I paused the Act 2 video after the first weighing:
How much might each bottle and glue stick weigh?
A bit slow to start, the students nevertheless went on find a variety of possible answers. To some groups who were still stuck after a few minutes, I suggested a weight for a bottle and asked if they could go on to find the weight of a glue stick. Other groups decided to make some assumptions on their own: e.g., that the weight was distributed evenly between the 4 bottles and 5 glue sticks, or that all of the objects “by some miracle” weighed the same (“there’s lead in the glue sticks”). A few groups wrote an equation and made a table of possible values by assuming a weight for one of the objects and finding the corresponding weight of the other; one group solved the equation for one bottle, and used it to argue that it could have a wide range of weights, depending on the weight of a glue stick.
Although a small number of groups wrote equations right away, recognizing a system and solving it to find the weights, many had trouble getting started. I suggested writing equations as a start to several. A few groups did something I didn’t anticipate: they assumed that the two weights, 679 g and 680 g, were basically the same, which led to some really great reasoning. I was blown away by the following, for example:
The two equations are 5x + 4y = 679 and 12x + 3y = 680. We noticed that when you add 7 glue sticks and take away 1 bottle, you get virtually the same weight. So, 1 bottle is approximately equal to 7 glue sticks. By the second equation, 33 glue sticks = 680 g, so 1 glue stick is about 20.6 g and 1 bottle is 20.6(7) = 144.2 g.
How awesome is that? The two groups who made this assumption used an algebraic approach (5x + 4y = 12x + 3y) to find that 1 bottle ≈ 7 glue sticks (y = 7x) and used this information to find the respective weights. The difference between the weights found using the different strategies was essentially negligible, which served to validate the responses.
I absolutely love moments like these, when students surprise me with reasoning that I did not anticipate, and I’m reminded of the importance of giving students space to use their own strategies to solve problems. Yes, there is an argument to be made for efficiency, or precision, depending on the situation, but this is a discussion that can occur once there are several strategies to compare. (This is also what happened during last week’s trashketball task, where students slowly came on the algebra train once they were tired of making tables of values. We all eventually agreed that algebra was more efficient and gave a more precise answer.) It also gives me the confidence that students don’t need to be spoon-fed procedures before they can “apply” them to “challenge” problems at the end of a homework set. On the contrary, a good problem can function as a sandbox where procedures are developed and refined.
On the other hand, during the same period, I was reminded of how a lesson can fall flat on its face when students are forced to use procedures that don’t arise from their own work and reasoning. Some backstory: for the past few days, my students have become quite confident in solving systems of equations by solving both for the same variable, then setting them equal to each other: e.g.,
5x + 4y = 679 → y = 679/4 – (5/4)x
12x + 3y = 680 → y = 680/3 – 4x
→ 679/4 – (5/4)x = 680/3 – 4x, etc.
This strategy arose from the trashketball activity. They’re looking for when x, y are the same on both lines, so solving for y (or x) and setting the two equations equal to one another makes sense to them. It also has the advantage of making it easy to compare slopes and determine how many solutions the system has. The only “disadvantage” I see is that it often results in equations that have fractions (which many of my students struggle with – something we need to work on), and I wonder if this is why the methods of substitution and elimination are privileged in textbooks.
However, I felt obligated to teach substitution because of its prominence as a strategy for solving linear systems. I anticipated the conceptual difficulties students would have, which is why I held off until today, but nevertheless I persisted in demonstrating the procedure for the above problem, after students had shared their own solution strategies, and having the students practice it for a set of problems. A few students were able to catch on, but the majority – especially the strong students – were lost. Why can you just substitute the expression into the other one, without solving for the same variable?, they wondered. I floundered in my attempts to help students understand, because I didn’t have a good model to refer to. Near the end of class, I told students to just use a method that made most sense to them.
I think elimination will be easier to tackle because balance scales provide a good model for building conceptual understanding of the method, but I’m left wondering whether substitution is something I should bother with going forward, when students are comfortable with a strategy that is arguably no less efficient and conceptually easier to work with. Consulting the curriculum, I notice that it only states that students should learn how to solve systems “algebraically,” which suggests that I should follow my intuition – to privilege understanding over rote procedure. But am I doing a disservice to my students?
This episode brings to my mind an article I recently read by Love and Pimm (1996), titled “‘This is So’: A Text on Texts'” (it has one of the best opening lines I’ve ever read: “In the beginning was the logos and the logos was the vehicle for mathematical expression.”) In the article, Love and Pimm describe how, and by what means, the textbook has come to dominate both perceptions and practices of school mathematics. In particular, they argue that the “immutability and perceived authority” of a textbook, which inherits some of its status from the quality of being “designed by experts,” tends to limit experimentation and encourage subordination to at least some extent, sometimes even in spite of the author’s intentions (p. 400-402). Although I don’t feel particularly bound to the textbook, which I rarely refer to during lesson design, I recognize this as being part of the tension I’m feeling: a tension between the perceived need to conform to a mandated resource and what I feel, based on my observations and interactions in the classrooms, is a more appropriate strategy for my students (at least, for the time being). In this particular tug of war, I think the latter will win (but please, if you have very strong feelings about teaching substitution, comment below).
Of course, despite a rhetoric of “inquiry” and “discovery,” conformity is not something we can escape within an education system that mandates certain outcomes for all students. This is the reality teachers must learn to operate in, but I wager that most (me among them) wouldn’t favor anarchy in its place. However, it’s important to recognize when we do have freedom to move within a set of constraints (which can be productive, as Davis & Sumara’s notion of “enabling constraints” suggests). And as I’ve found, sometimes when you reach out and expect a boundary, you only find more space to explore.
Davis, B., Sumara, D., & Luce-Kapler, R. (2008). Engaging minds: Changing teaching in complex times (2nd ed.). Mahweh, NJ: Lawrence Erlbaum Associates, Inc.