Most problems have multiple solution paths. Some solution paths are more efficient and some less so, but forcing a particular solution path down a student’s throat denies that student the chance to make sense of it in his or her own way. I think that part of the power of exploring problems and different solution paths is the sense-making that is inherent therein. I also think that some of the power lies in the chance to see how someone else thought about the problem, to have the chance to think about it another way. In thinking about a problem more than one way and trying to make sense of these divergent paths, there is also a moment when one begins to grapple with the idea of the efficiency of one’s solution path. At that point, after he or she has had the chance to make sense of something, the student can choose a path that is the most efficient for him or her (a supposedly efficient path is not efficient if a student can’t apply it correctly because it doesn’t make sense to him or her).
This idea of allowing students to find their own best path was one of the things in the forefront of my mind as I was putting together a foldable to summarize markups and markdowns this afternoon. I like to use foldables as my summary for a lesson from time to time because they bring together a lot of the ideas addressed in class into a single coherent document that students can use to study. Additionally, I have a twice-exceptional student who has difficulty with the physical act of writing. Giving him a foldable to complete makes it possible for him to be more successful.
In planning the design of the foldable, I wanted something that would give a whole/part/whole picture of the concept. This is really important to me because one of the students who will be using it this year is a twice-exceptional student who is pretty extreme on the conceptual end of the conceptual vs sequential continuum of thinking. He needs to see the whole of a concept to make sense of it and gets lost in the parts if he doesn’t see the whole first In order to give him the “whole”, I decided to start with the big idea. Next, I decided to compare/contrast markups and markdowns at each stage in the foldable. The use of this compare/contrast mechanism seems to really help him to make sense of things. Marzano’s research in Classroom Instruction That Works shows that it can have a big impact for all students (27% gain). Finally, I included two different solution methods for a markup and a markdown. These are the solution paths that I am anticipating students will have taken as they explored the problems.