[Kristin Umland]

A couple thoughts:

* First, remember that because something is not in the standards doesn’t mean teachers can’t address it. The main point about these topics not being in the standards is that the assessment folks should not be writing items that test whether students can apply the vertical line test or pick out sets of ordered pairs with a particular property. These are procedures that aren’t very interesting when extracted away from their reasoning purpose.

* However, students should be able to look at a graph of x=y^2 and note that, for example, the value x = 1 corresponds to y = 1 and also y = -1, so it is not the graph of a function. Note that this is not the same kind of argument as applying the vertical line test, because it connects back to the definition of a function. The argument, “The line x=1 intersects the graph in two places so the graph is not of a function” is a black-box explanation for most students–they are told that you do such-and-such, and you interpret the results in some way–it is like reading tea leaves or consulting the oracle, but does not constitute mathematical reasoning.

The problem with standard questions about functions that ask students to employ the vertical line test or to look at ordered pairs is that students almost never realize that these are fundamentally the same kind of investigation: if you took the list of ordered pairs and plotted them in the coordinate plane, applying the vertical line test amounts to the same thing as inspecting the ordered pairs and looking for x-values that correspond to different y-values. In other words, for most students, these are completely disconnected procedures rather than different manifestations of the same kind of mathematical argument, one that relies on the definition of a function to determine if a relation is a function. We want students to be able to reason from the definition of a function to determine if a relation is a function; we don’t care if they can enter the correct letter when prompted, “Apply the vertical line test and mark y or n for whether the graph shown is the graph of a function.”

[Kristin Umland]

Here is how I interpret the standard: Students can solve two-step word problems. They can also represent such problems with equations that use a letter to represent the unknown. That doesn’t mean they are required to represent them like this at all times (in fact, some times they might solve them simply be reasoning about the situation and not write an equation at all). I would imagine that teachers would work students up to such representations, possibly using question marks or boxes at first and then having them graduate to letters. So by the end of third grade, the ability to represent word problems with algebraic equations is one of the tools that students have in their tool belt, but like all tools, it doesn’t get used for every single construction project.

This doesn’t solve two issues related to the illustrations of this standard at the Illustrative Mathematics website that I think your post brings out. First, the set of tasks that illustrate this standard need to include some tasks that appropriately address the symbolic expectations of the standard–I don’t think the task that is there needs to have that representation, but we need some tasks that do. Second, in looking at this task, it has been moved around some and also it has been rated down a couple of times (I wish people would post comments explaining what they don’t like about a task so we could fix it), and to be honest, it isn’t my favorite task for illustrating this standard anyway (although I like the task fine). So for this reason also I think that having this task as the only task illustrating the standard is a mistake.

I can imagine a task that would have some story problems and some equations and might ask students to match them might be good. I’d like to see story problems that match to multiple equations (maybe, for example, a problem involving subtraction that can be represented by an equation involving addition and another involving subtraction to help emphasize the relationship between those operations). What do you think of that idea?

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