How We Think

A Theory of Goal-Oriented Decision Making and its Educational Applications

Alan H. Schoenfeld

• Editorial: Routledge
• Año: 2010
• ISBN:
  9781136909788
• Edición: 1
• Categoría: Pedagogía
• Categoría: Educación general

Teachers try to help their students learn. But why do they make the particular teaching choices they do? What resources do they draw upon? What accounts for the success or failure of their efforts? In How We Think, esteemed scholar and mathematician, Alan H. Schoenfeld, proposes a groundbreaking theory and model for how we think and act in the classroom and beyond. Based on thirty years of research on problem solving and teaching, Schoenfeld provides compelling evidence for a concrete approach that describes how teachers, and individuals more generally, navigate their way through in-the-moment decision-making in well-practiced domains. Applying his theoretical model to detailed representations and analyses of teachers at work as well as of professionals outside education, Schoenfeld argues that understanding and recognizing the goal-oriented patterns of our day to day decisions can help identify what makes effective or ineffective behavior in the classroom and beyond.

Part I
Overview of the Theory

1
From Problem Solving to Teaching and Beyond

Introduction

This book focuses on how and why people make the choices they make as they engage in a wide range of knowledge-intensive activities. A main emphasis is on studies of teaching, my goal being to offer a theoretical account of the (not necessarily conscious) decisions that teachers make amid the extraordinary complexity of classroom interactions. A full theoretical account of teaching would characterize not only the “big” decisions such as the structure of a lesson, but the small ones (for example, how the teacher will answer a particular question) as well. I believe that if you can fully explain decision making during teaching, then you can explain decision making in just about any knowledge-intensive domain.

The goal of this chapter is to provide the context for the current book. My early research was on mathematical problem solving. My major goals were to understand problem solving, and then to use that understanding to help people get better at it. In fundamental ways I view teaching as a (much more complex) problem-solving activity, and my goals for this book parallel the goals for my problem-solving work. The better we can understand a range of complex knowledge-intensive activities, including teaching, the better we can help people become effective at them. Here I explain how the current work is an outgrowth of the earlier work on problem solving.

The central theoretical contribution of my problem-solving research (see, for example, Schoenfeld, 1985) was what I called a framework for the analysis of mathematical problem-solving behavior. In simplest terms, I claimed the following:

If you want to know why people’s attempts to solve challenging (mathematical) problems are successful or not, you need to examine their:

• knowledge base—just what (mathematics) do they know?

• problem-solving strategies, a.k.a. heuristics—what tools or techniques do they have in order to make progress on problems they don’t know how to solve?

• monitoring and self-regulation—aspects of metacognition concerned with how well individuals “manage” the problem-solving resources, including time, at their disposal; and

• beliefs—individuals’ sense of mathematics, of themselves, of the context and more, all of which shape what they perceive and what they choose to do.

My argument was that those categories are necessary and sufficient for understanding problem-solving success or failure. The categories are necessary in the sense that if any of them are left out of an analysis of someone’s problem-solving attempt, the analyst runs the risk of missing the key factor that explains why the individual did or did not succeed. That is, there are situations where mathematical knowledge is the make-or-break factor in a problem solution. There are situations where the use of heuristic strategies brings an otherwise inaccessible solution within reach. There are situations where the effective use of available resources puts a problem solver in a position to obtain a solution, and situations where the inefficient or ineffective use of time or knowledge results in failure to solve a problem that the individual “should have” been able to solve. And there are situations where people’s beliefs (for example, about their capacity, about what is considered a “legitimate” approach in a particular context, or about the amount of time and energy that should be spent on a problem before declaring it impossible) either propel them toward success or guarantee failure. In sum, each of the categories listed above is necessary for analysis. I argued as well that the four categories are sufficient, in that every root cause of success or failure will be found within them.

The framework, the data, and the arguments have stood the test of time.¹ Over the past quarter century a substantial amount of research in mathematics education and in education more generally has emerged to confirm them. So what else is there to say?

There are two main things to say. The first is that Mathematical Problem Solving offered a framework for looking at problem solving, but not a theory of problem solving. A framework tells you what to look at and what its impact might be. A theory tells you how things fit together. It says how and why things work the way they do, and it allows for explanations and even predictions of behavior. In my earlier work I could account for success or failure by describing the impact of the knowledge and decisions made by the problem solver. What I couldn’t account for was how and why the problem solvers drew on particular knowledge or strategies, or how and why they made the decisions they did. That’s the focus of this book.

The second main point has to do with the scope of the phenomena that the theory covers. My earlier work dealt with mathematical problem solving. Most of the analytic work was conducted in my laboratory, a quiet place where I gave people challenging mathematics problems to work on.² But that’s too narrow, along a number of dimensions.

The first dimension is content. I studied mathematical problem solving because I was conversant with mathematics. But what would one expect a framework or a theory of problem solving in physics to look like? From my perspective a framework for examining success or failure in physics problem solving would have to examine the following:

1. the knowledge base (in this case the understanding of relevant physics content);

2. access to problem-solving strategies, some of which are tied to physics content and some of which are more general;

3. monitoring and self-regulation; and

4. relevant beliefs.

Surely that seems plausible: physics and math are pretty close. It’s likely that the same would be the case for other quantitative domains. But what about a different kind of problem-solving domain—say writing?

Writing is clearly a problem-solving activity, in that one can conceptualize any writing task as a problem-solving exercise: the problem (or the goal) is to produce a body of text that achieves a certain purpose. That purpose might be to demonstrate one’s knowledge of Russian literature, to obtain a job interview, or to explain a theory one has been working on for thirty years. For all of these purposes, the following holds:

• What the writer knows, at the level of having something to say and at the level of being able to produce suitable and grammatical text, is critically important.

• Writers use scads of heuristic strategies, which range from small-scale (“use topic sentences”) to large-scale (“make an outline before producing text”) to more broadly suggestive (“to make sure people get your point, tell them what you’re going tell them; tell them; then tell them what you told them”).

• Almost everyone has had the experience of writing for a while and suddenly realizing that the direction of the text has changed or the intended audience has been lost, and that a lot of text will have to go in the trash can or be repurposed. This “loss of audience” is a failure of monitoring and self-regulation, and evidence of its importance.

• As anyone who has read student essays can tell you, students who believe that writing is just putting down on paper what’s in your head will produce very different text from those students who believe that writing is hard work, requiring multiple refinements in order to convey one’s ideas!³

In sum, there is good reason to believe that the factors that shape successful or unsuccessful efforts at writing are the same as the factors that shape successful or unsuccessful efforts at problem solving in mathematics or physics. Arguably, the same is the case in all problem-solving domains. Take cooking, for example. Producing a meal can be seen as trying to achieve something, and thus as a problem. Like all other problem solving, it is goal oriented. Knowledge, strategies, and techniques (not to mention material resources) are just as important in cooking a fine dinner as they are in solving a mathematics problem. With a bit of reflection, one sees the critical importance of monitoring and self-regulation (timing and coordination are critical) and beliefs (“fungi are fungi and liver is liver,” thought a friend, substituting brown mushrooms and chicken livers respectively for truffles and foie gras in what was intended to be a rather elegant dish!). In sum, there is a plausibility case, buttressed by a quarter century of literature in a wide range of fields, that the framework for examining the success or failure of attempts at mathematical problem solving presented in Mathematical Problem Solving is really a framework for looking at the factors that shape success or failure in any problem-solving activity.

A second dimension is that of social interactions. Early problem-solving research in most fields was done in the laboratory, away from other people. However, the vast majority of decision making and problem solving involves or is influenced by others. A widely applicable theory of decision making should explain how people make decisions in often highly interactive social contexts.

A third dimension involves the dynamic character of the environments in which people make decisions. Mathematics problems rarely change their character while you’re working on them. But the real world offers loads of surprises, and a broad theory of how people act should explain their behavior amid dynamically changing circumstances. My aim in general has been to explain the choices people make in a wide range of knowledge-intensive, highly interactive, dynamically changing environments. Teaching scores high on all of these dimensions, and it is obviously important. Thus, my goal was to explain teaching—but that was too large a goal to bite off at once.

Toward Greater Generality

A logical step in my research program was to study situations that were socially dynamic but not reflecting the full-blown complexity of the mathematics classroom. Thus, I moved to studies of one-on-one mathematics tutoring. In a tutoring session the problem solver (the tutor) is working with what appears to be a straightforward goal: trying to help someone else learn some specific mathematics. Yet the situation as understood by the tutor—the perceived problem state—can and often does change dramatically in the midst of problem solving. Here are two typical examples.

Example 1. Imagine a student and tutor working together on a calculus problem. All seems to be going well. The student gets to a point where some algebraic manipulation is called for, and writes

(a + b)^2 = a^2 + b^2.

Ouch! The tutor had assumed that the student’s algebraic foundation was solid, and now sees that it is not. This calls for a major decision: is it preferable to make a simple correction (“Watch it, the square should be a² +2ab + b².”) and continue with the calculus problem, or is the algebraic error important enough to warrant serious attention? Either way, the tutor’s perception of the problem state has changed dramatically in just a few seconds and something needs to be done about that.

Example 2. Perhaps later in the same session, the student says something that is ambiguous or that sounds slightly odd, suggesting a shaky understanding of an underlying idea. The tutor asks for a clarification and the student slumps visibly in response. The tutor has just crossed some line, although it may not be clear what that line is. Again, the tutor faces a major decision. Should he or she persevere at the risk of student disaffection, or back off and return to the issue when the student seems less vulnerable? If the tutor decides it is preferable to back off, what’s the next direction to pursue at this point?

Although the catalytic events differ in these two examples—the first was related to the student’s content understanding and the second to an affective issue— something similar has happened in theoretical terms. The tutor had certain high-priority goals and was working toward achieving them, implementing resources that had been selected to that end. Something happened that called for a re-evaluation of the problem state. With that re-evaluation, a series of decisions was called for. Should the tutoring session continue along the same path or should the tutor modify or abandon the current goals? If the same goals are maintained, should the tutor persevere in the same approach or try an alternative approach? If the tutor modifies the top-level goals, how does he or she choose what to do next?

I claim that in general such decisions are based on the tutor’s resources,⁴ goals, and orientations,⁵ as they play out in the particular context. Does the tutor think the student’s algebraic error is a slip or an indication of a serious misunderstanding? How much time is left in the tutoring session? Is there a test coming up? Is there time to remediate the algebra misconception if it turns out to be a deep-rooted misconception? What tools does the tutor have available to address it? Does the tutor feel a need to deal with issues as they arise, or does he or she feel comfortable letting certain things go? All of these factors shape the tutor’s decisions regarding what to pursue and how to pursue it. The same holds with regard to the tutor’s potential response to an affective issue. Is the tutor focused on content, affect be damned? If so, he or she will persevere in getting a clarification, although perhaps with a bit of cajolery. Does the tutor sense that the student is on the edge, and that persevering would be risky? If so, the tutor may back off, even if some of the student...