Thinking errors?

School’s tough. Maths is especially tough.

‘Maths anxiety’ causing fear and despair in children as young as six
Children as young as six feel fear, rage and despair as a result of “mathematics anxiety”, a condition which can cause physical symptoms and behaviour problems in class, according to a study.

Report examines origins and nature of ‘math anxiety’
A report out today examines the factors that influence ‘maths anxiety’ among primary and secondary school students, showing that teachers and parents may inadvertently play a role in a child’s development of the condition, and that girls tend to be more affected than boys.

More info on the research from the Nuffield Foundation…

Understanding mathematics anxiety
Learning mathematics can be challenging; however, not all mathematics difficulties result from cognitive difficulties. Some children and adults have mathematics anxiety (MA) which severely disrupts their performance.

… and from University of Cambridge’s Centre for Neuroscience in Education.

What is Mathematics Anxiety?
Does mathematics anxiety affect mathematics performance? When trying to figure out how Mathematics Anxiety relates to mathematics performance, we are faced with a problem similar to that of the chicken and the egg … which comes first? What we know is that people with higher levels of mathematics anxiety tend to perform more poorly on assessments of mathematics skills whilst those with better performance in mathematics tend to report lower levels of mathematics anxiety. What we don’t know is which causes which.

And here’s a link to the report itself.

Understanding Mathematics Anxiety: Investigating the experiences of UK primary and secondary school students
Abstract: The project investigated individuals’ attitudes towards mathematics because of what could be referred to as a “mathematics crisis” in the UK. Evidence suggests that functional literacy skills amongst working-age adults are steadily increasing but the proportion of adults with functional maths skills equivalent to a GCSE grade C has dropped from 26% in 2003 to only 22% in 2011 (National Numeracy, 2014). This number is strikingly low compared with the 57% who achieved the equivalent in functional literacy skills (National Numeracy, 2014).

This all looks far from straightforward. Here’s a very interesting, critical look at what seems to me to be a overly simplistic response to these issues — the growth mindset theory.

thinking-errors-1

The growth mindset problem
According to the theory, if students believe that their ability is fixed, they will not want to do anything to reveal that, so a major focus of the growth mindset in schools is shifting students away from seeing failure as an indication of their ability, to seeing failure as a chance to improve that ability. As Jeff Howard noted almost 30 years ago: ‘Smart is not something that you just are, smart is something that you can get.’

Despite extraordinary claims for the efficacy of a growth mindset, however, it’s increasingly unclear whether attempts to change students’ mindsets about their abilities have any positive effect on their learning at all. And the story of the growth mindset is a cautionary tale about what happens when psychological theories are translated into the reality of the classroom, no matter how well-intentioned. …

Growth mindset theory has had a profound impact on the ground. It is difficult to think of a school today that is not in thrall to the idea that beliefs about one’s ability affect subsequent performance, and that it’s crucial to teach students that failure is merely a stepping stone to success. Implementing these ideas has been much harder, however, and attempts to replicate the original findings have not been smooth, to say the least. A recent national survey in the United States showed that 98 per cent of teachers feel that growth mindset approaches should be adopted in schools, but only 50 per cent said that they knew of strategies to effectively change a pupil’s mindset.

The truth is we simply haven’t been able to translate the research on the benefits of a growth mindset into any sort of effective, consistent practice that makes an appreciable difference in student academic attainment. In many cases, growth mindset theory has been misrepresented and miscast as simply a means of motivating the unmotivated through pithy slogans and posters. A general truth about education is that the more vague and platitudinous the statement, the less practical use it has on the ground. ‘Making a difference’ rarely makes any difference at all. …

All of this indicates that using time and resources to improve students’ academic achievement directly might well be a better agent of psychological change than psychological interventions themselves. In their book Effective Teaching (2011), the UK education scholars Daniel Muijs and David Reynolds note: ‘At the end of the day, the research reviewed has shown that the effect of achievement on self-concept is stronger that the effect of self-concept on achievement.’

Many interventions in education have the causal arrow pointed the wrong way round. Motivational posters and talks are often a waste of time, and might well give students a deluded notion of what success actually means. Teaching students concrete skills such as how to write an effective introduction to an essay through close instruction, specific feedback, worked examples and careful scaffolding, and then praising their effort in getting there, is probably a far more effective way of improving confidence than giving an assembly about how unique they are, or indeed how capable they are of changing their own brains. The best way to achieve a growth mindset might just be not to mention the growth mindset at all.

What’s your number?

Another maths curiosity from the Futility Closet:

Fortuitous numbers
In American usage, 84,672 is said EIGHTY FOUR THOUSAND SIX HUNDRED SEVENTY TWO. Count the letters in each of those words, multiply the counts, and you get 6 × 4 × 8 × 3 × 7 × 7 × 3 = 84,672.

Here’s something I’ve (pointlessly) struggled with for a long time, now. Can you complete this sentence?

Written as words, there are _____________ letters in this sentence.

Use Excel’s LEN() function and AutoSum and try it like this, writing it out one word at a time.

number-1

So, forty three letters so far, with those two empty boxes. If you were to write forty three into those boxes, the total would obviously be more than forty three. A little trial-and-error, and we get

number-2

the answer fifty three. Well, that was fairly straightforward. Let’s try a slightly different sentence.

number-3

Maybe this isn’t so difficult, after all. One more?

number-4

That’s not right, there are forty nine letters in that sentence, not forty eight.

number-5

But now there are forty eight. Is it not possible to accurately complete that sentence, then?