High Flat Skill Profile

An interesting piece of popular science with a spurious pedigree is the idea that brain function is determined by hemispheral activity. We say that the left hemisphere is for quantitative, precision thinking such as mathematics and spatial reasoning, whereas the right brain deals with holistic activity, recognising patterns and governing intuition. So we talk about somebody being either left- or right-brained, with right-brained people being creative, disorganized and impulsive while the left-brained are fastidious, numerate and boring. This belief isn’t just imposed on others, but governs self-belief as well.

It turns out that none of this holds much weight. Many of us go through our education emphasizing either the arts or sciences, with students who consider themselves more quantitative shying away from softer tasks like writing essays or analysing literature, and those who find painting a picture or writing poetry more interesting than solving equations tending to ignore the more rigorous subjects. There’s an idea that you’re good at either one or the other of these, and like many prominent ideas it’s gained a degree of accuracy simply by virtue of how seriously it’s been taken. Since we tend to believe these things and allow them to govern our life decisions, they can determine what we end up learning and being good at.

But it’s not true that you can only be good at one or the other. It turns out that, in terms of actual ability, whether innate or developed, those of us who are able to understand complex mathematical subjects also do well in tests of verbal intelligence, and vice versa. There’s a strong correlation between these abilities, but often, by the time we’ve completed our formal education, we’ve learned much more of one of these than the other. Testing students who’ve graduated or are attending university, you find that most of them favour one of these areas more than the other. Those of us who do okay at both have what’s called a “high flat” profile, scoring in the top deciles for both mathematical reasoning and understanding of language.

To me, this isn’t much of a surprise. So far as I’m concerned, both of these categories of tasks are variants of symbol manipulation. They each involve the construction and modification of meaning based on a set of underlying rules. The rules of natural language may seem inconsistent compared to the hard and fast rules of mathematics and formal logic, but the degree of fuzziness is something which can itself be precisely quantified. Linguistics as a science is one of the most rigorous disciplines you can study, and along with mathematics and information theory is one of the “formal sciences,” which collectively provide the framework with which the rest of our knowledge can be described.

So what does all this cold, hard analysis have to do with those of us who enjoy the musicality of poetry and prose, or the transformative imagery of our favourite fiction? To me there’s a richness available in both mathematics and language, a complexity and elegance which combine to create something beautiful. Although to many people mathematics seems dry and too much like hard work, much of what it reveals can be not only useful but surprisingly aesthetically pleasing.

So what educational impact does this have? I do think individual students demonstrate divergent interests and skills, and that some of them respond more readily to English or mathematics instruction. But this doesn’t mean they’re completely resistant to one or the other. For children with a proclivity for the order and neatness of mathematics I find I can appeal to them by focusing on the structural elements of grammar and the logical consistency of the underlying rules from which meaning is constructed. For students who are more interested in the arts or who have a phobia of numbers, mathematical instruction has to be flexible, and I keep in mind the multiple possible approaches to solving the same problem. Each student develops their own understanding of a mathematical topic, and the beauty of mathematics comes from its consistency, such that divergent approaches to the same question converge on a single set of correct answers. So long as a student can understand what they’re doing and how they find the solution to a problem, there is a surprising amount of room for innovation, unique approaches, and the integration of wider knowledge.

Cognitive Load

I heard a story once about Albert Einstein, who, after a hard day’s work revolutionizing physics, left the university and went on his way, but ended up lost and confused, unsure where to go. It was like his brain had been working so hard that he couldn’t find his own house anymore; the supreme difficulty of whatever thought process Einstein was engaged in at work meant he had almost nothing left in his gas tank for the performance of even the most simple mental tasks, like recalling a piece of information he used every day. There’s no evidence that this story is true, and it’s about as extreme an example as you can get, but the phenomenon of cognitive load is clearly real and relevant to those of us who are interested in education.

Cognitive load is like the mental energy we use to perform difficult tasks. This can take the form of  solving  a mathematical problem, forcing ourselves to do routine household chores when we’d really rather watch TV, or biting our tongue when somebody annoying is provoking us. Apparently, all of these tasks take the same kind of mental energy. It’s effectively the same thing as stress; all of these situations act like a work-out for our brains. Just like you need a rest between physical resistance exercises if you want to avoid damaging your muscles through over-training, your brain can only take so much mental stress before its performance drops and the effect on your development becomes negative.

There are a number of strange phenomena which seem to be related to this concept. Apparently, if people are put in a difficult situation and have been making stressful ethical decisions, for instance making sacrifices for the benefit of others, they are less likely in the immediate future to resist their impulses. It appears that putting yourself second for a long time makes it harder for you to do the apparently ethical “right thing” next time. This works the same if you were doing some other kind of mentally difficult thing, like trying not to eat a tempting sugary treat, working on a complex intellectual exercise, or waiting for something. In any of these situations, people were less likely to exhibit restraint if they were tested immediately afterwards.

So it looks like our brain only really has one gas-tank, and it fuels pretty much any difficult thing we do. This relates to how much effort it makes sense to put in each day. It seems like today’s society glorifies the state of being busy, the rushing from one engagement to the next. Our lives are over-scheduled and we’re expected to be productive much of the time. It looks like our genetic heritage might not be entirely suited to this. Our brain can probably only manage about four hours of useful work a day, for most of us, and the rest of what we do is largely on autopilot. It’s important not to over-exert ourselves mentally, and to save our energy for the important stuff so we don’t get drained working on all those other things that take up our time and cognitive capacity. A healthy balance is required, such that we experience sufficient stress to promote growth, and we exert effort in areas which will reinforce our learning and help us continue to develop, but without imposing expectations on ourselves which could end up being counter-productive.

Growth Mindset

The idea of intelligence as an innate quality which determines an individual’s ability to learn new skills and perform well on cognitive tasks has gained some widespread popular acceptance. When we find ourselves impressed by somebody’s performance on an intellectual task we often say things like “Wow, you’re really smart,” with the implication that their performance is evidence of intellectual giftedness. IQ tests purport to measure this idea of general intelligence and are pretty robust, statistically, giving relatively consistent results between tests and reliably ranking one individual compared to another. But the idea of general intelligence is problematic and in some ways potentially counter-productive.

Developmental and academic testing has shown that there are two competing ideas of intelligence, and the one outlined above represents what’s called a static, as opposed to a growth, mind-set. The static mind-set defines performance in terms of ability and sees success as evidence of talent. The growth mind-set is focused not on ability but effort, and sees success as evidence of growth. While the jury’s still out on exactly how much of intelligence is based on innate ability and how much can be modified by effort, most evidence points to effort being significantly more important, and there is another benefit of thinking this way. It turns out that if you believe in the importance of effort, you’re likely to gain a wide range of cognitive benefits. Individuals who favour a growth mind-set enjoy a boost in performance on cognitive tests, and children who are praised in terms of effort seem to develop more readily than those who are praised in terms of ability.

Multiple tests have been performed and replicated on children, and the results are quite striking. One of these tests involved children drawing pictures, and then being praised either for their apparent talent or for their effort. If they were told “Wow! What a great picture; you must have worked really hard!” then they were more likely to continue drawing for fun, trying new techniques and developing their skills than children who were told “You have a natural talent for drawing.” Another experiment involved a series of more formal tests, which began with something relatively easy, designed to encourage the students to do well, and continuing with the option for children to select more difficult tests or continue at the same difficulty level. If, after their initial success, children were praised based on their ability, then they were less likely to choose more difficult tasks later on. They tried to play it safe and stick to things they knew they could do well. Children who were praised based on their efforts tried to take on more advanced tasks and challenge themselves.

The results of these experiments and others seem to indicate a pretty clear overall trend. Children praised based on personal qualities such as intelligence, while feeling good about achieving well, also viewed their failures as evidence of stupidity. They were more afraid of performing poorly and more likely to stick to easy tasks. Children praised according to effort were more interested in doing well, not just looking good, and had an understanding that they could change what they were capable of by applying themselves through increased effort. They were also less likely to compare themselves to the other children.

These developmental trends are pretty clear, and the significance for all of us is obvious. Beyond the implications for children, pretty much anybody who thinks this way is more likely to succeed. It may not always be true that intelligence is malleable and you can grow from your effort, but believing in it and acting like you can change what you’re capable of acts like a self-fulfilling prophecy. For children who are still developing, or for those of us who are interested in doing the best we can in whatever tasks we set ourselves, the growth mindset offers significant benefits.

R & D

Over the last month I have been doing a few different experiments, with the Raspberry Pi and Mathematica, to develop a number of different lessons/experiments for our students. I don’t have much time  for educational theorists so I take an empirical approach to teaching and believe it is important for teachers to be counted among “those who can do”.

Theory often motivates many changes to the educational system in this country. Consider recent changes to the New Zealand Maths Curriculum, which introduce more emphasis on statistics. The changes seem sensible. Students now gain a better understanding of the statistical process and the variability inherent  in sampling and data.

Nevertheless, there are problems with the implementation of this new approach. The usual problems of NCEA , where teachers spoon-feed students and teach directly to the  assessment they deliver, are common. Technically speaking, it seems that students gain the understanding that  Statistics is a narrative activity (a form of story telling). I  am not in favor of the narrative  approach, since many flaws in statistical reasoning are caused by our tendency to tell stories and  join the dots. I encourage my students to withhold judgment, unless there is strong evidence, and to also consider base rates when making  inferences. Story telling seems particularly prevalent when it comes to time series analysis.

Maths and statistics students should develop a maker mentality and that means they need to code. Essentially, code is maths brought to life. While report writing and analysis has it’s place, it is  a bureaucratic activity, while  making  and experimentation leads to new ideas, technology and growth. The thoughts detailed in this video do match my experience when it comes to using maths.

My post,Wi Fi to the Beehive and Raspberry Pi,in the Wolfram Community shows how running a few experiments leads to some interesting new ideas.