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.

How will you organize your classroom of the future?

This past Wednesday was the last day for my Teaching and Learning Middle Grade Mathematics course. It was also the end of our unit on teaching mathematics, which meant developing an anchor chart to represent our current understanding (much like we did for doing math and learning math). Instead of having them summarize the content, I asked them to consider what their future classroom might look like given what we knew about doing, learning, and teaching mathematics.

The following video sums up the activity. This group of preservice teachers considered the role communication (learners placed in groups) and technology would play in their classroom but then they stopped and thought about their future learners. The result was surprising to me (the reason I like open-ended activities like this).

Cynically, I might think that they were trying to avoid the work, but they had already done the work. The final vision of their future classroom represented an authentic concern about meeting learners where they are at. The other groups agreed and I felt good about what they had learned this semester.

We spent the rest of the class watching a TED Talk and discussing the future of education. It wasn't the end-of-the-year party that a lot of classes were having, but it was a celebration. I love my job!

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.

Why were they placed in pairs?

I wrote before how GVSU's Teacher Assisting semester is a fairly unique program. The Mathematics Department has gone one step further by placing pairs of Teacher Assistants [TAs] in classrooms (or as a cohort at a school). We have found that this approach has supported the TAs in numerous ways. The biggest benefit is that it helps them to break from the culture of isolation so prevalent in American teaching.

Today, the TAs did their final presentations. I wanted to try out a new app, MindNote, and created a concept map showing the connections made as they reflected on their experiences this semester. As you can see below, collaboration was important to many of the TAs.

What was interesting was the number of TAs who came into the semester skeptical of being placed with a peer. They thought that their experience would be diluted if they had to share a classroom. Instead, they found it a powerful and positive opportunity to improve their practice.

In a recent speech, Education Secretary Arne Duncan said we need to do more to attract great talent to teaching. Listening to the passion and commitment of these future teachers, I would respond that we do not need better teachers; we need to support better teaching. The best and the brightest are already here.

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.

What is job-embedded professional development?

The following is from the talk I gave at the NCTM 2011 Annual Meeting an Exposition. The title of the talk, Developing Teachers’ Algebraic Reasoning through Job-Embedded Professional Development, refers to our attempts to design a professional development experience based on teachers' immediate identified needs. This is the description we wrote almost a year ago:

Presenters will provide a model of job-embedded professional development used to support the teaching of algebraic reasoning in middle school.  We will explore how the use of context and representations enhanced content and pedagogical content knowledge to help support learners as they move from the concrete to the abstract. 

My co-presenters (Esther Huntzinger Billings and Mary Scott) and I struggled in developing the final presentation because things had not gone according to plan since we submitted the proposal last spring. Still, we were able to talk about what worked, what didn't, how we adjusted, and where we are headed. Under the circumstances, I think it went well. Here, see for yourself:

If you require more information about specific slides from the presentation, please use the comments to share your request.

How many now?

Hierarchical inclusion is the idea that whole numbers build by exactly one and that they nest within each other by this amount. (Fosnot & Dolk, pp. 36 - 37). Understanding hierarchical inclusion leads to the big ideas of decomposing & composing numbers and part-whole relationships. Consequently, hierarchical inclusion is an important concept learners need to experience and explore early on.

A few years ago, I wrote the book In Flew One More as a way to introduce hierarchical inclusion to early elementary learners. The book is a series of poems and illustrations that leverage learners desire to find patterns. Through reading and discussing this book, the idea of numbers building on by one is explicitly explored.

Given that April is national poetry month, I thought it was an appropriate time to share my book.

And here I am reading In Flew One More in my wife's first grade class this past Tuesday.

Finally, these are some cards related to the book that I made for a learning center on hierarchical inclusion.

I hope you enjoy National Poetry Month!