One of things I’ve been thinking about recently is the metaphor of problem-solving as “traversing a path”, and the ways in which the metaphor influences instruction. I don’t want to spend too much effort defending this claim, but I think it’s somewhat defensible for me to say that it’s one of a few primary metaphors we use to conceptualize problem-solving.
- For example, there are problem-solving “steps” (and even missteps) and “paths” / “pathways”.
- We might say to students, “You’re almost there!” or perhaps remark how “you got there a different way than me,” or “I think we’ve gotten off track somewhere,” or “You took the long way,” or “That’s an interesting short cut.”
- It implies there’s some starting point, ending point, and the goal is to construct a path that gets you from the one starting to place to the one end place.
- This metaphor allows us to bring in language such as dead ends, obstacles, and impasses.
- We make use of this metaphor I think when we want to emphasize to students that it’s not the answer that’s important, but the solutions. We are saying, “It’s not the end point that matter as much as the path that gets you there.” In some sense, this makes problem-solving metaphor includes “route-finding” not just “path traversing”, and like even “orienteering” since you don’t even always no where you are and where you are headed.
Problem-Solving as “Map Making”
I guess I’ll start this blogpost by putting up front a slightly different alternative metaphor–one that is very Skempian, I suppose. I want to suggest here that something more akin to “map making” needs to be a part of the collection of metaphors that accompany “route-finding” and/or “path traversing”. I’ll try to motivate it this way. If the mindset we want students to avoid is being to focused on “answer-seeking” (getting to the destination) and instead focus on “route finding” (constructing and traversing path), I might suggest that we as instructors might be making a similar mistake at a different level. We are focusing on problem-solving as “route finding” when we might be better served as seeing the activity as “map making”.
So here are some of my thoughts:
I think one of the concerns I have with “path traversing” is that there is often no landscape to even traverse, at least not initially in problem-solving. One has to explore the territory and then actually construct the landscape. The landscape is not simply there to be traversed. Now, I think we do kind of have language for this, like the whole metaphor of a “problem-solving space”, but the path-traversing metaphor draws attention to the path as the end-product. And the path is sequential in its nature. Certainly, in order construct that path you may wander around and explore, but the wandering and exploring is not the point. And so we admit, as experts, that it may take a non-sequential process in order to produce an end-product that is sequential (a traversed path).
I guess I should state that I don’t think I’m really adding anything new about what we know about problem-solving. We know that problem-solving is typically taught poorly, etc. Perhaps, what I am trying to add is is the idea that problem-solving is taught poorly (in part), because it’s wrapped up in this implicit metaphor. What I suspect might be true is that changing the way we teach problem-solving requires a new metaphor. And I don’t mean abandoning the old metaphors, because I’m sort of just using the same metaphor and taking a broader perspective. Students are focused on destinations, we want them to be thinking about paths. We are focused on paths, when we might be better served by thinking about maps?
So how might the map-making metaphor help? I think that map-making may help draw attention to a different end-product. There’s still, in the metaphor, a landscape to explore and routes and such, but the point of map-making is not to construct paths. It’s to know the terrain well enough to make a map–maps (I think) are useful in that they show relationships among parts of the landscape and they also foreground/background certain parts. Maps are not sequential, they are relational Any given path on the map is sequential. Hence, my reference to Skemp’s instrumental and relational.
I should probably get more concrete with this idea. So here is a student solution that I think is more “map like” then “path like”.
It shows a lot of relationships. And certainly, in this map, you could probably route a path (or paths) of how they got from some beginning point to the final point.
So here what’s I’m interested in continuing to think about:
- Is this the useful twist on the metaphor? Or if not new metaphor, metaphorical extension? How so? Why not? What other metaphors may be useful?
- How might awareness of this metaphor help instructors to work from a different vantage point on “problem-solving” that can support students? Perhaps it can shape the way we model, ask questions, or set the stage for problem-solving?
- How might the adoption of the metaphor influence the way we as instructors think about assessing student work? Perhaps the path metaphor draw our attention to (show all the step), and with map-making it might draw attention to (shows relationships).
- If students think they are constructing maps rather than traversing paths, what different attitudes about problem-solving might develop? Will they talk about their work differently? What upsides / downsides might there be?
I want to say again that I don’t think that this “map-making” metaphor that I’m proposing is necessarily novel, nor do I think that it will necessarily generate any new /novel instructional practices on problem-solving. But, it may be a useful way to think about (and package) practice that are already “map-like”. My practices with problem-solving were changing before I had the metaphor, and I’m curious if the metaphor will allow me to further hone my practices and or better communicate the practices.
I’d be happy to know any of the following, plus anything else:
- How I am getting this metaphor all wrong?
- Examples of your students’ work that you think is more “map-like” or “path-like”?
- Pointing me to places that help me see where I’m just reinventing the wheel?
I can’t believe we are week into July.
Have you read any of the Fosont and Dolk “Young Mathematicians at Work” books? They each have a “Landscape of learning” chapter which has a lot of useful exploration of similar ideas. Get yourself a copy of the one about place value, or about multiplication and division (don’t get the algebra one, which came last and in which things get a little goofy)
I have not. I’ll order it today!