Homesteading as an Analogy for Math Curriculum

In my previous post, I commented on the transitional state of math curriculum in the district of my dyad placement. After years of an "investigational" type curriculum, the district is in the process of implementing a more traditional math curriculum.

My perspective on the issue is different after having spent several days in classrooms still using the investigative style and others piloting traditional texts. As a parent, I must admit to being critical of my daughter's "Discovering Geometry" text from a couple of years ago. I especially disliked the fact that geometric conjectures and formulas were stated with blanks where critical information would go. Because it was designed to be an investigative approach, students were not simply given formulas and such. Instead, they were to discover them and then, as we have also seen in our dyad placement, record important ideas in a separate notebook, commonly called their Toolkit. This Toolkit becomes the student's reference book.

Now I think I better appreciate the advantages of an investigative approach. In an effort to represent the differences between the two curricular approaches with a non-mathematical situation, I came up with the following analogy.

Suppose four families set out to "homestead" in new territory. They've brought along a few supplies, but they will largely be starting from scratch. One of their first tasks is to build simple houses. The homesteaders work together, and through a process of group collaboration and trial and error, they construct simple, functional houses. Although none of them has previously built houses, they have some prior construction knowledge and "discover" what is necessary to build a sturdy home. The first home doesn't have an adequate foundation, but once the problem is identified, the group develops a process for building a foundation using local materials. Similarly, the first chimney does not draft properly, but through analysis of the situation and more trial and error, they create a chimney that functions well. After much discussion, they decide to create sloped roofs, even though they are accustomed to flat roofs, because they fear the effects of northern snow. Each homesteader now feels that he or she understands the basic principles of house-building and could apply them in future situations.

Another group of homesteaders arrives nearby. They, too, bring basic supplies, but they have also brought with them copies of plans for building a house. They each choose a home site, and then refer to the plans to see how to proceed. They follow the plans, and the houses go up fairly fast. One foundation starts to wash away in the first heavy rain, and the homesteaders refer to the plans for help, but the plans don't tell how to fix a damaged foundation. They shore it up as best as they can. Another chimney doesn't draft well, and the house is often smokey, but the settlers aren't sure what to do about this either. They learn to live with it. Snow piles up on the flat roof, but this is, after all, what was called for in the plans. However, one homesteader has a natural, intuitive grasp of building principles, and his home is especially snug, sturdy and well-finished. Unfortunately for the rest, that winter the plans burn up in the drafty fireplace, and the homesteaders hope that they can recall enough to build new homes should they move again.

Investigation is more work. It requires meaningful collaboration. It takes a relatively long time, which sometimes means that students do not move on to the most computationally challenging problems. It usually results in internalization of concepts. Concepts can be more readily applied to new situations because they are understood at a deeper level.

Traditional math can move more quickly, as the teacher introduces a method for solving a particular type of problem, and the students follow the method to solve similar problems. Therefore, advanced students will move on even more quickly to more difficult problems. Group work is possible, but not essential as it is with investigations. Students who fall behind may find it difficult to keep up, because problem types build sequentially, one upon the other. Some students will internalize concepts, while others will memorize methods. Students may have more difficulty applying the concept to new situations.

So much in education can be visualized as a pendulum, and math curriculum is no exception. The one thing we can count on is that five or ten years from now, districts will be piloting "new and improved" curriculum, which will almost certainly resemble something seen before.