### As maths educators, we want our students to be able to closely analyse and evaluate various mathematical representations.

We also want students to reflect on their own mathematical strategies, especially in relation to other possible methods. Looking in detail at two representations, two strategies, or even just two images — like the fingerprints below — can help students to understand more about how mathematical concepts and representations work.

We’ve found that asking students “What’s the same? What’s different?” can help to stimulate this close observation in an open-ended way. This line of questioning should allow every student to offer an observation. And it is always useful to hear what students are seeing. You may be in for some surprises!

On this page we use our materials to model three ways in which “What’s the same? What’s different?” can be helpful in classroom discussion. Our examples focus on how these two questions can help students to:

• practice close observation,
• make connections across given representations,
• analyse their own and other students’ solutions and strategies.

### Practicing close observation

The “fingerprints” slide (pictured above) is taken from the final section of For Every One, PR2. This slide asks students to identify two closely related images and name their similarities and differences. As part of a module on proportional reasoning, these questions draw on students’ intuitions about distortion. Later in the module this sense of distortion will gradually be formalised into the mathematical concepts of similarity and scale factor.

In the video above, Sue and Kate model how you might get started with the fingerprint slide. They discuss how the PR2 Teaching Guide supports teachers in expanding on students’ ideas. Sue points out that when first deploying open-ended questions like “What’s the same? What’s different?” teachers sometimes feel uncertain about what students will say. She highlights how the section of our teaching guides labelled “What the student might do” can help you prepare for a discussion about distortion that gets learners to articulate their visual sense for scaling shapes (and, eventually, quantities).

### Making connections across representations

Asking students to “compare” two representations can sometimes bring out snap judgments — “This one is better.” “That one’s wrong!” These statements can close discussion down and prevent students from looking more deeply at mathematical relationships. Asking students to weigh up both “What’s the same?” and “What’s different?” can help them to slow down and pull out connections, rather than corrections or quick appraisals.

This slide from Sorting It Out (D1) asks students to explicitly name “What’s the same?” and “What’s different” between two mathematical representations: a tally chart and a stem-and-leaf-plot. Previously in this module, students have studied and organised a raw data set about the age at which British Prime Ministers came to power. Here, students examine two representations of the data set. In the video below, Sue and Kate model how students might work through this slide.

In the video, you’ll see that Kate’s observations start out naming same/different. Gradually, however, she becomes more interpretive as she looks back and forth between the data representations. This is likely to happen in classroom discussion as well.

To help students dig deeper into these data representations, two more questions appear on this slide, after “What’s the same? What’s different?” These questions follow another discussion strategy —  “Where can you see … in … ?”. When worked together, these discussion devices help students to build strong connections between these data representations.

### Analysing students’ solutions and strategies

In a final example, Sue and Kate look at an interesting problem from Challenging Gradient (A2). Here, they discuss how “What’s the same? What’s different?” can give you insight into how students are developing their understanding of a novel mathematical concept – in this case, the Cartesian plane. This questioning technique encourages conversations between peers about their choices and representational strategies, as well as open-ended self-reflection.

At the end of this video, Kate summarises all that she’s learned in talking with Sue about these examples. She discusses how “What’s the same? What’s different?”:

• promotes a culture of thinking independently and listening to each other
• is helpful in enabling learners to look across two representations and notice detail
• can produce less judgmental responses than when you ask students to “compare”
• is supportive of teaching mixed attainment and serves as an assessment for learning (AFL)