Friday, November 8, 2013

A Characteristic of IBL Teaching: Mediating the Interaction Between Students and Mathematics

A question that comes up frequently is, "What are some of the main features of IBL teaching?"  One of the main characteristics of IBL teaching that I'd like to highlight in this post is mediating the interaction between students and mathematics.  

Education is full of monikers for active, student-centered teaching.  We say things like, "Guide on the side," "Mentor in the middle," "Coach," etc.  These are nice ways to think about the nature of IBL teaching, and it's important to unpack what these things specifically mean.

In IBL Math there exists a particular role for instructors, which is to set-up and mediate the problem-solving process.  Students are engaged in exploring the mathematical landscape via well-chosen, logically ordered sequences of problems.  As students engage in their explorations, the instructor works to support the class so that students' interactions with the mathematics results in learning.  

Here are some ways IBL instructors mediate the interaction between students and mathematics.  Instructors
  1. design/adapt appropriate curriculum.  The problem sequences must be matched to the goals of the course, must be logically consistent and coherent, and meet the needs of the specific students in the class.
  2. keep students going without getting overly stuck.  That is, struggle is good if the struggle is fruitful.  Being stuck is part of the learning process, and students come in with a "tolerance threshold" for being stuck.  It's important to stay within these tolerances so that students stay in the game.  One positive outcome of an IBL course is that tolerance for being stuck can increase. It's something that can be improved upon.
  3. provide scaffolding.  This idea is related to the previous point is being able to control the "being stuck phases" by offering "bread crumbs" (hints, lemmas, sub problems, deploying group work, etc.) in just the right type and size to keep the cognitive level of the task high, while simultaneously avoiding student shutdown due to being overly stuck.  This is where calibrating to specific classes comes into play, and making good choices depends on having good data about students.  Listening, observing, and interacting regularly with students forms the foundation of this piece.  With good data in hand, it's often clear what scaffolding is needed.  "Oh, students are having trouble thinking about supremum.  Students may need to work on a couple of problems about upper bounds…"
  4. set the class structure and environment for positive discourse.  Class presentations are of no value if the class environment is not setup appropriately.  IBL instructors set the roles, expectations, and procedures of class discourse, which is used to share and validate ideas.  The ethos of the class must include inquiry, discovery, mutual respect for students and learning, and valuing "mistakes" as important discoveries.
  5. provide structure of the material being covered, so that students know where they are in the mathematical landscape.  This is an excellent place to inject small lectures and/or organizational tasks, where students are not necessarily solving new problems, but organizing the information they have recently studied.  
The central focus is on the math and how students are working on the math.  The first major component is creating problems that guide students through the mathematical landscape.  Deploying these tasks determines the nature of the interaction students have with the mathematics.   As students work on solving problems, help or direction is needed.  The instructor provides enough assistance to keep students in the learning zone, without taking away all of the fun and exploration.  When students make progress and discoveries, the instructor provides a forum for the ideas to be shared, vetted, and learned by all.

This is the notion of mediating the interaction between students and the mathematics. It is a central component of effective IBL teaching, and a healthy perspective for instructors to embrace.

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