Republic of Mathematics blog

Especially one who likes it, but who is turned off by repetitive drill …

Jeanne Bernish has a 14 year old son who is  very good at math ( accelerated) but just doesn’t like it. He is into gaming, and is tired of drills. She feels he will probably be so turned off by mathematics in school that he probably won’t pursue it in college…. and that makes her sad.

Some people might ask where is the problem here? This boy has found other interests, has decided mathematics is not to his liking, and will pursue other avenues. Is that really a problem?

What I understand Jeanne’s issue to be is that she is sad her son is being turned off a  subject he is good at as a result of the way it is taught. No romance, no excitement, no challenge, no adventure, no fun! Just repetitive, boring drill.

A young man I know is currently a freshman at the University of Connecticut. He went there, as a Massachusetts resident, because he is good at mathematics, likes it, and could see a potential career as an actuary. U Conn has a very strong mathematics program. Last semester he was bored with mathematics, didn’t like the deadly dullness of how it was being taught, and wanted something more interesting in his life. His other love is sports. Luckily his mother and her partner put together a plan to get him to  look into sports broadcasting with MSNBC. He is now planning to major in statistics and journalism, and is thrilled to know that he can pursue a career in two areas he loves – math and sports.

So what should Jeanne do to sustain her son’s interest in mathematics so that at college he will, at least, have the opportunity to continue with mathematics if he wants?

Ms. Walsh tweeted “forget math as math. just start playing online offline or ds games for whatever skill you want them to know” and “coordinate geometry = battleship; ordering numbers = war; any dice = addition ie candyland/monopoly/yahtzee”.

I think there’s a lot to be said for this tack, especially to forget math as math homework and chores and to focus, at least a little, on games.

My own son, who was very good at mathematics, loved computer games, of all sorts and we always encouraged him to play. Jeanne’s son is interested in gaming, but this probably involves games of a different level of sophistication than those Ms. Walsh mentions. Modern computer games use a high level of sophisticated mathematics – differential geometry, for example – that is pretty much out of the range of expertise or ability of a 14 year old.

Ira Socol mentioned the Special Ed blog post Real World Math which focuses, productively, on what I would call quantitative literacy. This, in my experience, is an excellent way to get kids involved in mathematical thinking in a way that empowers them to deal quantitatively with their everyday world. My own take on this is that mathematical experiences ought to be playful, fun, engaging, and empowering.

Often, as Jeanne is experiencing, this is not how mathematics is presented in school. “No Child Left Behind” is not the cause of that, but it doesn’t help, either, with the heavy focus on high stakes tests. The fact is – sad to say – many teachers of mathematics are not trained in mathematics, and have no love of the subject. Hard for them, therefore, to present mathematics as fun and challenging.

On that note,  I think it is worth looking at the computational exercises of Cleve Moler: Experiments with MATLAB. Cleve started MathWorks which markets MATLAB, a professional level computational engine. A student version is available at a low price. Cleve’s book is an introduction to computational aspects of mathematics. This will not excite every mathematics student, but it won’t disappoint  any of them. It’s worth taking a look at as a way of stimulating interest in real mathematics.

The way I see it is that Jeanne’s story is all too common. Kids turned off mathematics by deadly dull repetitive drill and nothing else.  It seems to be a systemic problem involving teachers, principals, superintendents, school boards, publishers, policy makers and politicians.

Jeanne and others like her need to stand up and  show this system how a better way is possible. We owe it to our kids not to dehumanize them with boring drill, but to enliven and enrich their spirit and minds through the playful study of mathematics.

This is what the Republic of Mathematics stands for.

Postscripts

(1) After thinking about Jeanne’s issue overnight, it seems to me that it might be worth her getting William  Dunham’s book Euler: The Master of Us All.

This book tells a series of stories, relating how the master mathematician Euler made significant progress on a number of mathematical problems.

Euler writes in a way that exposes his thinking. It’s a great book, approachable in chunks by an intelligent kid.

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(2) The following is an excerpt from the Kenya Sunday Nation:

Maths is easy, just change that attitude

By John Maken,  Posted Friday, January 1, 2010

From the time she started primary school Fiona Musanga liked counting and numbers and solving problems. But as she moved towards Standard Seven, she found her interest slipping and boredom creeping in. She was on the verge of joining the legions of girls who hate maths.

But then she found herself in Mr Abdul’s maths class at Nairoi’s Lavington primary school and there was no turning back. “He made it really interesting and even fun,” said the 16-year-old who is now in Grade 10 at Aga Khan High School where she continues to be fascinated by the subject and does very well.

“I love numbers. Working out and solving problems is fun,” said the student, who actually looks forward to doing maths homework in the evening. “Most of my girlfriends don’t like mathematics. Even boys say it is a difficult subject,” she said, adding that the way mathematics is presented and taught in schools often puts students off.

Read rest of article …

Today I am posting something written around 1995 by my friend and colleague Nigel Smith, who taught mathematics at Hordle-Walhampton School in Lymington, and at Twyford School in Winchester, England.

Nigel died of brain cancer, aged 44, about 7  years after he wrote this piece.

I hope to post very soon an account some of Nigel’s work at Twyford School, highlighting his experimental, thoughtful and creative approach to teaching mathematics, and his clever use of technology.

How and What Shall We Teach?

by Nigel Smith, Hordle Walhamption School, Great Britain
Most mathematics teaching, I would contend, is fundamentally dishonest. One has only to consider the nature of mathematics to realize that much of what is learned in classrooms is but a pale reflection of mathematical activity. The main problem is that many teachers regard mathematical knowledge as an immutable collection of facts that should be passed on from generation to generation.

This point of view has been under attack ever since Godel first demonstrated that a mathematical statement can be true but not provable. Unfortunately it has only been relatively recently acknowledged that this view has important consequences for the way that mathematics should be taught. Mathematics is a very human activity and so is bound by its historical, social and cultural contexts.

Historically, mathematics has developed and advanced through the medium of problem solving, primarily within a socio-cultural context. Sometimes this problem solving has been collaborative (e.g., Hardy and Ramanujan) and at other times the result of debate and argument (e.g., Frege and Russell). It is through mathematical activity such as problem solving that mathematical concepts begin to acquire their meaning.

As Wittgenstein and others have observed, the meaning of a mathematical concept is defined by its use. From a constructivist standpoint, individuals build meaning for a concept from their interpretation of its use in this context. This “negotiated meaning” would, I suggest, provide an excellent focus for school mathematics. From an early age, children should be made explicitly aware of the nature of mathematics. It is not wrong to tell them that mathematics is fallible, that its meaning is negotiated and that it is bound within its historical context.

By creating an open, questioning approach to learning, mathematics becomes a much more rewarding and uniquely human activity. Recently, many mathematics educators have argued forcibly that a problem-oriented curriculum would help promote understanding of fundamental ideas in mathematics. However, one has to ask whether a problem-oriented approach to learning will by itself provide sufficient motivating force for change to be considered viable. Such a curriculum would have to be “uninhibitedly speculative,” giving children a key role in the solution process. But how is this to be delivered?

Three Examples

From a simple starting point the process of problem solving can, through multistage development, build up the meaning and applications of many fundamental concepts. For example, the concept of a moving point initially expressed in coordinate form can be built upon in stages to ultimately encompass a large network of interdependent concepts including equations of lines in a plane, parabolas, circles, ellipses etc. It may also serve as the geometric basis for the development of variable and function.

Or considers an example from a Japanese lesson: developing the concept of linear equations with two variables. The initial starting point here is the rolling of two 12-faced dice. The question posed is “In what cases will the sum of two times die A plus B equal 15?” From this initial starting point, can be built the concept of ordered pair solutions of an equation with two variables.

Finally, an example from my own classroom. I begin with the concepts of “odd” and “even” number, introduced both numerically and using squared paper to give a visual interpretation of both concepts. I move on in stages to consider the consequences of simple operations (e.g., odd + odd = even; even + odd = odd), initially in purely numerical terms. This is then extended to multiplication by considering pairs of factors from 1 to 40. From this the class develops evidence of why it is that even numbers have potentially more factor pairs than odd numbers.

These examples demonstrate the potential of multistage problem solving and how it can be used within a socio-cultural context. Indeed, one of the most exciting possibilities is the idea of nested concepts. The odd-and- even number example, for example, leads naturally to many related concepts including implications of the ratio of factor pairs of even numbers to odd numbers being 2:1. This sets the stage for a discussion of infinite sets and the idea of parallel infinities being analogous to parallel universes.

Solving Problems as a Means to Understanding

Many critics believe that mathematics teachers should return to teaching well-grounded basic concepts using an algorithm-dominated approach. Problem solving has, critics believe, significantly impoverished children’s acquisition of basic number skills. Unfortunately, many seem unable to make the distinction between problem solving per se and problem solving as a technique for establishing a network of interdependent concepts.

This is a crucial distinction if we are to adopt a radical reform schedule. Typical goal-oriented problems can actually hinder learning because they place a high cognitive load on the problem solver. Although this is an entirely separate type of problem-solving activity from that described above, many people believe that this is the only type of problem solving– something to occupy children on a rainy Friday afternoon. They fail to see that problem solving can be a powerful method of introducing and reinforcing essential mathematical ideas.

Some influential industrial groups seem to want in a workforce that can follow mathematical rules without necessarily understanding how they are derived. Consequently they are primarily concerned with the lack of familiarity that many school leavers have with the rules of arithmetic and their applications in the workplace. If only they could see how a problem-solving approach, properly implemented, could actually strengthen the understanding and application of these rules, they might agree with the radical reforms that I am suggesting. From my own experience, I am convinced that algorithms are more readily accepted in a problem-oriented curriculum than in a rule-dominated one because they are seen as a means to an end, not merely as the end in itself.

In my opinion, mathematics teaching is much more than just teaching mathematics. At the school level, it is one of the few subjects that can instill a sense of the worth of mental discipline, the ability to think for oneself, and the validity of always asking “why?” If taught in a way that challenges a child to respond, then surely this must give that child the chance to reflect, not only on what mathematics means, but also on how the process of thinking can help the child negotiate the perilous course that we label adulthood.