Everyone is immersed in algebra because algebra is the expression and manipulation of generality. Whenever a generality is present, algebra is present too; wherever a generality is manipulated or varied, algebra is taking place.
Whenever you purchase something, you are immersed in algebra. As a customer, you are interested in the final price, whereas the entrepreneur has to take VAT into account. While the specific VAT rate is fixed for some period of time, the method of calculation is fixed for all time and can be expressed as a formula, that is, algebra. The entrepreneur states a price, but has to know what VAT is owed to the government from that price and this involves algebra. Since the VAT charged to the customer is offset by the VAT paid by the entrepreneur to her suppliers, another calculation is involved.
Where a discount might be involved (for example three for the price of two, on sale at 20% off, 10% off for a dozen or more), the entrepreneur formulates the discount according to assumptions made about the number of extra sales the discount might lead to. So, although the income derived from individual sales may decrease, the number of those sales may go up to compensate. Not only are the policies actually expressions of generality, but the marketing involves manipulation of those generalities so as not to make a loss.
Combining discounts and VAT leads to the somewhat surprising result that, as far as the customer is concerned, calculating a discount before or after calculating the VAT makes no difference (Try it!). It does make a difference to the government, so they have the rule that VAT is always calculated last (Why?).
Every mathematical procedure (for example multiplying fractions, long division, factoring numbers, using scientific notation for very large and very small numbers) is actually a generality. No one expects children to memorise all possible three-digit subtractions; rather, children are inducted into a general procedure that enables them to perform all such possible tasks. Often these are stated in words (“You take this digit and subtract it from that digit and if …”) because people assume that words are easier than formulae.
For the operations of arithmetic, they are probably right: words are easier than formulae. However, there is plenty of evidence that young children can use letters to express general relationships, and can make sense of general expressions expressed using letters.
Working with linear and quadratic equations makes little sense to learners who have not themselves expressed relationships between varying quantities. So taking every opportunity to recognise and acknowledge a generality, to express that generality in words and/or symbols, sometimes by the teacher, but increasingly by learners themselves, is a step towards algebra.
Suggestion 1
Take something like a litre of milk. Get children to find out how much the farmer actually gets, and what their costs might be; work out the mark up the supermarket makes and consider their possible costs.
Suggestion 2
Try to catch as many of the mathematical and non-mathematical generalities that are expressed in a single lesson as possible. Once you become aware of mathematical generalities, consider pausing briefly and drawing attention to that generality, so that learners become aware of them and begin to articulate them. Over time, you may then find opportunities to manipulate those generalities fruitfully, and to get learners to manipulate them.
All the best,
John Mason
John Mason
John Mason worked for the Open University Mathematics Department, where he designed two of the mathematics summer schools, contributed to numerous courses, and then helped form and run the Centre for Mathematics Education. He has written numerous books and booklets, as well as research articles and book chapters, the best known being ‘Thinking Mathematically’ (1982, 2010).