I’ve spent the last two years teaching maths – to Year 7 & and Year 10. Finding a good model to help students grapple with algebra has often been a sticking point, especially once we started performing operations on letters beyond collecting like terms; for some students this level of abstraction is incredibly difficult. The idea that a letter can represent a number that we then perform operations on is not intuitive to lots of students.
I’d used various devices to represent ‘unknown number’ but often this was unsuccessful. Simply asserting that ‘a’ is a ‘mystery number’ is too hard for some to hold on to once we start performing operations. If you use shapes or other objects it’s also problematic. eg saying 2 lots of (Apple plus Orange) = 2 Apples and 2 Oranges, is getting very convoluted in terms of models. What have apples got to do with numbers?
A colleague at HGS, the brilliant Head of Maths, David Shemoon, suggested that a jar of sweets works really well. He had previously brought in a real one for his students – but I tried it with images and found that this also worked.
The idea is that a jar of sweets -as often used in a ‘guess the number’ school fete stall – contains a fixed but unknown number of sweets. We can add more sweets or take some away (eat them!) so we can know how we’ve changed the original number without ever needing to know what that number is. Students can see the individual sweets; they can visualise that there is a number – it’s just that we don’t know its actual value. It could be 80 or 137 – or whatever; we’d just have to count them.
The sweet-jar visualisation helps with simple expressions and combinations of them:
Students can see that eating 4 from one jar and 5 from another means we have eaten a total of 9 from two jars. This deals with the tricky ‘plus minus’ issue – which often confuses students. We’ve taken away 4 AND taken away 5. – 4 + – 5 = – 9
I found adding a second type of jar of the same sweets helps with different variables. Again each jar has a fixed but unknown number of sweets. Students can visualise it:
The biggest effect I found was with dealing with brackets. In the example here, it is clear that if we have three lots of (Jar plus 4), you are going to have three jars and three lots of the extra four. The common error of not multiplying all terms in the bracket can be seen to be wrong.
This works with two variables as well. If each letter represents a whole jar, it is harder to lose track of them.
Obviously, this has limits. The number of sweets can’t be negative in a meaningful sense and you can’t talk meaningfully about more complex operations – like a x b or a-squared. It can be convoluted to start talking about subtracting negative numbers – unless you get into the concept of owing sweets and paying back debts; that works better with money and temperature analogies (ie taking out ice cubes raises the temperature.)
This worked a treat for several of my students. I only introduced it after having already tried a more standard approach and lost a few; I wish I’d used it from the start.