Which representations will help?

A few years ago, John Golden and I co-taught Teaching and Learning Middle Grades Mathematics. During the first unit on Doing Mathematics, we noticed that the preservice teachers were struggling to unpack how they used various representations to support their problem solving. We wondered what would happen if we limited the representations they could use while looking for a pattern in a sequence. That is how this workshop came to be.

Doing Math Workshop (Representations)

Schema Activation: What representations did Carl use?

We have been talking about writing Metacognitive Memoirs as a way to make our thinking visible as we solve problems in mathematics. Here is an example from Carl that we found on the internet. Please identify all the representations he uses in his write-up.

Focus: Representation Standard

This comes from the NCTM's Principles and Standards for School Mathematics:

In order to make our think alouds explicit, as it relates to this Process Standard, we need to highlight where we are using these algebraic representations:

• real; pictorial; verbal; written; numeric; tabular; graphical; and symbolic (recursive and explicit relationships)

These are the images we use to understand and communicate mathematics. This is analogous to mental images used to enhance reading by immersing the learner in the subject.

Activity: Limiting Representations for Thinking

The students are asked to avoid using any of the representations not explicitly identified on the worksheet. In this first case, they are limited to using only pictorial and symbolic representations. After a couple of minutes, I allow them to use verbal so that they can share with one another their efforts thus far.

The next worksheet limits the students to using verbal, written, and symbolic representations. This semester, one of the students expressed a great deal of discomfort with the fact they she was not going to be able to use the representations with which she was most comfortable. We talked briefly about how this might mirror how students who struggle could feel regarding the representations we expect them to use in our classes. Again, I added another representation, this time manipulatives (real), after a few minutes.

On the last worksheet, the students are limited to using tabular and symbolic representations. One student said she was relieved to be back in familiar territory. When I offered to add another representation of their choosing, they asked for verbal so that they could talk over the problems. It turned out that tables were not enough to help them feel confident in their answers.

Reflection: Journal Jot

How did limiting the representations you could use affect your thinking?

Do you really understand?

A version of this post was originally written for my Learning Museum blog. As I continue to consolidate my blogs, the timing seemed right to move this post here. You see, it's ArtPrize time again in Grand Rapids. Also, two of my classes are currently focusing on the differences between making sense of something and truly understanding it.

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Grace Chen wrote a very thought provoking post called, Making Sense of Understanding. I was reminded of her points as I attended an ArtPrize 2011 Sneak Peak at the University of Michigan's School of Art & Design and SiTE:LAB. As I reflected on what I made up as I watched the the movement of the piece in the movie below, it became clearer the distinction between making sense and understanding.

When I first saw it moving, I thought that it was a mobile and that the wave motion was the result of the wind. That didn't make sense, however, as we were indoors and there wasn't that much air movement. My wife, Kathy, thought that it might be moving to the music. But when the music stopped, it kept undulating. Then I looked up and saw that levers were moving up and down which resulted in the wave-like motion. Mystery solved - the piece moved in some preprogramed way. This "made sense" to me and so I moved on to look at more of the art.

Before I got to far, Kathy exclaimed, "Cool!" I turned back to see her reading a description of the piece near by. "It's the motion of the ocean," she continued.

Huh? That didn't make sense. But then I read the description too:

Because it might be difficult to read, here is what it says:

The installation draws information from the intensity and movement of the water in a remote location. Wave data is being collected and updated from National Oceanic and Atmospheric Administration data buoy station 51003. This station was originally moored 206 nautical miles Southwest of Honolulu in the Pacific Ocean. It went adrift and the last report from its moored position was around 04/25/2011. It is still transmitting valid observation data but its exact location is unknown. The wave intensity and frequency collected from the buoy is scaled and transferred to the mechanical grid structure resulting in a simulation of the physical effects caused by the movement of water from this distant unknown location.

Now I understand. And understanding did make the piece even cooler.

What if they ask me a question I can't answer?

I often coach student teachers assigned to teach topics they have not worked with in a long time. The last time many of them had Algebra or Trigonometry was in high school and some "rust" obscures their understanding of the subject. They are understandably concerned about making a mistake while working through examples or demonstrating solutions to homework questions. 

I can empathize having had to teach College Algebra many, many years after my last encounter with the material. Before each lesson I had to reteach myself the topics. (It didn't help that my initial interactions with the College Algebra content were what Skemp would call instrumental.) And when it came time to address homework issues, I was always concerned that I would make a mistake in front of the class. Turns out I had my own issues with math-phobia.

While I am comfortable with the idea of making mistakes in my mathematics education classes, because it represents the learning process, I have found college freshmen and sophomores taking a required math course less understanding than my preservice teachers. Students in College Algebra often have certain expectations of what mathematics teaching looks like and when they encounter something different (mistakes), they can shut down. So early on I want to establish a certain level of confidence for myself and for them.

I share this story with preservice teachers so that they can feel comfortable with the fact that any nagging doubts they have about teaching a topic they need to relearn can be expected and dealt with. There are two suggestions that I make to build their confidence. The first is to avoid answering off-the-cuff questions until they feel more comfortable with the content and the classroom. And the other is to always have the examples they want to share written out ahead of time.

When it comes to questions that arise in class, if it's not something the student teachers have prepared for I tell them it's alright to write the question down and tell the class they will get back to it. This models that all mathematical questions don't have immediate answers and hopefully helps students to become more comfortable embracing the patience required to be successful in learning relationally (see Skemp). Regarding homework questions, I suggest that the student teachers set up a procedure by which students can identify which problems they want support on. If this can be done beforehand (I used Blackboard), then the student teachers can determine which items caused the most confusion and prepare a whole-class demonstration to support student learning. Items identified by a single student can be attended to through a written example or individual conferencing. If there is no way to gather this information beforehand, then I suggest collecting it at the end of the class period when it is expected to be completed and preparing the support for the next period. In either case, have the work written out ahead of time.

When they share their examples, at least early on, I encourage the student teachers not to recopy the work on the board but to use the overhead or document camera to project their work. This addresses several issues that teachers trying to build their confidence often face. First, it tightens up time management. Writing on the board is time consuming and can allow students to get off-task. (Believe it or not, many of them are not copying it down.) I would simply project my work and add my thinking as I go over each step. Second, projecting previously written work alleviates the potential for making a mistake and eroding confidence for those already dealing with this issue. Multi-tasking when stressed is difficult, and I have found that the combination of copying and talking through my thinking can lead to errors even though I have written everything out correctly. Finally, and this is huge for new teachers in front of a class of teenagers, using written work projected on the board usually means not having to turn your back on the class. If you've ever taught in a secondary math class, you'll understand what I mean.

Let me be clear, I am not suggesting that this approach become the norm; it is just the place to start when teaching a new prep that might require some relearning of topics. Once a classroom culture of trust and respect has been developed, I have found that I can go back to responding to questions more immediately. But I still reserve the right to say, "I'll have to get back to you on that one."

What are the Conditions of Learning?

Note: The original version of this post was written for my other blog, The Learning Museum.

I never teach my pupils; I only attempt to provide the conditions in which they can learn.

Albert Einstein

In this post, we explore the learning theory developed by Brian Cambourne from his research on language acquisition in natural settings. His book, The Whole Story: Natural Learning and the Acquisition of Literacy, introduced the idea that certain conditions were necessary in order for us to learn language. These conditions were further explored in this article, Toward an educationally relevant theory of literacy learning: Twenty years of inquiry. In both the book and the article, Cambourne describes the eight Conditions of Learning in detail. Below is a figure from the article representing the relationships that exist between the Conditions.

From Toward an educationally relevant theory of literacy learning

Cambourne's work focused on applying these Conditions to literacy instructions. Others, have sought to consider their application in other learning environments. Edmunds and Stoessiger wrote a book and article about their efforts to apply the Conditions to mathematics. Jan Turbill's doctoral research examined the use of the Conditions in teacher inservice. ReLeah Cosset Lent wrote Engaging Adolescent Learners: A Guide for Content-Area Learners using the Conditions as a framework.

From Engaging Adolescent Learners

And I wrote a guess blog post on how I used the Condition to learn to Tweet.

How do the Conditions of Learning fit into your practice?