Its analysis, however, makes a good exercise for teacher training courses.
Is this beginning of the end of the traditional model of mathematics education?
This advert for PhotoMath gone viral: and enjoys an enthusiastic welcome.
Mathematical capabilities of PhotoMath, judging by the product website, are still relatively modest. However, if the scanning and OCR modules (“OCR” here refers to “Optical Character Recognition”, not to the well–known examination board). of PhotoMath are combined with the full version of Yuri Matiasevich‘s “Universal Math Solver“, it will solve at once any mathematical equation or inequality, or evaluate any integral, or check convergence of any series appearing in the British school and undergraduate mathematics. Moreover, it will produce, at a level of detail that can be chosen by a user, a complete write-up of a solution, with all its cases, sub-cases, and necessary explanations (with slight Russian accent, but that can be easily fixed).
In short, smart phones can do exams better, and the system of mathematics education based on standard written examinations is dead. Perhaps, we have to wait a few years for a formal coroner’s report, but we cannot pretend that nothing has happened.
In my opinion, a system of mathematics education which focuses on deep understanding of mathematics and treats mathematics as a discipline and art of those aspects of formal reasoning which cannot be entrusted to a computer is feasible. But such alternative system cannot be set-up and developed quickly, it is expensive and raises a number of uncomfortable political issues. I can give an example of a relatively benign issue: in the new system, it is desirable to have oral examinations in place of written ones. But can you imagine all the complications that would follow?
PhotoMath gives a plenty of food for thought.
An embedded link to YouTube, http://www.youtube.com/watch?v=EHAuGA7gqFU :
An Open Letter: To Andreas Schleicher, OECD, Paris
Heinz-Dieter Meyer and Katie Zahedi, and signatories – 5th May 2014
[Reposted from multijimbo.]
I have been a teacher for many years now; in fact, we’re now rapidly approaching the point at which I’m thrice the age of my students, rather than merely twice. I teach 2 very different things. On the one hand (which I can write without violating the basic tenets of the Number Liberation Front, as I’m using ‘one’ not as a number but as an adjective, I suppose), I teach mathematics to undergraduate university students, and on the other hand, I am 1 of the instructors in the University aikido club. I’ve been thinking recently about the commonalities and differences between the 2 types of teaching.
There is an obvious difference between the 2. To practice aikido requires physical contact. Someone grabs me, or attempts a strike, and I need to do something rather quickly. (Or is it quite quickly? Having lived in 2 countries where the use of ‘quite’ and ‘rather’ is different, I am now very confused and can’t remember which is the current local usage.) The end of a good, active aikido session can be sweaty. This physicality leads to a directness in teaching. When I’m being thrown by a student, I have the opportunity to feel exactly what they’re doing, right and wrong, which I can then feed back to them immediately.
Mathematics, on the other hand, can be done in isolation. (And to follow a random train of memories, this brings to mind Ms Shearer, my 6th grade teacher, with whom we spent a session listening to Simon and Garfunkel’s I am a rock, I am an island, and discussing how people cannot exist in isolation, as they remain part of the cultural in which they grew up.) Also, mathematics rarely involves physical combat. Not never, mind you, just rarely. In terms of teaching, though, mathematics teaching is a bit more at a distance than aikido teaching. Part of this is that mathematics classes tend to significantly larger than the aikido classes I teach. Also, a good, active mathematics class rarely ends in sweat.
Even so, there are for me some deep and significant similarities. These are things that no doubt are similar to the teaching of many things, but hey, this is my meditation. The similarity I would like to focus on here is the lead-a-horse-to-water phenomenon that is regularly, and sometimes almost brutally, brought home to me in both teaching fora.
In both aikido and mathematics, there are some basic, fundamental ideas that underlie everything that we do, and that I try to bring out and illustrate as much as I can through my teaching. This is after all, in my mind at least, what a teacher should do. I have spent time studying how to do particular things, learning from my contemporaries and those who have gone before, and I can use the miracle of language to take what I’ve learned and provide my students with some short cuts, so that they can get farther along the path a bit faster than me.
In aikido, 1 of these basic, fundamental ideas is that at any moment in a technique, I should understand where my balance is and what is happening within both my own centre and my partner’s centre. The way I like to try and embed this idea into my students’ brains is to have them go slowly through a technique, paying attention throughout. But this requires that the student is willing to do the technique slowly, and alas not all of them are. So I talk, I demonstrate, I cajole, but in the end, I cannot force. Ultimately, I cannot teach anything. All I can do is to provide guidance for my students on how they might learn and provide them with an environment within which they can learn.
In mathematics, the basic, fundamental idea on which I like to focus is that each statement, each assertion, needs to come from somewhere. With each question, we have to start with things we know to be true and work out from there. Part of an undergraduate mathematics education, and indeed mathematics education before university, is to provide students with a collection of facts, procedures and processes that we know to be true. Mathematics does not come from nothing. Mathematical facts do not spring full-grown from the head of Zeus. Rather, mathematical facts are the product of accretion and accumulation (and this is where the sweat comes from). We have just come to the end of the semester, and as in all previous years, I have the evidence that some of my students listened, and some didn’t.
So, what to do? There is nothing to do besides persist. Some students listen and some students don’t, but I have come to believe that it is these larger things, these fundamental ideas, that are by far the more important things that I teach, far beyond the individual techniques of aikido or the definitions, theorems and examples in mathematics. And so we persist. As Samuel Beckett once wrote, ‘Try again. Fail again. No matter. Try again. Fail again. Fail better.’
From BBC http://www.bbc.co.uk/news/education-29342539 :
Low-level, persistent disruptive behaviour in England’s schools is affecting pupils’ learning and damaging their life chances, inspectors warn.
The report says too many school leaders, especially in secondary schools, underestimate the prevalence and negative impact of low-level disruptive behaviour and some fail to identify or tackle it at an early stage.
Source: Poll conducted by YouGov for Ofsted, http://www.ofsted.gov.uk/news/failure-of-leadership-tackling-poor-behaviour-costing-pupils-hour-of-learning-day
This is one of many low-level school issues that affect undergraduate mathematics teaching. In a mathematics lecture, weaker students are more prone to “loosing the thread” than in most other courses. Also, students for whom English is not the first language, in particular, most from overseas are more sensitive to the signal-to-noise ratio than natives, and, at a certain level of background noise, their understanding of the lecture becomes seriously degraded. In my opinion, this is one of many neglected issues of undergraduate mathematics education. I in my lectures always insist on complete silence in the audience (and usually start my first lecture with a brief explanation of the concept of signal-to-noise ratio).
Many mathematicians believe that that their brains continue to do mathematics during sleep. A paper
Kouider et al., Inducing Task-Relevant Responses to Speech in the Sleeping Brain, Current Biology (2014), http://dx.doi.org/10.1016/j.cub.2014.08.016
Proves that brain continues in sleep some mental activities of the day.
From the summary of the paper:
using semantic categorization and lexical decision tasks, we studied task-relevant responses triggered by spoken stimuli in the sleeping brain. Awake participants classified words as either animals or objects (experiment 1) or as either words or pseudowords (experiment 2) by pressing a button with their right or left hand, while transitioning toward sleep. The lateralized readiness potential (LRP), an electrophysiological index of response preparation, revealed that task-specific preparatory responses are preserved during sleep. These findings demonstrate that despite the absence of awareness and behavioral responsiveness, sleepers can still extract task relevant information from external stimuli and covertly prepare for appropriate motor responses.
The paper generated a huge response in mass media: BBC, New Scientist, NBC News. It is mentioned in this blog because the study of brain activity is relevant to mathematics education. A naive question: do our students get enough sleep?
A. E. Kyprianou: The UK financial mathematics M.Sc. arXiv:1405.6739v2 [math.HO]
Postgraduate taught degrees in financial mathematics have been booming in popularity in the UK for the last 20 years. The fees for these courses are considerably higher than other comparable masters-level courses. Why? Vendors stipulate that they offer high-demand, high-level vocational training for future employees of the financial services industry, delivered by academics with an internationally recognised research reputation at world-class universities.
We argue here that, as the UK higher education system moves towards a more commercial environment, the widespread availability of the M.Sc. in financial mathematics exemplifies a practice of following market demand for the sake of income, without due consideration for the broader consequences. Indeed, we claim that, as excellent as such courses can be in intellectual content and delivery, they are mismatching needs and expectations for such education and confusing the true value of what is taught.
The story of the Mathematical Finance MSc serves as a serious case study, highlighting some of the incongruities and future dangers of free-market education.
It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. The same pattern played out in the 1960s, when schools gripped by a post-Sputnik inferiority complex unveiled an ambitious “new math,” only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.
The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them. [Emphasis is mine -- AB.]
A sample KS2 test based on the official publication from Standards and Testing Agency,
2016 key stage 2 mathematics test: sample questions, mark scheme and commentary,
was published in The Telegraph. One question attracts attention. In The Telegraph version, it is
The answer given is £12,396.
And this is the original question from 2016 key stage 2 mathematics test: sample questions, mark scheme and commentary
In my opinion, both versions contain serious didactic errors. Would the readers agree with me?
And here are official marking guidelines:
And the official commentary:
In year 6 pupils are expected to interpret and solve problems using pie charts. In this question pupils can use a number of strategies including using angle facts or using fractions to complete the proportional reasoning required.
Pupils are expected to use known facts and procedures to solve this more complex problem. There are a small number of numeric steps but there is a demand associated with interpretation of data (or using spatial knowledge). The response strategy requires pupils to organise their method.