Zwaarddijk wrote:I see how numbers (and algebra, and so on) give methods of explaining things that I always have wanted to know how they work - e.g. why chords don't just sound like a bunch of simultaneous tones, but an actual ~sonority~ of its own, and lots of other things. That's why I like maths.

You like Mathematics on the basis of non-Mathematical things.
Need i say more?

Sia wrote:

Zwaarddijk wrote:I see how numbers (and algebra, and so on) give methods of explaining things that I always have wanted to know how they work - e.g. why chords don't just sound like a bunch of simultaneous tones, but an actual ~sonority~ of its own, and lots of other things. That's why I like maths.

You like Mathematics on the basis of non-Mathematical things.
Need i say more?

Notice the "I see how...", not "I like how...". In a way, this is the icing on the cake. Maths for the sake of itself is mostly a fascinating endeavour.

Sia wrote:

Zwaarddijk wrote:I see how numbers (and algebra, and so on) give methods of explaining things that I always have wanted to know how they work - e.g. why chords don't just sound like a bunch of simultaneous tones, but an actual ~sonority~ of its own, and lots of other things. That's why I like maths.

You like Mathematics on the basis of non-Mathematical things.
Need i say more?

God likes you because you agree with him, and you like him because he agrees with you.
Need I say more?

I hated maths at school too, but since I left I started to find it more and more interesting...

Perhaps they're just teaching it wrong...

There can be a bit of an over-focus in classrooms on skills rather than application, problem solving and play. However, some children enjoy the predictability skills practice and there can also be a focus on application and problem solving to the detriment of skills. For example, some children struggle to multiply or use fractions. It's hard to effectively solve problems when you cannot easily employ basic skills.

I think mathematical games/ simple puzzles and stressing mental mathematics (which is usually stressed in the UK but much less in North America) does actually do much for children's understanding of mathematics and numbers.

I think something that tends to come to adults that doesn't come to children is a good, solid understanding of what the basic concepts of math actually mean. For example, as a child, I never understood that fractions are in fact simply division that hasn't yet been completed. I didn't make the connection between the fraction-like way of representing division and what fractions look like. Once I understood that, suddenly things began to come together.

Understanding that maths isn't only represented numerically has been a very recent revelation for me. At school, maths was numbers and that geometry and other representations were sort of secondary. Now I have suddenly understood by reading about the history of mathematics that both are equally valid mathematical representations and that people such as Descartes and Feynman visualised mathematics as diagrams, not as numerical figures.

I was always good at visual mathematics and I feel that if I had made this connection as a child I would have been far more interested in mathematics and developed a much deeper understanding.

The real problem is, of course, that most teachers don't have a comprehensive understanding of mathematics because they themselves were taught in by people who *also* didn't have a comprehensive understanding of mathematics. As a teacher, I understand: primary teachers, especially, are required to teach many subjects and it is a bit much to ask every teacher to be an expert in every subject and then deliver it to children who also require a lot of social coaching for what (in most countries) is a unimpressive amount of pay.

Most people don't like maths because they were taught it very, very badly. This is the norm, as teaching mathematics well is very difficult. So the chances are that you don't like maths because you were taught it badly.

The link below, sent to me by a fellow maths tragic who found it on the internet, expresses what many mathematicians feel about the violence that is done to their beloved art in primary and secondary schools around the world.

A Mathematician's Lament

Very insightful. Unfortunately, it is one person's opinion (or 3) against a society that in the large has no immediate hopes of appreciating it.

andrewk wrote:
The link below, sent to me by a fellow maths tragic who found it on the internet, expresses what many mathematicians feel about the violence that is done to their beloved art in primary and secondary schools around the world.

A Mathematician's Lament

Thank you for this!

While that Mathematician's Lament makes a point, most classrooms and curriculums contain more puzzles and mathematical play than they formerly did.

However, I should note that not all children take to playing mathematically just as all children wouldn't take to playing musically. Not to mention, skills like adding, subtracting, multiplying and dividing are not always intuitively understood by children. In the world, people are required at the very least to do this, whereas people are expected to carry out musical tasks in this way whereas they might be expected to sing.

A lot of mathematicians approach mathematics very intuitively, and have always done so. They see that the triangle takes up half a corresponding rectangle themselves and, given the opportunity, that's brilliant. However, not all people are able to do this and because children are required to be able to use numbers in a basic way we do actually attempt to teach those who don't learn it intuitively the skills they do need.

We try to do this, nowadays, using as much visual and three dimensional play as possible, and with as little emphasis on drill as possible. But the beautiful dream of just play, while it would work for those people who take pleasure in mathematical patterns, would be fun but ulimately less effective, for those who take a different approach in life.

Just as an intuitive musician wonders how people have to learn to hear music notes in a chord in order to harmonize once they understand that harmony exists, or as an intuitive reader wonders why everyone can't just pick up reading from having books laying around and being taught a bit at the beginning, an intuitive mathematician wonders why someone cannot see, immediately and without instruction, that division and multiplication are the same operation performed in a reverse direction.

And of course, given enough directed play and puzzling, they can be made to see this. But this is a huge societal leap. Teachers are often working very hard to convey the joy of mathematics without had any instruction in it themselves. They buy games and dig out books of puzzles when their own instruction was pencil to paper, drills and skills. I've already said this: teachers are from a previous generation. Things are changing already.

What mathematicians who have a good idea of how to teach mathematics more intuitively have to do is get into schools, see what the reality is nowadays, and see if they can nudge the schools in the right direction. Provide a mathematical club (never been at a school where one exists), talk to teachers, and most importantly become teachers and curriculum advisors.

The writer's first example, that of cutting triangles up, I performed with my kids in my classroom using paper. It sort of worked and I would do it again with a few changes to improve it. But you know what one of the top kids' parents said when their child brought them to look at their mathematical triangle postery thing? "That's more art than mathematics, isn't it?" They didn't understand that seeing why b(h)/2 works is important, and these were highly educated people.

So, please. The violence was done in a previous generation. Teachers often teach the why and show the why... with the advent of SmartBoards, the amount of visualization and play has gone through the roof because suddenly everyone has access to mathematical games, if the school can afford it.

What you say probably seems to make a lot of sense, but I am not so sure that what you observe is an inborn capacity rather than an early-learned attitude in the majority of cases. I am not saying that there is no genetic component at all in mathematical predisposition, but I think entering school believing math is difficult, boring, or both plays a role. And of course, it is more important that children come to school without major nutritional, emotional, or sleep deficits than what curricula are. Poverty in the USA interferes with learning in a way that no combination of good teaching and school counseling could ever hope to successfully address.

Teshi, your comments are fair. I think the Mathematician's Lament diagnoses the problem well and presents it in a powerful manner (and therein lies the value of the essay), but I am doubtful that the author's proposed solutions would work. There is an inescapable tension between the desire of most parents (including me) that their children know their multiplication and addition tables from memory, and the desire that they learn the skills of pattern recognition and creative problem solving that are the real heart of mathematics. There are great difficulties in attempting both. It sounds like you're the sort of maths teacher a child would be lucky to have, and do your best to bring it to life. There are many primary teachers who teach mathematics in a horrible, dry, boring manner that would turn almost anyone off though. Probably because they were taught poorly themselves and see it as an unpleasant, necessary evil.

Perhaps it would help if maths were taught by specialist teachers in primary schools, as it is in secondary. That would make it more likely that the person teaching it had a taste for it and was able to convey some of the joy of the subject. It may be hard to do that in a budget-neutral fashion though.

Someone wrote:What you say probably seems to make a lot of sense, but I am not so sure that what you observe is an inborn capacity rather than an early-learned attitude in the majority of cases.

On a slight tangent, this is exactly how I feel about art. I absolutely hate it when people put it down to inborn "talent". It diminishes both the lifetime of work artists like me have put into their skills, as well as the capability of would-be artists who really just gave up before they even tried. I often feel that my ability to make art is largely the result of my parents buying me mountains of Crayolas and Play-Doh and construction paper and tape and glue when I was very young, instead of telling me to quit "wasting my time".

Anyway, I was never good at math either. I won't say I hate math (not anymore anyway, although if you had asked me that in highschool the answer would've been an emphatic "YES"), but I certainly cannot grasp it very well. I put this largely down to bad teachers (I think I had two good math teachers in my entire 12 years in the public school system), an extremely active imagination that was almost always more entertaining than the person droning at the front of the classroom, and my extreme difficulty in holding any number in my head for any length of time. I've suspected I may have some level of dyscalclia. Interestingly, like most people with dyscalclia, I actually did quite well at Geometry. Although I'm not sure if that's because I actually "got" Geometry because it had a very obvious connection to the physical world, or if it's because my Geometry teacher was simply a hilariously squirrely and exuberant woman that could keep even my ADD self's attention.

One very good example that should make one question just how much mathematical ability is innate comes from the world of chess. While it does concern chess and not mathematics, and it does deal with the high-end extreme of performance; the success of Laszlo Polgar with his three daughters--Judit, Szusza, and Sofia, in decreasing order of current ability at the game--has pretty much on its own demolished the conception that the high end would inevitably always be a masculine domain. There is probably about a 25% chance that a female will become champion (for a time) within the next 20 years, with Judit one of the people still having a fair shot (along with young Hou Yifan of China, some other currently active players, and many who have yet to touch their first chess piece). Again, this is about chess and not math per se, and more importantly it is about the top level and not about people near normal; but it should still help raise some doubt about what seem to be obviously true assumptions on mathematical ability.

What you say probably seems to make a lot of sense, but I am not so sure that what you observe is an inborn capacity rather than an early-learned attitude in the majority of cases.

In many ways, it is true that success is set before a child begins school. However, you need only to look at the siblings within a single family (who presumably got similarish exposure to ideas) in order to see that ability in anything does have a component that is genetic. A child who struggles to visualize things will struggle in maths. A child who memorizes more slowly will also struggle. It's not so much they're "bad at maths", it's their skillset makes learning maths more difficult and so naturally their interests and abilities turn a different way. However, an attitude towards maths can be learned. Many people come from parents who found mathematics difficult and/or boring and this attitude gets conveyed.

Someone, you mention a chess player who has taught his three daughters to play chess. Are these daughters adopted? If yes, you can talk about environmental factors-- exposure from a young age plays a huge part in success. However, if they are his biological children, then genetics could be a very strong factor.

Being a woman, I never had any illusions that women can't play chess. Are you saying that there were ideas out there that women physically couldn't?

On a slight tangent, this is exactly how I feel about art. I absolutely hate it when people put it down to inborn "talent". It diminishes both the lifetime of work artists like me have put into their skills, as well as the capability of would-be artists who really just gave up before they even tried.

Oh I recognise that practice is hugely important. But I also know that there are people who get it earlier and more innately. And I want to make sure that people know that. It's not that they're good "at maths", it's that they have a brain that visualises well, or one that memorizes quickly, or whatever little mini-skills make up being good at mathematical tasks. An artist may be someone who simply looks harder and therefore has become more interested in colours and shapes than someone who looks for information.

This is assuredly helped by hard work and practice, but you can have people who work really hard to master something and you have people who slack off and still master it with far less effort. Most things are achievable with effort and it is important that we as a society recognise this, but it's also a problem if we think that all things come to all people with the same level of effort and/or priming from an early age.

It's hard to explain. I do believe very strongly that our brains are radically shaped by our earliest years, but I also believe that even the best, most varied and rich childhoods produce children who are radically different in areas of ability. What I do believe is that most people are intelligent and empathetic, provided they are given a rich childhood starting at day 1.

What that intelligence looks like is going to vary from person to person. And just as some people are going to pick up a pencil and be able to draw more easily because their brains were inclined to looking more carefully, some people are going to understand multiplication more innately because their brains are inclined towards totals and since day one they've been carrying out pre-mathematical tasks.

I'm not a psychologist, so this is pure speculation.

Maybe better examples are the three New York African Americans (technically the term should be 'Afro-Americans', but the term used is standard) who have become US National Masters concurrently under age 14. (See http://www.nytimes.com/2011/11/13/cross ... &ref=chess ) Unrelated fellows. I think you are confusing matters slightly when you say that what is observed of a toddler's cognitive capacities is genetic simply because of intrafamilial differentiation. That would be opinion against opinion, neither being scientific. Obviously, hurdles have to be leapt all along the way from learning to suckle to gaining a Nobel Prize in some field. It does all a disservice to lock some into a state of backwardness based on temporally-limited observations. Economics is highly determinative in the question of whether to play a child toward his or her strengths or weaknesses, as yet another reason economics weighs heavily on educational outcomes. My perspective is that it's ideal to get children universally up to speed in understanding mathematics' usefulness in a broad general sense, for political reasons more than anything else. How this could be done would probably earn me three Nobel prizes though--Economics, Peace, and Medicine--so I have no idea, of course. It appears to simply be a vicious cycle. The uneducated cannot adequately educate themselves, and the educated really cannot understand why despite their education.

By the way, I am not the author of the 2nd quote above. This would perhaps not be clear to a person reading this thread in a partial or overly casual way. To be quite serious, educational psychology is a morass worse than climatology, with incorrect biases more influential in wrong-headed policy and public discourse.

To answer your second direct question to me, Teshi: No, of course all men have always known that women are innately as mentally competent as men.

Teshi wrote:Someone, you mention a chess player who has taught his three daughters to play chess. Are these daughters adopted? If yes, you can talk about environmental factors-- exposure from a young age plays a huge part in success. However, if they are his biological children, then genetics could be a very strong factor.

Being a woman, I never had any illusions that women can't play chess. Are you saying that there were ideas out there that women physically couldn't?

Check out this video: http://video.google.com/videoplay?docid=-6378985927858479238
.

Susan seems to have a knack for making herself look bad or allowing others to help her do it without effective protest. Nona Gaprindashvili was the first female to attain to Grandmaster status (not WGM, but GM) two decades before Susan and her considerably younger sister Judit did in the same year. I won't comment on the whole story. Anyone intersted in Susan Polgar can look her up. Phenomenal chessplayer, but 'first female Grandmaster'? Who writes, films, and edits such garbage; and how does Szusza let it pass? I think I actually watched this all the way through on a previous date. Perhaps there is a punchline I forgot. Please let me know if there is.

On the main topic, I think I might actually have a workable and possibly even saleable approach for the kids that are fortunate enough to make it to school ready to learn. 2 hours every day of academic and mental competition, with entire maths curriculum through highschool folded in and no classes in maths until university. Give exceptional problem-posing teachers on a national scale massive bonuses as well--a meta-competition of sorts. You essentially expand mathematics as a subject area to mathematics plus mathematics-like subject matter and also include plenty that is neither. You could take up another hour to hour-and-a-half on arts generally and do well by that too. Instead of grading work in a highly punctuated way (as with final exams, to clarify), better to just grade in a constant way. Then you tell the bottom half in each subject who they are, tell any who truly need remediation who they are, phase out need for remediation by increasing grade level through a focused approach, not actually give any real grades in the standard sense until the third from last year, and so on.