How might we deal with the mismatch?

Last week I heard Kat Holmes talk about her new book, Mismatch. I learned how designing for "normal" can miss the mark. As a result, many of us encounter mismatched experiences in our physical and virtual spaces. Effectively dealing with these mismatches requires inclusive design principles: 1) recognizing exclusion; 2) learning from diversity; and 3) solving for one - extending to many. In order to illustrate the second principle, Kat introduced us to Victor Pineda who made the following point:

There's a triangle of three different things that have to come together to really unlock human accomplishments for people with disabilities. And those involve assistive technology, personal assistance—somebody that's aware, understanding how to support you, and three is coping strategies. And so these three things sort of create a variety of tools.

Of course, I started connecting these ideas to teaching.

A curriculum is often designed for the normal/average student.

In reality, a student and the curriculum are typically mismatched.

To help the student to connect with the curriculum and be a contributor, the teacher might need to offer personal assistance, assistive technology, and/or coping strategies. And we can ask the student (learning from diversity, or as Dr. Emdin writes - co-teaching) to participate in the design.

Having planned for one student, the teacher must consider how to extend the design to many students.

After Kat's talk, I had the opportunity to watch a student-teacher deal with the mismatch between her students' experiences solving problems involving scientific notation [8.EE.A.4] and the curriculum used in her school. We talked about using a think-aloud (personal assistance) to create an anchor chart (assistive technology) that students could refer to (coping strategy) while solving 8.EE.A.4 problems. 

In the next lesson, she tried these ideas out. The lesson started with her making her thinking visible while solving an 8.EE.A.4 problem. Next, she asked students what they noticed in her thinking and added it to an anchor chat. She happened to put the anchor chart in the back of the room so it was obvious later in the lesson how many of the students were using this tool as they turned in their seats to see it. Because of her efforts, students were able to successfully connect to the curriculum.

I'm still processing a lot of this and would appreciate you sharing your thoughts in the comments.

[This blog post was written with the help of the Innovators' Compass. Check out my planning.]

Don't you want math to be better for your kids?

That's not the way I learned it! And if it was good enough for me, then it's good enough for my kid! (Along with either: "I was bad at math." or "I was good at math.")

That's how I interpret some of the posts trying to pass themselves off as examples of "bad Common Core math problems" (Google it and take a look at some of the images). Justin Aion has a great post that points out the problem with associating these examples with the Common Core State Standards in Mathematics (CCSSM). However, even if these examples are decoupled from the CCSSM, there's still the sentiment that these new math approaches are flawed.

Take this post, for example. The parent's letter says it all:

From Jeff Severt (some context)

"simplification is valued over complication" writes the Frustrated Parent. But is the parent's approach the simplest way to compute the difference between 4,000,002 and 3,999,999? As math educators, we encourage young mathematicians to build up a variety of computational tools so that they can attack any problem with confidence and phronesis.

Recently, my class explored the thinking inherent in the work of these third grade girls.

From The Big Dinner 

This was a Big Idea on the Multiplication and Division Landscape, Proportional Reasoning, that was new to nearly all of my preservice elementary teachers. Consequently, I followed up with a Think Aloud to reinforce the Big Idea and connected it to the CCSSM 3.OA.B.5.

Afterwards, one of the preservice teachers said, "I've never seen this before. Why?" Why, indeed. 

What happens when I add this?

My preservice elementary teachers are just finishing up a unit on patterns. Coming into the unit, one thing some of them struggled doing was writing an explicit rule for a pattern reflecting linear growth. In my opinion, this represents a gap in experience not a lack of ability. Since all of them have had some background in Algebra, the question is what experience are they missing?

Coincidentally, a post by Dan Wekselgreene on Linear Patterns in Algebra 1 showed up in my Twitter feed this morning. (Thanks MTBoS!) It is a good example of immersing learners in different representations and supporting them in making connections that can result in deeper understanding of linear relationships. I had my students do something similar (see the example on the right) but for some it was not enough. There was still something missing. It is for this reason that I add one more representation (an idea I learned from literacy instruction) - a recount.

From Mooney's (2001) Text Forms and Features:

Recounts: Why? To give a sequential and detailed account of an incident, a series of incidents or a conversation

What? A written record of recall of events, with attention to sequence and accuracy, and often to detail

Doing a detailed account of each step provides learners with an added experience for seeing the structure of the linear pattern. A recount is demonstrated in the think-aloud provided below.

This seems to provide many of my learners with an experience that was lacking from previous math classes - a missing piece to the puzzle that bridges some cognitive gap. However, I am also aware that they have had experiences that the typical freshman in Algebra 1 might be lacking. What do you think, would adding a recount to your linear pattern unit help?

How did you teach it?

The idea of MTT2K began when a group of preservice teachers could not wait until the end of a Khan Academy video to voice their concerns about the quality of its content. (You can learn more here.) Once the movement to critique Khan Academy videos gathered some momentum, it was suggested that teachers do more than nitpick (although, as this post explains, nitpicking is important). Consequently, a group of bloggers set out to make 101 alternative lessons.

Nine alternatives were scheduled as of today - the day before the MTT2K prize deadline. While this is less than we hoped for, it is a start. The energy behind projects like the mathtwitterblogosphere demonstrates how we all benefit when teachers make their own lessons available for others to use and improve on.

The lesson I want to share comes from that same day when the preservice teachers watched the Khan Academy video. We were focusing on NCTM's Communication Standard that day, and I wanted to share examples of people communicating the ideas associated with integer multiplication and division. Many of the area schools use PowerPoints in their math classrooms, so I wanted to model how I might use this medium to communicate my thinking.

The think-aloud focused on my approach to trying to understand some integer rules I memorized years ago. I used "I language" to remind the learners that I am making my thinking visible, not telling them what to do. Afterward, we debriefed about what they saw and heard and how what I shared might support their ability to understand the concept for themselves. (The demonstration is split into two parts because Jing only allows for 5 minute recordings.)

Part One

Part Two

Having watched this demonstration, how would you respond to these points:

• What did I do?
• What did I say?
• What did I think?
• Develop a consolidated recount regarding multiplication and division of integers.

And how might you improve on this demonstration?

Can you help me with subtraction?

[Note: Based on feedback from readers, it seems this post needs some context. This is part of a series on exploring a new number system in a course for preservice elementary teachers. The purpose of this unit is to provide these future educators with an experience of what it is like to learn concepts of number and operation using a conceptual approach. In this final post of the series I describe the days leading up to the unit's final assessment. You can see the entire series by clicking on the Wumania tag at the bottom of the post.]

As we near the Wumanian number system assessment, some of the preservice elementary teachers express concern about their ability to demonstrate their fluency in computing with multi-digit numbers. I suggest that they consider putting the expressions into context using one of the Cognitively Guided Instruction stories types presented in the article, Using Student Interviews to Guide Classroom Instruction: An Action Research Project (PDF). We also try connecting the stories to manipulatives (the blocks introduced at the beginning of the unit - see below) using the approaches presented in Teaching without Telling: Computational Fluency and Understanding through Invention (PDF) and Second Graders Cirumvent Addition and Subtraction Difficulties (NCTM). In the process, I explicitly point out my use of practitioner journals to inform my instruction.

Because not everyone needs support in this area, I am in the habit of posting on our class website PowerPoint Think Alouds of how I might go about solving some of the problems on this practice sheet.

Learners with access to the internet can download the Think Alouds and watch them at their convenience. This is how those first PowerPoints looked.

With current technological tools, like Jing, here is what I can post now.

These Think Alouds provide the learners who are struggling to apply the approaches they have read about a model of the approaches in action. It also models how teachers can use technology to offer different levels of support to learners. Some might call me the Wumanian Sal Khan, but I can assure you that these productions were not one-take affairs.

How can I use technology to make my thinking visible?

Last week I attended the NCTM 2011 Annual Meeting and Exposition in Indianapolis. Whenever I attend conferences of this size I always like to choose some theme in order to select sessions that work together. This year, I decided to focus on ways to use technology to make thinking visible. The idea of using think alouds to model mathematical practices is a major goal of my Teaching and Learning Middle Grades Mathematics (MTH 329) course but it has been hard to put into practice because of our lack of experience. The NCTM Conference provided me with more experiences to share with my teachers in training.

The first session that I attended was Using Technology to Transform Students' Problem-Solving Experiences and Perspectives. The presentation was true to the title with many good examples used by the presenters in their classrooms. (I will try to post about these in more detail later.) What I found most interesting was the idea of using the green screen feature in Photo Booth to embed learners into their problem-solving efforts.

I decided to try this for myself, and the result is provided below. Unfortunately, for some reason, my work in the background kept fading in and out. This was especially bad whenever I moved my arms to gesture at particular points. That is why I used my eyes and head to focus the viewer's attention in the video:

In the second session, Using Screen-Capture Movies to Assess Quadrilateral Constructions in Sketchpad, presenters showed how they used Geometer's Sketchpad and Jing to share learners' efforts to explore quadrilateral properties. This session was especially appropriate since I assigned my MTH 329 class a Shape Maker Lab before I left for the conference. The final unit of the course is on teaching geometry and now I had another way to make my thinking visible in this content area.

The Jing software only allows for five minutes of video so it took several tries to get my timing down. I like that this feature forces the person doing the think aloud to consolidate his or her thinking to a reasonable length, but it does require some planning and practice. Using Sketchpad's redo command helped to make the initial construction go more quickly. I did the construction and then undid it without Jing running. And then I hit "redo" multiple times while Jing was recording until the initial construction was complete. Only the dynamic exploration at the end was completely "live" during the video:

Given that these think alouds using technology are my first attempts, I am happy with the results. However, I know that with more experience and support they can be improved. In fact, that is what we will be considering in MTH 329 tomorrow - glows and grows for these reasoning recounts. As always, your comments are welcome as well.