## Monday, 20 April 2015

### Fill in the box algebra - not for me thanks!

Recently I have been talking about introducing algebra to pupils with a number of my colleagues. Most of them use a variant on the 'fill in the box' approach i.e. [  ] x 4 = 28, what number goes in the box? Some use other symbols such as the playing card suits, but the approach is the same. I have started to wonder if this approach doesn't lead to a couple of problems, namely:

1) Does this approach lead to or reinforce the misconception that there is only one value for a variable (the dreaded "what is x?" question)?

2) Does this approach start part-way down the journey by focusing on the equation before ensuring a proper understanding of the more fundamental idea, that of the expression, has been explored.

Forgive me for expanding on this second point, but to me the fundamental algebraic idea is that of the expression. By equating expressions with values or other expressions we arrive at equations and formulae; by evaluating at integer intervals we generate sequences and from here we can explore graphical representations and other algebraic constructs. By introducing algebra using equations I wonder if we start ahead of ourselves and then have to try and move pupils backwards before we go forward again. I think it might be better to approach introducing algebra with the expression rather than the equation and I have found a way that I think works; the think of a number problem.

Don't get me wrong, I don't mean the classic "I think of a number, times it by 3 and add 5, and get an answer of 14." This is just an equation in a different form. No I mean the magic number problems like the 2, 4, 8 trick - Pick any whole number, multiply it by 4, add 8, divide by 2, then subtract your original number. Your answer is 4." These sort of number tricks do tend to capture the imagination of pupils as they like the idea of performing tricks on people; and for me they give the perfect vehicle for exploring what happens to expressions. It allows for the reinforcement that algebra is used to deal with unknowns and relationships between them and allows pupils to explore what an expression looks like through different manipulations (multiply by 4 etc).

I have a compendium of Mathemagic tricks here which can be used at varying levels with pupils to explore expressions or to go as far as using difference of two squares in formal proof; feel free to use to introduce the idea of an algebraic expression as the fundamental algebraic representation.