# Monthly Archives: July 2013

(So I have started getting in the weird habit of going on runs to process big events and also to train for the Boston half marathon this fall. Not quite this fast,  but just to allow hard-to-process ideas to come into focus a little more. TMC 13 required a 14-miler, which worked out well since I missed my long run while in Philly.)

I decided to sign up for Twitter Math Camp when I was in a talk by Sam Shah about the MathTwitterBlogosphere almost exactly a year ago. The number of ways that hour changed my life and first year of teaching would be hard to enumerate exactly. All I know is that I could not wait for Twitter Math Camp.

The session that got me thinking most about my classroom was Dan Goldner’s session on structuring a problem-based curriculum. While that phrase means all sorts of things that can be hard to pin down, Chris put it best:

One of the best things about Dan’s session was that he ran it in a “problem-based” way. From the materials here, he generated an incredibly rich discussion, all while talking less than 10% of the time to the entire group (and not much more to smaller groups), simultaneously being super approachable.

The gist of the exercise was to take a selection of problem-based courses, map certain decisions to values, and then use that to map our own values back to concrete decisions about how we structure our classrooms. This is easier said than done, but I think we made a good start, and it generated a TON of questions that I need to answer (or at least start working on). Some of them are more general and perhaps obvious, some are specific to my teaching situation. Here they are:

1. Where do I get problems from? How do the sources/types of problems I use reflect my ideas about what is important? For example, classrooms run on Exeter problems versus Park Math problems send very different messages to students about your ideas of how mathematics is held together, even though both of those curricula are very similar media-wise. If I believe in the personal nature of mathematics, should I be writing a lot of the problems myself, and should I decide on problems way ahead of time or should I pick problems in response to my students’ learning?
2. How does this tie into assessment? Last year I ran a Dan Meyeresque 4-point SBG scale on approximately 20 standards per semester, using maxima for grade calculations and with 1-3 summative exams per semester which also counted for standards. Due to being a newbie, being lazy/pressed for time, and the culture of my school, all formal assessment on standards was in a traditional, individual, timed setting (but with extra time always offered at lunch). Does a problem done alone deserve a higher level of mastery? Maybe, but that would disincentivize collaboration, not a value I want to stifle.
3. (How) can I hybridize problem-based curriculum ideas into my current teaching situation, if internal or external factors don’t allow for a switch to a completely problem-based curriculum? My school is not run like a traditional high school, but more like a traditional small college. Most English classes are seminar style, most science classes are lecture plus lab, and many math classes are (to be fair: well-run, thoughtful, interactive) lecture with some work time thrown in at the younger grades. I also believe that there are important mathematical ideas that don’t necessarily lend themselves to solving problems and mathematical activity that does not fall under the problem-solving header. For example: discussions of infinity, making mathematical art, coming up with conjectures. I also believe there is a (quite small) place for lecture, just like any music student should spend most of their time playing music, but benefits from listening to performances.
4. How do I adapt PBC ideas to match the pace asked of me at my school? As a Texas native, my first exposure to proof-based mathematics was a Moore method Calculus course at a summer program, in which the entire course consists of students working through a list of theorems (called the script) with minimal help, class time dedicated almost completely to presentation and critique. In addition to not being for the faint of heart, it moves very slowly content-wise.
5. How do I help students transitioning from a very traditional math classroom to a problem-based learning classroom? My school has an application process that involves a test, an application, and an interview, so we get a student body who are mostly very good at school in the big scheme of things. That said, there’s a wide range of study skills, prior training, parental support, and ideas of what school looks like in our incoming freshman class, especially since we have ninth graders who come from literally dozens of high schools. This means that I am not only tasked with helping students adjust to a class structure that will be new to (almost) all of them, they will be coming from vastly different start points, in terms of how much responsibility and genuine mathematical work was expected of them. How do I help them transition to my classroom (and implicitly) my values while respecting their prior experiences, which may have been good or not so good?

Perhaps these are all obvious questions to the non-newbie, but this list is what I want at the front of my mind as I start picking problems and designing expectations for the fall.

I’m teaching the two sections of honors 9th grade math at my school next year, and since we want to allow students to flow between the honors and non-honors sections at the end of any semester in the first two years, I think adopting some of these problem based learning structures would be a good way to challenge my group by pushing them in weird and interesting directions instead of just pushing them “ahead.”

There were numerous other completely awesome things that happened at TMC, but this is the one I had to write down/write about immediatiely. There’s already a Global Math Department session coming up on September 10, dealing with Problem Based Course design and run by Dan Goldner, which I’m looking forward to, but in the mean time, I want to think about how I can implement some of these ideas going into this year.

Any advice, resources, or further questions to add to this list are most welcome in the comments! And if anyone is interested in thinking through these questions together (especially anyone with freshman experience, eep!) hit me up @compactspaces or dprice at bhsec dot bard dot edu.

(This is the fourth and final “official” post as part of Justin Lanier’s smOOC, some short thoughts as I struggle to process a reading.)

Yesterday I had the chance to see a strange movie about mathematics and mathematicians called Colors of Math.

One quote in particular stood out, from a Russian mathematician named Maxim Kontsevich.

At every level, even an abstract theory is some sort of space. Every notion and every action is about geometric objects.

This was at the front of my mind reading through the Seymour Papert article “An Exploration in the Space of Mathematical Educations” (link). The article had a lot of things I didn’t quite buy (high conclusion to evidence ratio) and a lot of ideas I didn’t quite understand, but I did like the idea of the space of possible math educations as a mathematical object itself, and here’s the primary reason.

Suppose we associate with the space ME a function called A(x), which takes a education x in the space ME and returns the awesomeness of that mathematical education.

Is this function even continuous, or are there points where an arbitrarily tiny change effects a qualitatively different result? If I am teacher 1 and you are teacher 2, is $\int_{ME}(A_1-A_2)$ a measure of how much we differ in our views?

Papert’s definition of the space ME is vague enough in the article that I’m not sure where I would sit, but the idea of ME as a mathematical space makes me think of local maxima versus global maxima. A local maximum of A in the space ME is a teaching and education style that is more awesome than all the other very slight variations that are near it in the space; finding one of these maxima seems hard enough.

But simply being at a local maximum means nothing about the existence, nearness, or value of A(x) at other maxima. Does this mean that the benefit of activities like this smOOC, like reading blogs or more “serious research” is to ensure we can better understand ME, and thus move more quickly (even discontinuously) through it in our own teaching? Alone, we would be stuck with finding maxima on our own; we could still definitely improve but the success of our algorithms would rely so heavily on our starting points.

(This post is part of the Math Munch Exploration writing assignment for Justin Lanier’s smOOC. Read more here. If you are here and somehow do not know of Math Munch, you should probably cancel all plans for the next few hours and go here.)

This post is about a piece of mathematics I really love, how I shared it in my classroom, and how it went wrong.

One of my favorite mathematical objects is the Koch Snowflake.  It is one of the first fractals discovered and actually predates the word “fractal” significantly. Start with an equilateral triangle and remove the middle third of each side. Stick an equilateral triangle one third as long onto the big triangle. Now you have a six-pointed star made out of 12 line segments, each a third as long as a side of the original triangle. Now repeat this process (remove middle third, add smaller triangle) for each segment.

Iterations 0-3 of the Koch snowflake.

Now imagine the perimeter of the original triangle is three (side length one). This means that the perimeter of the six-pointed star is four, and each remove-and-add-triangle iteration multiplies the perimeter by 4/3.

So here’s the weird thing. The Koch Snowflake is not any one particular step in this process. It’s the limit of this process, which means that, since the perimeter is being multiplied by 4/3 at each step, the Koch Snowflake actually has infinite perimeter, even though it has finite area! (Fun fact: no step in the Koch snowflake process can tile the plane except the equilateral triangle, but the snowflake itself can, using two different sizes.)

But wait! There’s more! Let’s think about what it means to be one-dimensional; if an object is one dimensional, say, a line segment of length one, it means that a scale factor of three increases the object’s “size” by a factor of three. If an object is two-dimensional, say, a square of side length one, it means that scaling the object by a factor of three increases the object’s “size” by a factor of 9, since 9 is three to the second power. If an object, say, a cube of side length one, is three-dimensional, it means that scaling the object by a factor of three increases the object’s “size” by a factor of 27, since 27 is three the third power.

Stages 1-5 of the Koch curve.

Now, let’s think about one “side” of the snowflake, sometimes called the Koch curve, starting with a single side of the original equilateral triangle. Let’s call that single segment Stage 0.  Now think about the four pieces of the Koch curve (which start life as mere segments in Stage 1). Each segment is not similar to the Stage 1 curve overall, since each segment is straight and Stage 1 is composed of four non-collinear segments. However, the Koch curve itself is “infinitely detailed,” so each of the four pieces is actually a 1/3-scale copy of the whole curve. In other words, scaling up a copy of the Koch curve by a factor of three increases its “size” by a factor of FOUR. This means that, since four is greater than three but less than nine, the Koch curve is more than one-dimensional, but less than two-dimensional! Weird. This was one of the first examples of what is called fractional dimension.

(This is an abstract version of the “Coastline Problem,” which you can read more about on Math Munch here, the gist being that the length of a coastline, due to its fractal nature, depends heavily on the scale at which it is measured. )

I actually got the chance to bring this up in one my classes this past school year. We were studying exponential growth and decay, having gotten to the idea that a simple exponential function f(x) with power x is increasing if the base is greater than one and decreasing if the base is less than one. I had some extra time and decided to try out looking at the Koch curve with one of my classes.

Everything was going well. Students whiteboarded the first few stages in groups and the whiteboards were great, since they allowed easy erasing of the middle of each segment. Many students were quick to figure out that the Koch curve has 4^n segments at the nth stage, and that each segment is of length (1/3)^n at the nth stage, resulting in a total perimeter of (4/3)^n at the nth stage. Everything was awesome; we were doing some real mathematics and hitting up some necessary skills at the same time. Students got into making their drawing as precise as possible and a few groups even made it to stage 5 of the curve.

But then the questions and comments started coming:

“Why are we doing this?”

“Is this a separate standard or part of the exponential functions standard?”

“I hate drawing. I wish we were doing regular math.” (No kidding)

I slowly started to lose my audience, but I dragged them through the discussion, made the mistake of trying to share the infinite perimeter idea by just telling them, and we never really came back to it.

What went wrong? I know the content cold, definitely brought my own enthusiasm for the subject, and I think that playing with the Koch curve has a lot of promise as a classroom activity that builds basic skills while also exploring something weird and cool. There were definitely some small rookie mistakes I made in terms of execution, being a first year teacher, but there are some bigger issues with my first-year classroom that this episode illustrated for me.

1. No matter how much I like something, that doesn’t make kids like it. Enthusiasm is a necessary but not a sufficient condition.

2. I love and value recreational/weird/off-the-beaten-track mathematics, but one of my biggest obstacles at introducing more of that sort of math into the classroom might very well be my own students’ opinions about what math is and what it is not.

3. What sort of message does it send to students when we only do “fun math” when there’s some extra time left over in one period due to a fire drill in the other class? Students are not going to shift gears suddenly, so exploring ideas like the Koch curve really has to be a regular part of class if it’s going to be included at all.

These all seem pretty obvious even as I write them now , but I was just so surprised by how the whole activity fell apart at the end.

One of my many resolutions for my second year of teaching is to work at introducing topics like the Koch curve, but without the assumptions I had the first time. My kids didn’t like it in some cases and didn’t see it as “real math” in many cases; this is not an excuse for dropping it from my teaching, but rather the reason why I should give them more experiences that are confusing, artistic, and weird.

(This is a proof for part of a problem given out as part of Justin Lanier’s smooc. For my fellow smoocsketeers, spoilers below.)

From the problem set:

Let’s call a number n “nice” if it can be expressed as a sum of two or more consecutive positive integers.
For example, the expressions 5 = 2+3 and 6 = 1+2+3 show that 5 and 6 are nice numbers.
Which numbers are nice? Justify your answer.

We claim that a positive integer is nice if and only if it is not a power of 2. John Burk has an excellent post explaining why powers of 2 cannot be nice here. We prove here that if a positive integer is not a power  of 2, then it must be nice.

Proof: Let n be a positive integer that is not a power of 2. This means that $n = 2^k\cdot m$, where k is some nonnegative integer and m is an odd integer greater than 1. Note that m/2 cannot be equal to $2^k$ since m/2 is not even an integer.

Case 1: $m/2 > 2^k$.

Then n is nice since we can make n by adding up the $2^{k+1}$ consecutive positive integers which are centered at m/2. In other words,

$(m/2+1/2-2^k) + (m/2+1/2-(2^k-1)) + \cdots +(m/2-1/2)+(m/2+1/2)+\cdots +(m/2-1/2+2^k)= n.$

Case 2: $m/2 < 2^k$.

Then n is again nice since we can add up the m consecutive positive integers centered at $2^k$. In other words,

$(2^k-(m/2-1/2)) + \cdots + 2^k+ \cdots (2^k+(m/2-1/2)) = n$.

QED.

Here’s the fun part to think back on. HOW did we use the fact that we had an odd factor? Well, in both cases we needed m/2 to not be an integer.

How did we use the fact that we factored n into it’s “even” part $2^k$ and its odd factor. Did we ever use the fact that m was the largest odd factor of n? NO.

From the problem set:

Some numbers are “very nice”, in the sense that they are nice in more than one way.
For example, 15 is very nice because 15 = 1+2+3+4+5 = 4+5+6 = 7+8.
Which numbers are very nice? Explain.

We claim now that numbers which have more than one odd factor greater than one are “very nice.” We prove this by modifying the previous proof.

Let n be a positive integer that is not a power of 2. This means that n = ab for some integer a that is either 1 or even and some odd integer b > 1.

Case 1: b/2 > a.

Then we construct a sum as in Case 1 of the previous proof. This sum has 2a terms, centered at b/2.

Same deal with Case 2: b/2 < a. This sum has b terms, centered at a.

Now, if n has more than one odd factor > 1, that’s like saying there’s more than one choice for b. It remains only to show that each choice of b yields a different sum.

Say we wanted to show that the sum formed by using b and the sum formed by using some other b’ are different. If they both fall into Case 1 the two sums have a different number of terms since the corresponding a = n/b and a’ = n/b’ will be different, resulting in sums of length 2a versus 2a’. If they both fall into Case 2, the two sums again have a different number of terms since they have b and b’ terms, and we assumed b was not equal to b’. If one falls into Case 1 and the other into Case 2, the two sums yet again have a different number of terms, this time since one sum has an odd number of terms and the other has an even number of terms.

Thus we have shown not only that the “very nice” numbers are those with more than one odd factor greater than one, but that the number of ways to express any positive integer as the sum of consecutive positive integers is at least equal to the number of odd factors > 1 of that integer.

For more thoughts on very nice numbers, see this.

(This is the second writing assignment for Justin Lanier’s smooc)

Preamble: So one of the great (but difficult!) parts of being part of Justin’s smooc is that, while there are assignments and structures, it’s pretty self-directed, with lots of options. This post wanders a bit, since it’s mostly intended to help me process some of the ideas we’ve encountered and discussed so far. Here goes.

Yesterday, a sad analogy popped into my head:

“school math”

math

Why exactly is Episode IV (the original Star Wars) so awesome?

• It just drops you into this awesome universe!
• It’s part of a fully-fledged universe, but doesn’t explain everything to you. You have to glean some information from the context of the story (like any well-written story).
• It fills you with a sense of wonder. You know it does.
• It’s kind of charmingly imperfect and messy.
• Yeah, there’s a story, and the plot is awesome, but the thing that makes the original Star Wars so excellent is the feeling of myth. (It’s such a strong example of the hero cycle that the reprint of The Hero with a Thousand Faces actually has Luke Skywalker on the cover.)

Why exactly is Episode I so lame?

• It’s not a very good story in itself. So much of the plot is just to set up what is going to happen in the original movies.
• In my opinion, the extra movies make the Star Wars universe seem smaller, not bigger. They force the whole, overgrown universe into an annoyingly tidy, linearized package.
• You’re told pretty much everything. How the Force works. Everyone’s motives for all their actions. BORRRRING. Every character is basically a wind-up toy. It’s too polished and everything fits together too well.

Now imagine you had NEVER seen Star Wars and were taken to see Episode I. Is it any surprise then that so many people are turned off by their image of math? UGH. You are told to sit through all this exposition so you’ll understand the really good stuff better, and this exposition has a bunch of totally useless fluff padded in. No matter how great the old Star Wars movies are, you’re probably not going to get excited about them.

This is why it’s so important to see at least some of the “amazing stuff” first. As part of the smooc, we’ve been thinking back on our own mathematical experiences, and while in my short mathematical life there’s been plenty of sometimes boring and sometimes necessary work to get through, many of the big motivating moments in my mathematical life have been “drops,” where a teacher, a friend, or a lecturer gave to me an idea that was weird, “over my head,” and amazing.

Here’s an example.

My first big experience doing math with other people outside of school came when I got into a residential math camp in 7th grade. This was life-changing for a number of reasons, but the first was a lunch over spring break that students and their parents were invited to, where we would work on problems, get to know the staff and other prospective students, and see a guest lecturer.

This lecturer was a middle-aged man, a professor at a university, but he wasn’t talking about math it seemed. He was talking about balls, and donuts, and coffee cups, and squishing coffee cups until they looked like donuts. He used a big word that sounded like he was talking about maps.

And then he tied his ankles together and flipped his pants inside out.

About half the parents stared in silence while the other half laughed and clapped (I’m proud to count my dad among the latter). And that was it!

(I found out much later that this guy was Michael Starbird, a math professor at the University of Texas known for writing a book called The Heart of Mathematics, which the American Mathematical Monthly called the best book for non-mathematicians it had ever reviewed. See this link for a video of a kid flipping his pants!)

These are silly analogies and examples perhaps, but they speak to an incredibly important part of my mathematical experience. I don’t believe that mathematics is a narrow and linear path you can just follow. That will cheapen “the beginning” and put “the end” so far away most people will never reach it. Of course there are some facts you have to know before you can fully understand other facts, and there are skills you need to have before you can learn other skills, but to me, this needs to happen along with the understanding of just how wide the mathematical world it is.

Another short story:

Last summer I got the chance to work with an amazing teacher as a TA for a summer program a lot like the one I went to. He said one time that it’s best to hand little “gifts” to students every now and then, amazing simple mathematical ideas like how to compute the 46th digit of 1/47 without dividing 1 by 47 to 46 decimal paces. I countered that students should be made to figure things out for themselves (fancying myself a hardcore constructivist), and he said something along the following lines: “Yes, just like you should earn most of what you have. But that doesn’t take away from the pleasure of a gift.” And of course, an unwrapped gift is not nearly as fun; a gift is that much better when it’s wrapped and you have to figure out how to unwrap it.

This pants trick was just such a gift. It was another eight years before I “learned topology.” Maybe this is self-centered, but I think this is the way that math should feel; that surprise, without explanation, helped to make me feel like I was right at the edge of some larger story (and I was).

(Note to self: Terence Tao in this reading for our smooc  seems to settle on a quality of good mathematics as a piece of “mathematics that is “on to something”, that it is a piece of a larger puzzle waiting to be explored further.” In other words, good mathematics strikes a balance between answers and questions. Is this related? He’s talking about the mathematical research carried out in universities, but criterion seems similar.)

I have been lucky that much of my mathematical experience has been old Star Wars and not new Star Wars. There are all sorts of other benefits of a good math education, but already specific facts (even at my young age) are leaking out of my brain at a faster rate than they used to. I always valued my problem solving skills and content knowledge (perhaps overvalued that), but only recently do I find myself valuing more and more that feeling of getting to witness and to be a very small part of something very big and nonlinear; I find myself valuing the aspect of wonder, the aspect of myth.