Task intent (part one)
I thought i’d start this blog by discussing a topic that will no doubt be a recurring theme here: task design in mathematics. Specificailly task intent, and the challenges of both communicating, and interpreting the author’s intention when teaching. Communicating the intent of a task is difficult, but it’s an element of design that needs careful attention if an author stands any reasonable chance of another teacher using their materials effectively. More broadly, we’re talking about the difference between the intended outcome, and the enacted outcome. Let’s first examine two different worksheets covering the same concept - the area of a triangle (apologies for such a predictable topic).
A quick google search brought this example to my attention:
For what it’s worth, this feels like a poor, computer generated worksheet - but we can analyse all that in another post. One thing that is clear is that it’s simple. We’re presented with six triangles, with what looks like 4 unique shapes (q1 and q5 look the same to me, as do q2 and q6) and we’re provided with only two measurements for each triangle, which are conveniently (and intentionally) the two that we need to calculate the areas (the formula is also handily printed at the bottom of the sheet). Well… fine, but let’s now consider the intent of the task. We could crudely assume that the intent goes no deeper than ‘we want students to be able to find the area of a triangle’, but let’s consider for a moment that we actually want to use this sheet, and that the author had thought it through (generous perhaps).
Observation 1: Mixing units
The task uses different units for each question (centimetres, millimetres, and giant triangles using metres). Remember, we’re not the author here, and this wasn’t designed for our students. So the questions this prompts are:
‘is the intent of this task to practice the area algorithm with a focus on unit accuracy?’
‘Is there an assumption that unit conversion was taught recently, and is being reinforced here?’
Again, we’re being generous - but these questions aren’t just about trying to understand the intent of the task, they also by extension help us determine the suitability of the task for our own students, and how much we may choose to adapt the task should we choose to adopt it.
To answer these questions, note that the units are actually provided in the answers as well, so there isn’t any need for the students to even pay attention to them. You could argue therefore that the units of measure are in fact superfluous here and distracting, or maybe we want to keep them and draw attention to the change between linear centimetres and square centimetres for example. In either case, we’re beginning to shape our own intent. It’s important to note that when considering intent, what a task is not doing is just as important. If we know, for example, that this task is not developing a student’s ability to convert units, then we know we’re going to have to do that elsewhere with a different activity.
Observation 2: Orientation and number pairs
Ok that’s two observations in one, but it’s clear that this worksheet will not develop any real depth of understanding of the area of triangles - all the triangles have a horizontal base (how do we know that students can cope with different orientations if we only use these examples?) and all of the triangles provide only two numbers for a calculation that needs… two numbers. So we will not develop understanding of the appropriate selection of a ‘base’ and ‘height’ when presented with choice.
So there’s a lot of things that this worksheet doesn’t do, and despite it being generally quite poor (sorry), that doesn’t mean it’s not useful at all (paper aeroplanes or origami for example). If our intent is for students to simply practice the algorithm to calculate the area of a triangle, then this worksheet does do that. Is that a bad intent? I’d say no, but it’s a really small part of teaching and developing understanding around the area of a triangle. What’s more important is the decision to determine what the point of the task is, what it does and doesn’t do, and how it might need to be adapted to better fit with your own intent. The things that it doesn’t test - where are those gaps going to be filled? For example, it’s entirely sensible to start with just the base and height measures, but when will I introduce the added complexity of having to determine from all three measures which ones to use? Is there space to teach this isolated skill or does it need grouping with others? All these decisions are personalised, and are unlikely to be resolved without some tailoring of resources.
That was just a warm up. I’m going to leave you with a more complicated example and a few prompt questions.
The above example is more complicated and nuanced. Each question is designed to progress understanding in some way - but how easy is it to determine what the point of each question is? Why are there grids, then no grids? Do the measurements matter? What is the author trying to draw attention to? How do we ensure that a teacher doesn’t just hand out the sheet and mark the answers, but instead considers purpose and pedagogy? If you have time, share some thoughts in the comments, I’d love to read them. I’ll discuss this more in a second post soon.
