Tuesday, 14 February 2012

Linear independence vs determinants

When writing my blog post on threshold concepts I read some interesting comments from Gordon Royle on linear independence versus calculating determinants:
In first year linear algebra (which is the current cross I have to bear), the concept of “linear independence” seems to meet these requirements. Many students struggle for weeks with the definition, but then suddenly (for the lucky ones) “the penny drops” and it all seems totally obvious. Once they have mastered linear independence, suddenly finding bases and calculating dimensions becomes easy and natural. On the other hand, calculating the determinant of a matrix is also something that students struggle with, but it is not a threshold concept. It’s just memorizing the rules and the signs that some students find difficult. Mastering it doesn’t really lead them anywhere except for being able to calculate the determinant of a matrix.
I agree that the mechanical operation of calculating the determinant of a matrix is not a threshold concept. Around 1990 Bruno Buchberger (RISC, Linz) wrote:
Many areas of high school and undergraduate mathematics, by now, are "trivialized" in the sense that their problems can be solved algorithmically by existing mathematical software like Mathematica. Well, if an area of mathematics is trivialized, why should students bother to study the area? Rather, shouldn't we just teach the students how to solve the main problems in the area by applying, in a reasonable way, the appropriate algorithms in, say, Mathematica? There are two dogmatic answers to this question. The puristic answer: Ban math software systems from math education! The pragmatic answer: Don't spend time in class on any trivialized area of mathematics!
Buchberger developed the white-box/black-box principle for math education using math software, which I subscribe to:
The 'white-box' phase is traditional teaching (study of underlying theory together with practice examples). Once a topic is understood, software may be employed as a 'black-box' , and so on recursively as the student progresses through a mathematics course.
So I would argue that little time should be spent on hand calculating determinants. However, I take issue with Gordon's statement that
The trouble with our existing teaching materials (for my first year unit) is that the time and number of pages spent on defining linear independence and calculating determinants is almost exactly the same. Each of them are simply defined and a couple of examples given. No student would have a clue (unless they were told) that one is a peripheral computational technique and the other is a critical conceptual issue.
Understanding what a determinant is, its properties, and how it can be used (applications) is, I think, definitely a threshold concept, not a peripheral computational technique. Examples (from my MATH2200 course):
  • If two rows (or columns) of a matrix are the same (or proportional), or a linear combination of two yields another row then, without any further computation, the determinant vanishes (an example of linear dependence);
  • Swapping a pair of rows or columns changes sign (no computation required); 
  • Learning the Laplace (or cofactor) expansion, along with the above observations, is useful: orthogonal polynomials can be defined as the determinant of simple tridiagonal matrices and properties such as their recurrence relation are easy to deduce from this;
  • The determinant, like the trace, is a matrix invariant equal to the product of the matrix eigenvalues. This is a useful check on the computation of eigenvalues;
  • The matrix invariants of A are generated as powers of x in the expansion of the characteristic equation, |A - x I|;
  • Computation of Wronskians;
  • Determination of equations of conical sections passing through a set of points;
  • Wavefunctions for Fermions in quantum mechanics are antisymmetric under particle interchange and are constructed using the Slater determinant;
  • Using the Levi-Civita symbol and Einstein summation notation is a "simple" way to view the determinant.

Thursday, 9 February 2012

As we may think

As We May Think is an essay by Vannevar Bush (see also his Wikipedia bio) first published in The Atlantic Monthly in July 1945, and republished again as an abridged version in September 1945—before and after the U.S. nuclear attacks on Japan. His essay still makes interesting reading and his theoretical machine—"memex"—is regarded as the pre-cursor of hypertext.

Note that Vannevar Bush is not related to George Bush. To quote from George Bush's presidential archive:
America's tradition of excellence has long been nurtured by a tradition of free inquiry aimed at the simple goal of better understanding ourselves and the world. In the 1945 report that led to the founding of the NSF, the National Science Foundation, Vannevar Bush—no relation—wrote that "As long as scientists are free to pursue the truth wherever it may lead, there will be a flow of new scientific knowledge to those who can apply it to practical problems."

Wednesday, 8 February 2012

Threshold Concepts

The School of Engineering at The University of Western Australia has an Engineering Thresholds Project (ETP) for designing curricula for new courses. To me a Threshold Concept (TC) is "common sense"; threshold implying something that you must cross before further progress is possible.

In March 2011 I attended the first UWA ETP workshop. I found the presentation by Jan (Erik) Meyer (now a Professor of Education at UQ) to be opaque, self-serving, and over-reliant on name-dropping; notwithstanding the fact that he edited Threshold Concepts and Transformational Learning with Caroline Baillie. Further, I found his manner to be vague, unhelpful, and condescending. However, not all ETP workshop attendees felt the same way, with John Bamberg writing
Today I attended a fabulous workshop on “Threshold Concepts in Engineering”, organised by Caroline Baillie and Sally Male, and it got me eager to jot down some ideas for pure maths.
John went on to write that
The idea of a threshold concept needs some pinning down, but it is vaguely something that our students need to master in order to progress to other ideas; a competent understanding of a threshold concept opens the door to many other concepts. We also think of it as something that is typically difficult for the student and is transformative. That is, once the student “gets it”, it can change their way of viewing previous notions, it could change the way they approach and do things, and it could change the way they see themselves as students of a particular discipline (i.e., a student of mathematics then regards themselves as a mathematician)
which all sounds like "common sense" to me, well understood by scientists and engineers, and used for over 50 years by writers of good text-books and in the design of tertiary curricula. So what, really, is new?

At the second workshop in December 2011 a list of Engineering TC was proposed. I came away with a feeling of dismay and dissatisfaction; the definition of a TC was too vague and flexible, meaning whatever one wants it to mean. While all of the topics listed on the discussion papers are important, only a few are what I view as TCs. Many are just concepts; some are difficult or "troublesome", some are "transformational", but few are universal. Without a rigorous definition, discussions tend to go round in circles and are dominated by bias and opinion.

When I returned to my office I Googled "a critique of threshold concepts". Many statements in the paper A Critique of the Threshold Concept Hypothesis (TCH) by Professor Rod O’Donnell resonated with me (please contact Rod directly if you would like a preprint of his paper as it is now under review for publication):
The degree of elasticity in interpretation of the hypothesis that followers have allowed themselves, and that the founding figures have sanctioned, is alarming in at least two respects.
Behind the TCH lies a large terrain of tacit, undiscussed assumptions. One key presumption is that a discipline has an established body of fundamental knowledge that is unlikely to change for some time. The aim is then to transform the intellectual perspective of learners in the discipline so that they ‘think like an x’, or are ‘transformed into’ an x, where x is an economist, biologist, engineer, sociologist, doctor, accountant, historian etc. These assumptions have some worrying implications.
Land et al (2010) confidently assert that ‘the empirical evidence for threshold concepts has been substantially increased’ in a multiplicity of disciplines in many countries. A great deal of empirical and applied work is undoubtedly occurring, but such work proceeds on the assumption that investigators have either identified a threshold concept or possess a sound methodology for doing so. Since neither assumption has any foundation, such work presupposes knowledge we do not have.
So, while I am sceptical of the TCH, I do believe in the importance of Threshold Concepts (the "common sense"definition); certainly I am interested in good ways of identifying and teaching TC for a range of fields, especially maths and physics.

It is now 2012 and New Courses at UWA commence on Feb 27. It will be interesting to see how things pan out ...

High school maths incentives don't add up

Some very sensible comments about education in Western Australia by Tertiary Education Minister Chris Evans in High school maths incentives don't add up :
...schools are too focused on getting students high university entrance scores.
...there seemed to be a narrowing of the system which was very dangerous in terms of education.
...too often the systems now are about the ATAR score, not about education
We are encouraging students to take lower level, easier courses in Years 11 and 12, which is actually preventing them from going to university
The state government does, I think, need to act urgently to rebalance those incentives that prevent students taking up maths and science in high school, and continuing that through to university. There is a narrowing at high school that I think is very dangerous in terms of education
Hopefully, someone from the state government is listening. Unfortunately, with the liberal-labor divide, I doubt anything will come of it.

Tuesday, 7 February 2012

Welcome to the university of the future

Some interesting quotes from Larry Summers in Welcome to the university of the future:
It’s slightly absurd that in the English-speaking world on 15,000 separate occasions each year a lecture is given parroting the basics of capitalism,” he said. “Surely watching a video of the expert on the topic would be better? Technology can be harnessed to create better learning experiences.
In 2008 a survey at Harvard found that students who used video lectures said they learnt subjects faster than by attending lectures in person. There would, too, be cost and times savings: if students watched top experts on a particular topic on video, their own academics would be free to give more time for further discussion.
Summers says universities should place more emphasis on collaboration, rather than perceiving it as a form of cheating. It is all provocative stuff—but then Summers does not always get things right and is no stranger to controversy. In 2006, he quit as head of Harvard after he suggested the paucity of women in science and maths might be down to innate differences in ability between the sexes.
Summers cites an experiment launched by an outfit called The Floating University. The media venture has videoed world-famous scholars including the physicist Michio Kaku, the psychologist Paul Bloom and the cognitive scientist Steven Pinker—and Summers himself—speaking about the newest ideas in their field. The videoed lectures are to be offered at Harvard, Yale and Bard college in New York state, the first time such a course has been presented at all three institutions.
The explosion of knowledge, and our ability to access it through computers, demands change in the way universities operate, Summers says. When facts can be checked rapidly online, individual acquisition of them becomes less important. Indeed, there is so much information available that it becomes difficult for any individual to master more than a fraction.
So what is Summers’s prescription for improvement? In his brave new academic world, universities would ditch lectures from local professors in draughty theatres in favour of the best experts in their fields delivering talks by video, potentially to students anywhere around the globe.
Another of his suggestions is that “international experience should be part of what happens to every college student”. This is not to learn languages, but to discover cultures that work will throw up in a globalised world.
I particularly enjoyed his response to Cameron and Tyler Winklevoss's claim that their classmate, Zuckerberg, had stolen their big idea for Facebook:
One of the things you learn as a college president is that if an undergraduate is wearing a jacket and tie on Thursday afternoon at three o’clock, there are two possibilities. One is that they are looking for a job and have an interview; the other is that they are an asshole. This was the latter case.

Wednesday, 1 February 2012

Books to Read

  • Wine Grapes—A Complete Guide to 1,368 Vine Varieties including their Origins and Flavours
  • Addiction by design: machine gambling in Las Vegas
  • Shrapnel by William Wharton
  • A Mathematician's Apology
  • Our Final Century
  • Dr Tatiana's Advice to All Creation
  • Interpreter of Maladies
  • The Phantom Tollbooth
  • The Archivist's Story by Travis Holland
  • A beautiful mind
  • The man who loved only numbers
  • The Victorian Internet
    • Fingerprints
  • The Fifth Miracle
  • Mendeleyev's Dream
  • Strange Beauty
  • Koestler by Michael Scamell
  • A Pair of Ragged Claws
  • The Cogwheel Brain
  • J.M. Coetzee: Slow man + Diary of a Bad Year
  • The Cardboard Crown, Boyd
  • The man who loved children, Christina Stead
  • Eucalyptus by Murray Bail
  • The Children's Bach by Helen Garner
  • Poppy by Druisilla Modjeska 
  • Lillian's Story by Kate Grenville
  • Einstein: His Life and Universe by Walter Isaacson
  • The Worry Doll (Graphic novel by Matt Coyle, Tasmania)
  • Authors to watch: James Bradley, Delia Falconer, Julia Leigh, Sonya Hartnett

Places to Eat: Adelaide


  • Vincenzo's Cucina Vera (08) 8271 1000 77 Unley Rd Parkside, 5063
  • The Manse 142 Tynte St North Adelaide 08 8267 4636

Places to Eat: Melbourne

  • Newmarket Hotel 34 Inkerman St, St Kilda (03) 9537 1777
  • Miss Jackson, St Kilda for excellent breakfast
  • Blondie Bar, Southbank
  • Portello Rosso, great tapas and good wine
  • Moroccan Soup Bar 183 St Georges Road Fitzroy North VIC 3068 Phone (03) 9482 4240
  • Ladro 162 Greville Street, Prahran 9510 2233
  • Ladro 224 Gertrude St Fitzroy 9415 7575
  • Rumi 116 Lygon St Brunswick East 9388 8255
  • Cutler and Co 55-57 Gertrude, Fitzroy 9419 4888
  • Cibi (breakfast/lunch) 45 Keele Street Collingwood 9077 3941
  • Sotano Hilton Melbourne South Wharf 2 Convention Centre Place South Wharf 9027 200
  • Melbourne Food and Wine Festival
  • Mae Daya (Isakaya) 400 Bridge, Richmond
  • Mountain Goat Brewery (good pizza), North and Clark, Richmond
  • Bar Lourinha (Portuguese/Spanish share plates)
  • Gigibaba Turkish Tapas 102 Smith St Collingwood 9486 0345
  • Botanical (69 Domain Road South Yarra, 9820 7888)
  • MoVida
  • Cafe di Stasio Mamaganoush (56 Chapel Street, Windsor 95214141)
  • Ginger Boy (cool Asian Hawker style in Crossley Lane)
  • ezard (mod-oz under Adelphi) Meyers place bar (20 Meyers place)
  • Waiters Restaurant (20 Meyers place)
  • Loop (20 Meyers place)
  • Yu-u (Flinders lane/Oliver lane — but book weeks ahead 9639 7073)
  • Cafe Segovia (33 Block place — good for breakfast)
  • Cookie
  • The Deanery (13 Bligh place, med/middle eastern)
  • Nobu (Crown)
  • The Brasserie (Crown)
  • Rockpool Bar & Grill (Crown)
  • Giuseppe, Arnaldo & Sons (Crown, no bookings 9694 7400)
  • Vue De Monde (430 Little Collins, 9691 3888)
  • Attica (74 Glen Eira Rd, Ripponlea, 9530 0111)
  • Sarti (6 Russell Place, 9639 7822)
  • De Bortoli wines, Yarra Valley
  • Il Bacaro (168 Little Collins, 9654 6778)
  • Bella Vedere (Badger's Brook Vineyard, 874 Maroondah Hwy, Coldstream, 5962 6161)

Places to Eat: Sydney


  • Porteño (02) 8399 1440, 358 Cleveland Street Surry Hills, 2010
  • Rosso Pomodoro 91/24 Buchanan St Balmain NSW 2041 (02) 9555 5924
  • bel Mondo, Chef: Andy Ball
  • Marque 4/5 355 Crown Street, Surry Hills 9332 2225
  • New-wave Italian in inner-Sydney: La Sala, Intermezzo, Lo Studio
  • Red Lantern (545 Crown St, Surry Hills, 9698 4355)
  • Yoshii (corner of Harrington and Essex Street, 9247 2566)

Interests, Movies, Music, and Books

As I've moved my blog to Google+, the following lists get removed from my profile. For the record,

Interests: Coffee, Wine, Beer, Cycling, Swimming, Volleyball, Tennis, Travel, Piano, Perth International Arts Festival, Modern dance, Modern Art, Contemporary art, Impressionism, Art history, Photography, Architecture, Design, Rottnest island, Mathematics, Physics, Mathematica

Favorite Movies: Moulin Rouge, Indiana Jones 1 and 3, The Blues Brothers, Trading Places, Casablanca, Thomas Crown Affair (with Steve McQueen), Trainspotting, Goodfellas, The Fugitive, The Mission, The Insider, Once Were Warriors, Movies by Stanley Kubrick, Movies by the Coen brothers 

Favorite Music: Roxy Music, The Corrs, Divinyls, Gipsy Kings, Hunters and Collectors, Ottmar Liebert, Coldplay, The Beatles, Deborah Conway, Depeche Mode, Evermore, Fatboy Slim, Federico Aubele, Fine Young Cannibals, Marianne Faithfull, Paul Kelly, Pink Floyd, The Police, The Pretenders, Red Hot Chilli Peppers, Robbie Williams, Simple Minds, Sting, Yes, Tangerine Dream, Tom Petty, U2, A lot of latin, A bit of classical

Favorite Books: Historical novels, Popular science books (e.g. Cosmos, Comet), Catch 22, and books by Robert Hughes, Peter Carey, Louis De Bernieres, Richard Rhodes, Milan Kundera, Simon Winchester, Dava Sobel, Josephine Hart, Jack Kerouac, Iain Pears, Robert Graves

Places to Eat: Singapore

Some restaurants to visit, next time I'm in Singapore:
  • Sin Huat Eating House (659/661 Geylang Road, off Lorong 35; 6744 9755)
  • Makansutra (1-15 Esplanade Mall, 6336 7025)
  • Muthu's Curry (138 Race Course Road 6392 1722)
  • House of Peranakan Cuisine (Meritus Negara Hotel, 10 Claymore Raod 6733 4411)
  • $$$ Iggy's (Level 3, The Regent Hotel, 1 Cuscaden Road 6732 2234)
  • $$$ Au Jardin (EJH corner House, Singapore Botanic Gardens, Cluny Road 6466 8812)
  • $$$ San Marco at the Lighthouse (Fullerton Hotel 6438 4404)
  • East Coast Lagoon Food Village (1220 East Coast Parkway):
    stall 6 cze cha [full-menu Chinese Street Food]
    stall 29 Cheok Kee Eeting House [Teachew soy-braised duck with star anise]
  • Maxwell Road food Center (cnr South Bridge and Maxwell roads):
    stall 10 Tian Tian Hainanese Chicken Rice
    stall 99 Cantonese won ton noodles
    stall 51 stir-fired Hokkein noodles
    stall 71 cha kway teow
    stall 64 ngoh hiang
  • Chinatown Complex Food Centre (335 Smith Street) 02-128 Claypot rice (with Chicken, lup cheong, salt fish)
  • Marine Parade Laksa (59 East Coast Road, Katong)
  • Warung Nasi Pariaman (738 North Bridge Raod 6292 2374, near Sultan Mosque) Nasi Padang, Beef Rendang