Tangible contexts for mathematics

Do contexts help students to understand mathematics or do they just make it harder for them to untangle the mathematics from all the extraneous information? I think the answer is yes – both of these happen on different occasions. So, what is it that gives some contexts the potential to be powerfully illuminating?

I think the answer is not ‘relevance’ to a student’s personal interests. Relevance might be motivating, possibly, but it doesn’t necessarily make the context more illuminating of the mathematics. That way lies a ‘learning styles’ kind of fallacy, that every student needs a different context that is just right for them, and the magical right context will somehow make everything clear to them. I don’t think that’s right. And anyway, students often seem more switched on by contexts that take them out of their existing worlds (e.g., spaceships, dinosaurs, unicorns) than those which merely reference things they are already familiar/bored with. So I don't think matching personal interests is the most helpful approach. I think it's more likely that generally most students are helped by the same illuminating, well-chosen contexts, and not really so much by others. 

Ratio and multiplicative/proportional reasoning

Let’s take ratio or proportional/multiplicative reasoning as an example. This is widely acknowledged to be a (or possibly ‘the’) central concept in lower secondary mathematics. And something that many students really have a weak grasp of. If you wanted a concrete context to help students make sense of this area, what would you pick? If you opened a textbook at the ‘ratio’ chapter, what contexts would you expect to find?

Of course, ratio can be applied to all sorts of contexts, and it is important to do this and let students see how ratio can be relevant and important in a wide range of areas. That is fine. But what I am thinking about here is contexts that are deliberately used to try to develop students’ understanding of what ratio is and how it operates.

The problem for me with, for example, money as context is that if the ratio of money spent by, say, Usha and Sam is 3:1, and the ratio of money spent by Dave and Priya is also 3:1, it is quite hard to capture in words (or in pretty much any other way) what specifically it is about Usha/Sam and Dave/Priya that is the same, given that these ratios are the same. The ‘same ratio’ is a highly abstract concept here. So, although I think that money might at some point be a worthwhile context for using ratio, I don’t think it’s helpful for understanding what ratio is. My test is that I need to be able to complete the sentence: “When the ratios are the same, the _____ is the same” with something highly tangible and familiar (not mathematical) going into the blank space. For this reason, I think that most discrete ratios (money, different coloured beads on a string, different kinds of animals on a farm, boys and girls in a class, etc.) are not so useful.

Tangible context: paint

Instead, I think the ratios of continuous quantities are much more useful to begin with, and, in particular, my go-tos are always drinks (Foster, 2007) and paint. The fact that most students probably never mix their own drinks, and even professional decorators rarely mix pots of paint together to make new colours (and when students mix their paint in art, this would be by eye) is irrelevant. The point of the context is not that it’s something students do every day, or even ever. The point is that it’s easy to imagine (what Realistic Mathematics Education calls 'realistic', and which means something closer to 'realisable').

The reason that I think these contexts are useful is that:

• “When the ratios of red paint to white paint, say, are the same, the paint is the same colour.” and
• “When the ratios of orange juice to lemonade, say, are the same, the drinks taste the same.”

And everyone knows what these things mean. This means that you can have a discussion about various hypothetical mixtures of red and white paint, or fizzy orange, and you can initially completely avoid the word ratio and any 'rules' about when 'ratios' are or aren’t equal. You can just ask: “Would they be the same colour?” or "Would they taste the same?", and everyone knows what you mean and can engage in the thinking that you want them to do.

With paint, I find that having the two colours as red and white is particularly useful, because you then have the word ‘pink’ available, in addition to talking about ‘redness’ and ‘darker/lighter’. This all helps the discussion to focus initially on the mathematical thinking, rather than terminology. Once students appreciate that 2:3 and 20:30 and 1:1.5 and 4:6 are all ‘the same colour’, then it is natural to try to capture this ‘sameness’, and we can use a word like ‘ratio’ to do so. But doing it the other way round, beginning by stating that 'We say that' 2:3 and 20:30 and 1:1.5 and 4:6 are all ‘the same ratio’ invites students to ask, “What do you mean?” And that puts the teacher in the position of having to do the justifying, whereas really you want the students to be doing this, based on something that they have already gained a sense of.

Tangible context: fizzy orange

For the same reason, making fizzy orange using orange juice and lemonade can be another really illuminating context (and you could possibly even do this one for real in the classroom, Foster, 2007). Lemonade is better than water, I think, not just because the mixture tastes better, but because then you can ask, “Which mixture will be fizzier?” as well as “Which mixture will be more orangey?” Really tangible contexts like these do a lot of the work for you. Every child knows that adding more orange juice won’t necessarily make the mixture taste more orangey, if you are also adding more lemonade.

I would often begin a discussion of this scenario by suggesting a few possible mixtures of orange juice and lemonade (as in the table below), and asking students which mixtures would taste the same, and which would taste different. For any ones that they think would taste different, I would ask them which would taste more orangey, and I find that that sometimes causes them to change their minds. You often get to a situation where they think one mixture would taste more organgey, but also more fizzy, and so that causes them to go back and think again.As the discussion progresses, further possible mixtures are usually suggested by the students, and I would add these to the list. The point is to avoid telling students whether they are right or wrong, but to draw on their common sense and life experience to let them figure it out. They know everything they need to know to do this. This then forms a really good basis for more formal teaching of ratio.

For example:

Teacher: Would any of these mixtures taste the same? Are there any you’re sure would taste different?
Student 1: D and E would taste different.
Teacher: Why do you say that?
Student 1: D would taste stronger than E because there’s less lemonade in it.
Teacher: But D and E have the same amount of orange, don’t they, so shouldn’t they be equally orangey?
Student 2: No, because the orange is spread out in more lemonade in E.
Teacher: Can someone else explain what S2 is saying?
...
Teacher: Would any of these mixtures taste the same as each other?
Student 3: A and B would taste the same.
Teacher: Why do you say that?
Student 3: Because they both have 1 more lemonade than orange.

Teacher: Are there any other mixtures with 1 more litre of lemonade than orange?
Student 4: Mixtures C and D.
Teacher: So, would mixtures A, B, C and D all taste the same?

Students: Yes.

It’s likely at this point that some student will raise some doubt, perhaps relating to C being ‘nearly fifty-fifty’. Multiplicative language or thinking tends to appear around this point, if it hasn't already, which can then develop into getting the students to order A, B, C, D and E by ‘orangeyness’.

If this doesn't happen, then the teacher can be more proactive:

Teacher: Suppose I took two containers of Mixture A. How many litres would there be in each?
Student 3: 5 litres.

Teacher: What would happen if I mixed them together?

Every student will appreciate that mixing identical mixtures will lead to twice as much mixture, but that it will taste exactly the same. So this gives us Mixture E. And students will have already agreed that Mixture E must be less orangey than Mixture D, so this provides the nudge for everyone to think more deeply. Mixtures A and D can't taste the same if mixtures A and E taste the same and mixtures D and E don't! The idea that mixing 'identically-tasting mixtures' (still avoiding the use of the word ‘ratio’) will lead to a new mixture with exactly the same taste is highly intuitive, and nobody will ever doubt this. And that kind of knowledge is all that is needed to develop all the necessary ideas of ratio through this kind of discussion.

Tangible context: chromatography

Finally, I think a really helpful science context is chromatography and retardation factor ($R_f$) values (Note 2). There could be potential for some cross-curricular practical work with chromatography paper and water-soluble marker pens. Different inks dotted along a pencil line at the bottom of a sheet of chromatography paper will move at different rates as the solvent soaks up the sheet (Figure 1). Each component will travel at a fixed fraction of the speed of the solvent, and the $R_f$ value of each is defined as

$$R_f = \frac {\text{distance travelled by the substance}} {\text{distance travelled by the solvent}}$$

Figure 1. Calculating the $R_f$ for the red substance.

This seems to me like a perfect, dynamic scenario for understanding ratio, because molecules of a substance are highly obliging, and obey the rules perfectly (unlike, say, two runners in a race, running at different speeds, who need to negotiate bends and are likely to get tired at different rates). Here, when the ratio is the same, the height above the baseline on the chromatogram is the same (and the substance is likely to be the same). I would be keen to hear from anyone who has used this context as a way to explore ratio with students.

Questions to reflect on

1. What examples of illuminating contexts do you use - for ratio, or for other topics? What is so good about them?

2. When do you feel that contexts do and do not work well? Why?

Notes

1. For a free lesson plan based on the fizzy orange idea, see https://www.map.mathshell.org/lessons.php?collection=8&unit=6230

2. People's recent familiarity with Covid lateral flow tests may also make this easier to grasp.

Reference

Foster, C. (2007, May 24). Make maths sparkle. SecEd, 12. https://doi.org/10.12968/sece.2007.5.902. Available at https://www.foster77.co.uk/Foster,%20SecEd,%20Make%20Maths%20Sparkle.pdf