Perhaps it could by of interest: MIT open mathematics courseware.
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.
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.
Contact: Anna Mikulak
Association for Psychological Science
Understanding fractions is a critical mathematical ability, and yet it’s one that continues to confound a lot of people well into adulthood. New research finds evidence for an innate ratio processing ability that may play a role in determining our aptitude for understanding fractions and other formal mathematical concepts. Continue reading
Classes given by Alexander Shen at Summer School “Vanechki” in August 2014 in Portugal, based on his book Geometry in Problems. It seems that the audience are children of Russian diaspora, classes are conducted in mixture of English and Russian. However, an English speaking teacher of mathematics may make a lot of interesting observations.
Laurent Schwartz, as quoted from A Mathematician Grappling with His Century, Birkhäuser Basel, 2001, pp. 30-31. [With thanks to Jonathan Crabtree]
I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Even when I was the first to answer the teacher’s questions, I knew it was because they happened to be questions to which I already knew the answer. But if a new question arose, usually students who weren’t as good as I was answered before me. Towards the end of eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time. Not only did I believe I was stupid, but I couldn’t understand the contradiction between this stupidity and my good grades. I never talked about this to anyone, but I always felt convinced that my imposture would someday be revealed: the whole world and myself would finally see that what looked like intelligence was really just an illusion. If this ever happened, apparently no one noticed it, and I’m still just as slow. When a teacher dictated something to us, I had real trouble taking notes; it’s still difficult for me to follow a seminar.
At the end of eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn’t have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant. Naturally, it’s help to be quick, like it is to have a good memory. But it’s neither necessary or sufficient for intellectual success.The laurels I won in the Concourse General liberated me definitely from my anguish. I won first prize in Latin theme and first access it in Latin version; I was no longer merely a brilliant high school student, I acquired national fame. The Concourse General counted a lot in in my life, by helping me to get rid of a terrible complex. Of course, I was not instantly metamorphosed, and I’ve always had to confront the same problems; it’s just that since that day I know that these obstacles are not unsurmountable and that in spite of delicate and even painful moments,they will not block my way to accomplishment, which research represent for me. Fortunately, I had an excellent memory. For instance, in twelfth grade, in math, I believe that at the end of the year I remembered every single thing I had learned, without ever have written anything down. At that point, I knew my limits but I had a solid feeing of confidence in my possibility of success.
This type of competition is an excellent thing. Many young people feel self-doubt, for one reason or another. The refusal of any kind of comparison which reigns in our classrooms as a concession to egalitarianism, is all too often quite destructive; it prevents the young people who doubt their own capacities, and particularly those from modest backgrounds, from acquiring real confidence in themselves. But self-confidence is a condition of success. Of course, one must be modest, and every intellectual needs to recall this. I’m perfectly conscious of the immensity of my ignorance compared with what I know. It’s enough to meet other intellectuals to see that my knowledge is just a drop of water in an ocean. Every intellectual needs to be capable of considering himself relatively, and measuring the immensity of his ignorance. But he must also have confidence in himself and in his possibilities of succeeding, through the constant and tenacious search for truth.
[With thanks to Jonathan Crabtree]
Happy New Year!
I’d like to share with you that the latest English version of Lines and Curves is now available on the Springer website (they made thousands of their books freely available online): http://link.springer.com/book/10.1007/978-1-4757-3809-4
A later addition, 07 January 2016:
The Department of Mathematics and Science Education at Stockholm University islooking to appoint two full-time PhD students for a four year, possibly five year with a 20% teaching load, project on the development of grade one students’ foundational number sense in England and Sweden. The project, which is funded by the Swedish Research Council, will involve interviews, with parents and teachers of students and, following those, the development and implementation of surveys for use with parents and teachers in the two countries. Finally, the project will involve video-based classroom observations. The project is being led by Professor Paul Andrews paul [dot] andrews >>at<< mnd.su.se from whom further information may be obtained, and Dr Judy Sayers. It is possible that one of the students could be based in England. Applicants must be fluent in English and at least one must be fluent in Swedish.
The announcements in English and Swedish respectively can be found at
The closing date for applications is January 15, 2016.
In other words, the game of sudoku is remarkably similar to the calculations mathematicians do. Why is it so difficult to teach mathematics then?
When I was in high school, we once had a visit of a few Chemistry students from the local university. They were visiting high schools all over the city to inspire young students to apply for university-level chemistry education. Their promo event went like this: they showed us chemicals. That was it. They mixed chemicals of various colours, and created colourful smoke, steam, and liquids in test tubes of various shapes. And then they went home.
This sort of presentation inspires non-scientists and gives the wrong idea of what science is! You should become a Chemist if you love doing Chemistry, not if you love the end product. Doing Chemistry means: these are your starting chemicals, and this is your equipment – which methods and in which order will you do them to arrive at the desired chemical end product? That is what scientists love, that is what gets you the best brains in the class; the colourful chemicals attract non-scientists.
And if you think about it, the same goes for mathematics: “there is an infinite amount of prime numbers” – may be true, may not be true, but let’s find out, let’s either prove or disprove it, we shall see for ourselves. That is exciting. The knowledge of the mathematical proof is much more exciting than the knowledge whether the statement was true or false.
And that is not what we teach, we teach answers, because answers can be graded. I sincerely believe that if the greatest mathematical minds there ever were were born today, they would be disgusted by today’s mathematical education, and would go on to pursue other fields. It is said that Gauss was a rather annoying pupil because he always finished early during the math classes, so his desperate teacher, in an attempt to keep him occupied for the rest of the class, told him to sum all numbers from 1 to 100 when he was 6 years old – he immediately came up with n*(n+1)/2. This formula is what some high school students need to make a “cheat sheet” for and hide it into their shirt sleeves during an exam, because they have not developed the skills needed to derive this formula by themselves – to them it is just a series of symbols.
So, to answer the original question, why is it so difficult to teach mathematics then? – I think the education of mathematics will not change until we find a way how to put an exam grade on mathematical creativity – which is something you can not grade. All we can grade is the “hard work” – reducing a fraction, deriving/integrating a function, all the tasks that no mathematician really enjoys, because it is calculation, not math. So, the only thing we could probably do, is to rename the high school subject from “mathematics” to “calculation”, as it used to be named just a few decades ago in the Czech Republic.
Sorry if I seem way to passionate about this subject matter. It is because I work at a photonics laboratory as a physicist, but the more I do this job, the more I regret I did not study mathematics instead. It was not my fault though – mathematics on the high school level is the most mind-numbing discipline imaginable. It was not until I went to the university when I first encountered what mathematics is about, but that was too late, I was already on Physics. As insulting to scientists as it may sound, from my experience in a modern high-tech physics lab, I notice that all sciences are nothing but subsets of mathematics. I choose the word “subset” very carefully – mathematics is without any doubt the broadest science (not a natural science though!) that defines pure reasoning, true wisdom in its purest form, that the human brain is capable of. Natural sciences, such as physics, take a part of mathematics and put it into the context of atoms. Not the other way around. It is clear that mathematics was always steps ahead than technology – it can not obviously be the other way around.
Please, feel free to disagree with me, I would be happy to be wrong here, as the modern state of math education is sadly not a happy topic. I have heard someone say that a major revolution in math education came during the Cold War, as both the Soviet Union and the US started training mathematicians as “soldiers” – it was believed it would be the mathematicians who would build the best nuclear bombs and win the war, not the infantry soldiers. That is when the strict military-like math grading was enforced. But I found no evidence to prove or disprove this explanation.
Tim Gowers starts a new journal:
This post is to announce the start of a new mathematics journal, to be called Discrete Analysis. While in most respects it will be just like any other journal, it will be unusual in one important way: it will be purely an arXiv overlay journal. That is, rather than publishing, or even electronically hosting, papers, it will consist of a list of links to arXiv preprints. Other than that, the journal will be entirely conventional: authors will submit links to arXiv preprints, and then the editors of the journal will find referees, using their quick opinions and more detailed reports in the usual way in order to decide which papers will be accepted.