Alan Schoenfeld: At the Boundaries of Effective Mathematics Thinking, Teaching, and Learning

An Interview with Alan Schoenfeld: At the Boundaries of Effective Mathematics Thinking, Teaching, and Learning



Alan Schoenfeld is the Elizabeth and Edward Conner Professor of Education and Affiliated Professor of Mathematics at the University of California at Berkeley. He has served as President of AERA and vice President of the National Academy of Education. He holds the International Commission on Mathematics Instruction’s Klein Medal, the highest international distinction in mathematics education; AERA’s Distinguished Contributions to Research in Education award, AERA’s highest honor; and the Mathematical Association of America’s Mary P. Dolciani award, given to a pure or applied mathematician for distinguished contributions to the mathematical education of K-16 students.

Schoenfeld’s main focus is on Teaching for Robust Understanding. A Brief overview of the TRU framework, which applies to all learning environments, can be found at The Teaching for Robust Understanding (TRU) Framework. A discussion of TRU as it applies to classrooms can be found in What makes for Powerful Classrooms?, and a discussion of how the framework can be used systemically can be found in Thoughts on Scale.

Schoenfeld’s research deals broadly with thinking, teaching, and learning. His book, Mathematical Problem Solving, characterizes what it means to think mathematically and describes a research-based undergraduate course in mathematical problem solving. Schoenfeld led the Balanced Assessment project and was one of the leaders of the NSF-sponsored center for Diversity in Mathematics Education (DiME). The DiME Center was awarded the AERA Division G Henry T. Trueba Award for Research Leading to the Transformation of the Social Contexts of Education. He was lead author for grades 9-12 of the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics. He was one of the founding editors of Research in Collegiate Mathematics Education, and has served as associate editor of Cognition and Instruction. He has served as senior advisor to the Educational Human Resources Directorate of the National Science Foundation, and senior content advisor to the U.S. Department of Education’s What Works Clearinghouse.

Schoenfeld has written, edited, or co-edited twenty-two books and approximately two hundred articles on thinking and learning. He has an ongoing interest in the development of productive mechanisms for systemic change and for deepening the connections between educational research and practice. His most recent book, How we Think, provides detailed models of human decision making in complex situations such as teaching, and his current research focuses on the attributes of classrooms that produce students who are powerful thinkers.  Schoenfeld’s current projects (the Algebra Teaching Study, funded by NSF; the Mathematics Assessment Project (MAP) and Formative assessment with Computational Technologies (FACT), funded by the Gates Foundation; and work with the San Francisco and Oakland Unified School Districts under the auspices of the National Research Council’s SERP project) all focus on understanding and enhancing mathematics teaching and learning.

Michael Paul Goldenberg: It’s my pleasure to welcome you to The Math Blog, Alan. I appreciate your taking the time to be interviewed.

Alan Schoenfeld: Thank you, Michael. It’s great to have the opportunity to speak with you.

MPG: You started your professional career as a research mathematician. What led you to become a mathematics educator and researcher?

AS: I became a mathematician because I loved math, plain and simple. I also loved teaching. Then, I read Pólya’s HOW TO SOLVE IT and my head exploded – he described my mathematical thought processes! Why hadn’t I been taught them directly? It seemed clear to me that using these strategies could open math up to a lot more people!

I asked around and people I spoke with (Putnam team coaches and math-ed researchers) said that the ideas might feel right, but they didn’t work. Now THAT was a problem worth working on. This was in the mid 70s, as cognitive science was beginning to form as a field, and I decided to make the transition from mathematician to math-ed researcher. If I could figure out how to make Pólya’s strategies work, I’d be doing challenging research and doing work that could have real-world impact.

Having that kind of impact was my lifelong career goal. I love mathematics, and it’s distressing to see how many people don’t. The big question was, could we understand thinking, teaching, and learning well enough that we could create learning environments in which students would become powerful and resourceful thinkers and problem solvers, and as a result come to love math? (How could you not, if you’re successful?)

MPG:  I would think there’s a rather different flavor to that sort of task compared with doing mathematics itself.

AS: Yes, definitely so. The major goal in doing mathematics hinges on proving. The challenge in making the transition to educational research was the loss of certainty that one experiences when leaving mathematics. I don’t mean “proof” in the narrow sense. Yes, having a proof means that there’s no doubt: this thing is true. But that’s the tip of the iceberg. Most proofs provide a sense of mechanism. They say how things fit together, and why they have to be true.

MPG: Could you give us an example?

AS: Certainly. Think of space-filling curves, say one of the standard maps from the unit interval onto the unit square. You start with a simple map into the square, then expand it in a way that’s essentially fractal. As you do, you can see the sequence of functions you create both filling things up and getting more and more dense. So if you understand why the uniform limits of continuous functions are continuous, you know the result is onto. Moreover, if you look closely, you see that each 4th of the unit interval maps onto ¼ of the unit square, that each 4th of those 4ths maps onto 1/16 of the unit square, and so on – the map is actually measure preserving! In this example, which is the kind of mathematical argument I like, the proof tells you much more than the fact that a result is true. It tells you how and why it’s true. (For what it’s worth, my early mathematical work was in this arena, working on general topological spaces.)

When I left math, I left that kind of certainty and explanation. (Henry Pollak once said, “There are no theorems in mathematics education.”) So, what do you replace it with?

Basically, the tools depended on the problem. In my early problem solving days, part of the proof was in the empirical pudding. That is: I came to realize that Pólya’s strategies weren’t implementable as he described them: a “simple” strategy like “try to solve an easier related problem. The method or result may help you solve the original” was in fact a dozen or more separate strategies, because there were a dozen or more very distinct ways to create easier related problems. But those dozen sub-strategies were all well enough defined to be teachable, and students could learn them; when they learned enough of them, then they could use the strategy. The proof? Both in lab experimentation and in my courses, students could solve problems they hadn’t been able to approach before. (see MATHEMATICAL PROBLEM SOLVING).

MPG: So proof does come into play?

AS: Yes, but that’s proof in the narrow sense. The results were documented, but I couldn’t really say what was going on in people’s heads. In my work on metacognition (or “executive control”) I could show that ineffective monitoring and self-regulation doomed students to failure, and that they could learn to get better at it; but I still didn’t have a theory that described how and why they made the choices they did. That took another twenty years.

Since my ultimate goal was to improve teaching, I turned my attention to tutoring (an “easier related problem” in the space of teaching) – the question being, why does a tutor make the choices he or she does, when interacting with a student?

Here’s where the idea of modeling comes in. It’s just not reasonable to make ad hoc claims about what someone is doing and why – you can explain almost any decision in an ad hoc way, but if you keep track of the rationales for those decisions, you’ll find that the rationale for decision #20 may flatly contradict the rationale for decision #11. I know many qualitative studies provide “blow by blow” explanations of events, but I found that profoundly unsatisfying.

MPG: How can you avoid falling into that trap?

AS: Modeling prevents you from doing that. If you’re modeling someone’s (a student’s, a tutor’s, a teacher’s) decision making, then you have to stipulate the elements of the model and how they’re related: these things matter, and under these circumstances, the model will act in the following way. You take the thing you’re modeling, stipulate which aspects of it get represented in the model, and run the model. There’s no fudging. If you’re modeling a teacher’s decision making, does the model you’ve constructed make the decisions that the teacher does? Almost certainly not, the first time you run the model – which means you missed something. It could be a theoretical element, it could be something you hadn’t noticed about the teacher. So you refine the model and see if it does better. As you do that, you’re not just testing an individual model: you’re working on the architecture of such models in general. You’re building a theory, which you’re testing with a variety of models. If you can model a wide range of examples – from, say, a beginning teacher to a highly-reflective and knowledgeable one like Deborah Ball – then there’s a pretty good chance your theory focuses on the right things. After some 20-25 years, I’d gotten to the point where those ideas were pretty robust. (See HOW WE THINK: A theory of human decision-making with educational applications) 

MPG: Did you find applications for this model beyond teaching?

AS:  Yes, the work on decision making connected to decision making in other fields (e.g., medicine, electronic trouble-shooting) as well as being an abstraction of the problem solving work,­ but it wasn’t enough. The ultimate question for me was, what is the nature of productive learning environments – environments from which students emerge as powerful thinkers and problem solvers? The modeling work I’d done had focused on individuals, but there’s the whole question of classroom dynamics.

I found myself hamstrung. I tried using the modeling techniques I’d developed for understanding teaching to approach the classroom discourse as a whole, and it was just impossible. Ultimately, we started afresh: we listed everything we could think of that mattered in classroom interactions (from the literature, from watching tapes, etc.) – a huge number! – and then created equivalence classes of those. It turned out that we could distill all of the consequential events into five equivalence classes: those related to the quality of the mathematics, to opportunities students had for sense making and for “productive struggle;” equitable access to the key content for all students; opportunities for students to interact with the content and each other that allowed them to develop a sense of agency, ownership over the content, and productive disciplinary identities; and making student thinking public so that instruction could be productively modified to “meet the students where they are” (formative assessment). We call this framework the Teaching for Robust Understanding (TRU) framework.

(Details of its development can be found in Schoenfeld, 2013; an intro to the framework and a large set of professional development tools can be found in Schoenfeld & the Teaching for Robust Understanding Project, 2016).

MPG: So where is this heading and how is it working out?

AS: Of course, having such a framework is having a hypothesis: these five things are what counts. (In mathematical language, the five dimensions of TRU are necessary and sufficient for creating powerful mathematics classrooms.) This in a way sends me back in time to the R&D work on problem solving. I now have an idea of what counts, but how do I get compelling evidence, and how do I help people use these ideas? Some of the evidence is correlational: the students who come from classrooms that score well on a rubric that assesses the five dimensions do better on tests of thinking and problem solving than students who come from classrooms with lower scores. Some are existence proofs: when Chicago adopted TRU and used it for professional development, Chicago’s math scores went up while the rest of Illinois’ scores went down. Some is yet to come: we’re building tools to help Teacher Learning Communities implement TRU, and we’ll do detailed studies of what kinds of changes there are in classroom dynamics and how those relate to student outcomes (see and for tools and evidence). There are very positive signals as we work, some at a level of mechanism, but the road to understanding – and compelling evidence – is long and hard. Ask me in a few years and I’ll let you know how far we’ve progressed.

MPG: I would love to have you back to fill us in. What you’ve described thus far is exciting and thought-provoking to me as a mathematics educator. Thank you, Alan: you certainly have an open invitation to visit again.

AS: It’s been a pleasure speaking with you, Michael.


Additional References

Schoenfeld, A. H. (2013). Classroom observations in theory and practice. ZDM, the International Journal of Mathematics Education, 45: 607-621. DOI 10.1007/s11858-012-0483-1.


Schoenfeld, A. H., & the Teaching for Robust Understanding Project. (2016). An Introduction to the Teaching for Robust Understanding (TRU) Framework. Berkeley, CA: Graduate School of Education. Retrieved from


The TRU Math Suite Page on The Mathematics Assessment Project web site:


The TRU Math Suite Page on the TRU-Lesson Study web site:

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  1. Don Byrd July 26, 2017
    • Michael Goldenberg July 26, 2017

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