Division and multiplication signs

It's the start of the academic year, so once again I'm mostly thinking about Maths teaching. Or at least a little more than in August. Perhaps this is a fortunate distraction from British politics, so hopefully this post can be for you, too. Or maybe a discussion about Maths teaching is too dull to act as a good distraction even in these times. Let's see.

One persistent problem children seem to have in early secondary school Maths is related to two common signs. The multiplication sign '×' and the division sign '÷' .   The first problem they have (starting in primary school) is with the order of operations. When performing a calculation like 5 - 3 x 2 you have to remember that multiplication happens first (you may remember learning something terribly tedious called BIDMAS). The second problem arises when children transition to secondary school and start learning algebra. Algebra is laid out in such a way that putting two symbols next to each other implies multiplication. So 3ab means 3 × a × b. The problem is that children will end up switching between the two ways of writing operations (for example when substituting a value into an algebraic expression). This can be particularly problematic when they try and rearrange certain equations, because they way in which they learn to rearrange them is heavily reliant on visual cues and metaphors which are first and most easily related to purely algebraic notation. So if you have an equation like 7x + 3 = 10 - 3x, you either learn to 'move' the bits of the equation around, or 'balance' the equation by 'doing' the same thing to both sides (I can add 3x to both sides of the equation and get the new equation 10x + 3 = 10. But this gets more confusing for some children you have an equation more like 7x +5  = (2x +3)/3
                                                             
They might learn to multiply both sides by 3, but write that as 7x + 5 × 3 =  2x + 3. Even if they add in brackets (7x + 5) × 3 =  2x + 3 it still looks confusing and unfamiliar. Or sometimes they have, or end up with some expression like  x × 3 = 7, and not see that the solution is x= 7/3, because they are used to only being able to make that step when the original equation is written as 3x = 7. 

Similarly, early on, it is often difficult for children to recognise that a fraction is equivalent to a division. So 7 ÷ 3 doesn't look the same to them as 7/3. This can make all sorts of steps in rearranging equations, dealing with algebraic fractions or even just answering basic 'every day' type numerical problems a lot more challenging.

So here is a modest proposal: why not get rid of these signs? The multiplication sign can easily be replaced with the dot notation used in many other countries. So 10 x 3 can be written as 10 · 3. This would make the difference visually between algebraic notation and the notation used in arithmetic a lot less pronounced. After all, you are just writing things next to each other with a dot in between. This also makes substitution a lot simpler. If presented with the task of evaluating 3x^2 - 10 when x = 2, they can just write 3·2^2 -10 rather than writing the more convaluted 3 x 2^2 - 10 which has more room for mistakes in the order of operation. You’d still for more advanced stuff need different multiplication signs to distinguish between say, the dot and vector product of two vectors, but that requires no change. 

Similarly, unless I'm being stupid, I don't see any reason for the division sign at all. Just write all division operations as if they are fractions. On a more basic level, this makes it easier to see simpler routes to division problems (through manipulating fractions). Combining this with dot notation, it also makes the order of operations a lot easier to see visually, and could make the transition to algebra later more straightforward.