Academic journal article The Mathematics Enthusiast

# If You Really Want to Get Ahead, Get a Bunch of Theories ... and Data to Test Them

Academic journal article The Mathematics Enthusiast

# If You Really Want to Get Ahead, Get a Bunch of Theories ... and Data to Test Them

## Article excerpt

Introduction

To motivate the kind of approach I take in this paper, I begin with a metaphor. Suppose we consider the "problem" of air travel. The analogous question one might ask is: What theories do you need to know, in order do a good job of operating an air travel system?

One might start with grand theories. What does it take to get a plane off the ground? Newton's laws aren't bad for a start. If you're building airplanes, it helps to understand about gravity. But that's just a start. What about air lift? There are issues of wing configuration, for example; Bernoulli's laws apply there. Note that Bernoulli's laws, which are more "local," are at a different level of grain size than Newton's. Both the grand theory and the second, more local theory are essential. Moreover, to manufacture various parts of the plane you need theories of: metal expansion and contraction; jet propulsion (if you're manufacturing a jet), and relevant theoretical frames regarding the construction of every system (e.g., braking, oxygen, radar, etc.) that the plane will use. In addition, and absolutely essential: All of the theoretical ideas MUST be backed by rigorous empirical work, preferably in dialectic with the theoretical. Component systems are refined and tested rigorously before they are used for commercial flights.

All that, of course, is just for the airplane. But what about the human and material contexts within which air travel takes place? Having put together the most advanced and capacious airplane won't do any good if the local airport has a dirt runway and no radar; the technology has to fit the context. Beyond that, there is the question of how one builds a robust infrastructure for dealing with normal and not-so-normal travel-related issues. Recent news photographs of people stranded for days and weeks at airports because of volcanic eruptions or unexpectedly heavy snow make the point that a broad range of systems shape "local" day-to-day operations.

In sum, to make progress on complex, multifaceted issues (and supporting teachers' professional growth is certainly a complex and multifaceted issue!),

We need lots of theories, at different levels of grain size;

The theories should be refined and tested empirically;

The systems and practices one builds must be context sensitive and adaptable.

This last point implies that there is no "one size fits all" solution to issues such as professional development.

Now, let us turn our attention to the issue of teachers' growth and development. I will argue that to address this issue in a reasonable way, one should be informed by (and contribute to the development of) theory related to:

* The dimensions of competency one would like to see develop;

* How things develop;

* A theory of change, and a plan embodying the theory;

All of which must be contextand material resource-sensitive, which means that it is also necessary to have

* A theory of individual learning and growth, and

* A theory of how ideas can be spread or squashed in a social (i.e., district, state) context.

In what follows, I will lay out some of our theoretical ideas about (mathematics) teachers' proficiency and its development, describe some current efforts to promote teachers' professional growth and say why I am worried about them, and frame the enterprise in terms of a larger, (prospective) data gathering effort. My hope is that the kind of effort one can envision based on the consideration of such ideas would be an appropriate way to approach teacher preparation and other heretofore intractable problems.

Theoretical backdrop

Theory, Part 1. Dimensions of teaching proficiency.

If one is engaged in supporting teachers' professional growth, it helps to have a theory of the dimensions in which one hopes teachers will grow. To sum up a chapter in a table, Schoenfeld and Kilpatrick (2008) offer a provisional framework for looking at the dimensions of teacher proficiency. …

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