While reading the Bransford & Schwartz, one section really stood out to me because I remember it so well in my public schooling experience. They mention how “concrete examples can enhance initial learning” because it can “be elaborated and help students appreciate the relevance of new information.” When I was in school, there were two things that happened often. Thinking of my math classes, the first would be seeing examples of real life objects tied to the math we were doing. My seventh grade geometry class would have a tin can and we would talk about surface area of this tin can. I saw first-hand how that helped students visualize a cylinder and it did, in fact, help with initial learning. But just like the article says, this contextualization can “impede transfer because information is too tied to its original context.” Basically, giving just a single context limits the concept of a cylinder and the formulae for surface area and volume of cylinders to tin cans. While some could transfer this knowledge to other real-life cylinders, it was certainly not true that all students could. Especially when it intersected with another shape (transfer of problem to problem). Imagine a water tower that looks like a cylinder with a half sphere on the bottom and a cone on top, students might have been able to do each individual shape, but combining them seemed to be too difficult. I remember students in my class were wondering if there was a formula for the volume of such a shape (which is a fantastic question that students should be asking), but instead of trying to reason through the problem, they asked the teacher for help. Some only wanted hints and then figured it out after one hint. But students who did this began with “can we have a hint?” They didn’t first try to reason abstractly (and quantitatively) – a common core standard. Instead, they wanted to know how to get the answer. A question a toddler is capable of asking.

     My question is, at what point does contextualization go from being a valuable tool for understanding to over contextualization? I’m also curious if there is a difference in value for understanding if a teacher provides the concrete example versus if a student discovers it on their own? If a student does understand better when they contextualize a problem for themselves, how would they be able use this to begin a topic (for Bransford claims it enhances initial learning)?

     I guess in my thinking, I can see it being beneficial in both ways. In my (limited) understanding of project based learning, a student can receive a project that begins with context and has a student explore (e.g. “What is the length of r1 and r2 of this belt?”

) or they can ask a student to provide multiple examples of context for a problem and explain why those examples are valid (e.g. different functions such as f(x)=sin(x), f(x) = x^2, etc. [which all of this would follow many Standards for Mathematical Practice]).