### Everyday contexts generate meaningful mathematical thinking.

Students who have learned in RME classrooms often refer back to the contexts that most impacted their thinking. In describing volume, for example, one student mentioned “cheese cubes” (Exploring Space, G2) – a context which provides imagery linked to intuitive concepts of filling, layering, and fast counting. These important associations guided him in making sense of why (and when!) to use multiplication in calculating volume.

In RME, the aim is for students to build connections between everyday mathematical intuitions and the formal conventions of mathematics so that they can make sense of the more abstract rules they eventually come to use. As long as the route back to context is not lost, they will always be able to make sense of those rules.

The device of going back to the context is something which students may do naturally. However, for the RME teacher, encouraging students to return to the context is a useful pedagogical device for helping them debate, justify, and make sense of ‘shortcuts’ and ‘rules’. Here, we look at examples from our materials where students are drawn back to context as a way of making sense by:

• situating themselves inside a mathematical representation;
• explaining formal mathematics inside a given context;
• generating their own context through drawing or visualisation.

### Situating yourself inside a mathematical representation

This slide from the final section of Drawing from Data (D2), presents students with data from a welly-throwing competition. The data appears in a sophisticated and contracted representation -– as a grouped frequency table. Students often find this highly processed form difficult to interpret.

Prompted to “say what they can (and can’t) see” by looking at the table, students must closely inspect the graphic. They are encouraged to read the text preamble aloud and develop their own meaning for the table of figures. In the next slide (below), students are invited even deeper into the context. Asking students to choose a Wellington boot and predict how far they might throw it encourages them to visualise themselves within the table of data. In doing so, they gain a much better sense of its meaning.

The final question on this slide asks students to say how realistic the distances are in the table. This prompts them to consider what a throw of 70 metres might look like. They might use the length of the classroom to estimate that a 70 metre throw is very long!  Going beyond merely reading the table, discussion around this question enables them to confidently talk about the inequality notation and make inferences about the distribution of distances.

### Explaining formal mathematics inside a given context

In this example from Knowing the Unknown (A1), students are encouraged to go back to the context they have been working in to remind themselves how and why we use brackets. In the build up to this slide, students have worked with traditional weighing scales to develop ideas which underpin solving algebraic equations. Gradually, some students will have shortened their drawings of weighing scale problems, coming to represent the scales with symbols associated with algebraic equations, as shown in question C19.

At this point, some learners may already operate with the symbols. They may not require the imagery associated with the context. However, as the symbolisation progresses to include equations with brackets, the question directs students to “Try to make sense of the shorthand by drawing a scales picture”. In other words, the students must return to the context of the weighing scales to (re)make meaning for the use of brackets in the expression 3(d + 2).  This reminder ensures that students don’t lose sight of the meaning of brackets, and why we deal with them as we do.

### Generating your own context through drawing or visualisation

Teachers and students experienced in using RME begin to recognise that drawing something can be a very helpful way of seeing a bare mathematical problem in a context. For example when asked to compare the size of two fractions such as 2/3  and  3/4,  students  may draw two same-sized pizzas and show that 3 out of 4 equal sized pieces is more than 2 out of 3 equal sized pieces. Alternatively, some students describe a drinks can which is only 1/4  away from being full compared with a can which is  1/3 away from the top.  In For Every One (N2), students are encouraged to draw something to help them think about ratio.

Although some students may want to draw a ‘realistic’ picture, some will choose something which is closer to a mathematised representation.  In the example below, taken from Fair Sharing (N1), a student has drawn a strip or bar model to represent a 36 km race in order to figure out how far is  5/9  of 36 km. Going back to this representation of the race provides a way of making meaning of 5/9 of a distance. To some this means a great deal more than trying to remember a rule that says ‘you divide it by 9 and times by 5’.

Where students have worked with an RME approach over time they start to draw in order to refer back to or generate their own context without much prompting as a natural step in trying to make sense of a problem.

### In general: asking questions which draw learners back to context

In this example, taken from Seeing It Differently (N2), Ozzy uses a ratio table to adjust the amount of pasta needed in his recipe for smoked salmon pasta.  The ratio table provides a flexible structure for generating and recording how much pasta will be needed to make the recipe for different amounts of people.

Although the labels on Ozzy’s table (‘pasta’ and ‘people’) refer to the context, students will quickly move from the context into the world of numbers — describing doubling, halving, and combining quantities. This level of fluency is helpful, as it allows learners to quickly compute other values. However, for many, the rationale for all this number crunching may get lost.

This is when the teacher can ask questions that encourage students to go back to the context and make sense of what these operations mean inside of Ozzy’s kitchen. For example, a student might say, “He doubled the number of people and doubled the ingredients”.  Asking “Why would he do that?” or “What is Ozzy doing in the kitchen to make the recipe for 16 people rather than 8 people?” enables them to visualise the context and develop the reasoning behind the mathematical operations. Some students might refer to making a pan of pasta for 8 people and then making a second pan for 8 more people, arguing that perhaps it wouldn’t all fit into one pan. By digging into the context in this way, students develop a grounded rationale for their proportional reasoning.

### Using mathematical landscapes as a guide to how contexts are used in RME design

Contexts in RME are not simply about motivation or application. Particular contexts are chosen for the way in which they underpin specific mathematical strategies — like the bar model or the use of brackets. In staying close to the details of particular contexts, RME teachers can support students in making connections across a range of representations and mathematical concepts. As they grapple with difficult ideas—like the percentage change problem discussed in this RME lesson — students will find that going back to the context can help them to make sense of what is going on.  Each module is built on a mathematical landscape which you can find at the beginning of the teacher guide — this is the landscape from Fair Sharing (N1).

Contexts make up the bottom layer of the landscape diagram for each module. They serve as the foundation from which formal ideas can grow. Contexts build on each other to generate the key models and ideas of each module. It is the careful selection of each context which makes going back to the context such a powerful technique.

Even when they move into more abstract maths — dealing with gradients or circumferences, for example — students are able to go back to contexts that help them unpick and explain formulae. As your students’ mathematical imaginations grow, mathematical contexts (pure numbers, geometric shapes and analytic functions) will become contexts in their own right.