Cake
• In this example, a 3 by 5 rectangle stands for the number 1. That choice makes it easy to represent the fractions we are adding. By examining the figure, we see that the sum is 13/15. Doing this once, of course, would not be very helpful, but discussing the dimensions of useful rectangles, leads to a powerful strategy which can be applied to comparing, adding, or subtracting fractions.

Have students make things. In some cases, a constructionist approach can yield much learning. Constructionism is a theory of learning espoused by the Logo movement in the 1980’s, and it is still going strong with Logo descendants such as Scratch and Snap, or with the current popularity of maker spaces and STEM. To enhance math education, what students make needs to have a curricular payoff. It can be a turtle geometry computer program, a GeoGebra construction, a tiling of the plane, etc.

For an example of the latter, using a plastic template and a pencil (or a computer), students can create tessellations using triangle or quadrilateral tiles. Here is one based on a scalene triangle:

• This figure offers a context to discover and discuss many basic geometry theorems: sum of the angles in a triangle, exterior angle, parallels and transversals, translations and rotations, ...

CONCLUSION

Pursuing any of these suggestions should enhance discourse in a math class: who talks? who listens? who does the thinking? The key is to keep the students at the center. We, their teachers, can help, but they are the ones who need to do the learning.

Here are some relevant posts on my Math Education blog:

Taking Notes vs. Doing Math

Random Groups

Retakes vs. Test Corrections vs. Neither

Understanding “Understanding"

“Enrichment"

• Since you’re retired from the classroom, perhaps this is no longer an impertinent question: for how many years after becoming an NQT (new teacher) did you “Aim for the middle” with your teaching?

• I don't know how to answer that. Making changes in one's teaching is a slow and gradual process, with many dimensions and components. Some milestones for me were, more or less in that order: switching to a collaborative learning model; the gradual introduction of many learning tools (manipulatives and technology); lagging homework and assessments. Each step along that path helped me reach a broader range of kids. But I am not able to put dates on any of it.