Is algebra necessary? Part III

This continues an earlier post and another post, Is algebra necessary? Yes, algebra is necessary.

Jennifer Ouellette writes in her post on Scientific American blogs, Make Us Do the Math:

Well, I’m a former math phobe. I hated algebra, and avoided all advanced math and science until well into my 20s. But I’m standing in solidarity with the fusty old scientific establishment on this one. […]

[T]here is a deeper, uglier under-current to Hacker’s article. What he’s really saying is that we should know our place in society, accept that we’re just not smart enough and don’t need to worry our pretty little heads anymore about anything that might interfere with our enjoyment of American Idol. It’s not like we were ever going to amount to much anyway, amirite? As journalist/science fan Jesse Emspak observed on Twitter: “The argument isn’t about math. It’s really about whether anyone but the privileged should be educated.”

Read the full post at the Scientific American.

A tale about long division

Anne Watson has continued discussion of the role of long division  by posting a comment to one of the earlier posts. It is awkward for me to talk about long division: I teach at university, it is difficult for me to have an opinion on at what age and at what Key Stage schoolchildren have to learn long division. But I believe in the educational value of written algorithms for addition, subtraction, and especially long multiplication and long division — because the latter is a tremendous example of all important recursive algorithms.

My approach to school level mathematics education is very practical: I teach a course in mathematics for Foundation Studies, to students who wish to study hardcore STEM disciplines, but have not taken, or dropped out, or failed mathematics A levels. I work at the receiving end of the GCSE. And I have to make sure that my students master long division (with remainder!) of polynomials.   Why? Because relatives and descendants of the long division, various versions of the Euclidean Algorithm (including the ones for polynomials) saturate information processing around us; for example, they are used every time when we pay in supermarket by a credit card. Of course, the user of a credit card does not need to know  Euclidean Algorithms, but the society needs some number of people who know how credit cards are working, and therefore understand long division.

I believe that we should give a chance to learn long division to every child. I do not know what is the best way to achieve this. But I make my modest contribution: I give my students a second chance to learn long division, this time long division of polynomials. And I start this  topic with a brief review of long division of integers, largely with the aim to alleviate fears and psychological blocks accumulated by many of my students in their KS 1-4 studies. I intentionally do that in a lighthearted and semi-improvised fashion, engaging students in a direct dialogue.

What follows is an example which I improvised for my students in my lecture in December 2011; it was published in my blog on 9 December 2011. Most likely, my example it is not suitable for use in school level teaching, but, judging by response from my students,  it appears to serve its purpose to help those students who learned long division at school, but forgot it, to refresh their memories and move to the next level of learning, to long division of polynomials. Also I think that the fact that we have to remind long division to university students suggests that we cannot avoid some discussion of its place in the school curriculum.

A fable about long division. An innumerate executor of a will has to divide an estate of 12,345 pounds between 11 heirs. He calls  a meeting and tells the heirs: “The estate is about 12 grands, so I wrote to each of you a cheque for 1,000 pounds.”

The heirs answer: “Wait a second. There is more money left” — and write on the flip chart in the meeting room:

“Ok” — sais the executor – “there are about 13 hundred left. So I can write to each of you a check of 100 pounds”:

“But there is still money left in the pot” — shout the heirs and write:

“Well,”–  says the executor, — “it looks as if I can give extra 20 pounds to each of you”:

“More! More!” — the heirs shout. “I see” — said the executor — “here are 2 pounds more for each of you”:

“I deserve to get this remainder of 3 pounds and buy myself a pint. And each of you gets 1122 pounds”:

After finishing my tale on this optimistic note, I commented that the whole calculation, which looks like that:

is usually written down in an abbreviated form:

And we say that

12345 gives upon division by 11 the quotient 1122 and the remainder 3

which means

12345 = (11 times 1122) + 3

As simple as that.

Magic Mathematics

A new book: Magic Mathematics: The Mathematical Ideas that Animate Great Magic Tricks, by Persi Diaconis and Ron Graham.

From the Grrlscientist blog at The Guardian:

Magical Mathematics reveals the secrets of amazing, fun-to-perform card tricks — and the profound mathematical ideas behind them — that will astound even the most accomplished magician. Persi Diaconis and Ron Graham provide easy, step-by-step instructions for each trick, explaining how to set up the effect and offering tips on what to say and do while performing it. Each card trick introduces a new mathematical idea, and varying the tricks in turn takes readers to the very threshold of today’s mathematical knowledge. For example, the Gilbreath Principle — a fantastic effect where the cards remain in control despite being shuffled — is found to share an intimate connection with the Mandelbrot set. Other card tricks link to the mathematical secrets of combinatorics, graph theory, number theory, topology, the Riemann hypothesis, and even Fermat’s last theorem.

Diaconis and Graham are mathematicians as well as skilled performers with decades of professional experience between them. In this book they share a wealth of conjuring lore, including some closely guarded secrets of legendary magicians. Magical Mathematics covers the mathematics of juggling and shows how the I Ching connects to the history of probability and magic tricks both old and new. It tells the stories — and reveals the best tricks — of the eccentric and brilliant inventors of mathematical magic. Magical Mathematics exposes old gambling secrets through the mathematics of shuffling cards, explains the classic street-gambling scam of three-card monte, traces the history of mathematical magic back to the thirteenth century and the oldest mathematical trick — and much more.

My first impression: This book explains the mathematics that underlie a special group of card tricks that aren’t magic at all. The book also explains how these mathematical principles are more than cute ways to shuffle cards, they are useful in the real world, too. Informative and filled with lots of photographs, this book is a delight. Reading this fun and fascinating book, I find myself wishing I had a maths teacher who had taken this book’s example to heart because I can easily imagine myself eagerly awaiting maths class so I could learn more card shuffles (and through that, advanced maths) from my teacher. Are there any teachers out there who use card tricks to teach maths to your students? If so, I’d love to hear from you!

Olympic Legacy

One of the justifications for the Olympic budget is the pious hope that people will be inspired to participate in sport more than hitherto.  No previous Olympic games achieved this, and it’s easy to see why.  The Olympics offer a model of sporting activity that is unavailable to most and unattractive to almost everybody.  Running 150 miles each week is not an option or an aspiration for all but a handful of talents.  If the powers that be really want to raise levels of participation, they should offer the models suited to the mass of the population, with facilities to match (proper cycle lanes, school sports fields, local swimming pools,etc.).  There are rewards that come from participating in sport at a very low level, but you’d never know it from watching the Olympics.

This matters to the DMJ because the same point applies to mental activity.  Tales of geniuses making astounding breakthroughs will not encourage kids into mathematics any more than Olympic gold will inspire sedentary Britons to take moderate exercise.  What we need are images of middling intellects getting something valuable out of mathematics.  This is especially important because in our assessment-driven system, children know from early on where they stand in the intellectual league tables.  The great majority know themselves to be middling intellects long before they make decisions about what to study.  We need stories about mathematics and illustrations of its value that speak to children thus informed.

https://sites.google.com/site/mathematicalcultures/

“23 Mathematical Challenges” and teaching of geometry

This list of problems was published in 2008 by the USA’s Defence Sciences Office; not surprisingly, they remain unsolved: each problem is a brief description of a whole new direction in mathematics, computer science, mathematical biology and equires raising a new army of researchers. What attracts attention is the obvious geometric nature of many of them. Where will all the geometers come from?

Mathematical Challenge 1:  The Mathematics of the Brain
Develop a mathematical theory to build a functional model of the brain that is mathematically consistent and predictive rather than merely biologically inspired.

Mathematical Challenge 2:  The Dynamics of Networks
Develop the high-dimensional mathematics needed to accurately model and predict behavior in large-scale distributed networks that evolve over time occurring in communication, biology and the social sciences.

Mathematical Challenge 3:  Capture and Harness Stochasticity in Nature
Address Mumford’s call for new mathematics for the 21st century. Develop methods that capture persistence in stochastic environments.

Mathematical Challenge 4:  21st Century Fluids
Classical fluid dynamics and the Navier-Stokes Equation were extraordinarily successful in obtaining quantitative understanding of shock waves, turbulence and solitons, but new methods are needed to tackle complex fluids such as foams, suspensions, gels and liquid crystals.

Mathematical Challenge 5:  Biological Quantum Field Theory
Quantum and statistical methods have had great success modeling virus evolution. Can such techniques be used to model more complex systems such as bacteria? Can these techniques be used to control pathogen evolution?

Mathematical Challenge 6:  Computational Duality
Duality in mathematics has been a profound tool for theoretical understanding. Can it be extended to develop principled computational techniques where duality and geometry are the basis for novel algorithms?

Mathematical Challenge 7:  Occam’s Razor in Many Dimensions
As data collection increases can we “do more with less” by finding lower bounds for sensing complexity in systems? This is related to questions about entropy maximization algorithms.

Mathematical Challenge 8:  Beyond Convex Optimization
Can linear algebra be replaced by algebraic geometry in a systematic way?

Mathematical Challenge 9:  What are the Physical Consequences of Perelman’s Proof of Thurston’s Geometrization Theorem?
Can profound theoretical advances in understanding three dimensions be applied to construct and manipulate structures across scales to fabricate novel materials?

Mathematical Challenge 10:  Algorithmic Origami and Biology
Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.

Mathematical Challenge 11:  Optimal Nanostructures
Develop new mathematics for constructing optimal globally symmetric structures by following simple local rules via the process of nanoscale self-assembly.

Mathematical Challenge 12:  The Mathematics of Quantum Computing, Algorithms, and Entanglement
In the last century we learned how quantum phenomena shape our world. In the coming century we need to develop the mathematics required to control the quantum world.

Mathematical Challenge 13:  Creating a Game Theory that Scales
What new scalable mathematics is needed to replace the traditional Partial Differential Equations (PDE) approach to differential games?

Mathematical Challenge 14:  An Information Theory for Virus Evolution
Can Shannon’s theory shed light on this fundamental area of biology?

Mathematical Challenge 15:  The Geometry of Genome Space
What notion of distance is needed to incorporate biological utility?

Mathematical Challenge 16:  What are the Symmetries and Action Principles for Biology?
Extend our understanding of symmetries and action principles in biology along the lines of classical thermodynamics, to include important biological concepts such as robustness, modularity, evolvability, and variability.

Mathematical Challenge 17:  Geometric Langlands and Quantum Physics
How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?

Mathematical Challenge 18:  Arithmetic Langlands, Topology and Geometry
What is the role of homotopy theory in the classical, geometric and quantum Langlands programs?

Mathematical Challenge 19:  Settle the Riemann Hypothesis
The Holy Grail of number theory.

Mathematical Challenge 20:  Computation at Scale
How can we develop asymptotics for a world with massively many degrees of freedom?

Mathematical Challenge 21:  Settle the Hodge Conjecture
This conjecture in algebraic geometry is a metaphor for transforming transcendental computations into algebraic ones.

Mathematical Challenge 22:  Settle the Smooth Poincare Conjecture in Dimension 4
What are the implications for space-time and cosmology? And might the answer unlock the secret of “dark energy”?

Mathematical Challenge 23:  What are the Fundamental Laws of Biology?
This question will remain front and center in the next 100 years. This challenge is placed last, as finding these laws will undoubtedly require the mathematics developed in answering several of the questions listed above.

Tony Gardiner: Observations on the LMS Response to Draft Programme of Study in Mathematics, Key Stages 1–2

A. D.Gardiner, Observations on the LMS Response to Draft Programme of Study in Mathematics, Key Stages 1–2, The De Morgan Journal  2 no. 3 (2012), 139–148.

Summary:

The general response to the draft primary curriculum has been highly critical in some respects. But all responses appear to accept the fundamental idea that there is considerable scope for ‘raised aspirations’. This is remarkably positive.

Many responses also appear to welcome the idea of a clearer focus on core ideas and methods. For example, a survey completed by 5500 primary teachers revealed surprising support (~55%) for delaying calculator use until late primary. And—apart from one or two interest groups—there has been surprisingly little special pleading for the idea of preserving ‘data handling’ as a separate Attainment Target: it would seem that many respondents accept the need for a reduced profile in Key Stages 1-2.

In short, the underlying balance of opinion is now clearer in some respects than one might have anticipated. So the criticisms alluded to in the first paragraph should not be classified as ‘obstructionist’, but as reflecting a desire to give the new curriculum a reasonable chance of succeeding.

The summary of these criticisms provided by the LMS has been widely appreciated and focuses on six main points:

  1. There is an official insistence that a curriculum should concentrate on ‘what’ should be taught rather than `how’ it should be taught. This makes sense but can be taken too far: in mathematics the way a topic is developed over time may be designed to remain as part of students’ mental superstructure. But the official line should make it even clearer to specify something even more basic than `what’—namely `how many hours’ are to be devoted to mathematics in each School Year (the time devoted to mathematics in English schools is low).
  2. A main-school curriculum represents an 11 year journey. One cannot assess an outline of the early years without a clear idea of the mathematical destination it is leading towards. Since the primary curriculum (and the associated `leaks’ about developments at secondary level) raise very awkward questions, one cannot assess a draft for KS1-2 in isolation.
  3. The current draft is insensitive to `the way human beings learn’—in that it fails to convey the way in which the `mental universe of mathematics’ emerges from practical engagement with measures, shapes and quantities.
  4. The current draft is too ambitious—with unreal expectations in Years 1-2, and forcing material into Years 5–6 that belongs more properly in Years 7-8.
  5. The current draft still `nibbles’ at the same material year-after-year, instead of preparing the ground well whilst delaying the formal introduction of hard ideas, and then making significant progress when they are eventually introduced.
  6. Like so much in education, the success of any change depends on maintaining the support of teachers. For it is teachers who must interpret and present the changes to parents, and who implement them in classrooms. This support will be difficult to generate and to sustain without delaying to allow a more realistic schedule, and without a clearer sense of the associated assessment, accountability, and training structures.

Read the rest of the paper

Stephen Huggett: Multiple choice exams in undergraduate mathematics

S. Huggett, Multiple choice exams in undergraduate mathematics, The De Morgan Journal, 2 no. 1 (2012), 127-132.

From the Introduction:

In addition to a rigorous practical test called the general flying test, candidates for a private pilot’s licence have to pass written exams in subjects including meteorology, navigation, aircraft, and communications. These written exams are multiple choice, which seems appropriate. The trainee pilots are acquiring skills supported by background knowledge in breadth not depth, and this can be tested by asking them to choose the right option from a limited list under a time constraint. It is not necessary, of course, for pilots to understand the underlying theoretical concepts.

In contrast, students of mathematics are certainly expected to understand underlying theoretical concepts. To a certain extent, this understanding can also be tested using multiple choice exams. Clearly, mathematicians need skills too, of which one of the most important is the ability to perform calculations accurately. This can also be tested using multiple choice exams.

Given that no one method of assessment is good for all of the understanding and skills expected of a student, one should use a variety of different assessment methods in a degree programme, including things such as vivas, projects, and conventional written exams. There is no claim here that multiple choice exams can do everything!

Read the rest of the paper. 

Debate around the Khan Academy, II

I decided to continue discussion started in the post Debate around the Khan Academy by drawing attention to a collection of readers’s responses in the Washington Post blog, Khan Academy: Readers weigh in. In my opinion, these responses demonstrate why public discussion of mathematical education is so difficult: even professional mathematics lecturers and teachers have no language and shared conceptual framework for description of what is actually happening in the intertwined processes of teaching and learning mathematics. Then what can we expect from laymen: parents, administrators, politicians?

Is algebra necessary, Part II

A lively discussion of Andrew Hacker’s op-ed Is Algebra Necessary?  in the New York Times (see previous post) continues.

Evelyn Lamb ‘s op-ed on the Scientific American website,
Abandoning Algebra Is Not the Answer, a quote:

What is algebra anyway? It’s a huge subject, but at its heart, it’s about relationships. How does a change in one quantity affect another quantity when they are related in a certain way? Hacker suggests that we need arithmetic but don’t need algebra. But it’s really difficult to separate these two skills. Algebra and geometry, another subject Hacker could do without, help develop logical skills and abstract reasoning so we can understand why we are making less money than before if we get a 20 percent pay cut followed by a 20 percent raise (or a 20 percent raise followed by a 20 percent pay cut—hello, commutative law of multiplication!) or how much merchandise we can purchase if we have $100 and a 25 percent off coupon.

 in Huffington PostAlgebra Is Essential in a 21st Century Economy. A quote:

One fallacy in Hacker’s reasoning is clear: Why single out mathematics? Yes, a knowledge of calculus may or may not help one negotiate through traffic or connect one’s computer to the Internet, but the same could be said for many other disciplines. How does knowing whom Hamlet killed accidentally help one be a better consumer? Does knowing the history of the Spanish-American War help one complete one’s tax return?

And see blogs by Rob Knop and RiShawn Biddle.