by The Open University
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‘Thinking mathematically? is something that everyone does. It involves:
problem-solving and decision-making;
logical reasoning;
communication (including using diagrams, charts, graphs and symbols);
making connections and recognising common characteristics;
using mathematical tools, including calculations and measures.
Much of the mathematical thinking done in everyday life seems to have little connection with the school subject which, for many people, seems to have concentrated on writing out solutions to practice exercises. Yet most adults appear to gain sufficient knowledge, both of facts and know-how, to function reasonably well. Somehow we acquire a general sense of number, measures and shape. We learn mathematical skills (e.g. how to measure) and work out when it is appropriate to use them. For many day-to-day activities, we use estimates and approximations. When we buy apples, we don't need to know the exact number we already have at home; ‘two or three? is often good enough. We know that a cereal packet is too large to fit into the cupboard, without having to measure the packet or the cupboard. It may be good enough, when trying to determine whether a piece of furniture will fit a space, simply to compare the relative sizes using a piece of string. However, we may have to get out the tape-measure when ordering an item for a particular room. We understand when it is better to overestimate and when to underestimate: better to overestimate the time it takes to get to the bus stop, and to arrive early to catch the bus; but not so good to underestimate the amount of petrol needed to complete a journey! It is often only necessary to do rough-and-ready calculations: when working out whether there is enough cash to buy three items at £3.99, the approximation of ‘3 × £4? will do.
But what actually happens when we ‘do? mathematics? In some senses, mathematics only happens ‘mentally? and what is said or written down is the result of that mental activity. ‘Thinking mathematics? is subtly different to ‘thinking mathematically?.
Do the following exercise mentally. As you do, try to capture the sensation of what you ‘hear?, ‘see?, ‘do? and ‘feel?:
Imagine a square.
Turn it round.
Describe what happened in words and/or pictures. Do you think that other people would have the same experience?
Now repeat the process with the following two questions:
What is 3 more than 4?
What is 19 less than 27?
The following are comments from people who were given the same tasks.
‘I felt uncomfortable ? ugh! Subtraction, take away nineteen in my head! Oh! Inspiration, more cheerful ? take twenty and add one is eight. Satisfaction.?
Representations like those outlined above can have varying features of:
Context sensitivity: a quality that all images have to some degree; imagining an image of Trafalgar Square when asked to think of a square may be appropriate in some contexts, but perhaps not when considering mathematics.
Ambiguity: an ambiguous representation is one that associates the same form with more than one meaning within the same context.
Precision: the amount of detail in an image may be sufficient for the person conjuring up the image, but may not be enough to convey meaning to someone else. For example, the description ‘one square half black, half white? may be precise enough for you to recall an image like the one below on the right, but someone else might conceivably imagine the image on the left.
Redundancy: content or meaning can sometimes be recovered from just a fragment of information. Normal spoken and written language has a high level of redundancy: it is often possible to reconstruct sense from a poor telephone connection, or make sense of shorthand such as mlk nd sgr (the vowels omitted from the phrase ‘milk and sugar?). Being able to reconstruct meaning from such compressions is particularly relevant to reading symbolic mathematics: ‘3 + 4 = 7? is a succinct (low redundancy) symbolic representation from which several meanings can be recovered, depending on the context (e.g. putting three apples with four apples makes a total of seven apples; or starting at three on a number line and moving four in a positive direction arrives at seven).
Do the following task slowly, step by step, trying to capture what changes as your mental image evolves:
Imagine your kitchen.
Where do you keep your spoons?
What sort of spoons do you have?
To start with, you may have had a general impression ? perhaps the view you get when you have just walked into your kitchen for no particular purpose. The next impression is likely to be more focused on a particular area ? perhaps a drawer or a storage pot. Then you would need to conjure up all your spoons ? teaspoons, soup spoons, slotted spoons …
In the final two parts of the activity, you were stressing some aspects of your kitchen memory and ignoring others. ‘Stressing and ignoring? is an important aspect of being able to think. Depending on the context, sometimes it involves being aware of ambiguities (i.e. what is the same, what is different); sometimes it is about having a detailed perception (a high level of precision); and sometimes it is only necessary to consider a fragment.
When asked to work mentally ? to imagine, to remember or to think ? the brain conjures up various experiences that are representations of sounds, sights, actions and feelings. These are commonly referred to as ‘mental images?, even though not all of them are visual. Mason (2002) comments that a mental image can be such things as:
an awareness of which side of your sink the cutlery can be found;
a fleeting sense of recognition as someone walks past you;
the vivid recollection of a specific incident, complete with pictures, sounds and feelings.
Images can rapidly become abstracted. The word ‘spoon? may not trigger the mental image of a particular spoon but more a sense of 'spoon-ness?. If asked to describe ‘a kitchen?, you are likely to access various memories of kitchens: you might work from a particularly vivid image of a specific kitchen; or you might find yourself working from a generalised or abstracted image, perhaps composed of fragments of images of many different kitchens. It is helpful, in mathematics, to have access to similarly abstracted or generalised images rather than be confined to images of particular triangles or additions.
People tend to have a preference for the way that they work with their ‘mental screen?: some prefer visual images, some prefer sounds (audio), some go for physical sensations kinaesthetic, and a few think in symbols. Rich mental experiences are those that combine more than one kind of mental image. Having several mental representations of the same mental experience makes it possible to obtain a detailed recollection from a fragment and, having recalled the whole, to stress and ignore different aspects as required. Good mathematical thinkers tend to be those who have developed a bank of rich mental images they can call upon when tackling a problem.
This is a mainly practical activity, asking you to notice the way that your brain works. Work through the attached tasks from Primary Mathematics by Heather Cooke.
Click on the link below to open 'Number and algebra' from Primary Mathematics by Heather Cooke.
View documentYou may have been surprised to see that the chapter was called ‘Number and algebra?; it goes on to show how seeing and expressing patterns in various ways is a route into understanding algebraic ideas. You will do further work on this towards the end of the unit.
The tasks that you completed for this activity involved seeking out and using visual patterns. Young children are also good at detecting generalisations, both visual and behavioural.
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