
Who or What Broke My Kids? - ColinWright
http://powersfulmath.wordpress.com/2014/04/30/who-or-what-broke-my-kids/
======
overgard
Nobody broke the kids, they were doing exactly what they were taught to do.

I'm not a fan of the education system in the least bit, but something about
this rubbed me the wrong way. She's probably exaggerating when she said she
had a "meltdown", but all the same, what are you supposed to think as a kid
when your teacher has a tantrum in front of you because you're doing what you
were trained to do? To the kids she's just part of the institution, and in
this particular case the institution is sending them very conflicted messages.
"Think for yourself! As long as you're thinking what you're supposed to be
thinking."

In any case, I think the real problem with the education system is it's
constantly trying to bullshit kids into "learning" things they really don't
need to learn. (And they don't _really_ learn it. Quick sanity test: what do
you remember about high school trigonometry? But you "learned" it right? Or
did you, really?)

Most kids aren't stupid. They're naive, but they're not stupid. They can tell
when their time is being wasted. They're out to give the right answer and not
think about it because when you're literally trapped in a system like that,
the most logical thing to do is minimize effort and not make waves until you
can make your way out. So that's what kids do, and then pompous teachers are
surprised that kids aren't "passionate" about these things. Well no kidding
they're not! Why should they be?

~~~
3pt14159
I remember everything about highschool trig. And chemistry. And physics. And
certainly the rest of math. We might waste the time of the kids that are going
to end up flipping burgers, but we don't waste the time of the brighter kids.
If anything we move too slow.

~~~
_pmf_
> I remember everything about highschool trig. And chemistry. And physics. And
> certainly the rest of math.

Report back when you're above 22.

~~~
bsder
Couple of decades.

Anyhow, the important part about math isn't the _memorization_. It's the
_process_.

Geometry and trigonometry aren't that useful _by themselves_. Even Algebra is
marginal. Probability and statistics are _way_ more useful. However, all of it
comes together when you hit calculus which is _VERY_ useful.

Can I integrate from memory anymore? No. Can I solve a differential equation
from memory? No.

I'm going to use Mathematica if it's a one off. I _will_ bone up on the
process if I'm hitting it too often.

However, I'm always doing Bayesian probability estimates for all manner of
things in programming, business, manufacturing, etc.

------
falcolas
I'm 36, and math in my youth was always about getting the right answer, and
proving how I got that answer, even up through college calculus and discrete
mathematics. Heck, that's one of the things I really enjoyed about math, when
compared to, say, English.

So, if math isn't about getting the right answer, it's been broken a long
time.

Even today, math as I've explored it (which admittedly hasn't been much) has
right and wrong answers. Sure, there's a bit of fuzz in the answers now, but a
probabilistic model which can't determine ham from spam some majority of the
time isn't kinda right, or even on the spectrum of right. It's just wrong, and
needs to be fixed.

~~~
Spearchucker
" _...and proving how I got that answer..._ "

That's the part that counts, and is largely missing today.

~~~
goodcanadian
I am not so sure you are talking about about the same thing as the article.
Usually, when "proving your answer," I found there was still an expected
procedure for you to follow, and you would lose points for deviating from that
procedure even if what you did was mathematically correct. What the article
talks about is learning how to reason about a problem even if your reasoning
is questionable and even if you don't arrive at an answer.

~~~
sanxiyn
That is unbelievably terrible. You would lose points for a mathematically
sound justification?!?!

For example, when solving quadratic equations, I often used trial
substitutions and the fact quadratic equations have at most two solutions to
justify answers, instead of using quadratic formula. Why wouldn't this get the
full mark? (I would actually argue this is a better procedure if you are
looking for integer-only solutions, for example.)

~~~
fredophile
What do you mean by "trial solutions"? Were you just substituting numbers for
x and seeing if they worked out? If that is what you mean I can't see why it
would be a better procedure than using the quadratic formula.

~~~
sanxiyn
Yes, I meant substituting numbers for x and seeing if they worked out. To get
integer solutions, it's enough to try divisors of the constant term. Try it
yourself. It's faster.

~~~
fredophile
Yes, it is faster for small integer values. Most numbers aren't small integer
values. Test questions are usually designed to give nice clean answers but the
real world isn't like that. They were trying to test your ability to solve
quadratic equations in the general case. Your approach doesn't demonstrate
that ability.

------
wirrbel
I firmly believe there should be two math subjects taught at school. One being
"Calculating" where students do the boring calculations and all the stuff,
calculus, algebra, etc. and another one where the interesting and challening
part of math is taught in an accessible and fun way.

There are a lot of topics in math and related to math that could be explored
in that second, new subject, probabilities, paradoxa, symmetries, basic set
theory, even concepts of linear algebra/groups, etc. that are normally taught
in university can be broken down to really great middle-school or high-school
lessons. The only problem with that in normal math classes is, that the angry
parent will quickly complain when that is talked about in a normal math class
because they think that their kids should be bored to death by calculations
that their smartphone does better already.

I was recently talking to a retired high school teacher who had taught
language classes (he was really not a maths guy). We started talking about
password security and piece by piece we discussed how easy those passwords
could be guessed and in the end he had a good understanding of the concept of
probability/information/entropy. This really made me think, if a really non-
maths person can enjoy and understand a topic that a lot of students who have
gotten STEM degrees directly refuse (something like "I do not understand what
entropy is, that is something about the my room being in order or not but I
have given up .....")

To repeat myself: There are great concepts and modern curricula available,
sometimes for decades. However mathematicians, math teachers and parents are
just not supportive in chaning anything substantially.

~~~
mathetic
> One being "Calculating" where students do the boring calculations and all
> the stuff, calculus, algebra, etc. and another one where the interesting and
> challening part of math is taught in an accessible and fun way.

Wow. That can start a flame war. I am a huge fan of set theory and probability
but since when calculus and algebra are boring? Sure if all you do is evaluate
bunch of integrals, it is but what if you are learning about Fourier
Series/Transform? It requires vector algebra heavily as well as calculus and I
think it is just as stimulating as anything else. Sure it involves
"calculating" but it is not a bad thing, calculating is at the heart of the
subject.

~~~
wirrbel
My point - and I could have expressed this more clearly, granted - was that
you have one class for doing the numbers (the boring stuff), and one for doing
more theory and explorative stuff. I did not want to imply that calculus and
linear algebra are boring by themselves. In practice, linear algebra is not at
all taught in school, people are just forced to train a few landmark
calculations (gauss algorithm, etc.) as fast as they can.

What I would very much like to see is that people get a good unterstanding of
essential concepts, kernel, image, projections, etc.

As you mention, calculating is important, and thats why I propose two classes,
one being about calculating, one being about the theory, in practice with the
focus on _results_ in school, the conceptual part is left behind.

~~~
mathetic
In that case fair enough. The problem would be that understanding higher level
concepts one way or another requires the underlying mathematical rigour and
mental agility to do the "boring" stuff. So it would not be dividing existing
curriculum into to but doubling the amount of lessons. I would have loved that
when I was in school but I know that that course despite all of its beauty
would be extremely boring to most other people.

~~~
wirrbel
I think there are many topics in math that are left out because students have
to train a few numerical algorithms. For example I was not at all exposed to
logic in my math classes (if I was, it was very informal and only on a side
note). However people would greatly benefit from this, probably even more than
sucessfully calculating the shortest distanc between a line and a given point.

The additional class would have room for some algorithms as well, some graph
algorithms maybe (DFS, Djikstra), maybe also Binary Search, etc. A lot of
those concepts do not require extreme amounts of calculation skills but they
train the mathematical mind.

We expose students in calsses on literature, geography, biology with various
concepts, yet we limit contact to math on a narrow subset (roundabout
requirements of an engineering college class for college-preparing courses).
If you do not end up in STEM or CS, you will probably only use the rule-of-
three after having left high school for 5 years.

Just discussing questions like: "You have a scale and 12 billiard balls. One
is lighter than the others. How many times do you have to weigh balls until
you find the lighter ball?" would be inherently beneficial to students.
Students afraid of equations can work on this as well and even have fun.

------
jameshart
My son's elementary school homework this week included, after a series of
arithmetic exercises involving adding pairs of even numbers, this sequence of
questions:

What happens when you add together two even numbers? Is that true for all even
numbers? Why?

Part of me was overjoyed to see an actual mathematical question being posed,
but... what exactly was my second grader son expected to write under the 'why'
part of the worksheet? There was a couple of inches of blank space left for
him to fill in his answer. Is the teacher going to read his answer there and
mark is right or wrong and hand back the homework? I don't know if I have much
hope that this will be used by the teacher to gauge each child's understanding
in order to facilitate further classroom discussion. Have the kids been taught
a particular explanation in class which the homework is expecting them to
reproduce? What is the teacher's goal in sending home this question as an
exercise?

The fact that two even numbers always add up to an even number, as do two odd
numbers, is an interesting insight into the shape and flavor of numbers. It
gives you a practical tool - you can use this knowledge to parity check mental
calculations and intuitively reject wrong answers - as well as an insight into
the idea that esoteric mathematical properties numbers have can have
interesting consequences when they interact, which is kind of the essence of a
lot of mathematical thinking.

Is that always true? Why?

are probably the most important questions in mathematics. But... I have no
idea how you teach that, except by having a one one one conversation with each
student about it. I don't think you can get much value out of asking them on a
single page worksheet and sending it home as homework, though.

~~~
j2kun
I really hope your child's teacher doesn't just mark it right or wrong, but
provides a counterargument for a flawed answer.

The question of "how do you teach one to answer 'why?'" is a difficult one,
but here's what I understand is the established way of doing it. You give the
student a problem, they think about it for a long time, and give their answers
as to why. You discuss their answers with them, and then after they revise
their answers, you present an elegant and correct answer. Then you repeat.

It's interesting, for example, that nobody asks how to teach one to answer
"why" when you're, say, analyzing the motives of a character in some piece of
literature. What do they do? They discuss, then they revise, and at some point
the teacher presents either a good piece of student work or some other
established ideas. Is mathematics really that different just because in the
end you can tell with certainty that an answer is correct or incorrect?
English teachers have one on one conversations with their students about their
work, and they make students have discussions with each other about their
thoughts on a daily basis. Why can't math teachers do the same?

I think part of the reason math is plagued by such questions is that most
people don't realize that a proof (aka the answer to "why?") is not _just_
correct or incorrect, but has aesthetic properties. Nevertheless, it's clear
that mathematics is set apart from this on day one. Just compare any high
school math syllabus to a syllabus in another subject [1]

[1]" [http://j2kun.svbtle.com/what-would-math-class-look-like-
if-i...](http://j2kun.svbtle.com/what-would-math-class-look-like-if-it-were-a-
fine-art)

~~~
NoMoreNicksLeft
> I really hope your child's teacher doesn't just mark it right or wrong, but
> provides a counterargument for a flawed answer.

His son's teacher isn't going to do this. First, she didn't design the
question... even if some public school teachers have the capacity to do this,
they don't have the time or resources to design curriculums. Second, if she
didn't design it, she likely does not have the wit to be able to argue about
it without doing research. Third, public schools aren't places where they make
time for each student trying to inspire genius, she really is just checking
boxes down a list all semester. Fourth, doing so won't improve his scores on a
standardized test. Finally, there was only two inches of blank space for the
child to write an answer, she definitely isn't going to write him a thesis on
the margins of the paper.

Public education is blind idiot god, handing down edicts from on high, some
dumb and some smart, some sane and others insane, some incomprensible and some
not. And often enough, all of these at the same time. The bell will ring, it
will be time for a new lesson, and in that one he will be taught to never
leave any multiple choice question unanswered because "they unanswered
questions are scored just the same as wrong ones, you might as well guess".

~~~
j2kun
I gave a two-inch "proof" elsewhere in this comment thread. You don't need a
thesis to reason about even numbers.

~~~
NoMoreNicksLeft
You're not a school teacher, are you?

~~~
j2kun
Your comments reek of pessimism and resignation. How can you hope to educate
anyone with that attitude?

~~~
NoMoreNicksLeft
I have two young children, they won't attend a public school, or a private one
for that matter.

I'm very optimistic about their education. However, there is no hope of
educating masses of children in large assembly-line public school systems,
because those systems aren't designed to educate.

------
netcan
Math gets a lot of attention and therefore a lot of criticism from people who
think math education is substandard _and_ from people who think it is
overemphasised to the detriment of creative thinking and such.

I think that overall, our math education is surprisingly good, possibly better
than any other subject. Especially if you consider the level of proficiency a
high school graduate at below average achievement has. Reading, writing and
arithmetic. Those are still the things our education system is best at.

I don't mean that the average person will use trigonometry confidently, but
they will use basic algebra. That in itself is an achievement. The same cannot
be said of foreign language education, most science classes or anything else
that requires cumulative learning. Most people that learn a foreign language
(apart from english) for years in school do not come out with even a basic
ability to communicate in it.

I think part of the reason is the very long tradition of teaching the subject.
But I think another reason we're good at teaching math is the sort of
fundamental property of the subject. It's unforgivingly right or wrong, which
helps cut through the cynicism of a youth that doesn't want to be in that
class. The problem with the "holistic" (scare quote intended) approach to
subjects is that it opens the door to nonsense of the bikesheding type. It's
hard to distinguish nonsensical but enthusiastic from genuinely thoughtful.

The hard right-wrong distinction in maths gives kids a feedback loop of
positive and negative reenforcement. At the end of the year it's glaringly
obvious if the student now has ability that he/she didn't before.

I don't disagree that the ability to reason is important. But, I don't think
teaching "hard" maths is debilitating to developing that ability.

I'm not sure exactly where I'm going with this. Maths gets a bad rap. Teaching
kids "real world reasoning" within the fake world of school is hard. No one
broke the kids. They are just reacting to you changing the rules.

~~~
j2kun
> It's hard to distinguish nonsensical but enthusiastic from genuinely
> thoughtful.

English teachers successfully do this every day, as do history teachers and
teachers in most other subjects. Why can't we let mathematics teachers do this
when their students provide proofs via creative thinking? Certainly anyone
with sufficient training in mathematics can do this.

But overall your comment seems to fall short of reality. Math education in the
US produces students that seriously underperform compared both to other
countries and in light of the skills and knowledge required for technical
careers. Contrary to your belief, high school students still generally can't
do algebra. In my experience teaching the subject they don't learn algebra
until they take pass calculus, and even then it's _because_ it's a cumulative
learning process, not in spite of it. Even if we teach reading, writing, and
arithmetic well, that's just not enough.

~~~
netcan
What I'm claiming is that English teachers both succeed and fail at this every
day. Of course, you can disagree. No one wants to tell kids their idea is
wrong when the subject matter is somewhat subjective (like English or History)
if they appear to be trying. Math has built in quality control because of the
objectivity. It creates a constructive feedback loop.

I'm not from the US, I meant "we" as a whole. Obviously some places are better
than others, but I think math education in 7-18 schools is better than
physics, biology, history and most subjects. Obviously it's hard to compare.

Your point about algebra-calculus an interesting one. I've heard several time
an interesting rule of thumb that to be a teacher you need to have studied
about five years beyond the level you teach. I suspect that to be able to
confidently use knowledge like maths knowledge, you need to have studied x
number of years beyond. So, a person who never studied math past year 12, can
confidently use maths learned around year 9-10 or somesuch.

In any case, I'm not claiming that we reading, writing, and arithmetic are all
we should teach, just that we're relatively good at teaching these.

------
abecedarius
[http://www.psychologytoday.com/blog/freedom-
learn/201003/whe...](http://www.psychologytoday.com/blog/freedom-
learn/201003/when-less-is-more-the-case-teaching-less-math-in-school)

"Benezet showed that kids who received just one year of arithmetic, in sixth
grade, performed at least as well on standard calculations and much better on
story problems than kids who had received several years of arithmetic
training."

I've wondered if it was a big waste of time to teach arithmetic in elementary
school, if kids that age mostly just aren't ready and they'd catch up fine if
you waited. This experiment seems to say that it's worse: it inculcates cargo-
culting operations until you get the 'right answer'.

(I haven't read the original paper; the reference came from
[http://slatestarcodex.com/2014/05/23/ssc-gives-a-
graduation-...](http://slatestarcodex.com/2014/05/23/ssc-gives-a-graduation-
speech/))

------
monochr
After reading most of the replies in this thread everyone who thinks that
maths is about answers needs to read A Mathematicians Apology:
[http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%2...](http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%20Apology.pdf)

Mathematics has nothing to do with answers. It has everything to do with the
most sublime and subjective beauty there is. The fact that this purely
aesthetic system has says anything about reality should be more shocking than
if we lived in a world where playing Beethoven's 5th symphony made it rain
food from the sky:

[http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html](http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html)

------
delluminatus
_When did we brainwash kids into thinking that math was about getting an
answer?_

That's what _school_ is about. It's not math in particular. Doing well in the
US mandatory education system is the art of finding the right answer.

~~~
pdonis
Where finding the "right answer" is really more like guessing the teacher's
password:

[http://lesswrong.com/lw/iq/guessing_the_teachers_password/](http://lesswrong.com/lw/iq/guessing_the_teachers_password/)

------
euank
He is talking about the same problem that is discussed in Lockhart's Lament
[0]. If you've never read this piece of writing, I encourage you to do so now
as it presents our math education system in a wonderful and critical light. I
can't speak as elegantly as Lockhart, so I won't damage the piece by
summarizing it or discussing it in the post; rather, I'll leave you to read it
and draw your own opinions.

[0]:
[http://www.maa.org/external_archive/devlin/LockhartsLament.p...](http://www.maa.org/external_archive/devlin/LockhartsLament.pdf)

------
edwhitesell
IMO, this is one teacher who gets it. There is a lot of math that is entirely
about answers that are right or wrong. Most of the time that includes how you
arrived there. Given the references to MAP and Common Core in the blog, I'm
sure there's a lot of that in the classrom.

However, there is so much of the K-12 education that arguably should be about
exposure. Discussion about probabilities of pictures and phrases on a number
line is a great example. Have kids make their case among peers regarding why
'It will rain tomorrow' is .9 in one group, but .75 in another.

For most, you don't get to pick your desired area of study/career until
College. So why not use K-12 to expose kids to all sorts of different things
without standardized tests as the only goal?

------
javajosh
When Feynman was a boy he'd _always_ relate the math problem to a real world
puzzle. With trig problems, he might imagine being given a riddle involving a
flag post and rope and calculating distance. One got the sense that he was, in
effect, _using math to prepare himself to solve future, real-life riddles_. Or
rather, math was giving him the tools to (correctly) answer questions he
otherwise could not have answered - and Feynman's great preoccupation was his
obsession with solving puzzles of all kinds, and he would enthusiastically
(perhaps greedily?) grab at all the tools he could. (Physics leans heavily on
trig almost constantly, so a total mastery of it would be quite handy for a
budding young physicist).

It's also interesting to me that he was, at an early age, concerned with the
_usability_ of math, and was unafraid to create his own notation that was more
comfortable for him (he liked the square root symbol and created analogs for
sin(), cos(), etc.)

Frankly, I think this is a fantastic way to approach learning. After all, it
feels good to solve puzzles; if you solve enough of them, the way they fall
together, the way they relate to each other (sometimes in unexpected ways)
become useful insights in themselves. With a large, solid core of puzzle
mastery, you might even be able to turn your attention to the more difficult
puzzles of "how to teach". (Of course, the greatest thing you can teach is the
love of solving puzzles!)

I can't hold back anymore: what a foolish teacher! To get emotional over kids
asking if their answers are right! In general, the yearning to be correct in
one's calculations (and ordering cards is a calculation) is a good instinct,
not to be beaten out of them.

------
oe
I'd say it's the parents. Listening to parents of elementary school kids they
seem to get confused or angry if there isn't one single right answer to an
assignment. For example drawing shapes one line at a time and listing the
lines as "commands" to a computer. There might be multiple ways to form a
shape.

No use trying to argue that it might be valuable to understand how to think
with algorithms, even though it might produce several different answers.

~~~
goldfeld
That just moves the goal post, doesn't it? Such a parent is as much a broken
kid as their offspring in this respect.

~~~
oe
The parents might be afraid that their child could "fail" such an assignment,
even if it's elementary school and the children aren't graded.

------
tokenadult
I've lost count of the number of times that Colin has posted an interesting
article on mathematics education on Hacker News over a weekend. This
submission here is particularly good. The problem indicated is quite stark,
and very commonplace. Most pupils in a mathematics lesson in elementary school
swiftly learn that "getting the right answer" is the point, and some check out
and soon begin to doubt their own ability to REASON to the right answer.

The author of the submitted article writes, about a seventh grade class, "The
basic premise of the activity is that students must sort cards including
probability statements, terms such as unlikely and probable, pictorial
representations, and fraction, decimal, and percent probabilities and place
them on a number line based on their theoretical probability." As the author
makes clear, the particular lesson arises from the new Common Core State
Standards in mathematics,[1] which are only recently being implemented in most
(not all) states of the United States, following a period of more than a
decade of "reform math" curricula that ended up not working very well. I am
favorably impressed that the lesson asked students to put their numerical
estimates of probability on a number line--the real number line is a
fundamental model of the real number system and its ordering that historically
has been much too unfamiliar for American pupils.

The author continues by elaborating on his main point: "When did we brainwash
kids into thinking that math was about getting an answer? My students truly
believe for some reason that math is about combining whatever numbers you can
in whatever method that seems about right to get one 'answer' and then call it
a day." I like the author's discussion of that issue, but I think she misses
one contributing causal factor--TEACHER education in the United States in
elementary mathematics is so poor[2] that most teacher editions of mathematics
textbooks at all levels differ from the student editions mostly just in having
the answers included[3] and don't do anything to develop teacher readiness to
respond to a different approach in a student's reasoning.

What I LOVE about the Singapore Primary Mathematics series,[4] which I have
used for homeschooling all four of my children, is that the textbooks
encourage children to come up with alternative ways to solve problems and to
be able to explain their reasoning to other children. The teacher support
materials for those textbooks are much richer in alternative representations
of problems and discussions of possible student misconceptions than typical
United States mathematical instruction materials before the Common Core.
Similarly, the Miquon Math materials[5], which I have always used to start out
my children in their mathematics instruction before starting the Singapore
materials, take care to encourage children to play around with different
approaches to a problem and to THINK why an answer might or might not be
correct. (Those materials, both of them, are very powerful for introducing the
number line model of the real number system to young learners, as well as
introducing rationales as well as rote procedures for common computational
algorithms. I highly recommend them to all my parent friends.)

I try to counteract the "what's the correct answer" habit in my own local
mathematics classes (self-selected courses in prealgebra mathematics for
elementary-age learners, using the Art of Problem Solving prealgebra
textbook[6]). I happily encourage class discussion along the lines of "Here is
a problem. [point to problem written on whiteboard] Does anyone have a
solution? Can you show us on the whiteboard how you would solve this?"
Sometimes I have two or three volunteer pupils working different solutions--
which sometimes come out to different answers [smile]--at the same time. We
DISCUSS what steps make mathematical sense according to the field properties
of the real numbers and other rules we learn as axioms or theorems in the
course, and we discuss ways to reality-check our answers for plausibility. We
don't do any arithmetic with calculators in my math classes.

[1] [http://www.corestandards.org/Math/](http://www.corestandards.org/Math/)

[2]
[http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf](http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf)

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

[http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-
ma...](http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma.html)

[3] When I last lived overseas, I had access to the textbook storage room of
an expatriate school that used English-language textbooks from the United
States, and I could borrow for long-term use surplus teacher editions of
United States mathematics textbooks. They were mostly terrible, including no
thoughtful discussion at all of possible student misconceptions about the
lesson topics or of alternative lesson approaches--but they were all careful
to show the teachers all the answers for the day's lesson in the margins next
to the exercise questions.

[4]
[http://www.singaporemath.com/category_s/252.htm](http://www.singaporemath.com/category_s/252.htm)

[5] [http://miquonmath.com/](http://miquonmath.com/)

[6]
[http://www.artofproblemsolving.com/Store/viewitem.php?item=p...](http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra)

~~~
slowmotiony
Homeschooling as in not sending your kids to school and teaching them yourself
at home? This is such an exotic concept to me as a european, would you mind
telling my why you'd do that?

~~~
mercer
I can't answer for the OP, but I was homeschooled and I was around a lot of
people from other cultures that were also homeschooled.

In the case of the Americans, they were homeschooled primarily because the
parents felt they could do a better job than the US school system, and because
they wanted to use Christian-friendly learning materials. For social contact
and 'real-life' social education, they'd put their children into a variety of
(sports) clubs.

In the case of my parents, it was a practical necessity, as we lived in a
developing country. The nearest international school was a long drive away, it
was expensive, and reintegration into the Dutch school system would have been
much more difficult.

Instead, their solution was the get a just-out-of-school teacher to have one
or more exciting years volunteering and practicing their craft on me and my
siblings, which my parents would augment with personal attention and long-
distance help from retired teachers back in Holland. The teacher would use a
Dutch 'long-distance learning' approach that was the defacto system for Dutch
expats. For high-school material, this meant a book that explained, day by
day, what should be read and what exercises should be done.

Now, I personally feel the results were so good, that I would do everything to
make it possible to either home-school my children (if I have 'em), or to get
them into some alternative school (montessori, etc.). Aside from some
difficulties adjusting to Dutch society, my siblings and I did quite well, and
I believe on the reasons we are all active learners and autodidacts is a
direct result of not having been part of the normal system.

The primary argument I hear against home-schooling when normal schooling is
possible is that it hampers a child's social development. I never bought that.
Kids get a lot of that from playing out in the street, or being part of clubs.
Furthermore, for a significant portion of kids I believe their 'normal'
schooling can have a negative social impact, through various degrees of
bullying, and the group pressure to believe that learning is stupid. I did not
have a clue that voluntary study was stupid and nerdy until I returned to high
school at age seventeen...

(of course, I also understand that for many people home-schooling isn't a
practical option. I think that's a shame. And I also understand that some
parents might not be any good at teaching, and that a mandatory educational
system can be a benefit in that case.)

------
bittermang
School did.

When I was a very young boy, 2 or 3, my parents did everything in their power
to ensure I could read. They would read books to me, I would read books to
them. It is the very reason I am so well spoken and intelligent to this very
day. Books were awesome. I loved books.

Note the past tense. Loved. I have not been able to will myself to read a book
on my own since the third grade. It was Lois Lowry's The Giver, that is the
last book I can confirm to you that I read, in whole, because I wanted to.

It was in the second grade that I had to do my first book report, and the
entire concept of reading a book not for joy, but for work's sake, was a
concept I could never rationalize. Even while I was reading The Giver, a book
I had selected out of the elementary school's own library through my own
volition, I received snide and discouraging comments from the library staff
and teachers. "He shouldn't be reading that book." "That book is too advanced
for his age." And on, and on.

Something changed. I lost my will and my zest for books. Even books I would've
very much liked to have finished and I found interesting, such as Mick Foley's
Have a Nice Day!, his autobiography about breaking in to the wrestling
business, I have not been able to finish. I have no will left inside of me to
crack open a book, to even lift one off of the shelf. I feel like my love for
books was steadily beaten out of me at a young age by the very institution
whose job it was to educate me with them.

At least this author seems to be on the right track. In all of my years of
watching society ask "What is wrong with the kids today?" so few have bothered
actually trying to ask the kids themselves.

~~~
ehsanu1
If you have trouble finishing books, try audiobooks. That's what has worked
for me, and I got subscription at audible.com, which is great.

------
SonOfLilit
In his “Intuitive Explanation of Bayes’ Theorem”, Eliezer Yudkovsky wrote:

It’s like the experiment in which you ask a second-grader: “If eighteen people
get on a bus, and then seven more people get on the bus, how old is the bus
driver?” Many second-graders will respond: “Twenty-five.” They understand when
they’re being prompted to carry out a particular mental procedure, but they
haven’t quite connected the procedure to reality.

I was awe-struck, so I asked a friend who sometimes teaches second-graders to
try this. 11/18 wrote “25″, 5/18 wrote “25 passengers on the bus” and 2/18
returned a blank note.

I think this is a big part of the explanation. If you’re taught addition as a
process that happens in a notebook, not in reality, then you have no way to
separate answers that make sense from those that don’t. You also have no way
to connect math to things you experience in your life, and I think the most
common way to develop an interest in something is to find out it’s related to
something else that you’re already interested in.

In the last Super Bowl, the Seattle Seahawks scored 8 points in the first
quarter, and 14 in the second quarter. Who won the match?

------
calinet6
3 + 4 = _____

There's your answer.

When kids are measured by "correct answers" and pressured to find them, then
that is all they will care about. They aren't, then, doing math— they're
generating correct answers.

When you add multiple choice computer-readable tests into the fold, the
problem compounds on itself.

That is what broke our kids.

------
mbuffett
So the author asks a room full of kids to arrange numbers on a number line,
then gets mad when the kids want to know if their answer is right? As far as I
could tell, it was an exercise with one right answer. Why get mad at the kids
for wanting to confirm whether they understand the material?

~~~
kedean
He's not upset because they want to know if they're right, he's upset because
they just grab some options, throw them together, and check with him. There's
no thought going into it and I'm sure if you ask the kids why they chose the
locations they did, they'll have no response.

I recently graded some of the math final tests for my wife's second grade
class, and the students who are doing well all did well because they extracted
the parts from the problem that were required (even simple things like how
many jugs of milk are shown below), while the students with a history of not
doing so good in math instead picked numbers off the paper and either added or
subtracted them, assuming they just needed an answer. There was no thought
about the process, just the end result. Got some numbers? Good enough.

------
goodcanadian
I must have been taught critical thinking by my parents as even as a child, I
could see that the education system was mostly about regurgitating not even
the "correct" answer, but the answer expected by the teacher. If you want to
see who "broke the children," you need not look much further than that. I am
glad that teachers occasionally realize that there is value to being able to
reason about a problem rather than just returning the answer by rote. It is
rare that grade school teachers even notice there is a problem.

~~~
nbouscal
A good post on the subject:
[http://lesswrong.com/lw/iq/guessing_the_teachers_password/](http://lesswrong.com/lw/iq/guessing_the_teachers_password/)

------
jowiar
If I were a teacher/professor, the words "Will it be on the test" would be
criterion for an an F, or a detention, or something along those lines. But
somehow we've codified that mentality into law? Jesus.

We've broken school by making it all about getting a piece of paper with a
number on it, rather than by being about being able to do things better each
day. Learning is about answering "Can you do more, or do things better today
than you did yesterday?" Repeat this process and eventually you're good at
things.

~~~
overgard
> If I were a teacher/professor, the words "Will it be on the test" would be
> criterion for an an F, or a detention, or something along those lines.

So I assume you're not giving tests then? Because jesus, if you are, and those
tests are consequential to the kids future, that's a pretty damn relevant and
reasonable question. (Also: seriously, you'd punish a kid for asking a
question?)

The only way that attitude even remotely makes sense is if your students are
there voluntarily, but lets be real: they're probably not. Either they're
forced to be there (K-12), or they're compelled to be there for job prospects
(college). You're probably lucky if even %10 are there because they want to be
there.

~~~
was_hellbanned
Not the parent commenter, but:

>Because jesus, if you are, and those tests are consequential to the kids
future, that's a pretty damn relevant and reasonable question.

The point of a test isn't to test every single detail of what you were taught,
it's to hit a random distribution of topics and examples of what you were
taught. This gives a pretty good indicator of how well you learned the subject
overall.

Asking "is this going to be on the test" shows that you have no intention of
actually learning everything you're taught, which is the _point_ of the class,
and instead you're only interested in memorizing the minimum to pass a limited
test, which is _not the point_ of the class.

~~~
ubercow13
Knowing what a test is going to be like is pretty useful if that test matters
at all. Otherwise there is a higher chance that the people who fare well on
the test will be those who happened to work more on the topics or skills which
come up. I'm happy my lecturer for group theory in university let me know
which of the 5+ page proofs of theorems from the course wouldn't need to be
known for our exam - it wouldn't have been very good use of my time to fully
internalise all of them. However they were interesting and I think it added
value that they were in the course.

I think you just have an idealised picture of how tests work. In the best
maths exams I have taken, you know all you could within reason about what you
need to know and what will be the content of the test, and yet the questions
still surprise you and cause you to think in new ways.

~~~
was_hellbanned
_Knowing what a test is going to be like is pretty useful if that test matters
at all._

Knowing and understanding the subject matter is likely to be pretty useful if
the test matters at all. This is why you study and understand the entire
course, which is the point of the course. However, being told explicitly which
portions you will be tested on _goes against this goal_.

 _which of the 5+ page proofs of theorems from the course wouldn 't need to be
known for our exam - it wouldn't have been very good use of my time to fully
internalise all of them_

What? If you're doing mathematics correctly, you are ably to derive the proofs
on the fly. You don't have to memorize "five pages of proofs". You understand
the concept, you understand the mechanics, and you derive the proof as
necessary.

 _I think you just have an idealised picture of how tests work._

I'll refrain from saying what I think about your "picture of how tests work".

~~~
ubercow13
Maybe I worded my post badly; I was trying to say that your idea of a good
test isn't in line with how tests are designed in _my_ experience of
education. If you agree with the sentiment that children should be punished
for trying to optimise for exams, then yes I think your picture of how tests
work (at the moment, in school especially and somewhat college) is wrong. I
see nothing wrong with a well designed test, made to cover a whole course and
to be fruitless to optimise for. At least in the UK in school, all STEM type
exams are not like that as far as I remember. Humanities exams are much closer
in style to this.

I do think GCSE and A-level science exams could be much better designed. At
college level though I'm not sure the way you describe is the only good way to
do things.

I'll refrain from saying what I think about your idea of how to "do
mathematics correctly".

------
stretchwithme
I think learning comes naturally to all of us when we are trying to find an
answer or solve a problem.

Perhaps a good way to interest kids in math is to offer them the opportunity
to make something that actually requires learning some math.

Let them try to empirically do something that can really only be done with a
formula. Show them the power! Without it, the world as we know it would not
exist.

If you're really motivated, you'll spend years seeking the answers. We should
be motivating without coercing, providing tools for discovery, then just stay
out of the way.

------
Kenji
If I have to solve exercises and after that, I have no way to know whether or
not I did the right thing, I don't learn anything at all. The point of
exercises is to learn something new and uncover your misunderstanding of the
subject. Your wrong solution clearly points to the things you have to work on.
No solutions = no learning. Mathematics is about getting answers. It's a
formal way to declare problems and seek for answers. Right answers. I'm glad I
wasn't taught maths by this person.

~~~
ronaldx
The article's point is perhaps that it's better to try and gain some intuition
for whether or not you have the right answer, rather than checking the answer
key. Relying on the answer key (or waiting for the teacher to say "you are
right!") is an extremely poor habit and we should avoid teaching students to
depend on that (in maths class or otherwise): in real life, there is no answer
key.

One of the philosophical ideas of maths is that there are multiple ways of
looking at a problem, and those multiple ways should feed back onto each other
to confirm your answer.

Maths is not just a bunch of rules that have to be followed without question:
in this case, maths is modelling something practical and the answer needs to
match reality. Without that, the probability model could not even have been
invented. If you can't get beyond the stage of do_question+check_answer, you
are never going to be capable of anything more than grinding out other
people's mathematical results.

I'm sorry you weren't taught maths by this person.

~~~
Kenji
Before you've gotten a firm grasp on the subject, you might be convinced that
your answer is right but it's not. Especially when the maths becomes more
complicated. If nobody tells you it's wrong, you will incorporate more and
more wrong things in your understanding.

I notice this horrible effect every time I'm solving exercises for uni and
there are no solutions. I think I've done good work. Sometimes, everything
really was right, other times, I made fundamental mistakes all across the
exercises, showing me that I didn't understand something important.

Of course, when you're in research, there's no key. That's why it's so
difficult. But it doesn't make sense to reinvent the wheel, as in, researching
things that are already well known.

------
sandworm101
Why? Because students today are focused on "doing well". That means getting
good grades. Look at your tests, be them locally-created or standardized. How
are they graded? Do students get good grades for debate or mechanically
computing the right answer?

Who? An American culture that constantly enforces a science v. arts dichotomy.
I say American because in no other country is this split enshrined in
language: Metric units for the scientists and English for everyone else. I say
American because few other countries produce so many millionaires with zero
science education. Look at your student's role models. Athletes. Celebs. Seth
McFarlin. Politicians. Donald Trump. The American system does not reward
scientific creativity in the same way it does social aptitude.

Change would be painful, so painful that it cannot happen for a generation or
two Just try suggesting highway signs be changed from mph to kph. And that's
easy part.

~~~
throwaway344
I think that the obsession with "doing well" is by no means an exclusively
American trait. Just look at the Indian obsession with IIT-JEE, or Koreans
with the CSAT, or the English with A-levels, or the French with the bac.
American students, in my opinion, don't really obsess more about our system
that anyone else.

------
apeace
I recommend a simple, fun solution that kids of all ages LOVE:

1\. Download DrRacket [http://racket-lang.org/download/](http://racket-
lang.org/download/) 2\. Show the kids cool functions from the 2htdp/image
library, like (circle 10 'solid 'red) 3\. Show them the animate function from
the 2htdp/universe library 4\. Leave them to their devices and instruct when
they have questions

It won't be long before they want to make a game, and they'll need to learn
all sorts of things about coordinates, shapes, colors, animations, and
functions. Kids love functions. They'll have no idea how much math they're
doing, yet at the end of the day they'll actually get it.

For probability, check out the (random) function.

------
timtas
Whatever broke the kids, it's fairly predictable that Common Core will break
them more.

Hey, let's do a probability problem? How many teachers will turn this lesson
on it's head as this teacher did transforming it from something useless to
students and painful to teacher? My guess is somewhere south of 1%. In the
spirit of the true lesson here, I welcome well-supported alternate guesses.

Centrally directed (er, suggested) curriculum will, I predict, relieve
teachers from their responsibility to get results. How can you fault the
teacher in the next classroom for not having had a meltdown and simply
suffering the through the banality of the exercise as formulated by elite
geniuses who have surely figured out the ideal pedagogy.

------
QuantumChaos
I think the point the author makes is fundamentally correct: the correctness
of answers comes from human reasoning, and is subject to discussion and
debate. It's not something completely external, and education shouldn't be a
game to guess what these external decrees are. As the author says, what should
matter is what the student _thinks_ is correct, not whether they think their
answer will be accepted as correct.

However, the author uses language that is belittling and rude. I think that
teachers who use students as an emotional outlet for their frustrations are
very harmful. I would rather see the teacher find another way to get the
students to see her point of view, than emotionally browbeating them.

------
DanielBMarkham
These conversation and facilitation skills among kids are really great, and I
agree the teacher should be encouraging them, but I have some serious
questions as to whether this teacher and I have the same definition of "math".

To me, math is a skill involving the manipulation of abstract symbols. It
doesn't involve sharing, being nice to other kids, helping each other out, or
any of that. Those things are great, but they ain't math.

Now "teaching math in a primary school setting" may involve the combination of
all of those things, and that's fine. So I guess we agree. But this essay was
terrible. If this teacher came to me with this definition of math as an answer
on a test, I'd fail them.

~~~
DanBC
> To me, math is a skill involving the manipulation of abstract symbols. It
> doesn't involve sharing, being nice to other kids, helping each other out,
> or any of that. Those things are great, but they ain't math.

Coincidentally the math stack exchanges are regarded as the most toxic parts
of that community.

~~~
DanielBMarkham
You know, the really sad part about this essay is, because the teacher doesn't
know what the hell they're teaching, they miss the bigger picture. These kids
have been learning-crippled. It's not just math. They're looking for a
mechanical process for everything. That's a tragedy. But it's lost in some
misguided rant about how math might be involved in what's going on.

------
xsmasher
>One student had seen the weather and knew there was a 90% chance of rain the
other had not seen the weather and though the probability was 50% since it
would either rain or not. They compromised and picked the middle but that’s
not the part I cared about, I cared that they had a reasonable discussion
about their thoughts.

Wow. One of those students was right and the other was wrong; if the teacher
praised the discussion instead of pointing out AND EXPLAINING the right
answer, then the teacher did a disservice to the entire class.

I've heard jokes about touchy-feely "Bill has five apples and John takes two,
how do we feel about that?" math, but I honestly believed they were jokes.

~~~
NPMaxwell
Neither student was right or wrong. This is a Bayesian problem. Each student
had a distribution of priors based on evidence. Each source of evidence had
its own level of quality which stipulated the variance of the two sets of
priors. The challenge is how to combine these priors to create a posterior
distribution.

Another way to look at it is that, because "random variation" is how we refer
to unknown influences, the perfect-knowledge probability was either 1 or zero.
Neither student had perfect knowledge.

~~~
xsmasher
> [One student] though the probability was 50% since it would either rain or
> not.

That student might have the right answer, but their explanation is dead wrong.
I hope that (common) error was pointed out to that student and the whole
class. The story doesn't say that it was.

I should not have said the other student was "right." Their answer is just
rote repetition of something they heard, without an understanding of the
methodology, but their answer is more likely to be correct.

If, in the end, they decided to average the two answers... then their
discussion was unproductive and did not arrive at the correct conclusion.

~~~
ubercow13
What exactly is the correct conclusion for them?

~~~
xsmasher
The right answer here is not a single correct number, but a handful of correct
technique or methodology. Counting the number of rainy days in the last x days
would be one possible method of prediction. There are many wrong ones too,
like "rain has four letters, so it's 4%" or "there are two choices, so it's
50%."

I hope the teacher taught the students which answers were right and which were
wrong. If they did then I retract my criticism, even though that wasn't clear
in the article.

------
darkhorn
She is a teacher but as you expect she is not a statistician. If she would
have studied statistics in statistics department for a year she would know
what to teach. First you start with a coin, you have 1/2 chance to get this
and that etc. This means 50%, then you repeat your move 100 times and you say
see, the output is 53/100\. Then you repeat and the output is 502/1000\. So
they start to get what 50% means. Then you do it with die and so on. However
she has started it in wrong way. And I'm sure that most of the kinds do not
know what 50% chance rain means.

------
MisterBastahrd
It wasn't until a 10th grade class in world history that I encountered a
teacher who wanted you to not only know the answer, but why the answer was
correct. He ended up being one of my favorite teachers ever.

------
drdaeman
> If You Can Type the Problem into Wolfram Alpha and Get an Answer You Aren’t
> Doing Math

If something can be computed with WA, it's somehow a waste of human time to do
by hand. I could only hope, eventually, someday, we'll teach those aspects of
math somehow combined with algoritmization and programming. Like, "Read on
algorithms that could solve this class of problems, understand those (how they
work and why), implement them, run the problem sets to see how it works in
practice."

------
Confusion
It's unclear to me what the problem is that this teacher encountered. The
students were given an exercise to learn something. They wanted to validate
whether they learned something, by validating their solution. This could be
achieved by reaching agreement with other students or by asking the teacher.
So the problem is that only that ne was unclear that they were supposed to
reach agreement instead of asking nir?

------
graycat
> Who or What Broke My Kids?

From the OP, sounds like heavily that teacher did. Let's see why:

> understanding that all probabilities occur between zero and one

Better would be, "a probability is a number between 0 and 1 with both 0 and 1
possible".

Try not to use "occur" here because if event A is "it rains" and it does rain,
then we say that "event A occurred". So, it is better wording to say that a
probability "is" than to say that it "occurs".

> differentiating between likely and unlikely events

Using the word 'differentiating' here is not good because (1) it is crucial in
calculus where it has a quite different meaning and (2) it is just too long
for the simple concept of, say, 'identifying' likely and unlikely events.

The worst is,

> The basic premise of the activity is that students must sort cards including
> probability statements, terms such as unlikely and probable, pictorial
> representations, and fraction, decimal, and percent probabilities and place
> them on a number line based on their theoretical probability.

I have no idea what is intended here! Probability theory was the main
foundation of my Ph.D. in engineering from my research in stochastic optimal
control, and I can't make much sense out of the teacher's statement.
Calculating probabilities of poker hands would have made much more sense. Or,
just for a start:

We shuffle a standard deck of 52 cards and pick the top card. What is the
probability

(a) the card is the queen of hearts?

(b) the card is a king?

(c) the card is diamond?

(d) the card is a spade?

(e) the card is a face card?

Suppose that first card was none of (a)-(e), and we pick the next card from
that deck. Now what is the probability of each of (a)-(e)?

Or:

We flip a fair coin five times. What is the probability we get Heads exactly
three times?

> where they can solve any computation problem with technology with no issue.

I wish that were so! I guess the teacher has not heard of the problem in the
set NP-complete and how common such problems are in business planning and
scheduling.

> the probability was 50% since it would either rain or not

Uh, can we place some real money on that!!!!!!

~~~
ronaldx
Your questions are on discrete probabilities with knowable objective answers
(cards/coins/dice), but the teacher also needs to teach the sense that there
are continuous probabilities with fuzzy subjective answers (weather/real life
events).

This may be one source of the teacher's frustration: most likely the students
are answering the question "what's the probability of rain tomorrow?" as
though they are guessing the teacher's password "umm... is it 0.5?", as the
teacher has effectively taught them to, when in fact they are being asked to
have their own opinion.

~~~
graycat
For your first paragraph, I was trying to teach a little about probability.
For the probability of rain, I mentioned that, and my mention implied the good
lesson there: We can assume that there really is a probability of rain, that
is, it definitely exists. Now, especially for deciding to place a bet, we
could get an estimate of that probability. The 50% is a first cut estimate
with some curious properties about 'total ignorance' but otherwise poor. So,
how the heck to get an estimate? Likely the best practical way was by the
student seeing a weather report on the news. Otherwise, for, say, the
probability of rain of any day in, say, August "We might, what, class? Anyone?
How might we get an estimate of that? ...?"

Or, "Billy, what company makes iPhones? Billy, you can't get the answer by
teasing Sandra. Mary? Right, Apple! Okay, suppose we buy 100 shares of Apple
stock today and sell it seven days from now. What is the probability we make
over $1000? Assume that this probability exists and we want to estimate it.
People especially good at such estimates can become among the richest in the
world, fairly quickly. So, how might such an estimate be made? Information?
Any ideas? Anyone remember the motorcycle, airport, and airplane in the movie
'Wall Street'? What was the role of 'information' there for an estimate of the
probability that the stock of the steel company in Pennsylvania would go up a
lot, soon? Anyone? Sandra?"

------
noonespecial
How could any student not be fall-right-over excited about learning standard
7.SP.C.5 ?

------
ugk
So many schools seem to "teach only towards the test" because of the heavy
reliance on standardized testing. I don't claim to have any answers, but
society has pushed for this and it's a logical result IMO.

------
nymanjon
John Taylor Ghatto & John Holt have already laid down in great detail of the
problems of government schooling and solutions to those problems as
illustrated in many of the comments in this thread.

------
jacquesm
Tom Lehrer on 'New Math':

[http://www.youtube.com/watch?v=UIKGV2cTgqA](http://www.youtube.com/watch?v=UIKGV2cTgqA)

Enjoy the 'octal' bit.

------
briantakita
> Who or What Broke My Kids?

Rigid Institutions, Rigid Hierarchies, Social Stratification, Judgement, Power
& Control, Special Interests, Lack of Freedom, Lack of Autonomy

------
mtsmithhn
You hear about this all the time these days. Is this issue prevalent in all
schools now from public to private and low to high socioeconomic status?

------
tks2103
What cost effective alternatives are there to public schooling in the U.S.?

Is it possible or feasible to home school your kid with success?

------
mydpy
You're calling your students "broken"? Can we start there?

