The reason that Dan Meyer’s blog is a must read for me is that his posts, in addition to providing examples of great teaching ideas, are filled with links to scholarly articles to back up statements he makes. I find myself spending hours reading his posts, because I check out the links and become immersed. He is a doctoral student at Stanford, so his world revolves around this type of reading. When I went through my doctoral studies, I would happily spend hours in the library reading scholarly articles and culling information from them to back up (or not) my hypotheses on learning. Dan is helping me to relive those wonderful days immersing myself in the ideas of others. Every once in a while, after much sifting, I would find a jewel tucked into an obscure scholarly publication. Dan is doing the sifting for me. Thanks, Dan!

On his Riley Lark’s Red Dot post, which gives examples of worthwhile tasks which I will write about in another post, Dan opens with “We know there are important steps[pdf] you can take to ready students for an explanation of key concepts.”

Of course, I had to check out the pdf link to the important steps that we know about. I expected to see a checklist of steps. Instead, I found a pdf of the article “Inventing to Prepare for Future Learning: The Hidden Efficiency of Encouraging Original Student Production? in Statistics Instruction” (click on above “important steps” link) published in 2007 by Daniel L. Schwartz and Taylor Martin from the School of Education Stanford University, where Dan is doing his graduate work.

The authors argue that “invention activities, when designed well, may be particularly useful for preparing students to learn, which in turn, should help problem solving in the long run.” The phrase “invention activities” is used to describe “activities that ask students to invent original solutions to novel problems.”

By “novel problems”, the authors mean problems which students have not yet been formally taught a conventional way to solve. Rather, students apply their existing knowledge and persevere to figure out how to solve the problem, in a sense “inventing” their own procedure. This struggle helps students to uncover the essence of the problem, the particular issues that make it truly a *problem*, and not just an exercise.

However, this is not where the learning ends. The teacher then follows with a lecture that introduces the conventional way to solve the problem, with exercises following. By struggling through the problem without being told the “shortcut”, students have a much better depth of understanding of the mathematical issues that the problem presents. Being then shown a more efficient way to solve the problem, students are much more likely to transfer the new learning by connecting it to a context that they understand as a result of their struggle.

The study found that this method of teaching and learning, rather than rote memorization of procedures, prepared student to do well on standardized tests. I don’t think that standardized tests are a true indicator of student knowledge, but they are a necessary evil, so it is heartening to find out that the research bears out what I have long suspected: student learning of a concept is enhanced when they have to struggle a bit to make their own sense of a mathematical situation. However, this does not mean that students create their own knowledge and then move on. This could result in chaos. Once they have struggled to understand the problem situation, tried out their own approaches, and hopefully solved the problem, the teacher presents a conventional way to solve the problem. The very act of struggling causes students to more deeply understand the situation, and it creates a fertile “question mark “(?) eager to be answered. The teacher’s intervention then has a ready place to be planted in the many interconnected ideas of a student’s conceptual map of mathematics.

The authors accept the fact that “sequestered problem solving” (standardized tests) are and will continue to be the way that most educators assess student knowledge. As a result, teachers present students with procedures and mnemonics that will help them to do well on that type of test. The paper aims to show that student invention activities are worthwhile and can also result in high standardized test scores. I would add that they also lead to deeper and more long-term understanding.

The study focused on a High School Statistics course, but its findings can be applied to any math course. The authors acknowledge that simply having students create their own learning without teacher guidance or without well-chosen activities is no panacea:

Not any student production will help—“doing” does not guarantee “doing with understanding.” For example, Barron et al. (1998) found that children, when asked to design a school fun fair as a math project, spent their time excitedly designing attractive fun booths rather than thinking about the quantitative issues of feasibility and expense. Therefore, it is important to design productive experiences that help students generate the types of early knowledge that are likely to help them learn.”

The authors give an example of 9-10 year old students asked to solve fraction problems that indicate 1/4 of 8. One group was given pictures of pieces and were to circle the correct answer. Another group was given pieces that they could physically manipulate. The group given pictures typically circled one piece or four pieces. The other group that was able to manipulate the pieces were able to “reinterpret the pieces” by pushing them around and making piles. This allowed them to reinterpret two pieces as being one group. When the children could manipulate the pieces, they were successful three times more often than when they were given the picture. The authors summarized this experiment as follows:

Production (which in this case took the form of manipulating pieces) seems to help people let go of old interpretations and see new structures. We believe this early appreciation of new structure helps set the stage for understanding the explanations of experts and teachers—explanations that often presuppose the learner will transfer in the right interpretations to make sense of what they have to say.

The authors go on to say “Of course, not just any productive experience will achieve this goal.” One example of a valuable productive experience in High School Statistics is to have students compare and contrast cases of small data sets.

…students learn to discern relevant features by comparing data sets. Contrasting cases of small data sets, by highlighting key quantitative distinctions relevant to specific decisions, can help students notice important quantitative features they might otherwise overlook.

In this post, I can’t do service to their examples of these design features. The examples used are related to High School Statistics and are worth checking out in the original paper.

Important to the learning process is connecting new learning to previous learning. For new learning to occur, *transfer* needs to happen. In a lecture situation, the transfer is a transfer *in* from an outside source (the teacher). The paper asserts that there is another type of transfer that can be accomplished via inventive production.

A process called IPL (Inventing to Prepare for Learning) cycle is introduced in the second section of the paper.

This figure is not very clear, but is included in the paper which is linked to above.

The authors assert that the hidden value of invention activities are that they “prepare students to learn” the content at hand rather than teach the students. The invention activities set the stage for further learning through traditional lectures by the instructor. However, after delving into the activities, the students have a deeper understanding of the structure of the concept and are better able to make connections that allow the transfer of knowledge to happen.

In Figure 3 above, there are two cycles of invention/presentation couples. There could be more or less depending upon the demands on the topic.

In the invention phase, students work in small groups and invent their own solutions and representations. The solutions are shared on the board. An interesting twist is that, rather than have a student from the group explain the solution, another student NOT from the group is chosen at random to explain the representation and solution “as if they had been part of the group.” This twist makes it essential that students develop “representations that stand independently” and encourages students to “develop more precise and complete representations” as well as alerts them to “the importance of communicable knowledge.”

During these presentations, the teacher’s role is “primarily to help student articulation and point out significant differences between representations…The only hard constraint is that the teacher cannot describe the conventional method.”

Wow! This type of instruction must lead to a lot of sore tongues as teachers bite them to keep from going back to their formal role as the disseminator of knowledge. The good news is that this dissemination of the conventional procedure will come, but not until after the students have grappled with the idea via the invention couplets. After the lecture, students then practice using the conventional method.

I am anxious to learn more about the IPL cycle. I definitely see the merits of this approach. It meshes well with the CCSSM Practice Standards. I encourage you to read the entire report. It’s hard to do it justice in a few pages.