Saturday, March 5, 2016

A Conversation with Jon Star

I am currently part of a cognitive science inquiry group with about ten other math teachers. The goals of the group are to learn more about what cognitive science research says about teaching and learning and also somehow apply this research to our classrooms through some sort of new or revised structure or routine. So far, we have read Make it Stick and Why Don’t Students Like School and  analyzed them through the framework of generating questions that are prompted by our reading and the relationship of the ideas to our own teaching experience.

I have all sorts of questions, but this week we posed three of our group’s bigger questions with Jon Star, who is an educational psychologist at Harvard who focuses on flexibility in problem-solving and acquisition of algebra. I came away with so much to think about because he was able to give specific suggestions for practice that were very obviously grounded in research.

Question 1: What can we do to support students, many of whom have identified disabilities, whose working memory has a smaller capacity or who have other challenges with working memory?

Answer: Here is a list of strategies, all of which are centered around the principle of modifications/accommodations that do not sacrifice the learning goals. These are strategies that can be considered “good teaching” because they can benefit all students. However, the negative consequences of not using these strategies hurt some students more than others.

  • Reduce arithmetic complexity. Harder numbers make problems more complicated, but does not necessarily require a deeper understanding. He suggests making things conceptually harder, not computationally harder.
  • Be aware of the amount and type of words in a problem.
  • Be smart and organized about how you are using illustrations and board space. Processing everything verbally is extremely taxing on working memory. Board space and other visuals can be used a surrogate working memory for students.
  • Reduce the need for dual processing (Ex: Expecting students to read and listen at the same time)
I really appreciated that he also clarified that awareness of when you are/are not doing any of these things is the more important than making sure to use all of these strategies all of the time. There are good reasons for not following each of these strategies when there is a specific purpose.

Question 2: How do we foster a culture where students are motivated to persevere through spaced, varied, and interleaved practice, even though it is harder and the results are not as immediately obvious?

Answer: It is important to complexify what cognitive scientists are saying about practice.
  • Nuance #1: Practice should differ in different phases of learning—the first exposure to a concept/procedure vs. solidification after basic mastery. Some blocked practice may be necessary in the initial phase for motivation and initial formation of knowledge.
  • Nuance #2: Elaborative recall is the main principle that drives recommendations about effective practice. The more opportunities to reconstruct/apply/elaborate/expand knowledge, the more solidified it becomes. With blocked practice, the concept/procedure gets too automatic and students don’t reap the benefits of the recall. Also, once students automaticity with a procedure, that knowledge is very stable and difficult to reexamine/extend/reconfigure. Thus if students gain automaticity before associating with concepts, it is extremely hard to do this later on. Ideally, we should be building automaticity and connections simultaneously.

Question 3: Ideally, we are building procedural fluency from conceptual understanding. Where does practice fit in this arc and should the practice look different depending on where in the arc you are?

  • Point #1: It’s a misconception that there is an order to how students should learn concepts and procedures. It is not true that conceptual understanding needs to precede procedural fluency. It is not true that procedural knowledge cannot develop conceptual knowledge. Order is arbitrary—instead it is more important that they are connected and developed iteratively.
  • Point #2: There are the same best practices for developing procedural knowledge as conceptual knowledge. We can define conceptual practice as an opportunity recall and reconstruct concepts (just as we would define procedural practice as opportunity to recall and reconstruct procedures).
  • Point #3: It’s hard to articulate what it looks like for conceptual understanding and procedural fluency to be intertwined. One possible way to “see” this is through flexibility: students should be able to solve a problem more than one way and identify which strategy is “better” (ex: more elegant) and what the criteria are that decides what makes one solution strategy different than others.

One final gem from Jon Star: 

When you are listening to a performance of a piano concerto, often the performance itself is evidence enough of mastery of the skills and concepts behind the music. You wouldn’t need to ask the soloist to explain why they made the decisions they made. The same should be true for math. A student does not always need to explain in order to demonstrate conceptual and procedural knowledge. Sometimes the work/thinking they do speaks for itself.

On my mind now:

My biggest questions coming out of this conversation are about Star’s points about the (as one of my coworkers put it) commutative nature of developing conceptual understanding and procedural fluency.  My whole teaching framework is built upon the idea of building procedural fluency from conceptual understanding. NCTM’s Principles to Actions articulates this as one of their math teaching practices: “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.” 
However, based on the challenges that I have experienced when trying to teach this way (When are they ready to move from concepts to procedures? How do I help students actually connect the procedures to what they have built conceptually? How do I encourage students to go back to the conceptual understanding in order to re-build procedures that they have forgotten?), I am definitely in a position where I want to read and think more about Star’s position. I think that a lot of the current emphasis on conceptual understanding is a direct response to a history of teaching procedures with no conceptual understanding. It makes sense to me that the most important part is that we are doing both and connecting them, not that there is an extremely delicate, perfect sequence to have procedural fluency build out of conceptual understanding. Star also stated that understanding a procedure is different than having conceptual knowledge of a procedure, which is something I would like more specifics on. What does understanding of a procedure look like then? So I’m waiting on some recommendations from Star for some further reading in order to help me incorporate (or not) this into my ever-changing math education philosophy.

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