The Quadratic Formula in Malta’s Learning Outcomes Framework

What I see as a deficiency of the Learning Outcomes Framework is that it does not specify learning outcomes in a usable way.

There are several references to quadratic equations in Levels 8–10, for example

Level 8
Number – Numerical calculations
18. I can solve quadratic equations by factorisation and by using the formula.

If a student from Malta comes to my university (and I have had students from Malta in the past, I believe), I want to know what is his/her level of understanding of the Quadratic Formula.

There are at least 7 levels of students’ competencies here, expressed by some sample quadratic equations:

(a) x2 – 3x +2 =0
(d) x2 – 1 = 0
(c) x2 – 2x +1 = 0
(d) x2 + sqrt{2}*x – 1 = 0
(e) x2 + x –  sqrt{2} = 0
(f) x2 + 1 = 0
(g) x2 + sqrt{2}*x + 1 = 0

These quadratic equations are chosen and listed according to their increasing degree of conceptual difficulty: (a) is straightforward, (b) has a missing coefficient (a serious obstacle for many students), (c) has multiple roots, (d) involves a surd, but no nested surds in the solution, (e) has nested surds in the answer, (f) has complex roots, although very innocuous ones, and (g) has trickier complex roots. Of course, another list  can be made, with the same gradation of conceptual difficulty.

I would expect my potential students to be at least at level (d); but LOF tells me nothing about what I should to expect from a student from Malta.

And one more comment: a comparison of the statements in the LOF Level 10:

I can solve quadratic equations by completing a square

and in the LOF Level 8:

I can solve quadratic equations by factorisation and by using the formula.

apperars to suggest that at Level 8 the Quadratic Formula is introduced to students without proof or proper propaedeutics which appear only at Level 10. In my opinion, this should raise concerns: at Level 8, this approach has a potential to degenerate into one of those “rote teaching”  practices that make children to hate mathematics for the rest of their lives.

Malta’s Learning Outcomes Framework: a Discussion


Malta’s new Learning Outcomes Framework for school mathematics 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.

The original post of 13 August generated more responses than it was anticipated, and it is useful to collect them all at a single page.

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

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.



Douglas Quadling

Douglas Quadling, who was one of the four inspirational drivers behind the School Mathematics Project (SMP) in the 1960s and 70s, and a fine mathematician,  schoolmaster, and author, died on Wednesday 25th March 2015.
His funeral is in Emmanuel College Cambridge on Thursday 9 April at 2pm.

Yagmur Denizhan: Response to Comments

Anonymous on 4 January 2015 at 22:49 said in response to my post:

Anon: My comments are on a few themes which appear within the paper. They are stand-alone, selected on the basis of curiosity, and do not necessarily present a coherent over-arching argument.
On Games:
…the winning strategies in such games were typically based on identifying the underlying algorithm instead of being “misled” by the story.
This is very true, and I view it as a result of the human tendency to simplify or reduce puzzles to their essence.

YD: I would prefer to say the “pragmatically relevant essence”… Yet there is also another tendency that is unfortunately systematically suppressed by the system that I am criticising: The tendency to comprehend and delve into the essence of anything/everything.

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