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.

Is direct instruction a better approach to teaching math?

I received the following in my email today:

An intriguing question that I often wonder about myself. Cambourne's research found that learning requires engagement, which means it depends on how engaging the lecture is to the learner. I clicked on the link hoping for some clarity. I was disappointed.

Here is a sampling from Paul E. Peterson’s article, Eighth-Grade Students Learn More Through Direct Instruction, reviewing the research:

As an instructor myself, I’ve had trouble making up my mind. I can cover a lot of ground in classes where lectures consume about two-thirds of the time. But those classes get less enthusiastic student evaluations than some smaller classes where students are encouraged to solve problems through discussion. I, too, like those problem-solving classes. They require less preparation and are easier to teach.

Before we more on, here are my reactions to this portion of the article. The first is nit-picky, but when Peterson says, “I can cover a lot of ground” it is a red flag for me. I, too, cover more material through lecture, but research shows that many students fail to cover the same ground or retain any memory of the landscape. Second, there is the statement, “I, too, like those problem-solving classes. They require less preparation and are easier to teach.” All I can say is that if planning a problem-solving lesson requires less preparation, then it is not really a problem-solving lesson.

Peterson continues:

So when Guido Schwerdt and Amelie Wuppermann of the University of Munich figured out a way to test empirically the relative value of the two teaching styles (see “Sage on the Stage,” research), it is worth trumpeting the findings. These analysts took advantage of the fact that the 2003 Trends in International Mathematics and Science Survey (TIMMS) not only tested a nationally representative sample of U.S. 8th graders in math and science, but also asked their teachers what percentage of class time was taken up by students “listening to lecture-style presentations” rather than either “working on problems with the teacher’s guidance” or “working on problems without guidance.” Teachers reported that they spent twice as much time on problem-solving activities as on direct instruction. In other words, U.S. middle-school teachers have drunk deep from the progressive pedagogical well.

It is important to note that the results are based on teachers' reporting instructional approaches and not direct observation. This is important because Stigler and Hiebert (1999) found that: “Although most U.S. teachers report trying to improve their teaching with current reform recommendations in mind, the videos show little evidence that change is occurring. Furthermore, when teachers do change their practice, it is often in only superficial ways.” (The Teaching Gap p. 12) This also comes from TIMSS data. However, there is no corroborating evidence in Schwerdt and Wuppermann’s study that supports teachers’ claims that they are using problem-solving approaches.

Furthermore, as the Learning Pyramid shows, not all direct-instruction methods are equal in their effectiveness. A demonstration (think aloud) would be a superior method to simply “covering content” through a traditional lecture. Again, Cambourne has shown that demonstrations are a necessary condition for learning but not sufficient. Learners must be given the opportunity to take responsibility for their learning and “give it a go.”

None of this seems to matter to Peterson who ends his review with, “Sadly, U.S. middle-school pedagogy is weighted heavily toward problem-solving.” In my opinion, what’s sad is that he would try to pass this research off as settling what is clearly a complex issue.

My reading of the Schwerdt and Wuppermann study suggests that it tries to answer the question, “Is traditional teaching really all that bad?” without considering how their methodology might misinterpret the data. I already discussed the problem with associating teacher reporting with using observable data. I am also concerned that they combined, “working on problems with the teacher’s guidance and working on problems without guidance” into the single problem-solving category.

Watch this lesson of a U.S. classroom where students are "working on problems with the teacher’s guidance" (from the original 1995 TIMSS research) and decide for yourself whether this teacher “drunk deep from the progressive pedagogical well.” (You will need to sign up for a password but it is free.) My answer is, “No,” but this would be categorized as problem-solving time in the Schwerdt and Wuppermann study.

“Is traditional teaching really all that bad?” – based on this research, the jury is still out.

Where do I turn?

Last week I attended the 2011 NCTM Conference in Indianapolis, Indiana. On Thursday, I learned how to use technology to make my thinking visible. And on Friday, I presented with some colleagues our efforts to offer middle school math and special education teachers job-embedded professional development. But it was the trip down and back that provided the teaching metaphor I wish to share in this post.

I picked up my colleague and co-presenter, Esther, at her house. She had one of those talking GPS gizmos and asked if I wanted to use it. I assured her that I had looked at the directions and it looked like a pretty straight shot. "Besides," I said, "those things annoy me. I don't want to be told where to turn all the time." That is how we began what was suppose to be a five hour trip.

We filled the first few hours discussing what was happening in our classes. I shared my struggles in getting my preservice teachers to make their thinking regarding the Process Standards explicit. In developing their think alouds, metacognitive memoirs, and reasoning recounts, they often miss opportunities to highlight whichever Standard they identify as the focus. Much of the time they express frustration with having to add this information because it interrupts the flow of their thinking. Esther reminded me that this was new to them and that they would need demonstrations and support. I agreed and shared my hope that the NCTM sessions I had selected would help with ideas that I could share with them.

Having addressed this issue, we checked to see how much longer it until Indianapolis. My Maps App said 3 hours, but that didn't make sense since we had been on the road for 3 hours already. It then became clear that my "pretty straight shot" had a major turn that I had missed near South Bend. Esther was gracious about my error and we made the necessary adjustment (with the help of the nice woman in the GPS gizmo with the Scottish accent).

We used Emma (the name Esther's girls gave the voice of the GPS guide) on the way home. Yes, she was annoying at times, but her purpose was clear: She interrupted our mindless progress with important information at critical points in the journey. This is where the teaching metaphor comes into play. Emma was what was missing from my preservice teachers' attempts to make their mathematical thinking visible. A mechanism to interrupt learners mindlessly following a teacher's thinking in such a way that they would take notice of how an expert made important decisions.

Please, do not take this metaphor too far. I am not suggesting that teachers take on the complete persona of a GPS guide - directing learners what to do at every turn. But when teachers are sharing their own thinking, describing their own learning path, it would not hurt to interrupt the flow of the lesson at critical points to ensure that learners do not miss an important point. Don't be afraid to be direct and annoying every now and then. It might help, though, to use a Scottish accent.

Does Test-Prep Make Sense?

If you drop by a Michigan middle school mathematics classroom this September, chances are you will find yourself in the middle of test-prep instruction. October brings the Michigan Educational Assessment Program (MEAP), our high-stakes standardized test, and schools want to ensure that their pupils are adequately prepared. The test-prep instruction usually entails one part content, one part test-taking strategies, and one part cheerleading.

This is not the relationship between instruction and assessment conceptualized in the Teaching-Learning Cycle. The role of assessment is to gather data about learners’ progress toward predetermined educational goals. Current educational policy stresses standardized tests, however, which stresses out administrators and teachers. Consequently, the MEAP becomes master of instruction rather than partner.

I rallied against this test-prep situation (and other standardized testing practices) for years, to no avail. The stakes were too high for anyone to consider an alternative. It was at a Michigan Reading Association Conference that I found a possible answer. (I learn a lot at this conference.) Patrick Allen led a session on developing deep, long thinkers in the time of standardized testing based on a book he had co-authored, Put Thinking to the Test. I thought that these ideas could reframe test-prep.

Put Thinking to the Test applies the research on proficient readers to taking tests. In other words, it treats standardized tests as a specific genre which readers can make sense of using well-documented comprehension strategies. Each strategy is addressed in a chapter demonstrating how it can be applied to a standardized test. From the table of contents:

• Ask Questions

• Create Mental Images

• Draw Inferences

• Synthesize New Learning and Ideas

• Activate, Utilize, and Build Background Knowledge (Schema)

• Determine the Most Important Ideas and Themes

• Monitor for Meaning and Problem-Solve When Meaning Breaks Down

Even in test-prep, teachers can focus on fostering thinking rather than memorizing facts or practicing skills.

As a result of reading this book, I now have concrete approaches that teachers can use to return instruction to its rightful place in the Teaching-Learning Cycle. I share these approaches in the classes I teach for preservice and inservice teachers and in professional development workshops. I would love to share it with you, but first I need to know what questions and concerns you have about what I have written thus far. After all, asking questions is one of the essential comprehension strategies propelling our thinking forward.

Why the Change?

The fourth key element in The Learning Network’s original Teaching-Learning Cycle was labeled “Teaching.” This was confusing to me and to the pre-service educators that I coach. Based on their experiences on the other side of the desk, they tend to simplify teaching to what happens in front of a classroom full of learners; this is certainly an element of teaching but not the entirety of the craft.

In order to be explicit about what teaching entails I modified the diagram. I changed “Teaching” to “Instruction,” but I continue to use the initial description: “Amount of support needed for new learning to occur.” Sometimes that means a demonstration, and other times it means an activity that allows learners to engage independently in a worthwhile learning task. Sometimes instructions falls somewhere in between.

Instructional decisions depend on the plans the teacher develops as a result of his or her evaluation of assessment data. That was the reason behind the change – to enable pre-service educators to recognize the role of each element in teaching and not just associate teaching with instruction. And so far the change seems to be working.