Perhaps it could by of interest: MIT open mathematics courseware.
From BBC: A few quotes:
Stoke-on-Trent is trying to radically improve maths standards in its schools, including by helping to pay off the tuition fees of maths teachers who come to work in the city.
The maths project is aimed at improving the chances for young people growing up in a city where many traditional industries have declined. […]
The £1m maths project, a form of educational urban regeneration, is aimed at attracting bright young maths graduates to a city struggling with industrial decline and academic underachievement. […]
The Stoke project is offering cash to attract new recruits – £2,000 per year for three years towards paying off tuition fees and a further relocation payment of £2,000.
Mind/Shift. A few quotes:in
Prodigies in piano or dance can study at schools like Juilliard to develop their musical or performing arts talent. By contrast, nothing like Juilliard exists for children who show great promise at math. But an ambitious experiment will soon change that: In fall 2015, a small, independent school that’s exclusively tailored for math whizzes will open in downtown San Francisco.
The new school takes its inspiration from math circles, an Eastern European and Russian tradition that spread to the U.S. starting in the 1990s. These weekly extracurricular clubs bring youngsters together with a mathematician who guides them in exploring numerical ideas and concepts in depth. It’s often a highly interactive conversation, with the kids avidly chiming in with questions and thoughts.
I heard a famous French illustrator on the radio this morning and one of the thing he said strongly resonated with me. There were several versions of his background circulating in the press, publisher blurbs, web pages. In some of them he was an alumni of a famous Art School and in some of them he never had any formal training in drawing or painting. When asked by the interviewer about it, he simply said he did not go to an art school. He remarked he was often asked about his training, for instance: “When did you start drawing ?” He usually turns that around:
“You see, most children start expressing themselves through drawings, from a very early age, at home, in kindergarten. I did that, too, that’s nothing remarkable. Usually during primary school, they don’t do it anymore. I just didn’t stop. I never stopped. I kept drawing every day, every kind of things, and it happened partly because my parents did not block me or frown upon this activity. And I still do it. So when people ask me that kind of question, I ask them back: when did you stop drawing?”
Hearing him, I recalled my own frequent feeling of powerlessness when I try do draw something or see the kind of work I would like to produce myself. I told a friend who was listening with me to the radio program: “I think that’s what I did with mathematics. I started early playing with numbers, object combinations, dots, lines, a compass, gridded paper, I never stopped, and I never asked for permission.”
That’s probably what I should have done with drawing. Thinking about my terrible mandatory middle-school art hours may give me an insight into what people experience in the ordinary math classes — and what disgusts them.
To all those people feeling bad about mathematics,
we could ask : when did you stop doing “it”?
I write “it”, because many activities that are profoundly mathematical are not recognized as such by teachers and family of young children, while art is more commonly seen as a continuum. Parents are more open to their child expressing themselves in pre-art activities, to the point it becomes a nuisance to everyone else.
If I follow the analogy with early childhood drawings, it suggests that when helping people who have a failed or non-existent relation with mathematics, most approaches start with an excessive level of sophistication, preconception and structuration. We take for granted cognitive processes, standard viewpoints, rhetorics and expectations most mathematicians have acquired unknowingly from many clues. We are the ones who “got it”. We expect to bring people to reconciliation and insight within a few hours of structured exposure, we do not help them practice some accessible, spontaneous, “proto”-mathematics that could be formative, nor do we really prepare and aim for life-long practice, enjoyment and learning.
I hope I could “restart” drawing, as if I was 3 years old, discovering the fun of playing with color pens and sheets of paper.
There are many entry points for mathematics, many of them we have yet to find.
There are several ways to relate to mathematics, many ways to excel at it. This is not so widely known. Alexander Borovik, in several of his books, describes people warming to mathematics very early or others in their late adolescence or young adulthood (entering the University). I am rather of the first kind. Furthermore, I tend to quickly identify and entertain ancient connections between what I am studying and doing now and what I felt and longed for when I was in my school years, even as a very young child. Part of it is probably a self-serving fabrication: I take pleasure into the sense of cognitive continuity it offers. Genealogy conforts me and provides valuable analogies and insights.
But another part is linked to the fact that academic published mathematics has always been to me an extensive, wonderful and bewildering area of mathematics, not the whole. Before I was initiated to the global mathematical culture, I accumulated a store of pre-mathematical facts, experiences, tastes, concerns, implicit problems and naïve research programs that are still nagging me today. The corresponding perspective in art is very common: art is not restricted to what you can see in museums or what is labelled or publicized as such. You grow a sense of aesthetics, you look at some art and you see something that you always wanted to see or feel that something is missing. You know that art can be found almost anywhere, with various degree of sophistication, and that many starting points exist, many of them we have yet to find. I wish it were a more widespread opinion about mathematics too, especially among mathematicians.
Many authors of articles in “From Russia with Math” are prominent
research mathematicians; Stanislav Smirnov, for example, is a Fields
Medallist. Not surprisingly, they tend to focus on education for people like them.
In my assessment, this is the key message of the publication:
in Russia, high quality academically selective mathematics education remains possible even after the collapse of the system of mass equal-for-all education.
NYT Opinion Page, 3 December 2915, by
The new math was widely praised at first as a model bipartisan reform effort. It was developed in the 1950s as part of the “Cold War of the classrooms,” and the resulting textbooks were most widely disseminated in the 1960s, with liberals and academic elites promoting it as a central component of education for the modern world. The United States Chamber of Commerce and political conservatives also praised federal support of curriculum reforms like the new math, in part because these reforms were led by mathematicians, not so-called progressive educators.
By the 1970s, however, conservative critics claimed the reforms had replaced rigorous mathematics with useless abstractions, a curriculum of “frills,” in the words of Congressman John M. Ashbrook, Republican of Ohio. States quickly beat a retreat from new math in the mid-1970s and though the material never totally disappeared from the curriculum, by the end of the decade the label “new math” had become toxic to many publishers and districts.
Though critics of the new math often used reports of declining test scores to justify their stance, studies routinely showed mixed test score trends. What had really changed were attitudes toward elite knowledge, as well as levels of trust in federal initiatives that reached into traditionally local domains. That is, the politics had changed.
Whereas many conservatives in 1958 felt that the sensible thing to do was to put elite academic mathematicians in charge of the school curriculum, by 1978 the conservative thing to do was to restore the math curriculum to local control and emphasize tradition — to go “back to basics.” This was a claim both about who controlled intellectual training and about what forms of mental discipline should be promoted. The idea that the complex problems students would face required training in the flexible, creative mathematics of elite practitioners was replaced by claims that modern students needed grounding in memorization, militaristic discipline and rapid recall of arithmetic facts.
The fate of the new math suggests that much of today’s debate about the Common Core’s mathematics reforms may be misplaced. Both proponents and critics of the Common Core’s promise to promote “adaptive reasoning” alongside “procedural fluency” are engaged in this long tradition of disagreements about the math curriculum. These controversies are unlikely to be resolved, because there’s not one right approach to how we should train students to think.
We need to get away from the idea that math education is only a matter of selecting the right textbook and finding good teachers (though of course those remain very important). The new math’s reception was fundamentally shaped by Americans’ trust in federal initiatives and elite experts, their demands for local control and their beliefs about the skills citizens needed to face the problems of the modern world. Today these same political concerns will ultimately determine the future of the Common Core.
As long as learning math counts as learning to think, the fortunes of any math curriculum will almost certainly be closely tied to claims about what constitutes rigorous thought — and who gets to decide. [Emphasis is by AB]
Adapted from StackExchange, Getting Students to Not Fear Confusion
QUESTION: I’m a fifth year grad student, and I’ve taught several classes for freshmen and sophomores. This summer, as an “advanced” (whatever that means) grad student I got to teach an upper level class: Intro to Real Analysis.
Since this was essentially these student’s first “real” math class, they haven’t really learned how to study for or learn this type of thing. I’ve continually emphasized throughout the summer that they need to put in more work than just doing a few homework problems a week.
Getting a feel for the definitions and concepts involved takes time and effort of going through proofs of theorems and figuring out why things were needed. You need to build up an arsenal of examples so some general picture of the ideas are in your head.
Most importantly, in my opinion, is that you wallow in your confusion for a bit when struggling with problems. Spending time with your confusion and trying to pull yourself out of it (even if it doesn’t work!) is a huge part of the learning process. Of course asking for help after a point is important too.
Question: What is a good way to convince students that spending time lost and confused is a reasonable thing and how do you actually motivate them to do it?
Anecdote: Despite trying all quarter to explain this in various ways, I would consistently have people come in to office hours who had barely touched the homework because “they were confused”. But they hadn’t tried anything. Then when I talk around an answer to try to get them to do certain key parts on their own or get them to understand the concept involved, they would get frustrated and ask “so does it converge or not?!”
It is incredibly hard to shake their firm belief that the answer is the important thing. Those that do get out of this belief seem to get stuck at writing down a correct proof is the important thing. None seem to make it to wanting to understand it as the important thing. (Probably a good community wiki question? Also, real-analysis might be an inappropriate tag, do what you will)
ANSWER from Ronnie Brown: Has anyone tried as an additional technique the “fill-in” method?
This is based on the tried and tested method of teaching called “reverse chaining”. To illustrate it, if you are teaching a child to put on a vest, you do not throw it the vest and say put it on. Instead, you put it almost on, and ask the child to do the last bit, and so succeed. You gradually put the vest less and less on, the child always succeeds, and finally can put it on without help. This is called “error-less learning” and is a tried and tested method, particularly in animal training (almost the only method! ask any psychologist, as I learned it from one).
So we have tried writing out a proof that, say, the limit of the product is the product of the limits, (not possible for a student to do from scratch), then blanking out various bits, which the students have to fill in, using the clues from the other bits not blanked out. This is quite realistic, where a professional writes out a proof and then looks for the mistakes and gaps! The important point is that you are giving students the structure of the proof, so that is also teaching something.
This kind of exercise is also nice and easy to mark!
Finally re failure: the secret of success is the successful management of failure! That can be taught by moving slowly from small failures to extended ones. This is a standard teaching method.
Additional points: My psychologist friend and colleague assured me that the accepted principle is that people (and animals) learn from success. Another way of getting this success is to add so many props to a situation that success is assured, and then gradually to remove the props. There are of course severe problems in doing all this in large classes. This will require lots of ingenuity from all you talented young people! You can find some more discussion of issues in the article discussing the notion of context versus content.
My own bafflement in teenage education was not of course in mathematics, but was in art: I had no idea of the basics of drawing and sketching. What was I supposed to be doing? So I am a believer in the interest and importance of the notion of methodology in whatever one is doing, or trying to do, and here is link to a discussion of the methodology of mathematics.
Dec 10, 2014 I’d make another point, which is one needs observation, which should be compared to a piano tutor listening to the tutees performance. I have tried teaching groups of say 5 or 6, where I would write nothing on the board, but I would ask a student to go to the board, and do one of the set exercises. “I don’t know how to do it!” “Well, why not write the question on the board as a start.” Then we would proceed, giving hints as to strategy, which observation had just shown was not there, but with the student doing all the writing.
In an analysis course, when we have at one stage to prove A⊆B, I would ask the class: “What is the first line of the proof?” Then: “What is the last line of the proof?” and after help and a few repetitions they would get the idea. I’m afraid grammar has gone out of the school syllabus, as “old fashioned”!
Seeing maths worked out in real time, with failures, and how a professional deals with failure, is essential for learning, and at the research level. I remember thinking after an all day session with Michael Barratt in 1959: “Well, if Michael Barratt can try one damn fool thing after another, then so can I!”, and I have followed this method ever since. (Mind you his tries were not all that “damn fool”, but I am sure you get the idea.) The secret of success is the successful management of failure, and this is perhaps best learned from observation of a professional.
PhD 2016 Competition Scholarships [mathematics related]:
- An interdisciplinary study of mathematics and the body. Closing Date: 21 March 2016 (9am)
- Informal Mathematics Learning: New Approaches to Learning Mathematics, Arts, and Crafts, in Museums, After-School Programs, and Community Centers. Closing Date: 21 March 2016 (9am) Interviews: Week beginning 4 April 2016
- Investigating new forms of teacher education. Closing Date: 21 March 2016 (9am) Interviews: Week beginning 4 April 2016
- The Aetiology of Maths Anxiety: examining the role of socio-cultural factors in the primary stage of the educational lifecourse. Closing Date: 21 March 2016 (9am)
Teacher shortages in England are growing and the government has missed recruitment targets for four years, the official spending watchdog [the National Audit Office] has said.
Quotes about mathematics:
Teacher training places filled against targets, by subject
Head teachers’ unions said the report echoed their own research.
“The acute difficulties recruiting in maths, English, science and languages are now extending to most other areas of the curriculum,” said Malcolm Trobe, interim general secretary of the Association of School and College Leaders.Proportion of trainee teachers with 2:1 degree or aboveAverage for all trainee entrants was 75%[Source: National College for Teaching and Leadership]
National Association of Head Teachers general secretary Russell Hobby warned of “a significant difference between official statistics and the perceptions of those in schools.
“We’d welcome the opportunity to sit down formally with the DfE… but as yet, they’re not willing to acknowledge the scale of the problem.”
Labour’s shadow education secretary Lucy Powell called the report “a further wake-up call for the Tory government who have been in denial and neglectful about teacher shortages“.
A Department for Education spokeswoman said the report made clear
“that despite rising pupil numbers and the challenge of a competitive jobs market, more people are entering the teaching profession than leaving it, there are more teachers overall and the number of teachers per pupil hasn’t suffered“.
“Indeed the biggest threat to teacher recruitment is that the teaching unions and others, use every opportunity to talk down teaching as a profession, continually painting a negative picture of England’s schools.“