Mathographies

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.”
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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.

sq1

sq2

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. 

sq3

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?

sq35

sq4

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.

sq5

sq6

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.

How I Taught Domain and Range This Year

I’ve plotted a point, and now my x-collection is 2.

dr1

Now what’s my x-collection? // -3

dr2

What’s my x-collection now?  // Cool discussion on the nature of a line as an infinite collection of points. Students expressed answer as all the values between -3 and 2, including -3 and 2. I suggested the notation -3 << 2.

dr3

How about now? // No change in the x-collection. But the “y-collection” (their words) changed.

dr4

How about now? // Again, no change in the x-collection

dr5

How about now?  // Students added 4 to the x-collection, alongside -3 << 2

dr6

From there, students learned that “x-collection” is formally called domain. Students were able to extend their thinking to the “y-collection”, aka range without much assistance.

This lesson allowed me to change things up on the fly based on my read on the students. Too simple? Introduce open circles and strange twisty curves. Too complex? Stick with straight lines.

Piecewise functions made their first appearance immediately. This isn’t usually the case. But I think students were able to reason about them successfully because the lesson was grounded in the already familiar concept of plotting points. This is in contrast to some externally imposed technique such as “take a ruler and hold it parallel to the y-axis; move the ruler right until the graph stops”.

Powers of Two

If you add the successive powers of two (starting at 2^{0}), you get one less than the next power of two. This is usually written as 1 + 2 + 2^{2} + ... + 2^{n-1} = 2^{n} - 1.

The first time I saw this, I thought to myself “Whoa, why does this happen?!” Better yet, I wanted to know “What does this idea look like?”

 

The picture below is usually used in a visual proof to show that \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \frac{1}{2^{4}} + ... converges to 1. The idea is that every cut produces two smaller, identical rectangles, and one of these rectangles can be cut into two even smaller, identical rectangles. Beginning with \frac{1}{2}, you add on \frac{1}{2} of a \frac{1}{2}, and then \frac{1}{2} of that \frac{1}{2} of a \frac{1}{2}, and so on. This can go on forever without ever exceeding the area of the square, which is equal to 1. Neat!

square1

Let’s look at this picture differently and ask a different question about it.

square2

Question 1: How many of the blue rectangle fits inside the bolded area?

Answer 1: If the blue rectangle is \frac{1}{2^{n}} of the entire square, then by definition there are 2^{n} blue rectangles in the entire square and 2^{n} - 1 blue rectangles in the bolded area. Done!

Question 2: How many of the blue rectangle fits inside the bolded area?

Answer 2: One blue rectangle fits inside one blue rectangle. Two blue rectangles fit inside the next largest. Four blue rectangles fit inside the next-next largest rectangle…

This keeps going until you ask, “How many blue rectangles fit inside the \frac{1}{2} piece?” Since 2^{n} blue rectangles fit inside the whole square, we can say that half as many fit inside the \frac{1}{2} piece. So 2^{n - 1} blue rectangles.

square3

Let’s add up all the blue rectangles we’ve counted in each of the highlighted rectangles above:

1 + 2 + 4 + ... + 2^{n - 1}

The same counting question (how many blue squares fit inside the bold area?) can be answered in two different ways. So although each answer yields a different expression, both expressions must be equal.

square4

Neat!