Posted by: Alexandre Borovik | November 7, 2011

Herman Cain: “My mathematical training”

On page 6 of this NYT article, search for words “my mathematical training”.

Magazine Preview:  On the Ropes with Herman Cain
By T. A. FRANK
The end of the Cain campaign has been at hand for months. And yet the end doesn’t arrive. And the end isn’t about to arrive now either. 

With thanks to muriel.

Posted by: Alexandre Borovik | November 6, 2011

Amnesia Principle

A book picked at random off a shelf in the library of Mathematisches Forschungsinstitut Oberwolfach and opened at a random page revealed a little gem, a paper by Pierre Ageron Logic without self-deductibility, in: Logica Universalis (J.-Y. Beziau, ed.), Birkhauser, Basel – Boston -Berlin, 2007, pp. 87-93.

The “law of self-deductibility” is a simple principle “if A, then A” which expresses the reflexivity of entailment; apparently, its formulation can be traced back to Stoic philosophers. Pierre Ageron comments that

Self-deductibility has a paradoxical status. It seems so obvious that, unlike the law of exluded middle [...] it has hardly ever a matter of controversy [...]. Also it seems of no use whatsoerver in the mathematical practice and totally sterile in terms of deductive power.

However, then Pierre Ageron points out that Charles Pierce in hisAlgebra of Logic (1885)

mentions in passing quite a remarkable interpretation of the law of self-deductibility. Pierce referes to it as the “first icon of algebra” and argues that it “justifies our continuing to hold what we have held, though we may, for instance, forget how we were originally justified in holding it”. This clearly suggests that self-deductibility works as an amnesia principle: once some statement is proved, it allows us to forget how it was proved. In mathematical practice, it is obviously a good thing that we can continue to trust our old theorems even if we are unable to reconstruct their proofs.

In my book, I briefly discuss Timothy Gowers‘ observation that, in his opinion, a “comprehensible” proof is not necessarily the shortest one, but a proof of small width. Here, width measures how much you must hold in your head at any one time. Alternatively, imagine that you write a detailed proof on a blackboard, carefully referring to all intermediate steps.

However, if you know that a certain formula or lemma will never be used again, you erase it and re-use the space. A “small width” proof is a proof which never expands beyond one (small) blackboard. In other words, a “comprehensible” proof is a proof produced by a systematic application of the “amnesia principle” and the humblest of all tautologies, “if A then A”.

[Imported from my old and now defunct blog]

Posted by: Alexandre Borovik | November 6, 2011

Andrew Wiles on doing mathematics

Andrew Wiles:

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were.

[Quoted from Did earlier thoughts inspire Grothendieck? by Frans Oort, who refers to the BBC documentary by S. Singh and John Lynch: Fermat’s Last Theorem. Horizon, BBC 1996.]

[With thanks to Thomas Riepe]

Posted by: Alexandre Borovik | November 4, 2011

The De Morgan Journal

The De Morgan Journal, the London Mathematical Society’s Blog of Mathematics Education and Education Policy,
provides the academic mathematical community with a forum for discussion of issues in mathematics education and education policy. The aim of the The De Morgan Journal is

  • to encourage academic mathematicians to reflect on current issues in mathematics education at all levels—from primary school to graduate studies,
  • to encourage them to explore links between higher mathematics and elementary mathematics,
  • to examine significant policy implications which may affect the wider mathematical, educational, or scientific community.
A few samples are already available on the site.  These illustrate, but are in no way exhaustive of, the intended range of themes. If you wish to submit a contribution or a full paper, click here for instructions.
Posted by: Alexandre Borovik | November 1, 2011

Curiosity Doesn’t Kill The Student

Source: Association for Psychological Science

 

Curiosity may have killed the cat, but it’s good for the student. That’s the conclusion of a new study published inPerspectives in Psychological Science, a journal of the Association for Psychological Science. The authors show that curiosity is a big part of academic performance. In fact, personality traits like curiosity seem to be as important as intelligence in determining how well students do in school.

Intelligence is important to academic performance, but it’s not the whole story. Everyone knows a brilliant kid who failed school, or someone with mediocre smarts who made up for it with hard work. So psychological scientists have started looking at factors other than intelligence that make some students do better than others. Read More…

Posted by: Alexandre Borovik | October 31, 2011

Connectivity of mathematics

Connectivity of mathematics

Connectivity of mathematics. With thanks to xkcd, http://xkcd.com/8/

Posted by: Alexandre Borovik | October 31, 2011

The De Morgan Journal: new blog on mathematics education

The De Morgan Journal provides the academic mathematical community with a forum for discussion of issues in mathematics education and education policy. Its aim is

  • to encourage academic mathematicians to reflect on current issues in mathematics education at all levels—from primary school to graduate studies,
  • to encourage them to explore links between higher mathematics and elementary mathematics,
  • to examine significant policy implications which may affect the wider mathematical, educational, or scientific community.
A few samples are already available on the site.  These illustrate, but are in no way exhaustive of, the intended range of themes. If you wish to submit a contribution or a full paper, click here for instructions.
Posted by: Alexandre Borovik | October 26, 2011

The Grapes of Math, homeschooling and EU disability legistation

In the  Math Future weekly series, today’s event is a webinar with Greg Tang, a popular book author and game designer. The advert says:

Ten years ago, Greg Tang was looking for a better way to teach math to his kids. He wound up creating a groundbreaking series of picture books that included the New York Times best seller The Grapes of Math. His books quickly became staples in school libraries and university teacher training programs, garnering numerous awards and selling over a million copies. The free online version of The Grapes of Math.

The Grapes of Math is indeed a lovely book, and immediately raises a very interesting question: it uses colour coding of information, which appears to be forbidden in the institutional use in this country by law (adopted from the EU legislation), and, I have to agree, for good reason: 1 person in 20  is colour blind or have significant deficiency of discrimination of colours. American homeschoolers, if their children are not colour blind, can of course ignore this issue and use colour coding, a powerful didactic tool. But I, as a lecturer to a class of 280 students, cannot ignore the issues of equality of access: 1 in 20 for me means 14 students, I cannot damage their chances to learn.

By the way, the Wiki article on colour blindness contain some tantalising snippets:

About 8 percent of males, but only 0.5 percent of females, are color blind in some way or another, whether it is one color, a color combination, or another mutation. [...]

Any recessive genetic characteristic that persists at a level as high as 5% is generally regarded as possibly having some advantage over the long term, such as better discrimination of color camouflaged objects especially in low-light conditions.[2][10] At one time the U.S. Army found that color blind people could spot “camouflage” colors that fooled those with normal color vision.[11][not in citation given][12]

Posted by: Alexandre Borovik | October 25, 2011

Some high tech parents and school without computers

From NYT, October 22, 2011:

LOS ALTOS, Calif. — The chief technology officer of eBay sends his children to a nine-classroom school here. So do employees of Silicon Valley giants like Google, Apple, Yahoo and Hewlett-Packard.

But the school’s chief teaching tools are anything but high-tech: pens and paper, knitting needles and, occasionally, mud. Not a computer to be found. No screens at all. They are not allowed in the classroom, and the school even frowns on their use at home.

Read the rest of the article by  .  However, a few further random quotes:

While other schools in the region brag about their wired classrooms, the Waldorf school embraces a simple, retro look — blackboards with colorful chalk, bookshelves with encyclopedias, wooden desks filled with workbooks and No. 2 pencils.

Andie’s teacher, Cathy Waheed, who is a former computer engineer, tries to make learning both irresistible and highly tactile. Last year she taught fractions by having the children cut up food — apples, quesadillas, cake — into quarters, halves and sixteenths.

Posted by: Alexandre Borovik | October 12, 2011

“Mathematical needs”

The ACME policy document “Mathematical Needs” is a case of missed opportunities. Interesting data collected in workplace interviews are compromised by careless and methodologically flawed analysis.

For an example, one can look at the following case study.

“6.1.4 Case study: Modelling the cost of a sandwich
The food operations controller of a catering company that supplies
sandwiches and lunches both through mobile vans and as special
orders for external customers has developed a spreadsheet that
enables the cost of sandwiches and similar items to be calculated.
It was necessary as part of this work to estimate the cost of onions
in hamburgers, which was done by finding out how many burgers
can be filled from one onion. The most difficult parameter to
estimate for the model is the cost of labour.”

This example illustrates one of the fundamental flaws of ACME’s approach: factually interesting case studies are interpreted via the the skewed prism of “modelling” agenda. Meanwhile, anyone who ever did a spreadsheet of complexity of a sandwich should know that the key mathematical skill required is a basic ability of manipulating brackets in arithmetic and algebraic expressions, something that Tony Gardiner calls “structural arithmetic” and Michael Gove calls “pre-algebra”. At a slightly more advanced level working with spreadsheets requires understanding of the concept of functional dependency in its *algebraic* aspects (frequently ignored in pre-calculus): if the content of cell B10 is SUM(B1:B9) and you copy it in cell C10, the content of this cell becomes SUM(C1:C9) and thus involves *change of variables*.

Intuitive understanding that SUM(B1:B9) is in a sense the same as SUM(C1:C9) is best achieved by exposing a student to a variety of algebraic problems which would convince him/her that a polynomial of kind x^2 + 2x + 1 is, from an algebraic point of view, the same as z^2 + 2z + 1.

ACME never tried to look at the actual mathematical content of workplace activities, and therefore their recommendations for education are based on entirely false premises. That the mathematical content is missing from their analysis is further confirmed by an important observation found on page 2 of the document:

“Employers emphasized the importance of people having studied
mathematics at a higher level than they will actually use. That
provides them with the confidence and versatility to use
mathematics in the many unfamiliar situations that occur at
work.”

ACME missed a chance to ask a correct question: why was indeed this happening? Instead, they appear to accept the employers’ vague hint that this is something about emotional maturity of their employees. But this is not about emotions; indeed, it is fairly obvious that a person’s “confidence” is directly linked to person’s *understanding* of what he or she is doing; meanwhile, the word “versatility” directly points to some mathematical skills involved in solving practical problems; the last point was lost (or even never even looked at) in ACME’s analysis.

Next, we cannot avoid commenting that the intellectual vacancy of the concept of “modelling” as it is used by ACME is obvious from

“6.1.2 Case study: Mathematical modelling developed by a
graduate trainee in a bank [...]
1. Modelling costs of sending out bank statements versus going
online.”

In 19th century there was of course no option of going online, but in a similar situation they would simply say “comparing costs of sending out bank statements by post versus hiring an in-house courier”.

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