# The Building Blocks of Learning

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#### Figure 1. Robot Head Area Task

JANE: What do you notice about this student's work?ALICE: The student only measured the heights of the two heads.ROBERT: I noticed that the student knows how to use a ruler correctly, but he doesn't seem to understand the concept of area.ROSA: It was interesting that the student said the second head was "bigger" and didn't use the word "area."JANE: What do you think this student was thinking about?ROBERT: The student doesn't seem to understand that to compute area, you need to measure both the length and width of the shape and then multiply.ROSA: This student might not even know what "area" means.JANE: What would you do next in your instruction?ALICE: I would review the concept of area and emphasize that it is the amount of space inside of the shape. Comparing only the heights of the two shapes doesn't give you a comparison of the area.

## Incremental Learning

## Measures of Student Understanding

#### Figure 2. Hippopotamus Area Task

#### Figure 3. Thomas's Response

#### Figure 4. Excerpt of Learning Progression for Area Measurement

## A Look Back at the First Task

JANE: What do you notice about this student's work?ALICE: The student only measured the heights of the two heads.ROBERT: The student knows how to use a ruler correctly, but he doesn't seem to understand that he needs to measure both height and width to compare area.JANE: What do you think this student was thinking about?ROBERT: The student seems to only consider height as a way to measure area. Isn't this a common misconception in the learning progression stage of "perceptual coordination of attributes across figures"?ROSA: Well, we are not sure about that. In the task, the two heads look as if they have the same width. Maybe the student noticed this, and that's why he measured only the height.ROBERT: True. We need to examine this student's thinking further with additional tasks.JANE: What do you think you might do next in your instruction?ROSA: I would give the student two shapes that have very different lengths and widths to determine whether he recognizes the two dimensions.ALICE: I would try to rotate the shapes and ask him to measure again.ROBERT: We could also ask him to cut the shapes and try to "fit" one into the other. That will help him to master the idea of "conservation of area."

## Learner-Centered Practices

##### References

Baturo, A., & Nason, R. (1996). Student teachers' subject matter knowledge within the domain of area measurement. *Educational Studies in Mathematics*, *31*(3), 235–268.

Clements, D. H., Sarama, J., & Battista, M. T. (1998). Development of concepts of geometric figures in a specially designed Logo computer environment. *Focus on Learning Problems in Mathematics, 20*, 47–64.

Deuel, A., Nelson, T. H., Slavit, D., & Kennedy, A. (2009). Looking at student work. *Educational Leadership, 67*(3), 69–72.

Heritage, M. (2013). *Formative assessment in practice: A process of inquiry and action.* Cambridge, MA: Harvard University Press.

Langer, G. M., & Colton, A. B. (2005). Looking at student work. *Educational Leadership, 62*(5), 22–27.

Lehrer, R. (2003). Developing understanding of measurement. In Kilpatrick, J., Martin, W.G., & Schifter, D.E. (Eds.). *A research companion to principles and standards for school mathematics* (pp. 179–192). Reston, VA: NCTM.

Popham, W. J. (2007). The lowdown on learning progressions. *Educational Leadership, 64*(7), 83–84.