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Speaking Mathematically

David Pimm:

We name things for reference, and hopefully for ease of reference, to draw attention to the thing named. But naming also classifies and hence causes us to look at the named thing in particular ways, the chosen symbol stressing some and ignoring other attributes of the named object. Naming something gives us power over it, particularly in algebra, as we can transform and combine expressions involving the unknown – to find out more about it (p. 127).

This is the strongest case for algebra. Your ability to speak, think, and use variable notation makes you powerful – particularly when you interact with computers. But how often do students think of variables in math class and feel powerful? Those experiences aren't simple to devise.

I read Pimm's excellent book over the holiday in preparation for my dissertation proposal. I've pulled out several pages worth of quotes and supplemented them with a) my analysis and b) some details about my upcoming study. Comments are turned on in the Google doc, so let's talk about it.

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David Lloyd:

Perhaps using words as the descriptor ("number of songs") instead of using X (as in "Let x = the number of songs on Dave's iPod") would be a step in the right direction? The same level of rigor without the confusion of what X equals.


How often do students name their own variables?

Dina Strasser isn't and wishes she was:

In June of this year, I turned down the most prestigious scholarship for doctoral work that my local, nationally recognized university had to offer. It was as generous as you could hope for: full tuition, opportunities for stipends and grants. The gracious professors there, and others who helped me with my applications, spent hours of their own time walking me through the process, writing recommendations; they said, to wit, you were born to be a Ph.D. And I knew it, because I had figured that out for myself in third grade. It was the only lifelong dream I have ever had.

Paul Franz was and now isn't:

So why give up the prospect of a cushy professorship for an uncertain career as an entertainer and artist? Because being a PhD student has made me miserable, and because I would rather be true to myself and take a chance at pursuing my actual passions than pursue a path which I know ends in unhappiness and cynicism.

Paul was a member of my grad program here at Stanford. Dina is the Terrence Malick of ELA bloggers. Both are thoughtful writers and you'll find lots to learn from about life and work from both their pieces.

How could we improve this task?

Fuller, Rabin, and Harel (2011) [pdf] define "intellectual need," "problem-free activity," and offer several ways to improve that task in one of the best pieces I read last summer:

When students participate in mathematical activities that stimulate intellectual need, we say that they are engaged in problem-laden activity. Unfortunately, many students are engaged in problem-free activity, in which they are driven by factors other than intellectual need and, as a result, do not have a clear mental image of the problem that is being solved, or indeed an understanding that any intellectual problem is being solved.

The piece features:

  • Dialog between teachers and their students that results in "problem-free behavior" and "social need." There's something in here for everybody. Everybody — myself included — will feel a twinge of recognition reading one or more of those exchanges.
  • Great suggestions for how to mend those scenarios, for queueing up intellectual need and problem-laden behavior.
  • Five categories of intellectual need. The need for certainty, causality, computation, communication, and connection. You can lean on any of those categories and watch several great lesson ideas fall out.

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mr bombastic:

The recursive part in the original question is especially annoying in that it sends the message that math is used to take something that is totally obvious (two more brick in the next row) and somehow make it seem complicated.

If all of this ladder of abstraction material has seemed soft, fuzzy, and opinionated so far, I'll offer up my summer project, A Literature Review of the Process and Product of Abstraction. Feel free to add comments or questions in the margins. I'll try to get in there and chop it up with you. If you have written more than a handful of literature reviews yourself, I'd be grateful for your feedback on the format.

This is ten months of grad school in ten minutes:

That video has me explaining the research from my qualifying paper, which is the culmination of a grad student's second year at Stanford's School of Education. It qualifies you, in the eyes of your advisers, to take on the much larger research project they call a dissertation.

The Experiment.

I showed two groups of students an image of a water tank. One group saw all the information and abstraction relevant to the question, "How long will it take to fill?" The other group just saw the question, "How long will it take to fill?" and had to request the information and develop the abstractions themselves. If you're remotely aware of this blog's obsessions, you can guess the research questions I asked about that experiment. (Watch the video!) Perhaps the most surprising outcome of the experiment (to me) was that the higher-achieving math students in the study really disliked not having all the information and abstractions in front of them.

If you'd like to read the paper, you can feel free. If you have some commentary or criticism that'd profit us here, you're welcome to the comments.

A few other notes about the qualifying paper, my second year of grad school, and my next year of grad school:

On criticism.

Stanford gives great feedback. The school of education has several schools within it. My school, Curriculum and Teacher Education, does a great job preparing its students for the qualifying paper. In the spring of your first year, you take an introductory course. In the fall of your second year, you take a doctoral seminar that builds to a proposal for the qualifier. In my particular case, I had a qualifying committee that was generous with feedback when I needed it. I also developed the study while taking a course with Alan Schoenfeld at UC Berkeley. So my ideas and writing had as many as eight sets of eyes on them, as needed. (And that's just faculty. My student-friends gave great feedback also.) That's amazing and, from my understanding, kind of rare in doctoral programs. That criticism was occasionally contradictory, however, which required a certain discernment I haven't really developed yet.

The criticism I remember most vividly: a) my weak review of the literature, b) the sense that I wasn't really taking myself anywhere new with the study, and c) a claim about equity that had me reaching beyond my data.

Great classes I took.

  • Accelerated programming in C++. I had no business in an accelerated course in anything related to programming but I had a scheduling conflict and they weren't putting the standard class online. It nearly ate me alive but spat me out a better programmer and granted me a great deal of sympathy for students who felt like idiots in classes that I taught.
  • Analysis of Social Interaction. With Ray McDermott, if that name means anything to you. If it doesn't, read "Can We Afford Theories of Learning?", which begins, "If American culture were an Internet, the domain name “learning” would be owned outright by the testing services that use it to feed the yearnings of parents and their schoolchildren." So a great quarter, basically.
  • Front-End Programming in Javascript, HTML, and CSS, another course I took from afar and watched online. Patrick Young is one of the best lecturers I've had at Stanford and certainly the best I've had in the CS department. Really an invaluable course. I called my final project, "Better Online Math" and it's the closest thing I have to a dissertation proposal.
  • Qualitative Analysis, with Pam Grossman (one of my advisers) and Sam Wineburg, who have taught together for decades, dating back to their time together at the University of Washington. Put these items under the heading "exceeded expectations": a) the four assignments, b) the syllabus, c) their respect for our time.
  • Directed Research, for practical reasons. Make sure you write that one down, class of 2016-2017. There's no excuse not to max out your units.

Papers I flagged as being particularly worthwhile.

What I'll be doing my third year.

I have this image in my head from a movie from my childhood that I've forgotten. A man stands with one foot on each of two rowboats that are side-by-side. It seems like a good, fun idea at first but then the boats start to drift apart. His weight bears down on both boats, pushing them farther and faster apart until he falls in the water and we laugh.

One boat is christened "Grad School" and the other is "The Other Stuff." The thing I can do to help myself right here is tie several cords from one to the other, committing myself to projects, papers, and talks that are researchable or that will, at least, inform my research. I just don't have the time for a long stay in grad school but I may not have the skill to get a dissertation done quickly either.

So for the third year:

  • I'm still designing tasks for and consulting with publishers in the US and elsewhere.
  • I'll be facilitating some workshops and speaking at some conferences.
  • I'll be taking the winter quarter off to work with The Shell Centre in the UK. (Did you guys know they pilot their tasks five times before they release them. What new questions do they ask in each new pilot? Let's find out this winter, okay?)
  • I'd like to submit a dissertation proposal at the end of this school year. Vegas oddsmakers are frowning at that one, though.
  • I've taken the required major coursework for the education doctorate but I need to complete several more courses in my minor emphasis in computer science. I'll be taking as many of those as I can this next year, online as much as possible. As I narrow in on a proposal, I'll take some appropriate methods courses also. (ie. if I plan to run an experiment, then something in experimental methods.)
  • I'll continue to develop 101questions into the tool I need to be.

Of course, all of this has been and will be more fun with you guys tagging along, chirping comments and critiques at me as we go.

2012 Sep 13. Elaine Watson posts some thoughtful commentary on my qualifier.

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Bruce James:

What would you do with a doctorate degree that you are not already doing?

My answer.

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