Schools’ exam seasons look very different this year, but revision videos are still a valuable resource. I thought I’d share a couple to help support those studying maths and French.
Photo Anthony Delanoix
John Horton Conway, a ‘magical genius’ in math, dies at 82 – The New York Times
During what Dr. Conway called his “annus mirabilis,” roughly 1969 to 1970, he discovered what’s known as the Conway group, an entity in the realm of mathematical symmetry that inhabits 24-dimensional space. He discovered a new type of number, “surreal numbers.” And he invented the cellular automaton Game of Life, which is among the most beautiful mathematical models of computation. He described it as a “no-player never-ending” game.
I first came across this strange no-player game some time ago, and was struck by how simple its rules were, yet how intricate the patterns could become.
John Conway’s Game of Life
Each cell with one or no neighbors dies, as if by solitude.
Each cell with four or more neighbors dies, as if by overpopulation.
Each cell with two or three neighbors survives.
Each cell with three neighbors becomes populated.
I knew that these simple rules could create many different kinds of structures…
John Conway, inventor of the Game of Life, has died of COVID-19 – Ars Technica
This video, for example, shows a gun that consists of several large structures that each generates irregular streams of gliders. A bunch of gliders—37 to be exact—collide with one another in just the right way to produce a complex spaceship called a 6-engine Cordership, which then proceeds in another direction. It takes 784 moves to produce a Cordership before the cycle repeats.
… but I hadn’t appreciated just how complex they could get.
First replicating creature spawned in life simulator – New Scientist
When Wade posted his self-replicating mathematical organism on a Life community website on 18 May, it sparked a wave of excitement. “This is truly ground-breaking work,” wrote a fellow Life enthusiast, Adam Goucher, on the website Game of Life News. “In fact, this is arguably the single most impressive and important pattern ever devised.”
My understanding of all this stops way before these videos.
I can follow along well enough to make one of those glider guns, but a digital clock? You’re kidding me.
How’s this for meta? The game of life built within the game of life. No clue what’s going on here.
Let’s leave the last word to the man himself.
How bad will this get? It’s a simple enough question…
Why it’s so freaking hard to make a good COVID-19 model – FiveThirtyEight
The number of people who will die is a function of how many people could become infected, how the virus spreads and how many people the virus is capable of killing.
Straightforward enough, but the trouble begins when you try to fill in the numbers. Look at the factors and assumptions within just the fatality rate, for instance.
Think of it like making a pie. If you have a normal recipe, you can do it pretty easily and expect a predictable result that makes sense. But if the recipe contains instructions like “add three to 15 chopped apples, or steaks, or brussels sprouts, depending on what you have on hand” … well, that’s going to affect how tasty this pie is, isn’t it? You can make assumptions about the correct ingredients and their quantity. But those are assumptions — not absolute facts. And if you make too many assumptions in your pie-baking process, you might very well end up with something entirely different than what you were meant to be making. And you wouldn’t necessarily know you got it wrong.
There are so many factors as play here. This is the model they end up with. It’s one version, at least.
Over the next few months, you are going to see many different predictions about COVID-19 outcomes. They won’t all agree. But just because they’re based on assumptions doesn’t mean they’re worthless.
“All models are wrong, it’s striving to make them less wrong and useful in the moment,” Weir said.
Six unknown factors in coronavirus models and how they could affect predictions – The Conversation
Since the global outbreak of COVID-19, researchers have scrambled to develop and share models which can predict how the virus will spread. This is inherently tricky, as we know so little about the disease, and a model is only ever as good as the information you put into it.
I almost forgot — today is 22/7, Pi Approximation Day, “a holiday for people who are GOOD ENOUGH, just not transcendental! They do their best! They get by alright in most situations – just don’t try to build a bridge with them, you know?”
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.
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.
Another maths curiosity from the Futility Closet:
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.
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
the answer fifty three. Well, that was fairly straightforward. Let’s try a slightly different sentence.
Maybe this isn’t so difficult, after all. One more?
That’s not right, there are forty nine letters in that sentence, not forty eight.
But now there are forty eight. Is it not possible to accurately complete that sentence, then?