which makes it more likely that knowledge in working memory will transfer to long term memory.

It’s also important to note these two aspects of our students are not complete. There are lots of things to say about our students’ relationship with technology, their use of social media, their dealings with bullying and peer pressure, and their participation in a dozen different discourse communities.

In short, students are multidimensional and teachers get better at teaching when they tend to the various aspects of their students’ being.

I keep thinking about the term intersectionality, which was first given its name by law professor Kimberle Crenshaw. It’s a way of thinking about how interlocking systems of power impact people who are marginalized by society. It considers how class, race, gender, age, disability, and sexual orientation are interwoven in ways that render certain classes of people invisible, even to movements that traditionally serve marginalized populations. The origins of its naming has an interesting backstory, which Crenshaw talks about here. The summary is that in 1976, a group of black women sued GM for discrimination. They couldn’t work on the factory floor because they were women, and they couldn’t get secretarial jobs because they were black. The court dismissed their claim because the court believed that black women should not be permitted to combine their race and gender claims into one. Crenshaw coined the term intersectionality to “highlight the multiple avenues through which racial and gender oppression were experienced so that the problems would be easier to discuss and understand.”

I think there is a kind of “intersectionality” that occurs in education. But it isn’t intersectionality, exactly. It’s more about all the physical, neurological, social, cultural, and political realities and practices through which learning and knowledge is mediated. In one sense, what I’m thinking about generalizes intersectionality by considering aspects of our students that are not necessarily exclusive to marginalized populations, such as cognition and youth culture. In another sense, it narrows intersectionality by focusing on aspects that commonly arise in education. This is not to say that I’m not also thinking about race and gender and sexual orientation, and I’m also not saying we can ignore issues of power and privilege. I’m saying our students are wrapped in layers of interwoven social, cultural, psychological, sociological, economic, and linguistic contexts, and it’s our role as teachers to create conditions where students can learn about/embrace/come to terms with/leverage the many dimensions of themselves.

Practically, this means that as teachers, we ought to be trying to make ourselves as versatile as possible. In my first year of teaching, I labeled myself as “someone who was good at math and read a lot of books on cognitive science.” Sure, I was allowed to have my own style, but I know I limited myself in my ability to connect with my students on multiple levels of their being. The reality is that teachers shouldn’t have a “shtick”, even though I know some of them say they do. I’ve learned that it’s not enough for me to have one or two or even three different ways of forming connections with students.

I used to play lots of D&D (Dungeons and Dragons, hello!) When you start off, you choose a race and class. Regardless of race, I preferred to be a Bard. Bards are part-warrior, part-mage, and part-thief with lots of charisma and musical ability. The common characterization for Bards is that they are “jacks of all trades and masters of none.” I think the best teachers are Bards. But I fear a lot of them want to be mages. That is, they want to be master technicians who are well-versed in pedagogical techniques. I used to think this way and I think it’s how my school views our “best” teachers.

Since then, I’ve learned the value of versatility and the viewpoint that teaching is a situated practice. Lots of it I picked up by listening to people on Twitter. People like Ilana Horn, who say  “teaching problems are locally defined”, have helped me see myself as someone more than a “technician of best practices.” David Coffey talks about trying “to prepare teachers to think like researchers [and] not just follow the research”. llona Vaschchyshyn and Michael Pershan had a nice conversation about how hard it is to resist generalizations and simplifications in education, and that ideas–no matter how well-researched–can’t take the thinking out of teaching.

Having read Barton and Emdin’s books, my goal is to continue to read about, think about, and talk about a wide range of educational perspectives. And of course, to bring them into the classroom, get to know my students better, and create an environment where they can learn about themselves better. Reflect, and then repeat.



Math Venns

To solve a “Math Venn”, one must generate examples for each of the regions in a Venn diagram (or explain why no example exists for that region), where each region is labeled with a math concept or constraint.  I found them through Craig Barton, whose website contains Math Venns created by himself and others.  Here’s a screenshot from the website:


I like the idea of asking students to generate their own examples.  The NCTM calls examples created by students “learner-generated examples” (LGE), stating:

“When students generate their own examples, they behave more like mathematicians, drawing on connections and taking ownership of the concepts. As a result, generating examples can be motivating for students at all levels. Students benefit by becoming better problem solvers and developing a rich array of example types.  Further, when students generate their own examples, they reveal information about their thinking that is not readily available otherwise.”

In the current climate of math pedagogy, students are being exposed to an increasing number of problems with multiple solution paths.  Still, it’s likely (and students probably know it) that their solution is probably shared by someone else.  LGEs have the benefit of increasing the likelihood that what a student comes up with is completely unique.  It’s their example which they’ve created.

I see Math Venns as a great structure to promote LGEs.  They provide just the right amount of constraint but also a good deal of intellectual space to explore.  I wonder if students need a healthy bit of domain specific knowledge before they can truly benefit from engaging with Math Venns; I’d be hesitant to use a Math Venn while introducing a topic.

Here are a few Math Venns I made, along with some discussion about the topics they cover and ways of raising the ceiling.

Graphs of Functions


  • Potential Areas of Discussion: This Math Venn might raise discussions about the definition of a “function”, domain and range, piecewise functions, equations of functions, asymptotes, and end behavior
  • Extensions: If students have already learned them, this could be made more challenging by prohibiting piecewise functions or functions that pass through the origin

Factoring Quadratics


  • Potential Areas of Discussion: This one could raise discussions about factoring quadratics and sums and products of even and odd numbers. It also brings up the question “what counts as factoring?”.  For example, does (x – 2.5)(x + 6) count?
  • Extensions: Change “b is even” to “b is prime”.  I haven’t thought much about it but I’m hoping it yields interesting results.

Continuous vs. Differentiable


  • Potential Areas of Discussion: This one is about the definitions of continuous and differentiable and which implies which (and which doesn’t).  H is particularly interesting because of nowhere differentiable functions.
  • Extensions: I can’t think of an extension here but if you do let me know: @ m3lvinp3ralta



  • Potential Areas of Discussion: This one raises common misconceptions such as students believing that the sum of 1 + 1/2 + 1/3 + … converges just because 1, 1/2, 1/3, … converges.  It also raises the relationship between convergence and boundedness
  • Extensions: I can’t think of an extension here but if you do let me know: @ m3lvinp3ralta

Problem Posing SSDDs

Based on his book “How I Wish I’d Taught Maths”, Craig Barton started a new website about problems that are identical on the surface but have different underlying mathematical structures.  He calls them SSDDs (same surface, different deep structure).

His motivation for calling attention to SSDDs:

By always presenting students with a series of problems that are set in different contexts (i.e. have a different surface structure), but which are all from the same topic (i.e. have the same deep structure), we are robbing students of the opportunity to develop the ability to identify the problem’s deep structure and hence identify the strategy needed to solve the problem.

I love the idea, and the website has taken off in a short period of time.  Having explored it, I can’t help but ask myself: can the idea of presenting identical images be used to promote problem posing?

I’ve been gaining an increasing interest in problem posing over the past year.  Problem posing means shifting control over the problem-generating process from authority figures (me, textbooks, even MTBoS) to students.  The few times I’ve tried it, I’ve seen that it puts the math we’ve been doing in a sharper light for them.  Sometimes a dish tastes better when you’ve cooked it yourself.

Craig Burton’s website gave me the idea to give my students what is essentially a blank SSDD and ask them to create a different question for each image.  I tried creating some images on my own for Algebra 1, and here’s the result:



I haven’t yet given this task to my students but I hope to do so in the near future.  My plan will be to pair them up so they can take advantage of one another’s strengths.  It wouldn’t be too shocking to discover that students were stronger at some topics over others.  I’ll also ask them to answer their own questions.  Maybe it’ll give them a sense of self-efficacy and the realization that they can answer a wide range of their own scholarly questions.  Now that I’m writing this, I realize it’s like Dan Meyer’s Act Ones, except there’s a greater emphasis on comparing and contrasting structures.

Maybe this problem posing version of SSDD can also serve as a type of formative assessment.  Ideally, it’ll root out misconceptions.  But more than that, it’ll show me what topics spring to mind and what topics they only remember through heavier prompting.  Too often assessment is about “do you know” vs. “do you not know”.  Having them craft their own questions might help me (and them) see what they’ve truly attended to.

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.