# “Where can you see … in …?”

One of the important ways RME design facilitates mathematical development is to encourage learners to make connections across various representations of a situation. 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 becomes a bar model representation; the stacks of newspapers representation become a bar chart representation and so on.

The question ‘Where can you see …. in ……? ‘ arose from a need to help learners explicitly make connections between one representation and another. In the case of the subway sandwich you could ask ‘Where can you see the sandwich in the bar?’. In the case of the newspapers, ‘Where can you see the newspapers in the bar chart?’

For teachers new to this idea, the question can seem puzzling. To some it may seem like a trivial question to be asking, almost too obvious. Yet once you start to use this question, you begin to see the power of it: firstly, it produces a range of responses, and secondly, it forces the learner to still try to see/visualise the context image even when it is no longer obviously there.

The first two examples of this question in action are taken from the Data modules:

In the case of the Prime Ministers context the question ‘Where can you see the Earl of Wilmington in each chart has been shortened to ‘Where is the Earl of Wilmington in each chart?’. To answer this question learners have to look back to an earlier representation of the data (the list of Prime Ministers with the age they came to power) and match it to the tally chart and stem and leaf plot representations.

With the maths test results the ‘where can you see ….’ question prompts learners to identify median and quartiles in an ordered list and in a dot plot. Learners do not necessarily need to know what the median and quartiles are, the ‘where can you see…’ question prompts them to figure out how they relate and come from the other representations.

The next example is from the first Algebra module, ‘Knowing the Unknown’:

A traditional approach to this balancing situation would prioritise solving the puzzle. Instead learners are asked to first describe what each student has done i.e to focus on their strategies not the answer. The ‘where can you see…’ question then pushes learners to move from the answer back to where this answer can be seen in the balancing context situation.

In Section B of Challenging Gradient, students work on the context of Stacking cups. They move from figuring out the height of a stack to looking at a sequence of stacks to a graphical representation:

Once the stack is represented as a series of points on a graph, the question ‘Where can you see …. in ….?’ is used to help learners link up the image of the stacked cups with the formal graphical representation:

Student responses to this question will vary, some will see the ‘goes up by 3’ along the vertical height axis. Some will indicate by moving their finger one across from the first dot and then go up 3 to reach the second dot.

The ‘where can you see…’ question inevitably elicits multiple responses; the purpose of the question is to help learners *go back to the context* and link context and representation.

**‘Where can you see … in …? has applicability to every lesson.**

As we have seen above the ‘where can you see….in….?’ question appears explicitly in these materials on the powerpoint slides and for ideas about the sorts of responses students give, you can look in corresponding ‘What the students do’ sections of the teaching Guides. It is also mentioned in the Teaching Guide under the ‘What the teacher can do’ sections.

Go back to the mathematising page for more techniques.