At the beginning of my teaching course, I had to fill in a box in an initial assessment that asked me about my maths skill. To be a good teacher of anything it’s good if you understand basic maths. I believe this to be true because I would like to embed maths learning into my teaching, so that the people I teach don’t have the fear of maths that I do. I hate Maths. I’m really bad at it. When I try to do it in front of people I start feeling sick and this panic curtain (imagine the fire curtain at old theatres, but instead of saying fire curtain it says MATHS STUPID) comes down and I don’t know what 4 + 3 is (even now, with no-one in the room, three fingers have to micro-twitch in turn).
So I wrote down that I needed to work on maths and then conveniently forgot about it for a bit. Then the other day I started researching dyscalculia, which is like dyslexia but with maths. I haven’t been tested for it but I’m guessing that if I did I’d have a middling to mild version of it (I think a lot of my problem now comes from the panic curtain and feeling stupid, but that came from somewhere, and a sense that things just didn’t connect right).
I found a website by a group of people who had designed a computer game for children with Dyscalculia here. They had this to say about how we learn about numbers and maths:
Our brain can process numbers in several different ways: visually as digits (“3”), verbally as number words (“three” – written or spoken), and concretely as a quantity (♥♥♥) or a position along a mental number line. Each of these is a different way in which the brain represents numbers, and there are specific brain circuits for handling each representation.
Different arithmetic tasks rely on different representations of number in the brain. For example, the digit representation is used when reading numbers written as digits or when writing them. The verbal representation is used when talking or listening to someone saying numbers, and also for storing multiplication facts in our memory (“three times five is fifteen”). The quantity representation is used to decide which of two numbers is larger, or to quickly approximate quantities.
Our brain can also transform numbers from one representation to another. For example, when we read aloud the number 5, our brain must understand the digit, transform it to its verbal representation, and instruct our speech system to say aloud the word “five”. At the same time, the brain also transforms 5 into a quantity, and we get a sense of how large the number 5 is.
The ability to handle the different representations of numbers is the cornerstone of numeric literacy. For example, if we could not transform digits into number words quickly and efficiently, reading digits aloud would be difficult for us. Being able to transform numbers into the quantity representation is especially important, because we usually see or hear numbers as digits or words, but it is the quantity representation that makes us understand the “meaning” of a number and have a sense of how large it is.
So there is a lot more going on in our brains than we are told about when we learn to handle numbers. I suppose most people don’t need to know exactly how we understand and translate the same concept into different forms. They pick up that translation skill just by being told that 5 is five is ooooo. But it stands to reason that some people are better than others at it, and if teachers knew that, they might be better at understanding why some people just don’t seem to ‘get it’. If I could go back in time 20 years and tell six year old me that this was a skill I’d have to really pay attention to, I think maths would have stressed me out a lot less during my life.
As part of my maths development I’ve started using Khan Academy. It has video tutorials and practice exercises that are mapped onto a diagram that connects different subjects from very easy to very difficult. Suffice to say, I began at the top, only avoiding the ‘telling the time’ cluster, thankfully I got that skill more or less down by the end of primary school. This ‘mind map’ way of understanding what I had to learn was great for my visual tangle-brain, but not the best thing about Khan Academy. My two favourite things are:
- A video tutorial cannot make a judgement on what you should know, but the practice exercises can assess what you do know.
- Video tutorials have pause buttons.
After realising this myself, I watched Sal Khan’s TED talk and realised that both these concepts have become important conscious elements in his work on Khan Academy (even though they may have been stumbled upon by accident originally). I recommend watching it.
These two factors overcome my two main obstacles to learning maths. Firstly, the ‘panic curtain’ I mentioned earlier comes from the inevitable millisecond surprised look even the most well-meaning people give me when I fail to do a simple sum, and the implication that I have internalised: that I’m maths stupid. So the panic curtain doesn’t come down nearly so hard if no-one is making a judgement of my skill against who I am (a fairly intelligent 26 year old). It does sometimes come down anyway; I can’t just erase my own judgement against who I am, but crucially, I can usually see through it enough to attempt the job at hand. Brilliant.
Related to the panic curtain is the feeling of falling behind and being rushed through everything. If I feel like I’m losing track, the panic curtain comes down pretty quickly. In maths classes my brain always worked slower than the speed the teacher explained things. I might understand eventually, but more often than not it wasn’t in the classroom, or if it was it was by accident a few classes later when I had to use the concept in something more complicated (that I obviously was going to struggle with!). On Khan Academy I can watch five minutes of a lesson, pause it and stare at the screen for 3 minutes, go back 2 minutes and play the bit I’m just beginning to understand again. No pressure. Brilliant.
This very process led me to really comprehend how to do long multiplication for the first time in my life. I could do it in high school, I had to re-learn the steps on Khan Academy, and I did with a bearable level of stress. But I just knew what I was supposed to do, not how it worked. I’m sure that’s how I managed all maths all through school. I decided to watch the tutorial again to see if I could make a bit more sense out of it. Half way through he said ‘but of course the two isn’t really a two, it’s a twenty’. I paused the video. I stared at the video. I stared at each of the numbers in the sum in turn. Then they became one number, not individual numbers. I said four hundred and twenty five in my head while looking at the number(s). I let the video play until the result was up on the screen and paused again. I read it out as a whole number, and looked at each of the digits in the sum in turn, even the little carried numbers, saying ‘twenty’ or ‘four hundred’ to myself instead of two and four. The digits began to join up in my head, the ones on the bottom linked with the ones on the top. It began to look like one of my tangled diagrams because I could ‘see’ the connections they had to each other. I pressed play and watched another few sums go by, pausing occasionally. I had comprehended, instead of accepted the foundation concept of our entire calculation system. Its called ‘base ten’. Now I get it, its kinda cool. Someone really smart must have invented it.
So. There was an important shift in my thinking that on the surface might not look all that different. The difference between accepting something to be true, and understanding why its true. Maybe knowing how to go through the steps of long division is enough to do long division. But what about the next step? It seems to me that to become mathematically literate, you need to comprehend each step, as they each represent important principles that you will later have to apply to different situations.
But how on earth do you assess the difference between acceptance and comprehension? I know my maths teachers would try. They would ask me over and over if I understood, and I would look vaguely confused and say yes. As I’ve said, I understood the steps. This was as deep a knowledge as I knew, and so as far as I knew, I did understand. I just kept getting things wrong. And the next step was impossibly hard, because I had to start over with the accepting, instead of using my set of useful principles. And my teachers would ask me to listen more carefully. Or they would go back to the previous step and I’d feel like they didn’t understand that I understood; that they thought I was just stupid. Looking back, I now know I had some brilliant maths teachers, even though at the time I hated them. But I must have been difficult for them. They didn’t understand that I didn’t understand, or how to teach in a way that helped me comprehend. But we found a way, and that got me my GCSE C, an achievement I’m still very proud of, purely because of how difficult it was.
I want to learn from this. I want to find a way to teach people concepts in a way they comprehend rather than accept. If only because it makes things easier for them. I doubt I’ll ever be a maths teacher, but I think all of this can be translated to most subjects. I would love to teach in a way that assesses comprehension accurately, and doesn’t expose anyone to concepts that rely on comprehension of previous lessons until they are ready.
Sal Khan taught me that time and space away from teacher is extremely important. My own experience taught me that understanding how I learn helps me to learn, and communicating this to teachers is the ultimate challenge. And of course, as a teacher this concept is important in reverse. How do I understand how individual learners learn?
Ok, back to the sums, I’m still getting them wrong.