A Critical Language for Problem Design Saturday, Jan 18 2014 

I am at the Joint Mathematics Meetings this week. I had a conversation yesterday, with Cody L. Patterson, Yvonne Lai, and Aaron Hill, that was very exciting to me. Cody was proposing the development of what he called a “critical language of task design.”

This is an awesome idea.

But first, what does he mean?

He means giving (frankly, catchy) names to important attributes, types, and design principles, of mathematical tasks. I can best elucidate by example. Here are two words that Cody has coined in this connection, along with his definitions and illustrative examples.

Jamming – transitive verb. Posing a mathematical task in which the underlying concepts are essential, but the procedure cannot be used (e.g., due to insufficient information).

Example: you are teaching calculus. Your students have gotten good at differentiating polynomials using the power rule, but you have a sinking suspicion they have forgotten what the derivative is even really about. You give them a table like this

x f(x)
4 16
4.01 16.240901
4.1 18.491

and then ask for a reasonable estimate of f’(4). You are jamming the power rule because you’re giving them a problem that aims at the concept underlying the derivative and that cannot be solved with the power rule.

Thwarting – transitive verb. Posing a mathematical task in which mindless execution of the procedure is possible but likely to lead to a wrong answer.

Example: you are teaching area of simple plane figures. Your students have gotten good at area of parallelogram = base * height but you feel like they’re just going through the motions. You give them this parallelogram:
Thwarting
Of course they all try to find the area by 9\times 41. You are thwarting the thoughtless use of base * height because it gets the wrong answer in this case.

Why am I so into this? These are just two words, naming things that all teachers have probably done in some form or another without their ever having been named. They describe only a very tiny fraction of good tasks. What’s the big deal?

It’s that these words are a tiny beginning. We’re talking about a whole language of task design. I’m imagining having a conversation with a fellow educator, and having access to hundreds of different pedagogically powerful ideas like these, neatly packaged in catchy usable words. “I see you’re thwarting the quadratic formula pretty hard here, so I’m wondering if you want to balance it out with some splitting / smooshing / etc.” (I have no idea what those would mean but you get the idea.)

I have no doubt that a thoughtful, extensive and shared vocabulary of this kind would elevate our profession. It would be a concrete vehicle for the transmission and development of our shared expertise in designing mathematical experiences.

This notion has some antecedents.[1] First, there are the passes at articulating what makes a problem pedagogically valuable. On the math blogosphere, see discussions by Avery Pickford, Breedeen Murray, and Michael Pershan. (Edit 1/21: I knew Dan had one of these too.) I also would like to believe that there is a well-developed discussion on this topic in academic print journals, although I am unaware of it. (A google search turned up this methodologically odd but interesting-seeming article about biomed students. Is it the tip of the iceberg? Is anyone reading this acquainted with the relevant literature?)

Also, I know a few other actual words that fit into the category “specialized vocabulary to discuss math tasks and problems.” I forget where I first ran into the word problematic in this context – possibly in the work of Cathy Twomey-Fosnot and Math in the City – but that’s a great word. It means that the problem feels authentic and vital; the opposite of contrived. I also forget where I first heard the word grabby (synonymous with Pershan’s hooky, and not far from how Dan uses perplexing) to describe a math problem – maybe from the lips of Justin Lanier? But, once you know it it’s pretty indispensible. Jo Boaler, by way of Dan Meyer, has given us the equally indispensable pseudocontext. So, the ball is already rolling.

When Cody shared his ideas, Yvonne and I speculated that the folks responsible for the PCMI problem setsBowen Kerins and Darryl Yong, and their friends at the EDC – have some sort of internal shared vocabulary of problem design, since they are masters. They were giving a talk today, so I went, and asked this question. It wasn’t really the setting to get into it, but superficially it sounded like yes. For starters, the PCMI’s problem sets (if you are not familiar with them, click through the link above – you will not be sorry) all contain problems labeled important, neat and tough. “Important” means accessible, and also at the center of connections to many other problems. Darryl talked about the importance of making sure the “important” problems have a “low threshold, high ceiling” (a phrase I know I’ve heard before – anyone know where that comes from?). He said that Bowen talks about “arcs,” roughly meaning, mathematical themes that run through the problem sets, but I wanted to hear much more about that. Bowen, are you reading this? What else can you tell us?

Most of these words share with Cody’s coinages the quality of being catchy / natural-language-feeling. They are not jargony. In other words, they are inclusive rather than exclusive.[2] It is possible for me to imagine that they could become a shared vocabulary of our whole profession.

So now what I really want to ultimately happen is for a whole bunch of people (Cody, Yvonne, Bowen, you, me…) to put in some serious work and to write a book called A Critical Language for Mathematical Problem Design, that catalogues, organizes and elucidates a large and supple vocabulary to describe the design of mathematical problems and tasks. To get this out of the completely-idle-fantasy stage, can we do a little brainstorming in the comments? Let’s get a proof of concept going. What other concepts for thinking about task design can you describe and (jargonlessly) name?

I’m casting the net wide here. Cody’s “jamming” and “thwarting” are verbs describing ways that problems can interrupt the rote application of methods. “Problematic” and “grabby” are ways of describing desirable features of problems, while “pseudocontext” is a way to describe negative features. Bowen and Darryl’s “important/neat/tough” are ways to conceptualize a problem’s role in a whole problem set / course of instruction. I’m looking for any word that you could use, in any way, when discussing the design of math tasks. Got anything for me?

[1]In fairness, for all I know, somebody has written a book entitled A Critical Language for Mathematical Task Design. I doubt it, but just in case, feel free to get me a copy for my birthday.

[2]I am taking a perhaps-undeserved dig here at a number of in-many-ways-wonderful curriculum and instructional design initiatives that have a lot of rich and deep thought about pedagogy behind them but have really jargony names, such as Understanding by Design and Cognitively Guided Instruction. (To prove that an instructional design paradigm does not have to be jargony, consider Three-Acts.) I feel a bit ungenerous with this criticism, but I can’t completely shake the feeling that jargony names are a kind of exclusion: if you really wanted everybody to use your ideas, you would have given them a name you could imagine everybody saying.

Dispatches from the Learning Lab: Partial Understanding Monday, Apr 30 2012 

So here’s another one that I suppose is kind of obvious, but nonetheless feels like big, important news to me:

It’s possible to only partly understand what somebody else is saying.

Let me be more specific. When you’re explaining something to me, it’s possible for me to get some idea from it in a clear way, to the point where my understanding registers on my face, but nonetheless the other 7 ideas you were describing I have no idea what you’re talking about.

<Example>

I am a 9th grader in your Algebra I class. You’re teaching me about linear functions. You are explaining to the class how to find the y-intercept of a linear function, in slope-intercept form, given that the slope is 4 and the point (6,11) lies on the line. You explain that the equation has the form y=mx+b and that because we know the point (6,11) is on the line, that this point satisfies the equation. Thus you write

11=4\cdot 6+b

on the board. At this point I recognize that we are trying to find b and that we have an easy single-variable linear equation to solve. My face lights up and you take mental note of my engagement. Maybe you even ask for the y-intercept, and since I recognize that this must be b I calculate 11-24 = -13 and raise my hand.

Meanwhile, I have only the vaguest sense of the meaning of the phrase “y-intercept.” I have literally no understanding of why I should expect the equation to have the form y=mx+b. I have a nagging feeling of dissatisfaction ever since you substituted (6,11) into the equation because I thought x and y were supposed to be the variables but now it looks like b is the variable. Most importantly, I do not understand that the presence of the point on the line implies that its coordinates satisfy the equation of the line and conversely, because on a very basic level I don’t understand what the graph of the function is a picture of. This has been bothering me ever since we started the unit, when you had me plug in a bunch of x values into some equations and obtain corresponding y values, graph them, and then draw a solid line connecting the three or four points. Why am I drawing these lines? What are they pictures of?

Occasionally, I’ve asked a question aimed at getting clarity on some of these basic points. “How did you know to put the 6 and 11 into the equation?” But because I can’t be articulate about what I don’t understand, since I don’t understand it, and you can’t hear what I’m missing in my questions because the the theory is complete and whole in your mind, these attempts come to the same unsatisfying conclusion every time. You explain again; I frown; you explain a different way; I say, “I don’t understand.” You, I, and everyone else grow uncomfortable as the impasse continues. Eventually, you offer some thought that has something in it for me to latch onto, just as I latched onto solving for b before. Just to dispel the tension and let you get on with your job, I say, “Ah! Yes, I understand.”

</Example>

This example is my attempt to translate a few experiences I’ve had this semester into the setting of high school. The behavior of the student in that last paragraph was typical of me in these situations, though it would be atypical from a high school student, drawing as it does on the resources of my adulthood and educator background to self-advocate, to tolerate awkwardness, even to be aware that my understanding was incomplete. Still, often enough I ended up copping out as the student does above, understanding one of the 8 things that were going on, and latching onto it just so I could allow myself, the teacher and the class to move on gracefully. Conversations with other students indicated that my sense of incomplete understanding was entirely typical, even if my self-advocacy was not.

The take-home lesson is two-fold. Point one is about the limitations of explaining as a method of teaching. Point two is about the limitations of trusting your students’ (verbal or implied) response to your (verbal or implied) question, Do you understand?

The basic answer (as you can tell from the example) is, No, I don’t.

Now I myself love explaining and have done a great deal of it as a teacher. I fancy myself an extremely clear and articulate explainer. But it couldn’t be more abundantly clear, from this side of the desk, how limited is the experience of being explained to. I mean, actually it’s a great, key, important way to learn, but only in small doses and when I’m ready for it, when the groundwork for what you have to say has been properly set.

I am somewhat chastened by this. I am thinking back self-consciously to times when I’ve explained my students’ ears off rather than, in the immortal words of Shawn Cornally, “lay off and let them fucking think for a second.” It’s like I was too taken with the clarity and beauty of the formulation I was offering, or in too much of a hurry to let them work through what they had to work through, or in all likelihood both, to see that more words weren’t going to do any good. Beyond this, I’m thinking back on the faith I’ve put in my ability to read students’ level of understanding from their faces. I maintain that I’m way better at this than my professors, but I don’t think I’ve had enough respect for how you can understand a small part of something and have that feel like a big enough deal to say, and mean, “Oh I get it.” Or to understand a tiny part of something and use that as cover for not understanding the rest.

Partial Illumination for the Chords-of-an-Ellipse Problem Tuesday, Aug 31 2010 

Back in July, Sam Shah described a beautiful and haunting problem he had encountered at PCMI:


Put n evenly spaced points on a unit circle, with one point at (1,0). Then draw chords from this point to all the other points. Then multiply the chords’ lengths. What do you get?

Mimi Yang drew some great illustrations of the problem, and Mr. Ho made a very slick geogebra applet that illustrates and gets data at the same time.


Sam mentioned an extension problem:

Scale the circle vertically by a factor of \sqrt{5}. Scale all the chords too. What is the product of the lengths now?

Mr. Ho outdid himself by creating a geogebra applet for this one too. In fact, he lets you scale the ellipse in all kinds of ways.

It’s really this extension that’s the subject of this post. But first…

Background on the original problem

If you haven’t bumped into this problem yet and it’s not obvious to you how to solve it, you might want to experiment a bit, if nothing else then with Mr. Ho’s applets. I am going to talk about the problem’s results, which are awesome, so I’m warning you now that if the problem is new to you but you read on, I’m going to steal some of your fun.

People posted solutions in the comments to Sam’s original problem statement. The solutions fell into 2 categories:

1) Direct calculation of the product using trigonometry. These methods were able to produce a surprising conjecture about the product of lengths for n points, but were not able to prove it.

2) Recognition that if the points are treated as numbers in the complex plane, they are precisely the nth roots of unity, followed by exploitation of the algebra of the nth roots of unity. These methods were able to prove the conjecture. (For example: Andrew’s comment at Sam’s original post; gasstationwithoutpumps has his/her (?) own post on the subject.)

This is as far as I can get before stating the result; so – SPOILER ALERT.

For n points on the circle, the product of all these chords, most of which have irrational lengths, is a positive integer. Not any positive integer, but n itself.

Here is the roots-of-unity proof, which I’ve done my best to render in a way that’s accessible even if you’ve never worked with roots of unity before. You do need to be familiar with the geometry of complex numbers though.

If you interpret the n equally spaced points on the circle as complex numbers, then they are precisely the nth roots of 1: \omega, \omega^2, \dots, \omega^{n-1} where \omega is the first of the numbers you find if you go around the circle counterclockwise after you leave the positive real axis. (Think about what happens when you multiply this number by itself n times, to convince yourself of this.) Since the point from which all the chords emanate is the number 1, the product of the lengths is the absolute value of the product of the complex numbers (1-\omega), (1-\omega^2), \dots, (1-\omega^{n-1}). This product has a startlingly elegant form, which can be seen by noticing that this product is exactly the product (z-\omega)(z-\omega^2) \dots (z-\omega^{n-1}) evaluated at z=1; and that this product is exactly the monic polynomial that has \omega, \omega^2, \dots, \omega^{n-1} as roots. What are they roots of? Oh right, unity. In other words, they are all zeros of the polynomial z^n-1. This polynomial also has 1 itself as a root (because 1 is an nth root of 1 too), so you have to divide it by a factor of z-1 in order to get the polynomial that has only the omegas as roots.

z^n-1=(z-1)(z-\omega)(z-\omega^2)\dots(z-\omega^{n-1})

(z-\omega)(z-\omega^2)\dots(z-\omega^{n-1})=\frac{z^n-1}{z-1}=z^{n-1}+\cdots+z+1

And now, evaluation of this at z=1 gives you 1+1+…+1 = n. The product of the chord lengths is the absolute value of this, but this is a positive real number so its absolute value is itself. Thus the product of the chord lengths is n. QED.

The ellipse extension

So everybody who produced a proof to the original problem did it by using the algebra of the nth roots of unity. The first thing that grabbed me about the ellipse extension is that while it is obviously closely related to the original problem, it immediately destroys the linchpin of this method. As soon as the circle gets stretched up to the ellipse, the points are no longer nth roots of unity!

The next thing that grabbed me about it was Tom’s comment on Sam’s original post:

I created my own Geogebra applet to investigate the problem with the ellipse (as you have it written I think). The products turn out to be very interesting:

The lengths of the cords are as follows for certain n:

n length
2 2=2*1
3 6=3*2
4 12=4*3
5 25= 5*5
6 48=6*8
7 91=7*13
8 168= 8*21
9 306=9*34
10 550=10*55

So the conjecture would then be

n n*F_n

where F_n is the n’th Fibonacci number.

WHAT???

(As an aside, the type of reaction I had to Tom’s comment is something we should be cultivating in students. Observation of an unexpected pattern leads naturally to a feverish search for an explanation. This is why it’s so important not to treat a pattern noticed as an established fact: this kills the students’ natural wonderment about the why of the pattern.)

Mr. Ho confirmed Tom’s calculations with his own applet. He had a look at other scale factors besides \sqrt{5} but didn’t find anything. At this point, the problem officially had itself ensconced semi-permanently in the back of my brain. How (the hell) are the frickin Fibonacci numbers arising? If you know something about the Fibonacci numbers, you know that they’re related to the golden ratio, which is related to \sqrt{5}, so that kind of makes sense, but c’mon now, only kind of. It’s not like the connection is jumping out at me.

Still, I couldn’t justify spending a lot of time on the problem. I try to stick with math problems that fit into a program of study I’ve given myself, and this one didn’t obviously do that. Nonetheless, it was lodged in my head firmly enough for me to describe it to my excellent colleague Japheth Wood (News from the Math Wizard). The next day, he sent me the following in an email:

Update on the stretched diagonals problem. I wrote a short program in
Python to do some calculations, and I found out the following
patterns, when sqrt{5} is replaced by sqrt{4a+1}:

Conjecture: The product of the diagonals of the stretched n-gon is P_n
= n*T_n, where T_n is an integer sequence defined by:
T_1 = T_2 = 1 and T_{n+2} = T_{n+1} + a*T_n
* This is true in the case of the non-stretched circle: a = 0, and
this recurrence gives P_n = n * 1, which is known.
* This seems true in the PCMI case. 5 = 4*1 + 1, so a = 1, and the
recurrence is Fibonacci.
* The other cases where a is an integer seem true by data I’ve collected.
* The conjecture even seems true when a is not an integer.

WHAT??!!?

Okay, now the time had obviously come for me to give this problem some serious love. Japheth had just a) exploded onto a completely new level my sense that there’s a lot going on here, and b) given me enough of a direction that I knew some real elbow grease would pay off. I spent quite a few hours this weekend with the problem. What I found is still nowhere near a proof, but it strengthens Japheth’s conjecture and suggests some lines for further investigation. Here’s what I did:

First I found some closed forms for Japheth’s recursive sequence T_n:

Closed form #1: Using a standard method involving encoding the recursion T_{n+2}=T_{n+1}+aT_n into a matrix and then diagonalizing the matrix (which I learned – like all the linear algebra I know – from Michael Artin’s Algebra, chapters 3 and 4), I found that

T_n=\frac{u^n-v^n}{u-v}

where u and v are the two roots of the polynomial x^2-x-a. I didn’t end up using this result but I think it’s very pretty. I’m sure it’s well-known but it’s new to me.

Closed form #2: I got a different formula by directly calculating the first few values of T_n in terms of a, and looking for patterns:

T_1=1
T_2=1
T_3=1+a
T_4=1+2a
T_5=1+3a+a^2

I put this in a table for ease of calculation and pattern-searching:

n T_n= 1 a a^2 a^3 a^4
1 1
2 1
3 1 1
4 1 2
5 1 3 1
6 1 4 3
7 1 5 6 1
8 1 6 10 4
9 1 7 15 10 1

This is just a downward-slanted Pascal’s triangle! (Look at what the recursion T_{n+2}=T_{n+1}+aT_n does to a pair of rows to get the next row; this tells you why.) It follows that

T_n=\displaystyle\sum_{k=0}^{\left \lfloor (n-1)/2 \right \rfloor} \binom{n-k-1}{k} a^k

The next thing I did was to reason as follows: in the original circle problem, all the power of the roots-of-unity method came from knowing the polynomial x^n-1 that has these numbers as roots. Now that the circle has been stretched to an ellipse, the points no longer represent roots of unity; but if I could find the polynomial that had these numbers as roots…

I assumed that a would be such that \sqrt{4a+1} would be a real number, so that using it as the vertical stretch factor would make geometric sense. I put no other restrictions on a. Then I just started calculating:

n=2: The points are at 1 and -1; no change when you stretch. The polynomial is g(z)=(z-1)(z+1) = z^2-1.

n=3: The points are at 1 and \frac{-1 \pm \sqrt{3} \mathrm{i}}{2}. Vertical stretching by a factor of \sqrt{4a+1} makes the latter two into \frac{-1 \pm \sqrt{4a+1}\sqrt{3} \mathrm{i}}{2}. So, multiply:

g(z)=(z-1) \left ( z- \left ( \frac{-1 + \sqrt{4a+1}\sqrt{3} \mathrm{i}}{2} \right ) \right ) \left ( z- \left ( \frac{-1 - \sqrt{4a+1}\sqrt{3} \mathrm{i}}{2} \right ) \right )

When all is said and done, this simplifies to

g(z) = z^3 + 3az - (3a+1)

And so on. I made it to n=8. Let me tell you, n=7 was a dirty job. I looked for ways to make things easier – for example, I wrote the points as \cos(2\pi k/n)+\mathrm{i}\sqrt{4a+1}\sin(2\pi k/n) so that I could take advantage of some symmetry, cross-cancellation and trig identities, and for n=7 I found the polynomial that has \cos(2\pi k/7) as roots, which then gave me some symmetric expressions in the \cos(2\pi k/7)s that showed up in the final product, without needing to actually calculate the value of each \cos(2\pi k/7). But still. That case alone had to have taken me at least 2 hours. I messed up several times, thankfully in ways that contradicted each other. But I got my data:

n g(z)
1 z-1
2 z^2-1
3 z^3+3az-(3a+1)
4 z^4+4az^2-(4a+1)
5 z^5+5az^3+5a^2z-(5a^2+5a+1)
6 z^6+6az^4+9a^2z^2-(9a^2+6a+1)
7 z^7+7az^5+14a^2z^3+7a^3z-(7a^3+14a^2+7a+1)
8 z^8+8az^6+20a^2z^4+16a^3z^2-(16a^3+20a^2+8a+1)

Clear patterns are emerging in the structure of these polynomials, what with how
a) the powers of z are falling by 2 and of a are rising by 1
b) the constant term is the negative of the sum of the coefficients of the other terms

Together, these mean we could really regard the whole thing as a sum of terms of the form

\mathrm{coefficient}*a^k(z^{n-2k}-1)

The last thing to do is figure out what’s going on with the coefficients. The lead coefficient is always 1, the next (provided it contains a z) is counting numbers, and the next (again, provided z is involved) – the sequence 5, 9, 14, 20 – is increasing by counting numbers. This is an echo of the Pascal’s Triangle, so I looked for an expression for these numbers in terms of chooses. I found one: the k+1th coefficient of the nth polynomial (i.e. the coefficient of the a^kz^{n-2k} term) seems to be

\dbinom{n-k+1}{k} - \dbinom{n-k-1}{k-2}

So, without further ado –

Conjecture:

* If you take the unit circle and place n evenly-spaced points along it, the first at (1,0)
* And you then scale everything vertically by a factor of \sqrt{4a+1}, where a is any real number such that \sqrt{4a+1} is real
* And you then regard each of these scaled points as a complex number,
* The monic polynomial g(z) with these numbers as roots will be equal to

\displaystyle\sum_{k=0}^{\left \lfloor (n-1)/2 \right \rfloor} \left [ \dbinom{n-k+1}{k} - \dbinom{n-k-1}{k-2} \right ] a^k (z^{n-2k} - 1)

If this conjecture is correct, Japheth’s conjecture follows. (For those of you who like that sort of thing, how does it follow?) So proving this conjecture would settle all the outstanding questions (e.g. it would prove the Fibonacci pattern in the \sqrt{4a+1}=\sqrt{5} case).

But I have no idea how to prove it. Why are the powers of a showing up? How about the chooses? What’s going on here?

Thoughts?

(Darryl Young and Bowen Kerins, the writers of the PCMI problem set, are invited to give the rest of us hints!)

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