Hello Everyone!
I have kind of fallen off of the “one post every two weeks” wagon, but I will keep trying to get posts out as regularly as possible! Today’s post is related to a blog I found via the MathEd Twitter. This blog, written by Matt Felton-Koestler, discusses the use of context in math problems (the blog can be found here. In general, the “context” of a math problem relates to how it is being asked (is it simply 10 times 9, or is it related to buying carpet for a remodeled room). As I have stated before in many blog posts, the context is an important factor in students’ construction of knowledge. The more meaningful and related the context, the more students will take from the lesson. Now, looking back at Felton-Koestler’s blog, I will not rehash everything he said in his post, but his main points boil down to:
- Why is context important
- Real-ish problems (problems that have a real world connection that students can relate too, but are also not completely meaningful, such as splitting 5 pizzas between 3 people)
- The problem with psuedocontexts (contexts that are so grossly unrelated that they convince students that math is unrelated to real-life)
- The problem with real-ish contexts (students start to randomly manipulate numbers in the problem even if they are unrelated, for example, the Shepard Problem).
This blog resonated with me because it connected to two different ideas I have been thinking about recently. One of these ideas is the IMP curriculum we use at ARHS (see my blog post titled “Interactive Mathematics Program”). Since I started at ARHS, I have often thought about the context of IMP problems. Many IMP activities have meaningful and productive contexts. For example, one sequence of tasks has students investigate the perimeter and area of different polygons to determine fencing on a ranch. This context is meaningful, engaging, and can be adapted to non-ranch contexts as well. However, some activities have contexts that are closer to the dreaded psuedocontext. For example, another series of activities involve using permutations and combinations to determine the different combinations of ice cream scoops in an ice cream shop. This context is unrelated to the problem and could be easily stripped away. Therefore, I think IMP has great meaningful problems, but I believe there is much work to be done to make sure all activities have a productive and engaging context.
The other idea that came to mind while I read Felton-Koestler’s blog is the context of mathematical modeling problems. This is an idea that has come up in my UMass course work. While reading the NCTM Annual Perspectives on Mathematics Education 2016 book on modeling, my class came up with a conjecture: “is a modeling task defined as being a task that requires the context to be solved”? In other words, for a task to be a modeling task, the context cannot be stripped away. Therefore, the ice cream shop example would not be a modeling task, because it can be solved by either stripping away the context, or using a completely new context all together. However, in the fencing task, the context becomes more integral to the problem as a whole. All these ideas bring me to a few questions: what is the connection between real-ish problems and modeling problems? Are they the same? Are they different? Which one is more effective for our students’ learning?
As a pre-service teacher, this is an issue I am struggling with. Since I have less than two years of experience under my belt I am trying to lesson plan, grade, and think of how to anticipate, problem pose, and sequence lessons. Adapting curriculum to include these meaningful contexts is something that is a little over-my-head at this point. That being said, of course this is still something I should be thinking about and trying to work on. In fact, on occasion, my mentor and I have adapted some IMP tasks to be more relevant to our students. For example, instead of looking at the ice cream shop problems, we used the idea of combinations and binomial probability to look at potential racism in jury selections from famous court cases. This task seemed to hold students’ attention and required them to think critically about the world around them. However, these adapted tasks are few and far between. Nevertheless, these ideas of context, real-ish problems, and modeling are ones that I will be battling with for most of my teaching career.
As always, I would love to hear what others think! Please comment any ideas, thoughts, or suggestions.
See you next time!