The use of the term ‘Expected Frequency’

The June 2015 GCSE Subject Level Conditions and Requirements for Mathematics includes (P3)

“relate relative expected frequencies to theoretical probability, using appropriate language and the 0 – 1 probability scale”

and this leads to questions like

“If you rolled a die 600 times, how many sixes would you expect to get’.

which is taken from the CIMT MEP Pupil’s textbook on probability, and is given the answer

‘You would expect to get a 6 in 1/6 of the cases, so 100 sixes’.

This seems a confusing and misleading term. What exactly is an ‘expected frequency?’ The obvious meaning is the frequency that you expect. But we are trying to support the concept of a random variable, with ideas that a random variable is unpredictable in terms of value, that values do not form patterns or sequences, and can only be forecast and predicted in some general ways.

If you roll a die 600 times, I do not expect any value for the number of sixes. That is the most significant aspect of a random variable.

The implied sub-text is that

Expected frequency = probability X number of trials

So that, for example, if we toss a fair coin 100 times, what is the expected frequency of heads? Well, 50. So does that mean we expect to get 50 heads? This is a Bernoulli trial, and the probability of getting precisely 50 heads in 100 tosses is about 0.08. So we would need to say to a pupil

‘The expected frequency is 50; but it is unlikely that you would get 50 heads’

which hardly makes sense.

The probability of 51 is about .078, and 52 is .074. So, of course, 50 is the most likely frequency.

The phrase ‘most likely frequency’ is straight-forward, makes sense, and says what it means, unlike ‘expected frequency’.

Please can we stop using the phrase ‘expected frequency’?

The Great Mystery of Malta’s Learning Outcomes Framework

Important update below: it is no longer a mystery.

Malta’s new Learning Outcomes Framework is an important case study of the European Union’s approaches to implementation of its education policies in member countries. For that reason the Framework deserves a close attention.

An attempt to study the official website

immediately leads to a question:

Who had actually developed the Framework?

According to Wikipedia, population of Malta is about 445,000. When compared with the City of Manchester (about 514,000), it becomes clear that development of the Framework is a job beyond capabilities of a small nation.

So, external consultants were hired, some institutions or companies from English speaking parts of Europe. Taking into consideration traditional cultural connections, this part of Europe is likely to be the UK.

Added 24 August 2015:  Indeed I could not locate contractor’s names using advanced Google search on,but  serendipitously discovered their logos in the document Joint Venture Presentation dated 28  Jan 2015:

IoE_24Aug15Outlook Coop is a company on Malta specialising in project management with expertise in EU funded projects. 

East Cost Education Ltd is a small private company based in Northumbira with specialism, judging by their website,  concentrated mostly in vocational education and training. In recent years, they worked on Malta on several projects in vocational training.

Institute of Education, London, is

the world’s leading centre for education and applied social science.

Outcome Based Education

In 1990, South Africa regarded Outcome Based Education (OBE) as its preferential educational paradigm, and designed Curriculum 2005. The South African Department of Education was very influenced by William Spady — an American proponent of OBE, who visited South Africa as a consultant on the issue. The National Qualification Framework went into effect in 1997 with great expectations, but these expectations were not met. It became evident even to the most vocal OBE-proponents that the educational approach gave inculcate skills not conducive to pursue any university education in mathematics and science. Since then, the curriculum underwent several corrections, and now is at stage of Curriculum Schooling 2025. Meanwhile, William Spady distanced himself from the South African version of OBE, describing it as a professional embarrassment:

“So now, with a decade of confusion about OBE behind us, I would encourage my South African colleagues to stop referring to OBE in any form. It never existed in 1997, and has only faded farther from the scene since. The real issue facing the country is to mobilize behind educational practice that is sound and makes a significant difference in the lives of ALL South African learners. Empty labels and flowery rhetoric are no longer needed; but principled thinking and constructive action are.”

Educational experts may argue whether it was Outcome Based Education, or some kind of Education Based on Outcomes. These experts may further argue on the terminology, but the fact remains it was supposed to be transformational OBE. A close look at their mathematics curriculum reveals that it is not so different from the proposed new Learning Outcomes Framework (LOF) for school mathematics in Malta, and in some aspects is even better. What is however completely identical in both is the educational utopia of outcomes coming from nowhere.

Essential mathematical skills are not just about a computational answer, for it is not the answer that is of the greatest importance to school children’s mathematical development. Rather it is children’s ability to apprehend mathematics as a conceptual system. Many education systems are emphasising on this, here is an excerpt from the Secondary Mathematics Syllabuses in Singapore:

“Although students should become competent in the various mathematical skills, over-emphasising procedural skills without understanding the underlying mathematical principles should be avoided… Students should develop and explore the mathematics ideas in depth, and see that mathematics is an integrated whole, not merely isolated pieces of knowledge.”

Unfortunately, in Malta’s case the design falls far short of such goals. Here is an example from level 5:

(COGNITIVE LEARNING) 16. I understand that multiplication is repeated addition.

Accordingly, a factor can only be added to itself a counting number of times. In Singapore’s Primary Mathematics Syllabus, multiplication and division are conceptualised gradually, and still on that level are introduced area and various square units. In contrast, square units are not present in Malta’s LOF for school mathematics. In fact, the proposed LOF is teeming with conceptual deficiencies. For instance, there is some kind of misconception between “equation” and “function”. Equations were never related to unknown variables, while functions are assumed to be somehow equations between the variables “x” and “y“. Use of radian measurement is not present, but learners are supposed to “plot graphs of trigonometric functions”.

Perhaps, Malta can learn from Singapore’s remarkable success since independence and the policies underlying its achievements in mathematical education.

Response to “Malta: new Learning Outcomes Framework”

Thank you Alexandre for taking an interest in the curriculum being developed for the Maltese schools. (As a matter of information, this curriculum is being developed by a consortium of foreign “experts” supported by a European Social Fund grant. What is shown on the website is work-in-progress, and one hopes that the final product will be a more coherent curriculum and banalities like the one you pointed out will have been removed.)
So, let me share my answers to the same question you ask, basically why does this draft curriculum contain such a statement: I can use equivalent fractions to discuss issues of equality e.g. gender. I agree with your two responses, namely mis-use of vocabulary and the strictures imposed by an Outcomes Based (OB) curriculum. But allow me to elaborate further.
In my view, the above statement would be banal whether one uses the term “equivalent fractions” or “similar fractions” or any other notion which extrapolates from 1/2=2/4=3/6=etc to anything having to do with gender equality. The problem, in my opinion, is that some people do not realise that, in science, we expropriate a word from everyday vocabulary to use in a context which does have some similarity to the everyday use of the word, but whose meaning becomes something technical which cannot be exported back to the everyday sense of the word.  I sometimes taught classes of Arts students who felt they needed to use some mathematical jargon in their essays (a few years ago the fashionable thing to do was to drop the words “chaos” and “fractal”). One of my usual examples of how wrong this is involved the use of the word “work”, as used in science and in everyday life. Translated into the context of curricula, the analogous banal statement could be something like: I can calculate the work done by a given force moving an object through a given distance and I can use this to discuss the conditions of work in factories and industry. 
What surprises me when statements such as the one on gender equality are made is that while the ambiguity of language is appreciated outside science, in fact it can be a wonderful tool in the hands of a good writer, when transporting scientific vocabulary back into the everyday world, this variegated meaning of the same word in different contexts is sometimes forgotten. I have no explanation why this happens.
But another problem with curricula written in OB style and which could have a bearing on such wording is the necessity that the statements should be written in a way that the learning child would write them, for example, by starting the description of each outcome with “I can…” That sentences such as the one you quote about gender issues crop up is not, in itself the main problem, in my opinion. Such sentences can be edited out when reviewing the curriculum. The problem, as I see it, is that this style excludes the possibility that the curriculum contain concepts to guide the teacher but which the student would not likely be able to express. So take your improved statement of how mathematics can help understand social inequalities:
I believe in the power of mathematics and I am convinced  that comparing numbers (for example, salary)  reveals a lot about gender inequality (and other, frequently hidden,  inequalities in the world — just recall the Oaxaca Decomposition and its role in fight against discrimination of any kind). 
It might be reasonable to expect a Level 5 student (aged 7-8) to express such a statement up to “gender inequality”, but hardly the rest of the statement, although the writer of the curriculum might very well want to make a reference to the Oaxaca Decomposition to give the teacher an example of a highly non-trivial use of mathematics in this context.
This OB format, I believe, betrays a fallacy about the teaching of mathematics, namely that teaching elementary mathematics to 7-year olds, say, does not involve deep knowledge of mathematics, certainly not deeper than what a 7-year old can express.
I look forward to reading other comments, especially by readers of this blog who are more familiar with OB curricula than I am.

Malta: new “Learning Outcomes Framework”

Malta published the new Learning Outcomes Framework for school mathematics

In my opinion, it is representative of current trends in mathematics education around the world and deserves a wider open discussion.

A random bit from Level 5:

31. I can use equivalent fractions to discuss issues of equality e.g. gender.

I believe in power of mathematics and I am convinced  that comparing numbers (for example, salary)  reveals a lot about gender inequality (and other, frequently hidden,  inequalities in the world — just recall the Oxaca Decomposition and its role in fight against discrimination of any kind). But equivalent fractions? 1/2 = 2/4 = 3/6? How are they related to gender issues?

I am a teacher of mathematics; when I hear a strange statement from my student, my first duty is to try to analyse my student’s way of thinking.

I found that the  “Learning Outcomes Framework” triggers in me the same Pavlovian reflex of trying to figure what the authors of the “Framework” have meant.   In this particular case, I cannot come up with anything better than a conjecture that perhaps the authors of  “Learning Outcomes Framework” associate the words “equivalent” and “equality” a bit too closely. Every teacher of mathematics  knows that mixing similary sounding terms is one of more common stumbling blocks for weaker students. The standard pedagogical remedy is to help the student to separate the concepts by asking him/her a splitting (or separating) question, for example

Equivalent fractions are also known under the name “similar fractions”. Why does the learning outcome

31. I can use similar fractions to discuss issues of equality e.g. gender.

appear to be less coherent and less convincing?

My main concern about “Learning Outcomes Framework” is that an official governmental document of a souverign nation of proud historic past has to be analysed using didactical tools (such as “separating questions”) reserved for work with struggling students.

Malta is a small country, and contributions to the debate from mathematics education experts from around the world might happen to be useful to our Maltesean colleagues. Please post your comments here.

Alexandre Borovik

David Singerman: X + Y the movie

[To appear in the LMS Newsletter]

There are now an increasing number of movies where mathematics plays an important role. Usually we are let down by the parts featuring the maths because the makers of the film have little knowledge about our subject. So it is a real pleasure to review x+y a beautiful film where the mathematics is carefully done but not in a way that will put off a non-mathematical audience. The director is Morgan Matthews who also made the BBC4 documentary Beautiful Young Minds about the Mathematical Olympiad and the film is clearly based on this documentary. This documentary can be seen on Youtube.

The main character is Nathan. From the BBC synopsis

Preferring to hide in the safety of his own private world, Nathan struggles to connect with people, often pushing away those who want to be closest to him, including his mother, Julie. Without the ability to understand love or affection, Nathan finds the comfort and security he needs in numbers and mathematics.

Even though there are similarities between this film and the documentary, the main story line is totally fictitious. Near the beginning, Nathan, who has Asperger’s syndrome, is involved in a car crash which kills his father to whom he was very close. He is then mentored by his maths teacher Martin Humphreys, who when young had taken part in the Mathematics Olympiad. He was diagnosed with multiple sclerosis but also has other problems to do with self worth and soft drugs and ended up being a secondary school teacher.

Humphreys recognizes Nathan’s abilities and persuades him to enter for the Olympiad. He goes to the preliminaries in Taipei.

One of the scenes where there is actual maths is when Nathan is brought to the board to explain how to solve a problem. This involves playing cards which can be face up or face down.

Nathan’s solution is to model this with binary arithmetic involving 0s and 1s and he then turns the problem into an arithmetic one which is easy to solve.

In Taipei he meets Zhang Mei, a girl on the Chinese team. The film concentrates on two relationships. One between Nathan and Zhang Mei and the other between Martin Humphreys and Julie.

The scene moves from Taipei to Cambridge where the Maths Olympiad takes place.

There is real pathos in the final scenes. One where Nathan finally opens himself up to his Mother, and another when Nathan and Zhang Mei while travelling back from Cambridge by train see a rainbow and the viewer feels that their relationship will last. At last, Nathan feels and understands love and affection. Some critics have thought that this ending is too soapy, but if you see the documentary on which this film is based, the rainbow really was there!

One should also mention the excellent cast. Nathan was played by Asa Butterfield, Martin by Ralf Spall, Julie by Sally Hawkins and Zang Mei by Jo Yang. A lovely film where mathematics plays a central role.

Geoff Smith on X + Y

Reposted from the UKMT’s Newsletter:

In March 2015, the film  X + Y  will appear in cinemas all over the UK. This is a romantic drama, and explores a collection of intense personal relationships. One of the main characters is a teenaged boy (played by Asa Butterfield) who competes enthusiastically in UKMT competitions, and who dreams of going to the International Mathematical Olympiad. Several leading actors decorate the cast (Sally Hawkins, Eddie Marsan, Rafe Spall, Jo Yang). The film was made with the co-operation of UKMT and the IMO, and logos and flags appear accordingly. The film has secured international distribution contracts, and will be seen in many countries, and on airlines.

This film grew out of the BBC2 documentary “Beautiful Young Minds”, and the common director is Morgan Matthews. If UKMT were to make such a film (an exceptionally bad suggestion), the emphasis would be much more on the mathematics and less on the relationships. Morgan Matthews has become very interested in the way people on the autistic spectrum can prosper in mathematics. There has been a natural concern in the maths community that portraying some mathematicians as being less than socially fluent is dangerous, because it could lead to the misapprehension that mathematicians are all strange.

My personal view is that the prefix “mis” in the previous sentence can be deleted. All mathematicians are strange because they place such an exceptional value on thought, ideas and understanding. I think that the maths community should be proud of the way it embraces people on the basis of their enthusiasm for and interest in mathematics. University maths departments are happy places, where the socially adroit rub along in harmony with people who live in more private spaces. The trick is mutual respect and affection. This is equally true of UKMT maths camps. Most students are relaxed and outgoing, with the full set of skills that allow them to prosper in the teenage social maelstrom. Some others are not, but everyone gets along almost all of the time, united by a passion for ideas and ingenuity. We all know maths people who sometimes appear confused and nervous, but who have beautiful mathematical insights.

Things would be even better if women and all racial groups were richly represented in the maths community, and UKMT has done excellent work on the gender issue by founding the European Girls’ Mathematical Olympiad and running the annual talent search examination, the UK Maths Olympiad for Girls. The mentoring schemes make an excellent education in mathematical problem solving available to all social groups. However, while social inclusion is very much “work in progress”, the incorporation of people on the autistic spectrum into the wider maths community seems to be a great success, and in my view, a cause for celebration.

Geoff Smith, Chair of the BMO and the IMO, University of Bath.

Disclaimer: Geoff was involved in assisting to make X + Y, so his views are not impartial.

Book Review: “What the Best College Teachers Do” by Ken Bain, 2004

Book review by Richard Elwes:

Open a typical book on the theory of pedagogy, and all too often one is confronted by a morass of impenetrable and, one often suspects, unnecessary jargon. So it is a particular pleasure to read Ken Bain’s “What the Best College Teachers Do”. The book is the outcome of a fifteen year study in which Bain and colleagues identified and analysed around a hundred excellent teachers at US Colleges and Universities. Through extensive observations, discussions, and interviews with the teachers and their students, Bain arrives at a range of conclusions regarding the practice of good teaching. His findings are laid bare in a series of straightforwardly entitled chapters: “How do they conduct class?”, “How do they treat their students?”, and so on.

Few of his discoveries come as complete surprises, yet many are genuinely enlightening. For instance, the best teachers “have an unusually keen sense of the histories of their disciplines, including the controversies that have swirled within them, and that understanding seems to help them reflect deeply on the nature of thinking within their fields”.

Many of the insights within this book derive from the removal of extraneous and superficial aspects of education. How do good teachers speak to their students? Obviously, there are countless possible answers. But what do these approaches have in common? “Perhaps the most significant skill the teachers in our study displayed in the classroom… was the ability to communicate orally in ways that stimulated thought.”

The author often allows his educators to speak for themselves, and as one might expect, they are a thoughtful and frequently amusing group. Thus we read the Harvard political theorist Michael Sandel opining that teaching is “above all… about commanding attention and holding it… Our task… is not unlike that of a commercial for a soft drink”. On the other hand, Jeanette Norden, professor of cell biology at Vanderbilt University, “told us that before she begins the first class in any semester, she thinks about the awe and excitement she felt the first time anyone explained the brain to her, and she considers how she can help her students achieve that same feeling.”

The teachers analysed come from a wide range of Colleges and academic disciplines; some teach only elite students, others specialise in assisting strugglers; while several are eminent researchers, a few have no research publications at all; they deploy a variety of educational techniques. Among this diversity, the conclusions that Bain avoids are as interesting as those he draws. “[P]ersonality played little or no role in successful teaching. We encountered both the bashful and the bold, the restrained and the histrionic…. We found no pattern in instructors’ sartorial habits, or in what students and professors called each other. In some classrooms first names were common; in others, only titles and surnames prevailed.”

All the same, some common traits are apparent. “Exceptional teachers treat their lectures… and other elements of teaching as serious intellectual endeavors, as intellectually demanding and important as their research and scholarship.”

Particularly important, Bain argues, is the fostering of a “natural critical learning environment”. This is the closest the book ever comes to jargon, but that judgement would be unfair: “‘natural’ because students encounter the skills, habits, attitudes, and information they are trying to learn embedded in questions and tasks they find fascinating… ‘critical’ because students learn to… reason from evidence, to examine the quality of their reasoning… and to ask probing and insightful questions about the thinking of other people”.

At this stage, the reader might worry that this catalogue of heroic deeds could be dispiriting to the rest of us. Not so. Whilst Bain is full of admiration for his teachers, he by no means deifies them. “Even the best teachers have bad days… they are not immune to frustrations, lapses in judgement, worry, or failure.” On the contrary, their ability to confront their own shortcomings is one thing which sets the best teachers apart from those others who “never saw any problems with their own teaching, or they believed they could do little to correct deficiencies”. Good teachers show humility and willingness to improve.

In comparison, the teachers identified as the “worst” by their students often appear to carry the attitude, as one of Bain’s subjects puts it, that only “smart men can possibly comprehend this material and that if you can’t understand what I’m saying, that must mean I’m a lot smarter than you are”. As the biologist Craig Nelson says “The trouble with most of us… is that we teach like we were god.” Contrast this to the view of Dudley Herrschbach, another of the teachers in the study (as well as being a Nobel Prize-winning Chemist) that “You have to be confused… before you can reach a new level of understanding anything.”

In summary, this short book is far more readable and entertaining than a text on educational theory has any right to be. It offers every Higher Education teacher an invaluable opportunity: to learn from the best.

Ken Bain, What the Best College Teachers Do, Harvard University Press, 2004. ISBN-10: 0674013255. ISBN-13: 978-0674013254.

David Mumford on Grothendieck and magazine “Nature”

Can one explain schemes to biologists

December 14, 2014

John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Now he is a hero of mine, the person that I met most deserving of the adjective “genius”. I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on “Existence theorems”. His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me.

So John and I agreed and wrote the obituary below. Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck’s work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology. Here’s what we came up with:

Continue reading

Andreas Schleicher: Seven big myths about top-performing school systems

A paper by Andreas Schleicher, , at the BBC website. The list of “seven big myths”:

  1. Disadvantaged pupils are doomed to do badly in school
  2. Immigrants lower results
  3. It’s all about money
  4. Smaller class sizes raise standards
  5. Comprehensive systems for fairness, academic selection for higher results
  6. The digital world needs new subjects and a wider curriculum
  7. Success is about being born talented

In my [AB] humble opinion,  this appears to be the case when the negations of myths are myths, too (with a possible exception of no. 7). School systems cannot, and should not, be compared without first having a close look at socio-economic, cultural, and political environments of their home countries.