
Teaching Kids to Love Math - joeyespo
http://online.wsj.com/article/SB10000872396390444914904577615690632669590.html?mod=lifestyle_newsreel
======
orangethirty
They dont need to learn how to love math, they already do. What you cant teach
them is to love a system designed two hundred years ago to teach very ignorant
people about math. Most of the people who "hate" math do so because they were
forced to "learn" it through an antiquated system. I hated my math classes,
yet was always the top score on the annual standarized tests. Its not about
math, but about how they teach it.

Best way to teach math? Same way as with other science. Use it. Dont teach
kids fractions because of fractions. Show them how fractions work in the real
world. My 10 year old niece was having trouble with the multiplication. Why?
Because her teacher made them say it in front of the whole class. I went ahead
and sat her down. Booted up ubuntu and taught her how multiplication works
with javascript. She sat for around one hour toying with the code I wrote for
her.

What do you think is more fun?

\- Standing in front of other kids who are making fun of you and saying the
multiplication tables.

\- Writing done the following code and tinkering with it and calculating what
the answer will be.

    
    
        var base = 5;
        var num = 4;
    
        var total = num * base;
    
        document.write(total);
    
        //She spent one hour with those four lines of code, and learned more about multiplication. She is now learning python. :)

~~~
001sky
_Best way to teach math? Same way as with other science. Use it._

\- Agree with this. "By doing, we Become"

------
stephengillie
The only people who love math are math majors. Most humans don't love math.

Teaching kids to love math is pointless. Math is a tool, like a hammer or a
DLL. Do you love your screwdrivers and methods? Let's teach kids about how
math is useful to them in their lives, and let them take it from there.

 _Ongoing research is shedding new light on the importance of math to
children's success. Math skill at kindergarten entry is an even stronger
predictor of later school achievement than reading skills or the ability to
pay attention, according to a 2007 study in the journal Developmental
Psychology._

This is like trying to become smarter by listening to classical music. Is it
ironic that those suggesting these ideas can't separate correlation from
causation?

~~~
lutusp
> The only people who love math are math majors. Most humans don't love math.

Oh, this is so wrong. Did you like Avatar? Then you like math. Avatar is one
long, beautiful mathematical expression, from beginning to end.

> This is like trying to become smarter by listening to classical music.

Granted the problem with the basic idea, this has it all over trying to become
smarter by listening to Country & Western music.

> Is it ironic that those suggesting these ideas can't separate correlation
> from causation?

Math is neither a cause nor an effect -- it is both.

I could argue this point in detail, but instead I will get Bertrand Russell
argue it for me: "Mathematics, rightly viewed, possesses not only truth, but
supreme beauty — a beauty cold and austere, like that of sculpture, without
appeal to any part of our weaker nature, without the gorgeous trappings of
painting or music, yet sublimely pure, and capable of a stern perfection such
as only the greatest art can show. The true spirit of delight, the exaltation,
the sense of being more than Man, which is the touchstone of the highest
excellence, is to be found in mathematics as surely as poetry."

Richard Feynman said, "To those who do not know mathematics it is difficult to
get across a real feeling as to the beauty, the deepest beauty, of nature."

I can't think of a place that equals America for complete and willful
misunderstanding of mathematics.

~~~
walrus
> Oh, this is so wrong. Did you like Avatar? Then you like math. Avatar is one
> long, beautiful mathematical expression, from beginning to end.

Can you elaborate? To me, the interesting part of mathematics is the process
of manipulating expressions, not the expressions themselves. For example, the
number pi would be pretty boring if it weren't for the theorems related to it.

~~~
5xz41s0P8T5N
That argument confuses 'loving math' with 'loving things made of math'. I
enjoy eating cake and playing video games, that does not make me a chemist and
developer.

Your parent comment missed the point of his parent.

~~~
lutusp
> That argument confuses 'loving math' with 'loving things made of math'.

That might be a fair objection if they could be separated. If I say y = cos(x)
* e^-x^2, when expressed in two dimensions I get this:

<http://i.imgur.com/MePPy.png>

And in three dimensions I get this:

<http://i.imgur.com/9gr4u.jpg>

But I'm still looking at y = cos(x) * e^-x^2 -- the picture only confirms the
mathematical identity. So it is with Avatar -- when we watch Avatar, we're
looking at math.

And much of nature is defined using math -- note that I said, not _described_
, but _defined_. Here's an example -- there are species of locusts (cicadas,
actually) that reproduce at 13-year and 17-year intervals. Until recently, no
on knew why. It turns out that both 13 and 17 are prime numbers, and this ties
into a survival strategy hatched by natural selection (quite by chance). All
explained here:

[http://arachnoid.com/prime_numbers/index.html#Mathematical_L...](http://arachnoid.com/prime_numbers/index.html#Mathematical_Locusts)

In other words, the locust survival strategy is math speaking out loud.

If you read a book in which a seashore is described, do you argue that the
description isn't germane to the thing being described? If the writer isn't
skilled, or the reader is lacking in the capacity for visualization, then that
is perhaps a legitimate objection, but for most people, words convey meaning.
So does math. Math is a language in much the same way that words are a
language.

The distinction between "loving math" and "loving things made of math" is
dubious at best. When P.A.M. Dirac wrote his now-famous equation that
describes how relativistic electrons behave, he noticed that it had two
solutions -- sort of like a quadratic equation.

At first he doubted that there could really be two kinds of matter, as his
equation suggested. But within a few years antimatter was observed, and Dirac
forever afterward wished he had been willing to take his own equation at face
value and predict antimatter himself.

My point? The math told him something about reality that no one knew, even
him. To put it another way, _the math was reality._

------
VSerge
How is learning maths playfully not insisted upon in this article or the
comments? 2 examples: \- many card games require players to count points
depending on the cards they won. lots of adding and mental calculation
training in a fun setting. \- more advanced games like DragonBox, which blew
me away from a purely game design point of view. FYI, I'm a game designer and
I had never seen a game teach players, 5yo and adults alike, how to solve an
equation for x in a matter of hours, while actually being fun, and WITHOUT
triggering math-phobic reflexes.. (<http://dragonboxapp.com/>)

------
lutusp
The article fails to mention the single most effective way to improve math
education -- _teach mathematics first, then arithmetic later_.

What do I mean by that? This is mathematics:
[http://www.hdwallpapers.in/walls/jake_sully_avatar_2009-norm...](http://www.hdwallpapers.in/walls/jake_sully_avatar_2009-normal.jpg)

And this is arithmetic:
<http://www.scielo.br/img/revistas/bjp/v37n2a/a07frma1.gif>

The second part (arithmetic) is required for the first, but it doesn't need to
be presented first.

The problem is easy to state -- we're teaching mathematics in the wrong order.

~~~
tokenadult
_The problem is easy to state -- we're teaching mathematics in the wrong
order._

Where would we go to find a real-world example of mathematics being taught in
the correct order to elementary-age pupils? There are many different schools
(and many different settings for learning outside school) in many different
countries. What is a good place to look for examples of demonstrably
successful practice?

AFTER EDIT: Thanks for the reply. The thing I would want to look at in such
intervention studies is how the teaching effectiveness is measured.

AFTER FURTHER EDIT: Oh, I see the video "Teaching Math Without Words, A Visual
Approach to learning Math from MIND Research Institute" is one I have watched
before, and have discussed among other mathematics teachers. Similarly, the
publication Mathematics Teaching in the Middle School is a publication I
subscribe to as a member of NCTM. I should start out in my reaction to both of
these links you've kindly shared by pointing out that my own bent in thinking
about mathematics is to think visually, and indeed that is why I like Sawyer's
book Vision in Elementary Mathematics (which leads off my longer reply in this
thread) so well.

But although I like visual approaches to teaching mathematics very well, and
use them in the mathematics classes I teach, I also like studies of
educational interventions to check results. So far, I can't find a publication
by independent researchers (after searching Google Scholar and the new
Microsoft Academic Search) that verifies the educational effectiveness claims
made in the video for the proprietary intervention described there. The
company's own website

[http://www.mindresearch.net/cont/research/re_publications.ph...](http://www.mindresearch.net/cont/research/re_publications.php)

provides only a very sparse set of links to published research on the issue.

Plainly, a more visual approach along the lines of the typical school
textbooks used in Taiwan, Singapore, and Japan would be a better presentation
of mathematics content than is found in typical United States textbooks. That
should be possible to deliver without any computers at all, as it was
delivered to pupils in Taiwan in my wife's generation, when Taiwan was still a
poor, Third World country.

~~~
lutusp
> Where would we go to find a real-world example of mathematics being taught
> in the correct order to elementary-age pupils?

Good question -- and it seems this idea is being put into practice. Here are
some examples:

"Using Art to Teach Fraction, Decimal, and Percent Equivalents" :
<http://mason.gmu.edu/~jsuh4/math%20masterpiece.pdf> (PDF)

"Teaching Math Without Words, A Visual Approach to learning Math" (TED talk) :
<http://www.youtube.com/watch?v=7odhYT8yzUM>

"Teaching with Visuals: Students Respond to Images" :
<http://www.edutopia.org/visuals-math-curriculum>

There are many other similar examples. Here's one from an article I wrote
("each member of the running sum of odd numbers is a perfect square"):

<http://arachnoid.com/example/index.html#Math_Example>

The graphic says it all.

> The thing I would want to look at in such intervention studies is how the
> teaching effectiveness is measured.

That will take much longer and is fraught with measurement difficulties (as
with all social science questions). We might end up with people possessed of
self-confidence about their math skills, but who can't balance a checkbook. (I
say this as the devil's advocate, not because I actually believe that will be
the outcome.)

> But although I like visual approaches to teaching mathematics very well, and
> use them in the mathematics classes I teach, I also like studies of
> educational interventions to check results.

Yes, a perfectly legitimate concern, and one especially important if one is to
openly advocate a change in public school curricula. It is equally clear that
there are no reliable data on outcomes at this time.

------
Afton
I find the most challenging part to putting this into action is the same
things that one finds challenging when designing a test for a class: getting
the challenge level correct.

For a while this summer I'd write out simple math problems for my daughter to
do when she got up. I stopped when I realized that I was doing harm by
consistently getting the difficulty of my questions wrong. These kinds of
articles make it seem easy, but I find it quite difficult.

For obvious reasons, I'm continuing to try to help my children with math (and
other things!), but like most things, it seems to require patience, careful
thought, and practice.

------
lmkg
Children are betting at learning than we are at teaching. You can try teaching
2+2, but if you're uncomfortable with math, then your children will not only
learn 2+2 from you, they will also learn "math is hard." And having to un-
learn that lesson is a very large, very difficult barrier to have to overcome.
When you teach your kids math, don't just teach them 2+2, also make sure that
they learn math is fun & easy.

------
5xz41s0P8T5N
As a math undergrad dealing with mathematical anxiety (how silly, I know), I
wish math education started with logic.

Is there evidence that learning discrete math first leads to better skill, or
that it is easier/more fun for kids? or even teachers?

I suspect I only feel this way because discrete math seems so foundational
now, but perhaps _what_ is taught matters less than _how_ it's taught.

I wish education was more empirical.

------
oob205
Worth noting that a way NOT to teach kids to love math is to inundate them
with arithmetic. My math education, like many in the US, was an endless
barrage of memorizing facts, solving equations, then memorizing some more
facts. It was not until college that I began to see the essence of math is
creativity. Kids are naturally creative and inquisitive and we should use this
to our advantage in early education. We should explore topics like topology
and infinity, topics that still blow my adult, math-major mind. We also need
to encourage students to be creators in math. One of the ways we learn to love
reading is by writing and making up our own stories. We can have students make
puzzles, write computer programs, and see that they can invent things with
math.

For a more informed perspective on how we teach math all wrong, I recommend
Lockhart's Lament: <http://www.maa.org/devlin/LockhartsLament.pdf>

~~~
joezydeco
I spent an _hour_ last night working on a math homework assignment with my
second-grader. The practice was adding up single digits to sums greater than
10.

The idea behind the problem (say 9+5) is to "make a 10" and then figure out
the sum from the remainder. So 9+5 -> 10+4 = 14. _But he didn't know which
numbers routinely added up to 10_. That was never memorized. The assignment
was so focused on strategy and technique that it neglected the fact that the
basic tools aren't there.

There's a point where you need to memorize _something_ to make the strategy
and discovery easier.

------
tokenadult
An interesting article, pointing out that mathematics anxiety on the part of
adults sometimes limits engagement with mathematics learning opportunities
among children. Mathematician W. W. Sawyer wrote about this quite a while ago:
"The proper thing for a parent to say is, 'I did badly at mathematics, but I
had a very bad teacher. I wish I had had a good one.'" W. W. Sawyer, Vision in
Elementary Mathematics (1964), page 5. Elementary school teachers in the
United States often fear mathematics themselves,

[http://news.uchicago.edu/article/2010/01/25/female-
teachers-...](http://news.uchicago.edu/article/2010/01/25/female-teachers-can-
transfer-fear-math-and-undermine-girls-math-performance)

[http://www.jstor.org/discover/10.2307/41192533?uid=3739736&#...](http://www.jstor.org/discover/10.2307/41192533?uid=3739736&uid=2&uid=4&uid=3739256&sid=21101177886117)

and from time to time regret the gaps in their own mathematical education.

<http://www.ams.org/notices/200502/fea-kenschaft.pdf>

"The teachers are eager and able to learn. I vividly remember one summer class
when I taught why the multiplication algorithm works for two-digit numbers
using base ten blocks. I have no difficulty doing this with third graders, but
this particular class was all elementary school teachers. At the end of the
half hour, one third-grade teacher raised her hand. 'Why wasn’t I told this
secret before?' she demanded. It was one of those rare speechless moments for
Pat Kenschaft. In the quiet that ensued, the teacher stood up.

"'Did you know this secret before?' she asked the person nearest her. She
shook her head. 'Did you know this secret before?' the inquirer persisted,
walking around the class. 'Did you know this secret before?' she kept asking.
Everyone shook her or his head. She whirled around and looked at me with fury
in her eyes. 'Why wasn’t I taught this before? I’ve been teaching third grade
for thirty years. If I had been taught this thirty years ago, I could have
been such a better teacher!!!'"

The last time I posted a link to this article on HN, another HN participant
kindly posted a link to what is surely the "secret" referred to by the
elementary school teacher,

[http://www.tech4mathed.com/MAT156/topics%20test%202/twodigit...](http://www.tech4mathed.com/MAT156/topics%20test%202/twodigitmultiplication.htm)

pedagogical content knowledge that would be very routine for any elementary
mathematics teacher in east Asia.

[http://www.amazon.com/Knowing-Teaching-Elementary-
Mathematic...](http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-
Understanding/dp/0415873843/)

(book link above, review links below)

<http://www.ams.org/notices/199908/rev-howe.pdf>

<http://www.math.wisc.edu/~askey/ask-gian.pdf>

So this advice for parents is good in helping parents provide a supportive
environment for their children's mathematics learning.

I have frequent occasion to write about mathematics education here on Hacker
News. My occupation is 1) providing supplemental lessons in advanced
mathematics to pupils from ten counties in Minnesota through a nonprofit
corporation I cofounded and 2) coordinating parent workshops and other aspects
of the summer program Epsilon Camp,

<http://www.epsiloncamp.org/>

perhaps the most advanced mathematics program of its kind for YOUNG learners
in North America.

To date, I recommend to my own children and to my clients in my own
supplemental mathematics education program that they also turn to ALEKS,

<http://www.aleks.com/>

which is a commerical online site (in which I have no economic interest)
delivering personalized instruction in mathematics through precalculus
mathematics. The ALEKS website includes links to research publicatoins on
which ALEKS is based.

I also recommend the Art of Problem Solving (AoPS)

<http://www.artofproblemsolving.com/>

(where I first took on the screenname that I also use here on HN) for more
online mathematics instruction resources, and I also share specific links to
specialized sites on particular topics with clients and with my children. I
should note for onlookers that the articles on mathematics learning on the
AoPS website

<http://www.artofproblemsolving.com/Resources/articles.php>?

are very good indeed, especially "The Calculus Trap."

My children make quite a bit of voluntary use of Khan Academy (both watching
videos and working online exercises) and I am gratified that my previous
suggestions to the Khan Academy developers here on HN

<http://news.ycombinator.com/item?id=2760663>

have been followed up as Khan Academy developers have communicated with me by
email about new problem formats available in their online exercises, which are
becoming increasingly challenging.

Besides that, I fill my house with books about mathematics, and circulate
other books about mathematics frequently from various local libraries.

I also recommend that all my students use the American Mathematics Competition

<http://amc.maa.org/>

materials and other mathematical contest materials as a reality check on how
well they are learning mathematics.

In general, I think mathematics is much too important a subject to be single-
sourced from any source. Especially, mathematics is much too important to be
left to the United States public school system in its current condition. I was
rereading The Teaching Gap: Best Ideas from the World's Teachers for Improving
Education in the Classroom (1999) last month. It reminded me of facts I had
already learned from other sources, including living overseas for two three-
year stays in east Asia.

"Readers who are parents will know that there are differences among American
teachers; they might even have fought to move their child from one teacher's
class into another teacher's class. Our point is that these differences, which
appear so large within our culture, are dwarfed by the gap in general methods
of teaching that exist across cultures. We are not talking about gaps in
teachers' competence but about a gap in teaching methods." p. x

"When we watched a lesson from another country, we suddenly saw something
different. Now we were struck by the similarity among the U.S. lessons and by
how different they were from the other country's lesson. When we watched a
Japanese lesson, for example, we noticed that the teacher presents a problem
to the students without first demonstrating how to solve the problem. We
realized that U.S. teachers almost never do this, and now we saw that a
feature we hardly noticed before is perhaps one of the most important features
of U.S. lessons--that the teacher almost always demonstrates a procedure for
solving problems before assigning them to students. This is the value of
cross-cultural comparisons. They allow us to detect the underlying
commonalities that define particular systems of teaching, commonalities that
otherwise hide in the background." p. 77

Plenty of authors, including some who should be better known and mentioned
more often by HN participants, have had plenty of thoughtful things to say
about ways in which United States mathematical education could improve.

A discussion of the Common Core Standards in Mathematics, "The Common Core
Math StandardsAre they a step forward or backward?"

<http://educationnext.org/the-common-core-math-standards/>

gets into further details of how mathematicians look at the general school
curriculum in the United States. It is not the worst curriculum possible, and
survivors of the system often have access to outside resources to supplement
school lessons, but the public school instruction in mathematics in the United
States still shows plenty of room for improvement.

The last time I posted about these issues, a reply asked what I think about
essay "Lockhart's Lament." I think it is an interesting read, but less
practical for reforming mathematics education than I had hoped. I wonder if
Lockhart's forthcoming book Measurement

[http://www.amazon.com/Measurement-Paul-
Lockhart/dp/067405755...](http://www.amazon.com/Measurement-Paul-
Lockhart/dp/0674057554/)

will be a successful attempt to teach mathematical reasoning to students who
have already lost confidence in learning mathematics, which would be a great
contribution to society.

~~~
lifeisstillgood
As a father of two (both under 3) I want to (obviously) supplement my
childrens official education with home encouragement - but resources and
kicking off points like that above are - well I am sure they are around, I
just don't know where.

I am half considering starting the Concerned Parent (tm) github repo

Is there any value in a github repo on recommended books and practises to help
polyfilla in the cracks in the school system?- different approaches could have
their own branches, commits could be discussed (Did you like that book The
Teaching Gap? Should it be recommended? etc)

No I am not trying to recreate mumsnet. How crazy is this?

~~~
SiVal
It's not crazy, but it's inefficient. Go hang out in home schooler forums
instead. People homeschool for many different reasons, but I think it's great
that it's still allowed in the US. The result of a diversity of people with
diverse needs meeting a private market that hasn't yet been eradicated by
"progressive" teachers' unions and the politicians they employ, is a
cornucopia of private resources available to parents and students who exercise
some initiative.

These resources are under constant discussion on homeschooling forums.

~~~
lifeisstillgood
Thank you - as a brit I had always assumed it was, well, nutjob territory.
Will look into it.

~~~
SiVal
Yes, the mainstream media and the educational establishments in both the US
and UK are dominated by the same leftist statists who, in order to promote
their social agenda, need to portray all those who don't surrender themselves
to the state programs as obvious nutjobs.

Those who leave the herd do so for all sorts of reasons, religious, academic
(my case--like many homeschoolers, I'm an atheist), political, special
educational need, lifestyle (e.g., frequent travel), or whatever. That's the
nature of independent individuals. That level of diversity means that you can
find plenty of people you'll consider nutty, which makes it easy for the media
to push the message that those who leave the state herd are mentally
defective.

But look at the effectiveness of the state schools and ask yourself, if the
state system were shrunk down to the size of the various small homeschooling
alternatives, would the state program be one you would pick for your own kids,
or would one of the alternatives seem much more sensible? If you suspect the
latter, maybe the state system is the real nutjob territory.

------
marcpaul22
I loved "Prof. E. McSquared's Calculus Primer" when I was learning Calculus
(look for used prices):

[http://www.amazon.com/gp/product/0913232475/ref=olp_product_...](http://www.amazon.com/gp/product/0913232475/ref=olp_product_details?ie=UTF8&me=&seller=)

Home page here: <http://www.math.sjsu.edu/~swann/mcsqrd.html>

Fun and funny, written in a comic book format. It illustrates the concepts in
such an entertaining style, but makes the point better than many textbooks.

~~~
calibraxis
Along those lines, you might enjoy _Thinking Physics_.
([http://www.amazon.com/Thinking-Physics-Understandable-
Practi...](http://www.amazon.com/Thinking-Physics-Understandable-Practical-
Reality/dp/0935218084))

------
diggan
That's great. Can you now teach me how I (~20 year old) can be interested in
math? I deeply want to understand it but it gets so boring after a while... Is
there any help for me out there?

~~~
GFKjunior
I recommend Calculus Made Easy by Thompson. This book is short but complete. I
read it at 22.

I gave up on mathematics after my first college course but found that for
everything I wanted to get into, Machine Learning & Bayesian stats, it was
essential . A year later I know math better than ever before even though I
haven't stepped inside a classroom in years.

You're probably getting bored because most math books are hundreds of pages
long and in addition to the material they include mathematicians bio's,
several solution methods, and the explanations of edge cases.

~~~
debacle
You can't justify a $200 price tag for a 100 page text book.

~~~
lutusp
That might sometimes be true, but the textbook the OP referred to is less than
$16.00:

[http://www.amazon.com/Calculus-Made-Easy-Silvanus-
Thompson/d...](http://www.amazon.com/Calculus-Made-Easy-Silvanus-
Thompson/dp/0312185480)

~~~
debacle
Which is why no one will ever use it, because you can't justify a $200 price
tag for a 100 page text books.

e.g. no college will ever endorse this book as a text, because they need to
sell 60k in calc 101 text books each semester so they can get their cut.

~~~
lutusp
> ... no college will ever endorse this book as a text ...

Yes, fair enough, however I think the discussion is not so much about
schooling as education. Not all education takes place in college -- and with
costs rising as they are, I predict a future with many more autodidacts --
self-educators.

"I have never let my schooling interfere with my education." -- Mark Twain

------
001sky
Numeracy is the foundation for math. That is what we should be teaching kids.
It makes it much easier to transition to more logic-intensive modes of
analyisis later on.

------
CeRRuTiTo
Math skills are essential to everything, so I like this!

------
csense
Most elementary schools have all subjects taught by one teacher. (At least
mine did; my K-12 education was entirely in the US public school system.) Kids
taking different subjects from different teachers doesn't happen until middle
or high school.

So I'd guess that most teachers of young kids probably chose to train as
elementary-education generalists, and math was hard and scary for them. There
are several mechanisms by which this might rub off on their pupils:

1\. If the teacher doesn't have an innate joy and passion for their subject,
the lack of enthusiasm may be contagious.

2\. The teacher may have no idea why things like the addition algorithm work,
and no idea how to explain it to their students.

3\. Most of the math classes in elementary school were largely spent reviewing
material covered in previous years. Having an expectation that students should
know material that's been covered in previous lectures is simple common sense.
I'm not against an occasional review class, mind you, but spending an entire
quarter or semester on nothing but previously covered material strikes me as a
colossal waste of everyone's time.

4\. High-stakes standardized testing is a recipe for disaster. The tests don't
accurately measure anything but the amount of time and effort teachers spend
teaching to the test. I was pretty far out of elementary school by the time No
Child Left Behind hit the fan, but I'd imagine even knowledgeable, passionate
teachers who want to motivate their students with the joy of learning to use
their imagination to manipulate abstract ideas get immense pressure from above
to turn their class into nothing but drill, drill, drill.

5\. The very name, "No Child Left Behind," implies is that the goal is to have
the entire class proceed at the pace of the least capable student. On the face
of it, this is deliberately aiming for poor results. For those of us who have
never observed the human race in action before, let me explain: People are
different; they have different levels of mental strength, motivation, talent,
and parental support; they have different areas of interest, different areas
of talent, put in different amounts of effort, and think in different ways. It
is inevitable, then, that some people will be better at math than others. But
we seem to be trying extremely hard to deny this simple and obvious truth, to
bring the least capable students to a level of competence that they may never
achieve, at great cost to taxpayers and immense frustration for everyone.
Meanwhile, the middle- and high-capability tiers of the class are bored,
frustrated and alienated by the endless repetition of very dull tasks -- which
is most assuredly _not_ what mathematics is about, especially since the
invention of cheap, ubiquitous computers.

