Metacognitive Memoirs - what are they?

At the recent Mathematical Council of the Alberta Teachers' Association [MCATA] Conference, I facilitated the following workshop: Making Mathematical Thinking Visible – Metacognitive Memoirs. Here is the workshop's description:

Metacognition is the awareness of one’s thinking. Memoir is a genre usually referring to a piece of autobiographical writing focusing on some problematic event. Together they represent a powerful tool for helping learners experience what it means to do mathematics by thinking about and communicating their efforts to others. In this session we explore how creative writing supports creative thinking in mathematics – certainly a road less traveled.

This was a breakout session based on a portion of my earlier keynote.

Schema Activation: Cambourne's Conditions of Learning [10 minutes]

In an effort to connect to their prior learning, participants looked over Figure 1 from Cambourne's article, Toward an educationally relevant theory of literacy learning, and identified what was important, what they could connect to, and what questions they had. Cambourne's Conditions were one of the themes of the keynote and I had promised that we would consider them in more detail during this workshop.

Focus: Metacognitive Memoirs [10 minutes]
Participants read the following information which allowed them to concentrate on the important aspects of this style of writing in mathematics.

Objective(s): The learner will develop the ability to monitor their thinking as they engage in mathematical problem solving. [According to the book, How Students Learn: Mathematics in the Classroom (National Research Council 2005), this is an essential principle to learning.]

Time: The first demonstration of the approach usually takes a class period. Shared practice, between the students and teacher, may require another couple of class periods. The goal is that after this initial period of scaffolding learners can use the metacognitive mathematical memoirs on a regular basis throughout the remainder of the school year. During the scaffolding, the mathematical content is still being addressed as it is the topic of the memoirs.

Activity: Have the learners look over a group of problems and identify one that represents a ‘just right’ problem – not too hard and not too soft; this might be a problem set they have worked on previously or it might be the first time they have encountered the problem. The idea behind selecting a ‘just right’ problem is that the learners want to be able to tell an interesting story of successfully solving a problem including struggles they encounter along the way.

The learners are asked to describe how they went about solving the problem and what they were thinking as working on it; this represents the metacognitive aspect of the activity. Because memoirs can include embellishments to make the story more interesting, the learners are encouraged to be creative (yet realistic) in describing the “challenges” they encountered along the way. In this way, they are anticipating problems associated with the content.

Pros: (1) I have found these memoirs support learners’ in both a better understanding of the mathematical content and a clearer picture of the process of problem solving; (2) The learners’ writing also provides insight into their thinking – something I often found lacking in their previous assignments, even when they were required to show their work; and (3) The memoirs have been much more interesting for me to read than a solution set – and I only read one or two instead of an entire set.

Cons: (1) Like any new approach, learners often balk at having to write in math class; (2) Also, if learners don’t have experience with metacognition, it can be initially difficult for them to write about their thinking beyond, “I just know it.” It will take time and modeling by the teacher for the learners to become more comfortable showing their thinking as well as their work; and (3) It takes longer to grade a memoir than correct a traditional assignment.

Assessment:  For simplicity sake, I often use these four Cs in evaluating learners’ efforts. Learners must have met the expectations at one level to move onto the next. There are opportunities for revision in order to support success.

D	C	B	A
The work is Clear – you have communicated your work with clarity.	The work is Correct – your efforts demonstrate an understanding of mathematical content and/or processes.	The work is Complete – all required aspects of the project have been addressed. There are no gaps in your thinking.	The work is Creative – you present a unique perspective and show you can extend your thinking.

Lessons Learned:  Do not try this alone – ask for support from our writing colleagues.


Activity: Choice [30 minutes]
Each workshop participant made a decision regarding how to proceed in order to further develop his or her understanding of Metacognitive Memoirs. Some people chose to look at existing models: High School Student (This is the result of a web search from many years ago. I would appreciate any help identifying the source so I can give credit where credit is due.), Preservice Elementary Teacher, and Preservice Secondary Teacher. Others worked on solving and writing their own Metacognitive Memoir using one of these problems: Which is 19/24? or Sowing Seeds.


Reflection: Author's Chair [10 minutes]
I asked one of the participants who had been working on writing his own Metacognitive Memoir to share with the group using an approach that I have adapted from literacy instruction.
What did he do?
He selected to work on "Which is 19/24?" because it related to struggles he was aware of that some of his current students were experiencing. He explained how he used what he knew about common denominators to narrow down the computations he would ultimately have to do. As it turned out, this approach arrived at a solution without having to do any fraction addition or subtraction.
So what did he learn?
While he wrote in great bit detail how he went about solving the problem using the common denominators as a sorting mechanism, he had not completely shared his thinking. Upon reflection, he recognized that he had thought about and dismissed using other approaches (compute all the items or use a benchmark of 1 to sort the items) because they would not be as efficient. In order to make his thinking visible, he realized that he needed to include this information in his writing.
Now what will he do with his learning?
Besides being more aware of his thinking (metacognitive) and sharing it with his learners, he also discussed how he might modify the problem so that his learners would have to do some computation. He acknowledged that he had done some deep, conceptual mathematics, but he wanted the problem to address some content skills as well. The other nice thing about using Metacognitive Memoirs is that you can tailor them to the needs of your learners.

How can we communicate our thinking?

The math teachers I work with often express frustration with their learners' inability to communicate their thinking when it comes to solving problems. If learners are stuck, they often struggle to articulate what they have tried. If the problem has been solved, learners have trouble explaining their efforts. Too often, learners respond to a teacher's question about their thinking with, "I don't know." This does not present teachers with the assessment data necessary to evaluate what learners can do or are trying to do, which makes it difficult to plan what comes next.


I encountered this same problem while working with fifth-graders on fraction computation. They were practiced in giving answers and even showing work but when I asked them to share their thinking they often said, "I don't know." This provided me with an opportunity to try a response suggested by Ellin Oliver Keene in a session at the 2009 MRA Conference: "Pretend that you did know - what would you say?" The fifth-graders found that this framing supported their communication efforts by freeing them to take a risk because they were "pretending."


While this got them talking about their thinking, they still needed more support to organize their efforts. I thought it would help to demonstrate what a reasoning recount might look like. I was introduced to the recount text form through Margaret Mooney's book, Text Forms and Features. Here is the model reasoning recount I wrote based on the prior efforts of the class to think about adding fractions.

Recently, I have been collaborating with Jennifer Brokofsky via Twitter and email about ways to connect reading, math, and writing. The figure below represents our current thinking.  I hope this vignette further demonstrates the link. We recognize that this is work in progress, and your support would be appreciated. Please share your thinking in the comments.

What's in a Name?

Learning, thinking, knowing and understanding are significantly enhanced when one is provided with opportunities for 'talking one's way to meaning'.

Brian Cambourne

I decided to blog after a discussion with a colleague in GVSU's College of Education. He explained that blogging gave him a platform to share those ideas that he found himself repeating in class after class. I could relate since I, too, have my favorite "stories" and "sayings" that I share every semester. Being a mathematician, I liked the efficiency associated with this approach to blogging. As I thought more about it, however, I realized that it offered more than that. A blog would provide me the chance to write my way to understanding.

That is where Scape comes from in my blog's name. I want to try to understand the educational landscape. I agree with Cambourne that trying to share my perspective with others consolidates my thinking and enhances my understanding. This blog is not intended to change the reader's thinking, just the writer's. (Here is an example.)

And that is where Delta comes in. Delta is the mathematical symbol for change. (It is also the Greek d, as in David - some things aren't that deep.) I am in the habit of beginning each class by apologizing to my learners. I'll teach the class better next time because of what I learn from my interactions with them and from their feedback. I remind them that they are free to take the class next year - when it is improved. No one takes me up on that, but it sets the tone that I expect to grow as an educator.

So that is why I chose Delta Scape as the name for my blog. Writing it will hopefully sharpen (and change) my perspective of teaching and learning. Feel free to peek in to see what I've learned. Or wait until next year, when I'm sure it will be better.