Would you rather have directions or a map? Can I have both please!
When a good friend of mine first moved to London he decided that he would take a bus and jump of at random stops, he would then attempt to make his way home. This allowed him to create a mental map of London. He claims he can walk home from anywhere in London, however he still prefers to use public transport. If we think of Skemps idea of relational understanding of maths as a map of the mathematical landscape it should suggest that we can work our way to the solution from the information we have. However that doesn’t mean we have to, often there is a faster way. Richard Skemp seems to suggest that relational and instrumental understanding are two separate types of maths I would argue that instrumental understanding is equal to only using public transport without knowing how it fits in to the greater mathematical landscape.
If we are stuck on public transport we may have to take a longer route and if we have only the map of London it could take us a long time to walk to where we want to go. If we have access to both the public transport and the map of London we can benefit from making our way from one point to another as quickly as possible. Surely combining the two is best?
Relational understanding seems to have some resistance against it, yet it is perhaps the best way to find our way around maths itself, is there a way to combat the resistance?
Perhaps the idea of different mindsets, promoted by Carol Dweck, could be a solution. If we can teach students to work on gaining a growth mindset, one where hard work is considered the solution to success, then we could impart how important learning relational understanding even though it may be more work. It also address this initial resistance which might be caused by the inability to get the right answer straight away, this mindset allows students to work on the problem and not give up as soon as it gets hard, people with a fixed mindset will tend to give up when things are tough. If we teach instrumental methods then the right answer can be found, but if we are always sat in our comfort zone our successes may mean a shorter lived confidence. Jo Boaler suggest that the pride gained from enduring and persevering gives students a longer lasting “high”.
To me the point that stuck out from both was the need for context, without it the interest in gaining a growth mindset is meaningless and the need for relational understanding is also meaningless.
Do we all need to push ourselves in to the unknown? Find ourselves in the middle of a mathematical landscape and find our way to the solution, not by finding the tube but by wandering along the streets to find an area we know and from there we can make our way home, or at least to the solution, then in future we can always use the tube.