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.

When does it work?

I have been sharing my experience teaching a group of fifth-graders how to problem-solve around fraction computation and using it as an opportunity to demonstrate the Teaching-Learning Cycle in action. Previously, I wrote about how I planned for a problem-solving lesson and then described my instruction during the following lesson. In this post, I want to discuss how I used assessment and evaluation to monitor the learners' progress and inform future planning and instruction.


We were using the clock model as a context for adding fractions and I wanted to gather data about whether or not the learners could determine when this model was an effective approach. I used an existing set of textbook items and asked the fifth-graders to: "Look at the expressions shown below - circle the ones that you think you could use the clock model to solve and place an 'X' through those you could not."

from Scott Foresman – Addison Wesley Math [5^th Grade]

Once the kids had completed this task, I asked them to solve one of the problems they had circled. As they worked, I gathered data on whether or not they were able to determine when the clock model could work.


In analyzing my observations, I noticed what the fifth-graders could do and what they were trying to do. First, they all recognized that fractions involving ninths and sevenths were poor candidates for the clock model. Those who chose to solve #2, #9, and #10 were also fluent in applying prior experiences with the time context. Some learners thought eighths could work (circling #4) and others struggled to see that fifths could work ('X'ing out #3, #5, and #8). These last two areas of approximation gave me some ideas about what to focus on next.


The last assessment I gave was intended to gather data about how the fifth-graders might apply the idea of context to a problem that could not be easily solved using the clock model. As a ticket out the door, I asked, "Now what could you do to solve a problem you put an 'X' through?" Based on my evaluation of these assessments, I was prepared to plan for future lessons.


What would you do next?

What support do learners need?

Last week I shared a lesson plan used to introduce the clock model for adding and subtracting fractions to a class of fifth-graders. This post focuses on the follow-up lesson, which concentrated on developing an anchor chart that learners could lean on as they solved progressively more difficult fraction computation problems. In particular, I will discuss how the instruction attempted to offer support so new learning could occur.

Learning is about moving from the known to the new. Therefore, we began by activating our schema regarding how we had used the clock model to add fractions in the previous lesson. Then we considered other fractions that could be represented using time as a context and began building an anchor chart based on our experiences with clocks in and out of the classroom.

An anchor chart supports learning by creating a record that learners can refer to as cognitive demand increases. In Reading with Meaning, Debbie Miller writes, "Anchor charts make our thinking permanent and visible, and so allow us to make connections from one strategy to another, clarify a point, build on earlier learning, and simply remember a specific lesson (p. 57)." This anchor chart offered the fifth-graders support both as a representation for thinking as they computed the fractions and a representation of thinking that they could point to as they communicated their thinking to their peers.

We worked through the number string together, with me starting the computation and learners offering advice as we went along. A number string is a series of progressively more difficult problems that build on the success of prior solutions. Being sure to use "I language," I started each problem by saying, "This reminds me of ..." As we went on, I asked more and more often, "What should I do next?"


From the perspective of the Gradual Release of Responsibility, my approach for this lesson would be considered Shared Practice [WITH]. (In the previous lesson, I had relied on Demonstration [TO] in order to support the introduction of the clock context - something new. Based on learners' efforts in that lesson, I was confident that they were ready for more responsibility.) With Shared Practice, the teacher supports learners by reinforcing how problem solvers get started, but there is still room for exploration and approximation as learners offer their suggestions for what comes next. This came in the form of the fifth-graders telling me what to do to complete the fraction problems. Not all of their suggestions worked, but we thought through them together and used our prior knowledge and anchor chart to get back on track. Any "mistakes" were used as an opportunity to foster a learning community that could support each other through difficult problems.

At the close of this lesson, I offered a final support - time to reflect. Without an opportunity to consolidate their experiences from the lesson, it is very likely that the learning would not last. I asked the fifth-graders to write in their journals a recount of the day using the What, So What, and Now What framework. Their responses would serve as a formative assessment used to inform future lessons. But that is for another post.

How do I plan for problem solving?

Almost a month ago, I wrote a post "Is direct instruction a better approach to teaching math?" that got a lot of attention (relatively speaking). My post was in response to an article which used one poorly constructed (my opinion) study to suggest that problem-solving or inquiry-based lessons were less effective than lecture-style instruction when it comes to standardized-test results. What seemed to get the most attention/ire was a comment by the article's author, Paul E. Peterson. 

"I, too, like those problem-solving classes. They require less preparation and are easier to teach."

This might be true for a tenured university professor who: (1) enjoys academic freedom; (2) has no "accountability" to a standardized, national test; and (3) does not believe in the problem-solving lesson as an instructional approach. But for the rest of us, a problem-solving lesson requires a great deal of effort. I want to share my process of preparation for such a lesson in this post.

Because I am also a university professor who enjoys those first two perks, I want to focus on a fraction lesson that I planned and taught a few years ago at a local elementary school. When the fifth grade teacher contacted me for help, she was very specific about the content that I needed to addressed in the unit. In Michigan, the driving force for most K-8 teachers is the Grade Level Content Expectations (GLCE) and for my series of lessons I needed to get at this standard: 

N.FL.05.14: Add and subtract fraction with unlike denominators of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 100, using the common denominator that is the product of the denominators of the 2 fractions.

My first step in planning was to gather data about my learners. Through a series of emails, I found out that this school grouped fifth graders by ability for math and that this teacher taught the lowest group. This included several learners with special needs supported by a special education teacher. While I am not a fan of ability grouping, this was not my fight. I was just grateful for the information.

Mathematics understanding is about experience not ability. It was up to me to plan a problem-solving lesson that offered learners an experience that would support their development of a relational understanding of fraction computation. Fortunately, I was familiar with an excellent resource that provides just such an experience. Planning a problem-solving lesson is not about developing activities from scratch (but if you have time and training, then this can work). Still, the planning does require effort in identifying appropriate resources and structuring them in such a way that they support learning.

Cathy Campbell wrote about this excellent resource on her blog. In particular, she discusses the clock model used for adding and subtracting fractions, which can be found in Minilesson for Operations with Fractions, Decimals, and Percents. Cathy does an great job describing this resource, so there is no reason for me to say much more except that I find its use of context and connections to prior successes very supportive for learners.

I am also fortunate to have the professional development packet that goes along with the series. This packet includes videos of teachers modeling some of the lessons. Before planning my lesson, I watched Joel teach the clock model, and it gave me some ideas of how to organize the lesson. In particular, it showed that he introduced the model in a whole-group setting.

Finally, I was ready to write out the plan. I decided to use a slight modification of a lesson planning framework Debbie Miller shared at the Michigan Reading Association Conference in 2008 and described in her excellent book, Teaching with Intention. You can view my plan here. As you can see, it is quite detailed, yet I do not consider it a script. I am a firm believer in Jon Stewart's approach to planning, "Creativity comes from limits not freedom ... When you have a structure, then you can improvise off of it..." (I still wish I had remembered to share that quote during my TEDx Talk). This detailed plan allowed me to make necessary adjustments as I taught the lesson, but that is for another time.

I hope this makes the point that planning for problem-solving is not easy. "Where's the problem-solving?" you ask. Let's compare the plan with the National Council of Teachers of Mathematics Process Standard for Problem Solving:

• Build new mathematical knowledge through problem solving;
• Solve problems that arise in mathematics and in other contexts;
• Apply and adapt a variety of appropriate strategies to solve a problem; and
• Monitor and reflect on the process of mathematical problem solving.

Please let me know in the comments if any of these are unclear in my planning.