Consistent use of images supports the understanding of mathematical structures. The outstanding examples of this in mathematics have become ‘canonical’, that is part of the mathematical canon. At school level these canonical images are: number line; function graphs (thankyou Descartes); 2-dimensional combination grids (thankyou Omar Khayyam and Cayley); Venn diagrams (thankyou Cantor and Charles Dodgson). You could say our systems of numerical and algebraic notation are also canonical, but these do not carry meaning in quite the same way as the images do – the symbols need translating and the grammar of combining them needs to be learned.
Then there are the images we use that are not part of the canon but connect mathematical meaning to quantity or structure in ways that school learners can recognise. Examples of this include:
- balances for equations,
- rods or bars for quantitative statements and relationships,
- tables of values,
- function machines,
- number base apparatus etc.
The best images also model mathematical meaning and show how elements combine. Learning is well-supported when teachers find out what images their students are familiar with and build on those. That is why good textbooks use a few images consistently throughout the school years, and also why one teacher’s ‘good way to teach’ does not automatically ‘work’ for another teacher.
There is another role for consistent images in teaching and learning, and that is creating situations that become familiar and can be extended and complexified as learners become more competent and knowledgeable about them. The world of puzzles has many of these. For example:
Tank problems
If you Google ‘tank problems’ you will find all kinds of ways to mend your toilet flush, clean out your hot water system etc. but you will also find families of mathematics problems. The simplest is to consider two connected water tanks, one higher than the other, and describe what changes and what stays the same as water flows between (height, total volume, partitioned volume, mass, cross-sectional area, range of possible values etc.) Some puzzles have networks of several water tanks asking which tanks will fill in which order and how far. Others offer a way in to using and reading graphs, (e.g. http://www.pmtheta.com/animated-situations.html). Then we have increasing difficulty by introducing taps that control flow, so that the rate of flow is no longer a matter for gravity alone. So I can imagine learners throughout a school having water tank problems offered as a way to think about a new-to-them mathematical concept, such as equality of volumes, rate of change, volumes by integration, building on what is already known.
Goat grazing problems
I know goat grazing problems from Dudeney’s collections of puzzles but they have been around since the 18^{th} century, having been published in the Ladies’ Journal. They may be older. A goat is tethered to a stake, what area can it graze? (Spot the locus content here). Suppose the tether is a ring that can slide along a straight rod; what area can it graze? Suppose a fixed stake is in the corner of a square field; what area can it graze? What about fields that are right angled triangles, or circular fields? (This last suggestion is not casual). And so on. Again, I can imagine these problems being posed to generate various geometric, mensuration and computation ideas throughout school.
Bus problems
When I visited Prague I had the privilege of spending time with Milan Hejny, who is one of those mathematics educators who spend a lot of time in schools working with learners, but also manages to be internationally known. He has spent about 40 years writing a textbook series that still isn’t finished, and every time he uses one of the tasks in classrooms he listens to learners and often rewrites elements to take account of something new he has learnt. The series is used in about 25% of Czech schools. At this point you might want to be looking up the relevant PISA results to see where the Czech Republic sits, and you may not want to read on, but I encourage you to think differently because of the 40 year development, the close attention to what learners actually do and say in classrooms, and that – as Milan says himself – many teachers do not use the teachers’ guides so do not fully use the potential of the series. That is true for all textbooks, and that is why teacher training in authoritarian countries often hinges on learning how to use the authorised scheme well, and the PD provided in relation to schemes in this country also focuses on use.
In Milan Hejny’s textbooks there are several consistent images that crop up year after year in increasingly complex ways. I was taken with the idea of his use of bus problems, which through imagination, role play, modelling, and arithmetical problem solving during the primary years introduce the additive relation, linear structures, and simultaneous linear equations in ways that enable learners to construct mathematical meaning and devise, adapt and extend robust methods of solving additive problems. Eventually learners need algebra to express what they can already do through reasoning. I know these appear elsewhere and are not exclusive to his books, but seeing them prompted me to think about the ways in which images can extend and even accelerate learning of early algebra. Algebraic representation of the unknowns and variables follows from wanting to describe the situations.
The basic model is about people getting on and off buses at bus stops, and comparing these quantities with the numbers on the bus. This may be about buses but it has NOTHING to do with ‘the bus stop method’ for division (or does it?). Diagrammatically it looks like this:
stops | A | B | C | D | E | F | ||
off | 2 | 3 | 2 | p | ||||
on | 1 | 2 | q | |||||
Start | 5 | 4 | 7 | r | s | |||
At stop A the bus arrives with 5 people on board, two people get off, one gets on, this means there are now 4 people on the bus. You can see that there are many possibilities for missing number problems that use various transformations of the additive relationship.
Cutting to the chase for early secondary work, there are various ways in which the relationships between the shaded cells can be expressed: r – p + q = s describes the actions at each stop;
s – r = q – p is relatively easy to explain but might involve negative numbers; s + p = r + q is less easy to explain and might prompt discussions about proof, since numerical demonstration from one or two cases does not guarantee that it will work for all cases. It is even worth discussing whether the third equation is better written as s + p = q + r as that reflects the algebraic manipulation, where the former expresses the status of the variables in the problem. The consistent structure of the situation at each stop creates both a need and the support for algebra to generalise demonstration of particular cases. How do the totals change if the starting number is increased /decreased by 3 but nothing else changes?
Playing with the givens creates interesting extensions. For example, supposed we do not know any of values for totals on the bus except for the starting value, or the finishing value: how could these be reconstructed?
There is also a related programming problem: what is the minimum amount of information needed to model the usage of one bus route?
Suppose two of the passengers are mathematicians (as often happens on my regular bus route in Oxford). One says to the other: ‘there are three times as many people after stop D as there were before stop B’. The other replies: ‘there were four times as many before stop D and there were after stop B’. What possible numbers could there be?
What problems could be posed if we change the bus stop labels to numbers?
As with any context, there are natural restrictions on the domain. We cannot use negative numbers for people, and bus capacity also sets constraints, but fortunately the whole problem type can also work for trains whose capacity is much greater.
It is not too fanciful to imagine that learners who have developed strong familiarity with the situation might find related problems more accessible. I have also posed problems for myself using the bus model where I found that using subscripts made the situation clearer for me; for example, suppose that during the rush hour the average number boarding increases by 10% at each stop and the average number leaving decreases by 20%.
You might have started thinking about similarities with bag arithmetic, in which teachers use bags containing unknown numbers of counters and do various things like adding more counters or taking counters out to nudge students into expressing algebraic relationships.
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.