An Open Letter: To Andreas Schleicher, OECD, Paris
Heinz-Dieter Meyer and Katie Zahedi, and signatories – 5th May 2014
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.
D. Edwards, The Math Myth, The De Morgan Gazette 5 no. 3 (2014), 19-21.
I’ve been concerned with what skills those who are working as scientists and engineers actually use. I find that the vast majority of scientists, engineers and actuaries only use Excel and eighth grade level mathematics. This suggests that most jobs that currently require advanced technical degrees are using that requirement simply as a filter.
[A version of this text appeared in the August, 2010 issue of The Notices of The American Mathematical Society.]
As an ex-schools minister I see value in the unions. But they are wrong not to join our battle against progressive educationalists. [...] This might seem like an odd thing for a Conservative MP and former schools minister to say, but teaching unions are not the problem with our schools. [...]
[...] who is to blame for our education system slipping down the international rankings? The answer is the academics in the education faculties of universities. This is where opposition to the use of phonics in the teaching of young children to read lies, despite vast evidence from this country and other English-speaking countries that systematic synthetic phonics is the most effective and successful method.
Within these education departments lie the proponents of so-called progressive education, which advocates that education should be child-led rather than teacher-led; many advocate a play-based classroom until children are seven years old. It is an approach that espouses learning by discovery rather than having teachers directly teaching children. For decades many education academics downplayed the importance of spelling, punctuation and grammar. Textbooks are regarded by many in the education departments as appalling teaching tools, and in the 1970s they virtually disappeared from primary schools. Progressive educationalists oppose testing and believe that a knowledge-rich education is pointless in the Google age.
It is challenging the hegemony of the education departments of the universities that must be the focus of any serious education reformer and anyone who believes, as Gove does, that the attainment gap between those from poorer and wealthier backgrounds needs to be closed. There are many in the teaching profession who share this view. There are many on the left who hanker for the type of education provided in the independent sector – largely untainted by the progressive ideology of the education faculties – but who want their children educated by the state. They, too, should be railing against these educationalists.
By Sean Harford HMI, National Director, Initial Teacher Education, Ofsted
Reposted from TES Connect.
In late February I was a member of a delegation representing HM Government that visited the three Chinese provinces of Shanghai, Beijing and Hubei with a specific focus on mathematics education.
I have waited until now to reflect on my visit to China because I wanted to go back into some English schools to test out the thinking I developed while there. The differences in maths outcomes for our young people between the two countries are stark and worrying for us, unless we act now to catch up – and I do not mean just in terms of PISA test scores. I am coming at this not only from an inspector’s point of view, but also from my background of being a physics teacher and so frequent user of maths, reliant on pupils being able to handle and manipulate numbers confidently. In this respect, Chinese children are streets ahead of ours, so the benefits of their high standards in mathematics go way beyond just this core subject.
As everyone knows, Her Majesty’s Inspectors are not concerned about the ‘how’ but ‘how effective’ with teaching. This approach requires a clear focus on the outcomes for the pupils and their response to the teaching, including crucially the evidence of learning and progress over time in their work books and folders. These were impressive in the classes we observed in China, and told a story of a consistency of approach and expectations that has led to the pupils being confident mathematicians, willing to have a go and able to tackle problems in different contexts.
For example, given this problem…:
X = 2√ (7/14 x 28/7 x 3/9 x 24/8 x 18/9)
… none of the 12-year-old pupils reached for the calculator; they couldn’t because they have been banned from their classrooms. They calmly looked for the potential to cancel and reduce the fractions, and spotted that this expression is really just the square root of 4. Not a job for the calculator; not for them at least. This was clearly not about them learning ‘tricks’ either. This problem was one of just 4 or 5 set by the teacher in a 5 minute burst of practice, to help the pupils master the concepts covered by her in the latest part of the lesson before they moved on confidently together to the next stage of increasingly challenging maths. The key was not the teacher’s ‘performance’ in this lesson, but the demonstration of the depth of the pupils’ mathematical learning over time and the impressive armoury of knowledge and skills they had built up to deploy as and when needed. Evidence of solidly knowing their times tables was absolutely apparent across the pupils, as was the ability to use efficient methods of calculation without having to really think. Their mathematical toolkit was there to be used as surely as a mechanic’s spanners, or a surgeon’s scalpel
Read the rest at TES Connect.