# Mathographies

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

# Joyous Math

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…

(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.

# How I Taught Domain and Range This Year

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

Now what’s my x-collection? // -3

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.

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

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

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

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”.

# Desmos and Visual Patterns

Combining Desmos and visual patterns opens up a world of possibilities.  Here’s my first crack at it: what is the 43rd step of the 43rd step?

https://www.desmos.com/calculator/zs6kouknvk

# 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!

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

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.

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.

Neat!