
A better way to teach math - dctoedt
http://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-math/
======
forkandwait
It might be paranoid, but I think the old style of teaching math -- a few big
examples, let the student find their way on their own -- is actually designed
to create an _artificial_ hierarchy and ranking. Rich kids and lucky kids
figure out they have to break things down into micro steps, either do that on
their or hire a tutor (have a parent) to do it for them, but in class they
just seem to be "bright". (I have been that tutor, and totally changed several
kids from "I'm/ he is just not good at math" to "Oh -- math is straightforward
if you break it down and work your ass off, like almost everything else"

This is basically how Law School Confidential thinks law school manages to
keep the supply of good lawyers low: fool the students into thinking that
talent and intuition teach you law, and let a few students who outline and
memorize like crazy "fall up" through the cracks.

I say "designed", but in that mostly subconscious way that we "choose" to do
most things -- it feels right, it is "obvious" because it is how we learned
originally, and it doesnt lead to weird results (a world full of talented math
users which might cheapen the supposed "talent" of the lucky ones).

In my experience, public school teachers absolutely love to categorize kids
into stupid and smart, with disastrous results unless you are one of the lucky
ones (which probably has as much to do with good looks, high-prestige parents,
and social skills). I wouldn't say there is NO bell curve, but it could be
much, much flatter if it we wanted it to be -- but it's no fun to be an
officer without a bunch of stupid enlisted men to boss around, and public
school wastes a lot of taxpayers money to _create_ those stupid people.

~~~
simpleTruth
To play the devils advocate: IMO, it's the focus on the micro steps that's the
problem.

HS level math is vary simple. You can write a single textbook that covered the
full range of math from preschool to calculus, but the focus on 3 days of
instruction, a day of review, and then a quiz or test slows things down. It
can be easier to skip ahead and then go back and review than try and approach
math in tinny nibbles. It's like spending a full year going over cement
foundations before you mention that the goal is to put a house on top of that
flat slab.

Personally, I used to do other classes homework assignments in my math
classes. I can recall getting in about 4 seconds a new topic that the class
spend a full week going though in minute detail. It was so bad I once
accidentally did the next chapters review vs the assigned homework and did not
even notice at the time.

PS: I am all for better instruction, but perhaps we could consider going a
little further. We could probably get the average 10th grader to really
understand Calculus, but I think the goal should be to dive into DifEq and
number theory etc.

~~~
hackinthebochs
>IMO, it's the focus on the micro steps that's the problem.

No, the micro-steps are the solution, not the problem. Those who "get" math
intuitively break problems down into smaller steps, often unconsciously. For
those that can't intuitively do this (most students), they need to be taught
explicitly the micro-steps. The micro-steps build the foundation for the
higher level steps and the problem solving.

Learning the micro-steps well is like learning an abstraction in a program.
Once you learn the abstraction to the point that your understanding of it is
unconscious, then your thought processes are lifted to a higher level. This is
true understanding of math.

~~~
simpleTruth
There are plenty of useful things to memorize that let you flat out skip
steps. Some of this stuff might feel like party tricks, but don't assume
people are using the same steps subconsciously.

~~~
hackinthebochs
I don't mean to say that people use the micro-steps all the time. What I mean
is that, on initially learning something, those that "get" it subconsciously
break it down into micro-steps that are just small enough for them. This
builds the bridge from what they know to what they're learning. Once the
abstraction or technique makes sense, then you apply it as a whole on further
usage.

I think one of the main things that separates the quick learners from the slow
learners is how much breaking steps into micro-steps can be done
unconsciously. The data lends weight to this. Teaching a concept in micro-
steps doesn't help the smartest people learn it better, but it brings the slow
learners up to speed. The difference is how small the steps have to be for the
students' unconscious to make the connections and thus reach "understanding".

------
happy4crazy
It sounds trivial, but the idea of breaking things down into micro-steps is
incredibly useful. I think it's fundamental to effective abstract thinking,
and yet most people (including me!) don't consciously think this way.

As an example, I was a physics major in college, so I'm used to thinking of
myself as being pretty numerate. And yet, I've noticed that I'm not a very
efficient learner of higher math; I enjoy it, which keeps me chugging along,
but I often find myself getting discouraged when my brain doesn't
automagically internalize new abstractions. Instead of approaching a new
abstraction as a bundled collection of less-abstract micro-steps, I think
"hmm, if I were really smart this would just sink in." Sometimes the new
abstraction does just sink in, but it often doesn't, and then I feel briefly
bummed about not being the radiant genius I thought I was.

This is a dumb attitude! An abstraction is like a steak; you can eat it, just
maybe not in one bite. And the incredible, amazing thing about abstraction is
that as you get better at it, you get to take bigger bites!

This all reminds me of something Kent Beck says in his TDD book:

 _It is not necessary to work in such tiny steps as these. Once you've
mastered TDD, you will be able to work in much bigger leaps of functionality
between test cases. However, to master TDD you need to be able to work in such
tiny steps when they are called for._

 _Being able_ to do things in tiny steps is a skill, and being _willing_ to do
things in small steps when you choke on a big step, rather than feeling dumb
and giving up, is the key to learning just about anything.

------
xal
The problem with math education is a lot more basic then everyone pretends:
there are two variables, mastery and time and we made the wrong one static.

This was a global decision born out of necessity. Because there aren't as many
teachers as students we had to make time static and mastery variable. You take
a class for a set period of time and then you get a grade based on how much of
it you understood. A B means you understand the topic about 80%. In an ideal
world every student would always get 100% but simply move at a different pace
with the best students simply consuming more material throughout their school
career (calculus, linear algebra, etc).

The tremendous news is that technology can and will turn this on it's head.
The Kahn Academy does this already with tremendous success and it's the single
most important thing that has happend to education in a long time.

For getting a full idea of the scope and vision, whats Sal's ted talk:
[http://www.ted.com/talks/salman_khan_let_s_use_video_to_rein...](http://www.ted.com/talks/salman_khan_let_s_use_video_to_reinvent_education.html)

------
shasta
Getting the students at the low end of the spectrum to "get math" is certainly
a noble goal, but in order to have such a low variance in math ability at the
end of the year, there's another required component - you have to keep the
high end students below their potential. As far as I'm concerned, anyone who
claims otherwise has a high burden of proof. Go take a remedial math class and
turn it into a winning USAMO team and then get back to me. High end math
education (at least in the US) is just as poor as the low end, but because the
scores are acceptable not as many people care. Talk of "evening things out" is
misguided.

~~~
raganwald
What if we discover some magic new program that improves every student's
scores, but it improves the poorest students the most and the gifted students
the least relative to our existing programs? Such a program wouldn't be
holding the gifted students back relative to existing education, it would
improve them, and it would still reduce the variance.

(My comment sonly apply to standardized education, not to streaming students
into different programs such as specialprograms for gifted students.)

~~~
shasta
I don't know why you have to postulate "magic". I didn't say that a program
that helps each kid progress as quickly as possible wouldn't reduce the
variance. It probably would, as the kids on the bottom have more room for
improvement. But I would still expect a large amount of variance. Actually, I
just noticed that the "after" graph is clustered near 100%, so it's possible
that there still is a large variance, but it's just not measured by the tests
that were administered. These are probably standardized test scores and don't
really address higher end performance.

So I guess my points are:

1\. I'm doubtful that the "paint by numbers" approach is what the young
Picasso needs.

2\. Maybe I'm wrong about (1), but we certainly shouldn't subject the advanced
students to this method until higher end performance has been measured.

3\. I find the attitude of the author, who used the phrase "even things out"
as if it was a good thing, concerning. Evening things out should be an
explicit non-goal of education.

~~~
raganwald
Expressions like "magic" are simply ways of making it clear that the notion of
what is or isn't possible is orthogonal to the question of what this specific
program does or doesn't do. But as far as your points go, the second one is
the most interesting. yes, we should have a good look at what happens when
"advanced" students are exposed to any method.

Some advanced students might even get _even better_ and increase the variance!
I can't comment on this program, I know nothing about tutoring. But I know
that I personally like breaking things down into little steps, and I think I
would have enjoyed a program running on these lines if I was allowed to move
at my own pace.

------
fname
The money quote:

 _Teachers tell me that when they begin using Jump they are surprised to
discover that what they were teaching as one step may contain as many as seven
micro steps_

It's about finding out how to simplify what some may see as the easiest step.
Think about the lowest common denominator and build from there.

------
scott_s
I took a similar approach when I taught programming to non-CS majors last
summer. The difficulty is that it required significant one-on-one time with
each student that needed help, and it required that they came to me. Luckily,
my students weren't shy and I made myself readably available to them almost
every day. But college students are generally better motivated to seek out
help than high school, middle school and elementary students.

What took so long with each student was, based on their original question,
systematically figuring out what their _real_ misunderstanding was. This could
be very time consuming, and took enormous patience on my part. It required
much back and forth with the student so I could build a mental model of
_their_ mental model. Then I had to figure out how to build a bridge from
their mental model to the correct one. That also was time consuming, because I
sometimes had to build several bridges before finding one that clicked with
the student.

Usually helping the Nth student on a project was quicker than helping the
first because by that time I recognized what were common problems and built a
bag of tricks to explain them. For example, it took me a surprisingly long
time to realize the students had no practice running through an algorithm on
paper, then translating that to code. (This was partially because the
algorithm was so simple I hadn't even realized it was an algorithm.) So one
technique I used was to, on paper, set up what was needed for the algorithm to
work (list of numbers, table of results, etc.), then make them tell me what to
do to get the correct result. I was acting as, basically, an intelligent
computer that could be programmed in English. After doing this a few times,
the students could finally "see" the algorithm, but it took a lot of time and
effort with each student.

------
asolove
Marvelous. I am glad to see someone emphasizing structure and practice in
early math curricula. While it is possible to overemphasize memorization and
rote practice, the pendulum has certainly swung too far the other direction as
a reaction against the Victorian knuckle-rapping methods.

~~~
patrickyeon
> While it is possible to overemphasize memorization and rote practice, the
> pendulum has certainly swung too far the other direction

Throughout my grade school and high school years (born mid-80s, Canadian),
near everything was memorization. Memorize multiplication tables instead of
learn to do arbitrary multiplications quickly. Memorize your table of elements
instead of emphasis on what it means. Memorize how to find the roots of a
quadratic function, the rules for arithmetic with fractions, physics
equations... (I'm pretty bad at memorizing, so this doubly infuriated me
because I had to look for the underlying relationships to do well). Even after
my province introduced a new curriculum which was supposed to stop all that
and focus away from memorization.

Anecdotally, this is only slowly changing if at all, as new teachers come in
and old ones who won't give up on rote memorization retire. I know this is all
just anecdotal, but in my (ongoing) university education, I still see a lot of
students put their emphasis on memorization, which suggests to me that they
were taught that is what it means to 'learn.'

~~~
hugh3
_Memorize multiplication tables instead of learn to do arbitrary
multiplications quickly._

How can you do arbitrary multiplications quickly without memorizing your times
table? You can't quickly compute 47*36 without being able to reel off "six
times seven is forty-two, seven times three is twenty-one, four times six is
twenty-four, four times three is twelve", can you?

I hated learning my times table (in fact I hated mathematics when mathematics
was just arithmetic) but a certain amount of memorization is necessary and
unavoidable if you're going to be able to do basic mental arithmetic, which
remains an important life skill even if you do have a calculator.

~~~
marklubi
> You can't quickly compute 47*36 without being able to reel off "six times
> seven is forty-two, seven times three is twenty-one, four times six is
> twenty-four, four times three is twelve", can you?

My mental process for solving your example equation was to reduce it to
50x36-3x36 which is IMHO a much simpler process.

I don't know where I learned that process (probably from my dad), but
simplifying the values to easily calculated approximate values and then
adjusting that value for more precision seems to make much more sense than
what was taught to me in school. It also caused a lot of frustration between
myself and my teachers who demanded that all of the micro steps be written out
instead of making what seemed to me as logical jumps.

Edit: formatting

------
ntoshev
I was also wondering recently why illiteracy carries social stigma, while
innumeracy does not. The article asserts everyone just considers math ability
to be innate, but I'm not sure this is a good explanation.

Our daily life seems to rely on literacy much more than it relies on numeracy.
You can only know how to add and subtract (mostly money) and you will still
function all right in your daily life and a great variety of jobs. Not so much
if you have difficulty reading. This seems to be slowly changing in the
future, but if you consider the learning curve, mathematics has a much steeper
one compared to reading and writing well.

Does anyone have a better explanation?

------
tokenadult
International comparisons such as the TIMSS and the PISA studies have already
been showing for more than a decade that school systems in North America (the
United States and Canada) have been underperforming and failing to serve most
students well. Chapter 1: "International Student Achievement in Mathematics"
from the TIMSS 2007 study of mathematics achievement in many different
countries includes, in Exhibit 1.1 (pages 34 and 35)

<http://timss.bc.edu/PDF/t03_download/T03_M_Chap1.pdf>

a chart of mathematics achievement levels in various countries. Although the
United States is above the international average score among the countries
surveyed, as we would expect from the level of economic development in the
United States, the United States is well below the top country listed, which
is Singapore. An average United States student is at the bottom quartile level
for Singapore, or from another point of view, a top quartile student in the
United States is only at the level of an average student in Singapore.

That the UPPER range of students in the United States is poorly served by
current school mathematics instruction in the United States is shown by a
careful analysis of the PISA studies of developed countries around the world.
PISA's own analysis refers to specific instructional practices in different
countries and other differences in country conditions that make a difference
in educational outcomes.

[http://www.oecd.org/document/2/0,3343,en_32252351_32236191_3...](http://www.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html)

Some bloggers in the United States persist in blaming these outcomes on the
ethnic diversity of the United States (ignoring the ethnic diversity of
Singapore and other countries that outperform the United States). Eric A.
Hanushek, Paul E. Peterson, and Ludger Woessmann point out in their analysis
of the PISA data, "U. S. Math Performance in Global Perspective: How well does
each state do at producing high-achieving students?"

[http://www.oecd.org/document/2/0,3343,en_32252351_32236191_3...](http://www.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html)

that the real problem in United States mathematics education is leaving behind
too many of the high-ability students, of whatever ethnicity, compared to many
other countries. A specific response about what is wrong with mathematics
teaching in United States classrooms comes from Patricia Clark Kenschaft in
the Notices of the American Mathematical Society volume 52, number 2 (February
2005).

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

Most elementary school teachers in the United States, repeated studies of the
issue have shown,

<http://www.nctq.org/resources/math/>

have poor mathematics preparation in their own higher education and little
mathematics knowledge when they enter the classroom. They then are directed by
their school districts to use textbooks that are ineffective for primary
mathematics instruction, so it is no wonder that most pupils in the United
States (and the same applies to Canada) finish primary schooling with poor
preparation for higher mathematics study. I speak and read Chinese and have
lived in various parts of the Chinese-speaking world. I have Chinese-language
textbooks of mathematics at home from more than one country. I am confident
that young people of any ethnicity in North America can learn math well if
they are taught with materials like those, because I am a math teacher by
occupation and my classes include a very ethnically diverse group of students,
who thrive in the classes and far exceed the meager expectations of United
States classrooms.

~~~
yummyfajitas
_Some bloggers in the United States persist in blaming these outcomes on the
ethnic diversity of the United States (ignoring the ethnic diversity of
Singapore and other countries that outperform the United States)._

Which bloggers?

Incidentally, I think "ethnic diversity" is a red herring. Any bloggers who
talk about "ethnic diversity" are merely trying to couch their conversation in
PC language to avoid ad-hominem accusations of racism.

The issue is not "ethnic diversity". The issue is that certain ethnicities
underperform. Specifically, African Americans and Hispanics (30% of the US, 0%
of Singapore) underperform. Nonhispanic whites (about 65% of the US and close
to 0% of Singapore) achieve mid-level performance. Asians (about 4% of the US
and close to 100% of Singapore) overperform.

These effects are HUGE in comparison to inter-country effects. Asian Americans
score about 10 pts below Singapore. The EU15 scores about 50 pts below Asian
Americans, and a couple of points above All Americans.

[http://super-economy.blogspot.com/2011/01/how-well-do-
above-...](http://super-economy.blogspot.com/2011/01/how-well-do-above-
average-american.html)

[http://super-economy.blogspot.com/2010/12/amazing-truth-
abou...](http://super-economy.blogspot.com/2010/12/amazing-truth-about-pisa-
scores-usa.html)

It is simply incorrect to pretend that ethnic gaps in education do not exist,
or to pretend that they do not explain a large portion of the gap between the
US and the rest of the [edit: first] world.

Also, it would be helpful if you were a little more specific in your
citations. Citing a gigantic PISA reports is much less useful than citing a
specific table or figure. I'd love to learn more, but I don't have time to
read the whole thing, and I have no idea which parts you are referring to.

[edit: wanted to clarify that I don't think the gaps between the US and poor
locations, e.g., rural inland China or India, are primarily due to ethnicity.
But gaps between wealthy first world countries do seem well explained by such
factors.]

~~~
kenjackson
Unfortunately, you can't disaggregate in the way suggested above. Blacks and
Hispanics are also far less wealthy than whites in this country (which has a
larger impact than even income). Once you control for wealth a lot of the gap
disappears.

Ethnic gaps may exist, but they probably are symptomatic of gaps in wealth
more than anything else. But who konws. There are so many other factors at
play for people of darker skin in general in society that I'd hesitate to
subscribe to one theory.

With that said, it does point to the fact that US public schools are probably
better than most believe them to be.

~~~
yummyfajitas
_Once you control for wealth a lot of the gap disappears._

I haven't looked at it extensively, but I don't believe this is true.

This study shows that SES (Socio Economic Status) doesn't explain much of the
gap.

[http://www.umich.edu/~rdytolrn/pathwaysconference/presentati...](http://www.umich.edu/~rdytolrn/pathwaysconference/presentations/craig.pdf)

This study (sorry, can't find a non-paywall version) shows that even holding
income constant, blacks do worse than whites at the college level. In
particular, low income whites perform as well as high income blacks.

<http://www.jstor.org/pss/2963200>

Also, even if you could explain the Black/white income gap via wealth, racism
or other environmental factors, how would you explain the White/Asian gap?

<http://en.wikipedia.org/wiki/File:Personal_income_race.png>

And how would you explain that in spite of wildly different environmental
factors (compare Singapore to Texas), there is a relatively small gap between
Asian Americans and Asians? And similarly, there is only a small gap between
white Americans and white Europeans?

(By the way, I'm not trying to claim there are no other factors at play. I'm
just pointing out that race does seem to be a biggie.)

~~~
kenjackson
The study you note focuses on income. There's another branch of research that
focuses on wealth that shows that wealth is more important than income when it
comes to things like college attendance.

See: <http://collegepuzzle.stanford.edu/?p=1590>

"“The Differential Impact of Wealth vs. Income in the College-Going Process”
finds that wealth and income affect the college choice process differently,
with wealth consistently being more significant in predicting who enrolls in
college, and the type of college they attend – even after controlling for
student differences in academic achievement, habitus, social capital, and
cultural capital."

More research clearly needs to be done here, but from what I've seen its clear
that wealth looks to dominate income.

And don't get me wrong, I'm not saying there aren't other differences. In fact
I suspect there are societal and cultural differences that will manifest in
one way or another. If you spend a lot of time in Asian households in the US,
you'll see they often resemble, culturally, canonical Asian homes more than
"US" homes.

And there's data about the success of African immigrants to the US. Who often
don't come wealthy (although some are), obviously are Black, but still do
well. So there's clearly a culture issue here too, but I don't think we can
really tease this apart without more data.

~~~
yummyfajitas
You could be right about wealth, though I'd be surprised - it would be really
weird if wealth effects in the US caused Asian Americans to perform almost
exactly as well in school as Asians, and white Americans to perform almost
exactly as well as white Europeans.

But this doesn't change my original point - US schools do only slightly worse
than Singaporean schools at educating people with an Asian-style home
environment, Asian genes, or whatever else being Asian is a proxy for. And
cultural factors in black America may cause black Americans to perform worse
than whites and African immigrants. Whatever the underlying factor is, it has
nothing to do with the school system. That's the only point I'm trying to
make.

~~~
lasonrisa
_Whatever the underlying factor is, it has nothing to do with the school
system._

This cannot possibly be true. School system are artificial constructions
created by men to serve men. They are not given laws of nature. The only thing
we can tell from the above is that the current school system does not serve
well black Americans. School system should adapt to whatever the "cultural
factors" of the populace they serve.

Whether this should be through prolonging schooling hours; providing stronger
and more persistent emphasis in personal hard work; correcting a harmful self-
image; getting rid of institutional racism; using different instruction
techniques; or any other method that may addressed the "underlying factor" can
only be discovered through research, experimentation and a willingness to
change and adapt.

It is easy to find examples of a schooling that adapts to the unique
circumstances of some children, for example: intense language training for
children from immigrant families, schools at hospitals or schools that follow
a circus around the world
(<http://www.guardian.co.uk/education/2007/jan/26/schools.uk>).

~~~
yummyfajitas
It may well be the case that one or both of US and Singaporean schools are
worse at educating blacks/whites/hispanics than hypothetical school system you
can dream up.

It is irrelevant when considering the question of whether Singaporean schools
do a better job than US schools. For all we know, Singaporean schools might be
_worse_ than US schools - i.e., they could education Asians equally well, but
blacks/whites/hispanics vastly worse [1]. But it wouldn't matter since
_Singapore doesn't have any blacks/whites/hispanics_.

[1] You actually see this effect (Simpson's paradox) when comparing US states.
Texas is better at educating blacks, whites and hispanics than Wisconsin, but
appears worse when averaging over the entire population.

[http://iowahawk.typepad.com/iowahawk/2011/03/longhorns-17-ba...](http://iowahawk.typepad.com/iowahawk/2011/03/longhorns-17-badgers-1.html)

------
keeperofdakeys
Relevant, <http://www.maa.org/devlin/LockhartsLament.pdf>

------
gersh
This goes against math tradition where classical math texts say things like
"this is obvious and left as an exercise to the reader". Implicitly, I think
many mathematicians believe students should be made to struggle for their own
good.

The problem is that this approach requires patience, persistence and hard
work. Do this help students in the long-term? If you learn math without
struggling, will you learn to think in the same way? Is this teaching to the
test over teaching you how to think?

~~~
icegreentea
You really don't see that type of text appear until university these days.
Most grade school, and highschool math texts that I have seen attempt to show
you where most things come from, and attempt to show the steps. Now, this
doesn't mean that they do so in an effective way (there's often that random
ass step which doesn't make any sense at all), but they do try.

The struggle mostly comes from situations where a) people just can't follow
the explanations at all, since the reasoning is not explained well enough, or
there are jumps too large, or b) people who have trouble adapting past
strategies and concepts to more novel situations. For example, they might know
how to solve some class A of word problems, but once you switch the word
problems to solving the equation in the other direction, they just get lost.

I think most of the 'left as an exercise for the reader' stuff appears once
you get to a high enough level that you a) assume that the reader is
proficient enough and b) that the reader actually cares enough about the
subject to be able to do so. For example, my friends taking a bunch of pure
math subjects get that crap all the time. "So we've proved X theorem for Y
case, Z will be left as an exercise", and they eat it up (partly because it
really is an exercise).

Now, I still think a lot of the time, it's inappropriate, and just used cause
the writer is lazy, or has used up too much space on diagrams (especially in
physics textbooks...).

~~~
gersh
In high level texts, the "obvious" problems often aren't easy. I think there
is a strong element of mathematician arrogance. If you can't figure it out,
you don't deserve to know. This may go all the way back to Pythagorus, who ran
a secretive cult.

There are actual exercises, where solving it is just a matter of applying some
concept. However, a lot of math books have things where you have beat your
head against the wall to figure them out. Although, the internet has made it
easier to lookup, nowadays.

School textbooks often try to make things simpler. However, when these texts
are written based off of older texts, which skip steps, they may inherit the
style of the older works.

------
RBr
This article is very interesting. However, while reading, it sounded like a
late night infomercial. There may be real value in this style of teaching
Math, but as my first introduction, this Times article feels like the
sponsored hooks used for products such as those baby reading flash cards and
acne medication.

I hope that the Jump system is real and that it solves the problems outlined
in the article.

------
rubergly
The philosophy of Jump reminded me a lot of the philosophy behind Khan
Academy, but with that philosophy used to adapt the curriculum instead of to
create a new way of digesting knowledge and exercising it. While they're
similar, the biggest advantage I see for Jump is that a school adapting their
curriculum to Jump would be much easier than adapting their curriculum to Khan
Academy.

------
crasshopper
Here's a peep at the JUMP curriculum:
[http://jumpmath.org/TM%20for%20Introductory%20Unit%20Using%2...](http://jumpmath.org/TM%20for%20Introductory%20Unit%20Using%20Fractions.pdf)

 _Emphasize the positive._

If a child gets three out of four questions wrong, I will mark the question
that is correct first and praise them for getting the correct answer. Then I
will say, “I think you didn’t understand something with these other questions”
or “You may have been going too fast,” and then I will point out their mistake
– or ask them to find it themselves! I’ve found that if I start by mentioning
the mistakes, a weaker student will sometimes simply give up or stop
listening.

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ph0rque
Breaking down the problem into steps so small that each is trivial to do: is
there anything this technique _can't_ solve?

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scotty79
Can you break into trivial steps the problem of how to break problems into
trivial steps?

~~~
ph0rque
it's easy if you understand recursion...

~~~
hugh3
What's the first step to understanding recursion?

~~~
discreteevent
The Little Schemer. Plus it must be one of the canonical examples of how to
teach by breaking things into the smallest steps you can imagine.

