“Where can you see …?”

RME encourages learners to make explicit connections inside and across multiple representations.

For teachers new to this idea, the question “Where can you see the diameter length in this circle?” can seem puzzling.

It may seem like a trivial inquiry. Yet, it is in coming to the board to trace out yet another diameter with their index finger, that students also begin to make sense of this abstract mathematical shape as a context linked to previous classroom explorations: swimming in a circular swimming pool, sewing circular designs in sequins, measuring the edges of circular mirrors, waste bins, toilet paper rolls, etc.

“Where can you see …?” is a question that supports students in explicitly linking up mathematical experiences. It aims to build on familiar ideas, by encouraging learners to seek out familiar relations in novel mathematical representations or strategies. Especially when paired with “What’s the same? What’s different?”, this questioning strategy can help your students to:

  • solidify and expand their conceptual “reading” of a single image/representation
  • work across representations, following the flow of information from one mathematical form to the next
  • compare student methods and ideas.

Looking again, but seeing more

As in the example above, this question from Fitting It In (G1) — “Where can you see a distance of 12 in this shape?” — will seem odd to students from the start. Interestingly, however, many students will answer it by circling the number 12 where it appears on the diagram. Even after pushing learners to really show where 12 is in the shape, they may get stuck after two rounds of answers. The top and bottom of the rectangle are not the only lengths of 12. Where else can you see 12 in this shape?

Although it may seem obvious to everyone in retrospect, “Where can you see a distance of 12 in this shape?” pushes students to search out and explicitly identify an infinite number of “12s” in this rectangle. Ultimately, this a conceptual framework for area that will still serve them well in calculus class. But, at any stage in their mathematical career, helping learners to “see” and literally draw out their tacit knowledge about dimension will be a huge asset, especially as they progress in geometry.

In fact, regardless of the mathematical domain, the question “Where can you see … in …?” allows learners to zoom in on mathematical concepts — diameters and lengths but also variables and number relations — without the interference of abstract definitions or teacher driven explanations.

Exploring new models and representations

This slide from Drawing from Data (D2) shows the maths test results for two Year 8 classes. After exploring the test result data on their own terms, students were asked in early slides to evaluate: Which class did better on the test?

In this slide, they are given an example of how two students worked together to answer this question (the table in blue). As you’ll see from the table’s label, Jack and Jill found the middle scores in each class, as well as the top score and lowest score. They also identified the “bottom 25% point” and “top 25% point.”

The slide asks students “Where can you see these results in the list?” and “Where can you see these results on the dot plots?” In doing so, it prompts learners to identify median and quartiles in both an ordered list and in a dot plot. But, to answer these questions, learners do not necessarily need to know what the median and quartiles are. The question “Where can you see…” encourages them to figure out how these two representations might be related. Building on familiar forms, toward knowledge of novel representations is a powerful application of this questioning strategy.

Making links between student strategies

Our last example of “Where can you see…?” comes from a slide in Knowing the Unknown (A1). A traditional approach to this balancing situation would prioritise solving the puzzle. Instead, on this slide learners are asked to first describe what each student has done. This requires them to focus on strategies for reasoning about equivalence rather that becoming absorbed in the solution to a particular problem.

The question “Where can you see…” pushes learners to move from the answer back to where this answer can be seen in the balancing context situation. This example emphasizes how “Where can you see …?” has applicability to every lesson. It serves as a way of focusing the classroom on a particular student’s mathematical strategy and to build stronger connections between student ideas.

The question “Where can you see …. in …?” arises from the need to help learners make explicit connections inside of or between representations. We can see the potential for these connections on the landscape diagrams where the contexts on the bottom layer of the landscape have a connection with the ‘models of’ those contexts on the second layer of the landscape and beyond. For example, the subway sandwich representation in Fair Sharing (N1) becomes a bar model representation. In this case, you might ask “Where can you see the sandwich in the bar?” In Sorting It Out (D1), stacks of newsprint become a bar chart. Here, we might inquire: “Where can you see the newspapers in the bar chart?”