A Practical Solution to "What We Say/What They Hear"

“In art intentions are not sufficient and, as we say in Spanish, love must be proved by deeds and not by reasons. What one does is what counts and not what one had the intention of doing.”  -- Pablo Picasso

It's my belief that teachers at all levels have good intentions and genuinely want their students to learn.  What this post is about is the notion that instructor intentions are not sufficient, and getting students to do what is needed for authentic learning is what counts.

Background
The starting point is for the ideas I want to get across from a the Launchings Blog of the MAA.  David Bressoud does a wonderful job of describing a recent paper by Kristen Lew, Tim Fukawa-Connelly, Juan Pablo Mejia-Ramos, and Keith Weber.

Below are links to Bressoud's twin posts.  They are worth the time!

"What We Say/What They Hear I"
"What We Say/What They Hear II"

The short version is presented in this diagram

Despite multiple passes through the proof and explanations, students in the study have a difficult time pulling out the instructor's intended messages in the proof and the instructor's spoken comments.  At first glance this makes sense, as the "information transfer" model doesn't work so well when the goal is "developing critical thinking."

As mentioned by Bressoud, Annie and John Selden and others have documented the difficulties students have with analyzing, proving, generalizing, packing/unpacking statements, etc. Learning higher mathematics is challenging, and most math majors struggle with learning proof.

Bressoud suggests options like flipped classrooms and clickers can work, but he notes that such methods have high initial investment in time and requisite knowledge and skill.  While these methods are learnable, many instructors may be in situations, where implementing them is not practicably feasible.  Other options are needed.

Another Option Exists!
Now to the main point of this post. There exists an "easy entry, high upside" method to help students come away with the intended messages.  Put simply, instructors can take their list of intended messages and turn them into math tasks. These tasks can be deployed via small group work, homework, etc.

Let's take a closer look. The intended messages of the instructor in the study are

1. Cauchy sequences can be thought of as sequences that “bunch up”
2. One can prove a sequence with an unknown limit converges by showing it is Cauchy
3. This proof shows how one sets up a proof that a sequence is Cauchy
4. The triangle inequality is useful in proving series in absolute value formulae are small
5. The geometric series formula is part of the mathematical toolbox that can be used to keep some desired quantities small

Each of these five points can be turned into "after the proof" problems.  They can be reformulated as

1. Explain using sentences and diagrams why Cauchy sequences "bunch up."
2. True or False and Explain:  One can prove a sequence with an unknown limit converges by showing it is Cauchy.
3. In the proof, find the part that proves the sequence is Cauchy.
4. In the proof, find the part where the triangle inequality is used, and then identify a general strategy based on this specific instance that you can put in your mathematical toolbox.
5. Explain how the geometric series is part of a mathematical toolbox to keep some desired quantities small.

One way to deploy these in the classroom is to present the proof on the board or pass out a handout with the proof, and then have students work through some of the tasks in pairs and then share (i.e. Think-Pair-Share).  If time is an issue, doing one or two and assign the rest for homework.  Alternatively instructors could ask students to read the proof before class, and the instructor could highlight the proof and spend more time on student-centered tasks in class.

Several advantages exist with this option compared to flipped classes or using clickers.  The first advantage is that it requires the least experience and least amount of pre-class planning or classroom equipment.  One could be in classrooms like I sometimes teach in with no technology, old boards, and desks built 50 years ago.

Another advantage is that it does not require deep knowledge of common misconceptions.  Instead, the instructor asks students to explain their thinking (in one way or another) and that's how the instructor gets insights into student thinking.  That is, use an activity and gather formative assessment.

A third advantage of this method is that it does not deviate from the conventional class prep process used by most instructors. Preparing presentations of a proof with explanations is something that instructors have done many times. The method presented here tweaks the process by transforming the intended messages that instructors would normally say to students into concrete mathematical tasks for students to work on. This change is practically feasible and doesn't require a significant alteration of an instructor's workflow.  It's also a step towards active, student-centered teaching and can be built upon over time into forms of teaching that more deeply engages students, such as IBL.

Instructors have good intentions and intended messages. I claim that it is how the intended messages are deployed in class that can be addressed in practical and effective ways. We can turn those amazing insights into amazing learning experiences, and "let problems do the talking!"