The Inspection Paradox is Everywhere

From a brilliant blog by Allen Downey:

The inspection paradox is a common source of confusion, an occasional source of error, and an opportunity for clever experimental design.  Most people are unaware of it, but like the cue marks that appear in movies to signal reel changes, once you notice it, you can’t stop seeing it.

 A common example is the apparent paradox of class sizes.  Suppose you ask college students how big their classes are and average the responses.  The result might be 56.  But if you ask the school for the average class size, they might say 31.  It sounds like someone is lying, but they could both be right.


The problem is that when you survey students, you oversample large classes.  If are 10 students in a class, you have 10 chances to sample that class.  If there are 100 students, you have 100 chances.  In general, if the class size is x, it will be overrepresented in the sample by a factor of x.
That’s not necessarily a mistake.  If you want to quantify student experience, the average across students might be a more meaningful statistic than the average across classes.  But you have to be clear about what you are measuring and how you report it.

Parents’ Math Anxiety Can Undermine Children’s Math Achievement

From: news@psychologicalscience.org
For Immediate Release

If the thought of a math test makes you break out in a cold sweat, Mom or Dad may be partly to blame, according to new research published in Psychological Science, a journal of the Association for Psychological Science.

A team of researchers led by University of Chicago psychological scientists Sian Beilock and Susan Levine found that children of math-anxious parents learned less math over the school year and were more likely to be math-anxious themselves—but only when these parents provided frequent help on the child’s math homework.

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Malta: new “Learning Outcomes Framework”

Malta published the new Learning Outcomes Framework for school mathematics

http://www.schoolslearningoutcomes.edu.mt/en/subjects/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:

COGNITIVE LEARNING
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

Evgeny Khukhro: George Boole exhibition opens in Lincoln

From Algebra in Lincoln, a blog maintained by Evgeny Khukhro:

An exhibition celebrating the bicentenary of George Boole simultaneously opened in University of Lincoln http://library.lincoln.ac.uk/news/2015/07/06/george-boole-exhibition/ and in University College Cork, Ireland. The launch event in the University of Lincoln Library on 16 July was attended by Professor Alexandre Borovik, a Trustee of the London Mathematical Society, which awarded a “Local Heroes” grant for support of the exhibition in the University of Lincoln and Lincoln Cathedral. A short speech by Professor  Borovik can be found here: http://education.lms.ac.uk/2015/07/george-booleglobal-hero/ . The exhibition was formally opened by Deputy Vice-Chancellor Professor  Scott Davidson (who, by the way, mentioned how people in Law Department., when using one of the first computerized databases in 1980s, had to learn Boolean “and”, “or”, “not” and bracket arrangements). The University Librarian Ian Snowley outlined the story behind the exhibition and thanked all the parties contributing to its success.

Professor Alexandre Borovik

Professor Alexandre Borovik (speaking) and Professor Scott Davidson, Deputy Vice-Chancellor of the University of Lincoln

The Exhibition

The Exhibition

The LMS corner at the Exhibition

The LMS corner at the Exhibition

Professor A. Borovik, a Trustee of the LMS; University Librarian Ian Snowley; Dr Mark Hocknull, Canon Chancellor of Lincoln Cathedral; Professor Scott Davidson, Deputy Vice-Chancellor

Professor Alexandre Borovik, a Trustee of the LMS; Ian Snowley, University Librarian ; Dr Mark Hocknull, Canon Chancellor of Lincoln Cathedral; Professor Scott Davidson, Deputy Vice-Chancellor of the University of Lincoln

 

George Boole, Global Hero

[A.Borovik, Talk at the opening of the The Life and Legacy of George Boole exhibition in Lincoln, 16 July 2015.]

I am privileged to take part in this celebration and I am honored to represent the London Mathematical Society.

The LMS was founded 150 years ago by Augustus De Morgan, a colleague and close friend of George Boole, just a year after Boole’s untimely death. The Society continues the work started by mathematicians of George Boole’s circle.

Some people say that mathematicians are remote from everyday life.

George Boole was not.

Here, in Lincoln, he taught at the Mechanics Institute, fought for the improvement of working conditions of shop workers, founded a building society.

His famous book An Investigation of the Laws of Thought was very down-to-earth, it was a textbook of practical thinking. It was written for humans, not for machines—after all, computers remained non-existent for another century.

Let us take a look at his famous definition of the universe of discourse – a concept that you will immediately recognise as obvious, everyone-knows-it kind of things – but which was new, fresh, and perhaps paradoxical in Boole’s time:

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined.

The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself.

But more usually we confine ourselves to a less spacious field.

Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation.

Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse.

In short,

  • The laws of thought are global; but
  • they are applied locally, for example, at a board meeting of a building society.

Please notice George Boole’s words:

we imply (without expressing the limitation)  …

This is his warning against undeclared assumptions that can poison the discourse, his warning against

  • hidden bias,
  • hidden prejudice,
  • hidden phobia,
  • hidden hatred.

Boole’s time was the era of tectonic shifts in technology, in economy, and in social life.

The need for practical logic for everyday use, logic freed from medieval scholasticism, logic accessible to everyman—was in the air of the epoch.

The great contemporary of George Boole, Abraham Lincoln, used in his political writings and speeches the implicit logic of the Euclidean geometry:

One would start with confidence that he could convince any sane child that the simpler propositions of Euclid are true; but, nevertheless, he would fail, utterly, with one who should deny the definitions and axioms.

The principles of Jefferson are the definitions and axioms of free society.

And yet they are denied, and evaded, with no small show of success.
One dashingly calls them `glittering generalities'; another bluntly calls them `self-evident lies'; and still others insidiously argue that they apply only `to superior races’.

From these two quotes, it is hard to avoid the impression that both Boole and Lincoln were thinking in terms of what we now call “human rights”.

It is also difficult to avoid the feeling that for Boole and Lincoln, Logic was the Logic for the Masses; it was

  • Logic for Personal Empowerment,
  • Logic for Social Advancement,
  • Logic for Liberation.

Abraham Lincoln re-used mathematical thinking of classical geometry dated 2 millennia back in time.

But George Boole took an audacious step into the future. He created a new logic and a new mathematical symbolism which supported it.

He extracted the most basic and fundamental laws of thought, so simple that they are now used by computers. Everyone in this room has a mobile phone; in every mobile phone, microchips contain millions of logical gates carrying out millions of Boolean operations per second.

By discovering the algebra of thought – now implemented in computers and electronic devices all around us – Boole changed the course of human civilization.

George Boole is a global hero.

But he wouldn’t become a global hero, if he was not a local hero here – in Lincoln.

His life and work are the best justification of the dictum:

Think globally – act locally!


 

Acknowledgements. I use this opportunity to say my thanks to everyone involved in setting-up of the two consecutive exhibitions in Lincoln, in the University of Lincoln and in the glorious Lincoln Cathedral. My special thanks go to Ian Slowley, Mark Hocknull, Dave Kenyon, and Eugene Khukhro.

Disclaimer: The author writes in his personal capacity and the views expressed do not necessarily represent position of his employer or any other person, organisation or institution.

Homeschooling in England

An important legal case reported by the BBC on 16 July 2015:

Council drops home education case

Merryn Hutchings,  Exam Factories? The impact of accountability measures on children and young people. Report for NUT Full text.

From the summary:

Professor Hutchings finds that:

  • The Government’s aims of bringing about an increased focus on English/literacy and maths/numeracy and (in secondary schools) academic subjects, has been achieved at the cost of narrowing the curriculum that young people receive.

  • Recent accountability changes mean that in some cases secondary schools are entering pupils for academic examinations regardless of aptitudes or interests. This is contributing to disaffection and poor behaviour among some pupils.

  • The amount of time spent on creative teaching, investigation, play, practical work and reading has reduced considerably and there is now a tendency towards standardised lesson formats. Pupils questioned for this study, however, say that they learn better when lessons are memorable.

  • Teachers are witnessing unprecedented levels of school-related anxiety, stress and mental health problems amongst pupils, particularly around exam time. This is prevalent in secondary schools but also in primaries.

  • Pupils of every age are under pressure to learn things for which they are not ready, leading to shallow learning for the test and children developing a sense of ‘failure’ at a younger and younger age.

  • Pupils’ increased attainment scores in tests are not necessarily reflected in an improvement in learning across the piece. Teaching can be very narrowly focused on the test.

  • The Government and Ofsted’s requirement that schools target pupils on Free School Meals with Pupil Premium money is prompting some schools to take the focus away from special educational needs (SEN) children. Accountability is discouraging schools from including SEN children in activities targeted at Free School Meals children even when children with SEN need the support more.

  • Accountability measures disproportionately affect disadvantaged pupils and those with SEN or disabilities. Teachers report that these children are more likely to be withdrawn from lessons to be coached in maths and English at the expense of a broad curriculum. Furthermore, some schools are reluctant to take on pupils in these categories as they may lower the school’s attainment figures. Ofsted grades are strongly related to the proportion of disadvantaged pupils in a school.

  • Ofsted is not viewed as supportive. It is seen as punitive and inconsistent, with the ability to cause a school to “fall apart”. In their analysis of a school, the inspectors also have a tendency not to take on board the way that individual circumstances affect outcomes.

  • The legacy effect of past Ofsted requirements means that these practices are still “drilled in” despite no longer being measured or required. These include the focus on marking of pupils’ work in a standardised manner and the monitoring of lesson structure.

Paul Ernest on Douglas Quadling

[See also: Douglas Quadling]

In 1979 I joined the staff at Homerton College, Cambridge as a temporary replacement for Stuart Plunkett on study leave. This was my first job after school teaching as a teacher educator. I worked alongside Richard Light, Tim Rowland, Bob Burn and Hilary Shuard. Alan Bishop at the university department of education (a separate body then) organised a masters course in mathematics education which our students as well as his attended. I also sat in when I could as an introduction to the fledgling science of mathematics education. Douglas Quadling was around, possibly teaching at the 3rd body, the Institute of Education (where Angela Walsh worked too). I remember most vividly the 2 seminars he gave to the masters course. He was a modest but immensely knowledgeable man who described very clearly and with great insight the development of the mathematics curriculum of the previous 50 years or more, and the great growth of textbook schemes in the 60s and 70s, including, most notably, the SMP series. His seminars were deceptively chatty, but rich in content and atmosphere. He was very active in the Mathematical Association. He published many texts and I especially remember his insightful 1969 book, The same, but different : a survey of the notion of equivalence in the context of school mathematics / by D. A. Quadling (published by Bell for MA), an early acknowledgement of a critical notion in school mathematics. I’m sure many others have further deep and affectionate memories of this man and his contribution.