The article’s authors, Sol Garfunkel and David Mumford, argue strongly in favour of a maths curriculum that exists in relation to the science, engineering, finance (and so on) that it serves every day, and they say why it’s so important.

Garfunkel and Mumford make an analogy with languages. Learning French teaches both abstract grammatical skills and practical language, but you will never have the chance to use Latin to buy a loaf of bread. In this respect – they argue – the abstract skills that you might gain from a dead language are wasted if they aren’t learned in a context that you can use.

This chimes with recent suggestions from a review into maths education in the UK (led by the lovely Carol Vorderman) that our maths curriculum pre-A-level should be broken up into ‘practical’ and ‘formal’ maths.

In this curriculum every student takes practical maths, arming them for a life choosing between mortgages and interpreting medical test results. More able and motivated students can opt to take the additional formal, and more abstract, maths module.

Mumford and Garfunkel say that the new US Common Core State Standards, which I’ve helped Whizz align its maths lessons to, is “highly abstract”, and “simply not the best way to prepare a vast majority of high school students for life.” But I’m not sure they’ve got the point of that (or any) curriculum. A curriculum should tell teachers and students what they need to study, not necessarily how, or even why, except maybe to put a subject in context.

It’s the good teacher who draws the abstract and the practical together to greatest effect. Not every teacher – especially at primary level – may be confident enough to link the two, but by giving the teacher the opportunity to do this, rather than prescribing how to apply ‘practical’ maths, an ‘abstract’ curriculum can be as relevant as the teacher and her students want it to be.

Maths is often called the science of patterns. And if chemistry really comes alive when we blow something up, then maths should come alive when we use it to, say, spot the ‘golden ratio’ in Chartres Cathedral (see Marcus du Sautoy’s BBC Series, The Code).

But it’s a two-way street. Project Euler, which I wrote about last month, trains budding programmers by getting them to solve maths puzzles. Garfunkel and Mumford say that practical skills, like learning to code, are more useful than abstract, but in the case of Project Euler the abstract is successfully used to teach the practical, and with some success.

I’ll concede that the principle outlined by Mumford, Garfunkel, Vorderman, et al – of the difference between the maths you use to calculate change and the maths you use to describe fractal geometry – makes perfect sense. But it may be a false distinction.

Whizz Education US company president, Ben, has a skill which I envy – he can intuitively understand the relationships between numbers. For me, it’s not quite so easy, but I wouldn’t want to be denied this insight into abstract maths because of a lack of natural ability; I can still appreciate Chartres Cathedral.

To return to the NY Times’ authors’ analogy, I took Latin and Ancient Greek at GCSE – two subjects that are, in themselves, utterly useless in the real world. But I was enriched by learning them. I could spot some of the hidden histories and meanings in words with ancient roots, and read street signs in modern versions of the ancient Greek alphabet.

If creativity lies in making connections between apparently unrelated subjects then ‘pure’, or theoretical, maths must be vital for creative young science minds, wherever they are. To assume that they should only use maths that tells them how to calculate compound interest (a truly vital skill) is to assume that they will get no pleasure from discovering a subject for its own sake. And that is sad.

At Maths-Whizz we try to instil a sense both of the practical and the abstract – and with our 1200+ maths lessons there’s ample opportunity for the young Whizzer to link the two. As Hilary and Steve, two of our expert founding mathematicians, once pointed out to me – put a pound sign in front of a sum or subtraction with decimal values and it suddenly becomes easier.

So, here’s to producing mathematicians who know that the abstract and the practical are two sides of the same coin. It’s a coin that can teach them as much about pi – that most wonderful of irrational numbers – as it can about the price of a loaf of bread.

Alison Gopnik, writing at Slate.com, comments on the increasing tendency to teach pre-school children, rather than allow them to play. She suggests that this approach, at least for the youngest children, might be stifling not just creativity, but the ability to learn, and she cites two new papers that draw from computer science to prove her point.

Adults often assume that most learning is the result of teaching and that exploratory, spontaneous learning is unusual. But actually, spontaneous learning is more fundamental. It’s this kind of learning, in fact, that allows kids to learn from teachers in the first place.

The key seems to be the way children learn about learning. A child will learn about the teaching method in order to understand better how to learn. Two recent papers in the journal Cognition explore the nature of that process by giving groups of children toys with hidden features and starting them off with either an exploratory approach or a taught one.

The understanding is that a purely demonstrative, instructional teacher implicitly conveys the understanding that the student can only learn by doing as she does, following her instruction. Whereas, a teacher who explores with the student (or even no teacher at all) brings with her the belief that experiment and investigation are just as useful as instruction, and that not all knowledge is directed.

Both research groups found that the less-directed students discovered more, and explored for longer. The differences between being instructed and encouraged to explore in both studies were subtle – a teacher might demonstrate an action (or set of actions) that produces a result in a toy and tell children to do likewise, or the teacher might explore the toy to produce an action as if by accident and encourage the children to see what they can find for themselves.

The research Gopnik refers to is new, and devoted to pre-school subjects, so it may be inappropriate to extrapolate this to primary age children. But these findings point to what she describes as a common intuition amongst teachers: that play can be just as powerful a learning tool as instruction if not, at least for the youngest, more so.

So what, if any, conclusions can we draw for the primary and early secondary children who use our online maths tutor Maths-Whizz? The first might be that there is such a thing as too young to study – we tend to discourage parents who want to use Maths-Whizz for their more advanced four year olds.

Second, we built into our maths lessons an implicit understanding of a child’s learning style at different ages. Earlier lessons at ages five and six are more exploratory and playful, and are devoted to the concepts that underpin maths rather than the more ‘math-y’ material in ages seven and up.

Finally, our recommended usage patterns reflect the need for younger children to have a more varied and undirected experience, away from a computer.

Gopnik echoes indirectly the call for creativity in education, given memorably by Sir Ken Robinson at a TED talk some years ago:

…it’s more important than ever to give children’s remarkable, spontaneous learning abilities free rein. That means a rich, stable, and safe world, with affectionate and supportive grown-ups, and lots of opportunities for exploration and play. Not school for babies.

Maths-Whizz, as numerous awards, hundreds of testimonials, and tens of thousands of online users prove, is a fabulous and enjoyable tool for building numeracy and boosting confidence in maths.

But we know there is always room for play in maths learning and we encourage students to step away from their online tutor now and then. So, how can you encourage your Maths-Whizzers to make the most of their creative instincts? Without second-guessing the results of future research in this field, here’s an informal selection of ideas:

• Explore maths concepts with the Whizz Tools Board (accessible from the bedroom). Play with cards, dice, counters, shapes, number grids and number ‘machines’.
• Ask your child to teach you. This is a great way to encourage students to marshal their knowledge and creatively convey what they know.
• Find maths in the real world. Look for shapes and angles in the house, think about how to estimate the number of leaves on a tree, and make your own graphs and charts.
• Tell stories with maths. Imagine you’re a number or a shape, and describe your day!

With Maths-Whizz, a little play can go a long way!

But we want to know what ‘happiness in numbers’ actually means for you.

Is there Happiness in Numbers?

We’ve started by talking to some of our students, parents, and teachers about how they’re finding happiness in numbers with Maths-Whizz. You can watch some of these ‘Maths-Whizz Chat’ videos on our Whizz Youtube channel.

But happiness in numbers can mean many things. It can mean the feeling you get when you work out your change at the shop faster than the person with the cash machine. It can mean turning Delia’s recipe for four into a six-person feast. It can mean spotting that that ‘value pack’ isn’t such good value. Or it might just mean the pleasure you get seeing how the world works with numbers, like the excellent Hans Rosling.

We want to broaden the conversation, see how people (and not just Whizzers) find happiness in numbers. So get in touch!

Here’s how:

• Add a comment to this blog post
• Follow the WhizzProf on Twitter, or post with the hashtag #happinessinnumbers
• Subscribe to our Maths-Whizz Youtube channel and comment on any of our videos
• ‘Like’ our Whizz Education page on Facebook, or the brand-new WhizzProf page, and add a comment.
• Join the Happiness in Numbers LinkedIn group
• Or, just get in touch with us at customerservice@whizzeducation.com

We’ll let you know where everyone finds happiness in numbers, so watch this space!

[Image via the Mostlymath blog]

Maths teaching is not a profession we associate with glamour, or influence, let alone popularity. But maybe it should be.

I thought I’d follow on from a post by Ginny at the Seattle offices of Whizz US. The EdWeek article that inspired Ginny focused on efforts to improve maths teaching in the US and the difficulties finding new teachers who “know and love math”.

Ginny reminds us just how influential a good maths education can be:

In a world where math is inextricably tied to our everyday lives, why are other professions so alluring? As a math teacher, think about how explosive one’s reach can be on the world…

How often do we consider the impact a math teacher has on a student’s success in future career endeavors?

And the answer is: “Not often enough.” I remember most of my own maths teachers as the targets of youthful contempt – bespectacled and socially awkward – and remember too little of the fascinating world that mathematical understanding can reveal.

Hold the world in your hands - with maths

Some of my early maths education must have rubbed off, and not in ways that I appreciated at the time – I was a big fan of MC Escher’s tesselations, mathematically accurate mirror-image drawings, and his famous never-ending staircases.

So, as Ginny asked on the US blog:

…how do you capture that excitement and make it infectious? How do you transfer the teacher’s passion to the student…? That love of teaching, that love of math – how do we reach students and show them that teaching math is indeed a “noble profession”.

How, indeed.

And, to be honest, I don’t have the answer, except to say that we must reinstate teaching as a profession equal in status to medicine or the law, rid ourselves of the pernicious maxim that “those who can, do and those who cannot, teach”, and compensate teachers to a level that recognises the enormous responsibility they have.

And few of those have a greater responsibility than the humble maths teacher.

With maths, you can unleash the power of the elements, land men on the moon, become a musical satirist, predict elections, help defeat wartime enemies, make pumpkin pie, predict the future, win the premier league, program computer games, and more.

With good maths you’ll even earn more money.

But now an OECD report has indicated the importance of maths for a healthy economy.

The level of mathematical knowledge among the general population was found to have a direct relation to Gross Domestic Product.

In other words – learn maths, end the recession!

So it was disappointing to read a recent too-smart-by-half ’10 worst…’ blog post at nymag.com. The post identifies ‘Ridiculous-Sounding Math Classes Currently Offered at Liberal-Arts Colleges’. [A 'liberal arts college', for those unused to the term, has nothing to do with liberal politics, and tends to teach science alongside the arts.]

This post is a classic illustration of the maxim that “a little knowledge is a dangerous thing.” In their attempt to demonstrate their disdain for wishy-washy pseudo-science courses for maths dunces the authors have managed to highlight their own ignorance of maths and science. A warning about people in glass houses springs to mind.

Click through for the offending top ten, but it’s clear that the authors have misunderstood some of these course descriptions entirely:

10. Topology: The Nature of Shape and Space: “In geometry we ask: How big is it? How long is it? But in topology we ask: Is it connected? Is it compact? Does it have holes?”…

…6. Models of Life: “In particular, we will ask such questions as: How do you model the growth of a population of animals? How can you model the growth of a tree? How do sunflowers and seashells grow?

Topology is hard maths. Modelling nature (such as the hidden mathematics in natural structures) is fascinating, mathematically robust, and not for slouches. Other mocked subjects include probability, folding, patterns of natural numbers, and so forth.

The authors have got a bit of a pasting from some of the comments, many pointing out that their selection of courses says more about the authors’ ignorance than the courses themselves, and we’d be inclined to agree.

The NYMag post ends:

This is why Asia is winning, by the way.

…A reference to the fact that China and India are producing vastly more pure and applied-science graduates than the USA or Europe, and how their numbers may be giving them an advantage in the global race.

We absolutely want more maths and sciences taught in colleges and schools. In fact, the GoW would be inclined to force students to take some basic scientific courses at all levels of secondary and tertiary education.

But the courses mocked in this list sound like just the sort of things arts undergraduates should be studying (alongside the fundamentals of mathematics, of course) to give them an appreciation for how this beautiful subject is completely interwoven in the creative and and natural worlds.

Not all of the courses listed seem essential to maths studies, but those courses that deal with creatively exploring the world through maths (never mind those that deal with the ‘proper’ maths of topology, probability, modelling and the like) are hugely valuable. GH Hardy who, in the early C20, was one of the most famous mathematicians alive, said:

“I am interested in mathematics only as a creative art.”

And from the man who introduced the world to Ramanujan, that sounds like confirmation that students should have a greater appreciation for the creative elements of maths, not less.

[via NYMag]

I have therefore kept myself busy this last month reading the on-line publications of the press and their maths related writings, reading blogs and tweets from maths teachers and others connected with the teaching profession and I have read the ‘chat’ of home educators and parents and their thoughts and concerns about maths. It’s been fascinating!

Not surprisingly, most maths-linked subjects are talked about on-line, but one that really interested me was the chat about society’s general attitude to maths. Evidence of this is easy enough to find, but one example is this Pizza Hut advert.

Why is it considered acceptable to be negative about maths? Why do people think that saying they are useless at maths is something to be proud of and why do parents tell their children not to worry if they can’t do maths because they could never do it either?

As an extreme example, if you couldn’t read you wouldn’t suggest to your children that it was therefore alright for them not to learn to read, so why tell children that it’s alright not to be able to do maths just because you can’t do it yourself.

Over at the Schoolfamily.com blog, Patti Ghezzi stresses the importance of parental PMA:

“The most important thing you can do is have a positive attitude. If you hated math as a child, don’t mention it; if you emphasize your own struggles with math, your child might assume that she is destined to struggle herself. Or she might think that because Dad turned out OK, math is unimportant.”

(full article by Patti Ghezzi)

Maths, unimportant… Never!

…ome hapless guy got slammed with a “penhora,” which translates into English as “distrainment”—the seizure of personal property to enforce the payment or discharge of an obligation. In this particular case, the subject of the distrainment had suffered the seizure of 1/6 of his assets. He petitioned the court for relief, claiming that he was suffering grave economic hardship.

The court solemnly pondered the petitioner’s request, noting the necessity of proportionately balancing the petitioner’s well-being against his responsibility to discharge his legal obligations. Upon consideration, the court ruled that the distrainment of 1/6 of the petitioner’s assets had been too severe and ordered a relaxation of the order. The new order instead stipulated a seizure of 1/5 of his assets.

This would be hilarious, if it wasn’t somewhat less than funny for the recipient of the Judge’s generosity.

But it goes to show that maths is necessary for all sorts of careers, not least those in the distinguished legal profession…

The Gresham College is a venerable London institution devoted to providing free lectures and events for the public, in the best of traditions.

The venerable Gresham College

In the clip below, Barrow shows the maths behind bank numbers and what’s known as the ‘Luhn Test’.

View Barrow’s introduction to the lecture, and clips about the mathematical patterns behind everything – post codes, six degrees of separation, mobile phone IMEI numbers and all sorts of other things.

If the God of Whizz was of an uncharitable turn of mind he might suggest that the world of finance has recently honoured mathematical accuracy more in the breach than in the observance.

In this vein, the Financial Times’ Gillian Tett suggests that mathematicians must get out of their ivory towers to restore credibility to the idea of ‘financial mathematics’, whose reputation the Credit Crunch and international financial crises have done so much to sully.

As Gillian Tett suggests:

What really damaged the financial system in recent years was not so much “maths” or “economics”; instead the crucial problem was bad maths (and economics) that was used and abused. Now, more than ever, mathematicians need to get out of their ivory towers or back offices and state that loudly, not just for their sake, but for economists. And, of course, those bankers.

The God of Whizz agrees, and humbly suggests a course of primary maths tutoring for the wizards who did all that financial abusing.