In contrast to what I wrote in a previous post about disliking word problems relating to solving systems, I found myself returning to the topic with a new approach that I really liked. I've read through the presentations of the Center for Mathematics Education (CME) project, and have gotten an idea of what they do through the examples they present. I've been very interested in getting actual copies of their textbooks, but haven't gotten around to it both because of my location (no shipping to China as far as I know) and because, well, other things have occupied my time.
I really like the general theme of how mathematical thinking is closely aligned with the sort of logical thinking we already do. The concept of 'guess-check-generalize' makes sense especially in the context of what I always find my students doing anyway when I present them with a word problem. See this post on the CME blog to get an idea of what it's all about. In the past I have tended to use verbal problems, especially in the context of systems of equations, as a way of reinforcing solution methods of these systems. I have also found many students will naturally use a brute-force guess-and-check method of trying to solve them. I was consistently impressed when kids with low levels of number sense and arithmetic ability would fall upon a solution to a system of equations after a period of deliberate and focused trial and error. Why were these students so willing to spend ten minutes trying a bunch of solutions while being unable to sit and listen for five minutes on how to solve it methodically? Was what I was presenting so abstract and disconnected that the obvious method that made sense to them but was a lot more work was clearly the better choice? Clearly so.
My response early on in my teaching career was then to give systems of equations that they would NOT be able to solve by brute force. Systems with solutions that were decimals and fractions were much less likely to be figured out. Doing this though felt so arbitrary. If I have to modify the questions I was asking in a contrived way in order for my algebraic method to finally become the better solution to this group of students, there was something wrong with MY presentation and application of the mathematics, not with the students' method.
This is part of the reason I would get frustrated teaching verbal systems of equations as part of solving systems of equations. The situations that came up (in most textbooks that I read) were made in a way that they fit the solution methods that were simple to solve using elimination or substitution. A person could know almost no English and still figure out a system of equations that most likely solved the given problem.
The thing that seems different about the guess-check-generalize framework though is that it encourages the type of self-aware mathematical thinking that we want students to do. This was the first time I really presented a problem this way, but it seemed to work well, particularly in the case of some of the students that have demonstrated both weaker math skills and/or a limited English proficiency. I gave them a problem of this type:
A store has a sale on sneakers and shirts. Tyrone buys three shirts and two pairs of shoes for $225. Maria buys two pairs of sneakers and five shirts and pays $325. What are the prices for a pair of sneakers and a shirt?
When I asked students to guess a solution to the problem, one student immediately 'guessed' that the answer was "2x + 3y = 225". It was a great moment telling the student that if he went into a store and asked a salesperson how much a shirt and pair of shoes was, and the salesperson started spouting off an equation, that salesperson would most likely be smacked in the head and fired for being unhelpful. It makes no sense to respond to a verbal question with an equation, but that is what students (including mine, unfortunately) have been conditioned into doing. With that expected response out of the way, we could move on with the guess-check-generalize model.
I decided to call the answer a "model answer" instead of a guess - I have an ongoing battle with students about how much I hate the word "guesstimate" because people tend to use it to make a true guess sound more authoritative by connecting it to the very different word estimate. I asked a student what a possible answer could be to the question.
What was pretty interesting was that the rest of the creation of the mathematical system came naturally from this guess. What were the variables? Since what the question was asking us to find was the unknowns, the quantities found in the model answer were what we would probably model in a system of equations. There was no argument this time about how X was not equal to "SHIRTS" and Y equal to "SHOES" - instead it was plainly obvious from the model answer that we were guessing a price of a shirt and a price of a pair of shoes. Here was how the legend appeared:
The system of equations came just as easily. No teaching the formulaic way I once did that "for a cost type equation, you multiply the x-cost by the x variable, add it to the y-cost multiplied by the y-variable, and set it equal to the total cost." Instead, we just found what the cost would be using our model answer:
If the model answer had been correct, then the cost would have been $70. I targeted this question toward one of the students that I was more concerned might not understand the whole process, and it seemed to come naturally. Clearly the $70 was wrong, but the students were actually thinking about this fact rather than blindly putting together an equation. The calculation using the model answer not only did this for them, it screamed out to us what the actual equation had to be. Smooth as silk.
It was exhilarating seeing this work with my group. Granted, they are generally a pretty strong group, but verbal problems like these (especially given the international make up of this class) tend to make them all visibly uncomfortable. This worked much more smoothly than any of my previous lessons. I certainly have tried to get students to think this way before, but never explicitly used a guess to generate the rest of the equation. For those ESOL students, it seems like a non-threatening first step to come up with an example of what an answer to the question might look like. This idea could help all students that have a tendency not to read questions all the way through and guess what they are being asked to do.
It is very possible that I'm just late to the guess-check-generalize party and teaching using this method is obvious. If that is the case, I apologize to my students for getting it wrong for so long. I see a lot of the similarities between this and modeling, which I've really enjoyed using with my students through exploration of Newton's laws. Maybe the parity between them is why I'm suddenly so excited about the overall concept.
In the end, I do have some more tricks up my sleeve for how I want to use some actual, interesting, realistic, and authentic problems with this group. The robot crash went really well and the students enjoyed that activity. I go back and forth as to the benefit of giving them word problems like the one we worked on. They exist in math world, the world of math textbooks, but not so much in the reality of us as math teachers trying to teach what authentic mathematical thinking looks like.