
Marvin Minsky: What makes mathematics hard to learn? (2008) - joaorico
http://web.media.mit.edu/~minsky/OLPC-1.html
======
sago
I had dinner with Prof Minksy, about 15 years ago. He asked what I was working
on (evolutionary algorithms) we got talking about domains of knowledge.

He said all knowledge has a half-life, the time it takes for half of what you
know to be redundant or wrong. Math, he said, is the longest, measured in
centuries or millennia. One should feel sorry for neuroscientists: they can go
to the bathroom and half their knowledge will be out of date.

The only time I spent with him, but both the metaphor and his passion for math
will stay with me.

~~~
jrcii
>neuroscientists: they can go to the bathroom and half their knowledge will be
out of date

Web development seems to be even worse than neuroscience in this regard. I
feel like I have to perpetually study to be abreast of best practices.

~~~
sedachv
> Web development seems to be even worse than neuroscience in this regard. I
> feel like I have to perpetually study to be abreast of best practices.

The worst part is web development is a completely arbitrary made-up system
without any abstractions you can learn and use in a different field of work.
It does not even transfer to something as closely related as desktop/mobile
GUI toolkits. I view "knowledge" like that as a waste of time and avoid it
whenever possible. I wish there was a word for it, something to imply "garbage
knowledge."

~~~
samizdatum
Really? Functional reactive programming, immutable data, unidirectional data
flow, application state as a fold over a stream of actions... it seems like
web development is seeing a really high uptake of valuable, transferable
abstractions and ideas, far more than mobile GUI toolkits, for example.

~~~
sedachv
JavaScript programming is, CSS and HTML not so much. I would also argue that
the details of frameworks like Angular also fall in the "garbage knowledge"
pile.

~~~
samizdatum
Fair enough, you meant to refer to a subset of web development, rather than
the entire field. It's a significantly weaker claim, though. I could pick out
subsets of any field of work, and declare them non-transferable, and the
details of most systems are going to be less transferable than the concepts
undergirding them.

------
kazinator
While kids may find arithmetic to be boring and demotivating, they're able to
do it. It's when math gets symbolic that you get the permanent attrition. For
instance, most people never get to calculus.

When you progress from arithmetic to abstract, it hardly gets _easier_.

Moreover, memorizing. You go from memorizing multiplication tables to other
kinds of tables, like tables of equations giving various identities, rows of
coefficients in series, and the like. Contents of various kinds of matrices.

The need to be precise and avoid mistakes never goes away. Manipulating a
complex math equation is still a form of arithmetic. And it's harder because
the underlying semantics means that something which is _mechanically_ correct
at the syntax level (easy to check) could actually be meaningless and wrong.

The simplistic notations used in math don't "keep up" with the increasing
complexity of what is going on. They just get harder semantics. Notation which
_looks_ like multiplication or addition in such and such domain is just "sort
of" like it, but, oh, here are the ways in which it isn't.

~~~
crispyambulance
Disagree. Kids disengage from math for many different reasons. There is no
single pattern to it. If it were that simple, it would not have remained an
issue since schools were commonplace.

Minsky solidly makes the point that when math is taught as nothing more than
an endless series of tedious drills with no purpose in sight, of course it
demotivates kids. No one likes pointless Sisyphean tasks.

Many students, however, "rediscover" math when they get to their first class
that involves theorems and proofs (often in high-school geometry) and when it
starts being used in science coursework. It is then that they realize that
math is a way of thinking and this can transform their opinion and motivation
for math.

The best teachers find ways to relate mathematics to real life and real
purpose. Yeah, it is always going to be challenging, but having a purpose
creates motivation to get through the tedium.

~~~
kazinator
Humanity as such absolutely loves pointless repetition, and abhors thinking
about something hard, which involves focusing intensely on one subject in
which where no repeatable progression of simple successes is taking place,
leading to a mounting sense of discomfort.

Just look at various cultural traditions, popular (and even serious) music,
crafts, sports, ... _sneeze_ ligion. Mindless repetition is everywhere.

The same kid that hates doing arithmetic will play the same video game for
hours in which a small variation on the same set of events happens over and
over again.

The kids you're describing are the smart minority, of whom a portion will go
on into STEM type fields. They too will hit their mathematics ceilings. Most
will go as far as a couple of third year undergrad math courses and that's
where it ends.

~~~
marincounty
For myself, I found math difficult. Never understood, even now, the need to
spend so much time memorizing multiplication tables.

That said, I started over in a community college, and finished up to trig.
with ease. I found trigonometry, surprisingly straight forward. I went on to
finish a year of physics. I didn't need calculus for my major, and heard it
was really hard. Plus, I didn't want to ruin my grade point average with a
class I didn't need. I now regret not taking calculus, but it's so much easier
to learn something these days.

My struggle with math was two fold. I didn't care in high school, and I never
had a firm grasp on basic math. I look back , and once I truly inderstood
basic math, especially fractions, and percentages; it all became so easy.

I see a lot of kids still struggling with math. They get pushed along, and
avoid any class that has math in it. I do blame our U.S. teaching system in
this case. Keep teaching basic math until the kid can teach it to to their
classmates.

Then, and only then move on to algebra, and trig? As to calculus, I didn't
take it so I won't comment, but when I was applying to health professional
schools, I didn't find one that required Calculus. Probally for good reason.
My physics classmates biggest, silent stumbling block was they were horrid in
math, and looking back--they just didn't know the basics, so every step up the
ladder became more mysterious. Most of my classmates seemed to the solving
problems in physics by memorizing the homework; not truly understanding the
problems.

~~~
seba_dos1
For myself, I love math. I also never understood, even now, the need to spend
so much time memorizing multiplication tables.

In fact, I've never learnt it. From time to time I still find myself
multiplying on fingers; as a kid I used it a lot, now the table went into my
head automatically simply from continued usage. The way to multiply two
natural numbers from <6; 9> on fingers was one of the most important things my
mom tought me :)

There's also a way as presented in the article (8 _7 = 8_ 8-8), but I mostly
use it when double checking, as I find it slightly more error-prone when doing
it in head (and obviously very inefficient when done on paper). Works great
for bigger numbers though.

Once in junior high school my math teacher caught me on multiplying on fingers
and ordered me to learn the table for the next lession. Managed to pass it,
but quickly forgot in following days. I never cared - I was always doing very
fast and well on math classes anyway.

~~~
krmboya
I view memorization as a shortcut that allows one to operate on higher level
concepts with ease. Much like abstraction mechanisms in programming.

------
orlandohill
There are four other essays in this series.

Effects Of Grade-Based Segregation
[http://web.media.mit.edu/~minsky/OLPC-2.html](http://web.media.mit.edu/~minsky/OLPC-2.html)

Role Models, Mentors, and Imprimers and Thinking
[http://web.media.mit.edu/~minsky/OLPC-3.html](http://web.media.mit.edu/~minsky/OLPC-3.html)

Questioning “General” Education
[http://web.media.mit.edu/~minsky/OLPC-4.html](http://web.media.mit.edu/~minsky/OLPC-4.html)

Education and Psychology
[http://web.media.mit.edu/~minsky/OLPC-5.html](http://web.media.mit.edu/~minsky/OLPC-5.html)

------
thomasahle
> Anecdote: I asked a certain 6-year-old child “how much is 15 and 15”and she
> quickly answered, “I think it’s 30.” I asked how she figured that out so
> fast and she replied, “Well, everyone knows that 16 and 16 is 32, so then I
> subtracted the extra two 1’s.”

Wait, is this girl some kind of base-2 native?

~~~
elementalest
This is how i often work with things. I break them down into things that i
know how to work with. I search my knowledge for elements that most closely
represents the problems and then combine them.

For example, rather than remembering 12 * 6 = 72, i recall that 10 * 6 = 60
and 2 * 6 = 12, hence 60 + 12 = 72. I don't need to remember as many things
and for me i can do that much faster than trying to recall it.

My memory has never been very good, so perhaps this is why i take such an
approach, as i find it overall easier.

~~~
agumonkey
That's how I felt about math for a long time. The so-called good laziness. You
don't bother doing the hard way, you search for tricks. Symmetries, difference
from simple cases as the girl did.

Instead of memorizing algorithms, kids should be shown this. It's free
gamification and, after some years, I believe it's a core idea of what is, and
why math can be pleasing.

~~~
ddingus
Parents can do it and it can really help.

I actually was taught this and a number of other "tricks" in primary school.
Estimation was a defined thing, and the idea of getting quick math results in
your head was a real goal. Reasons given: job interview, pricing goods and
services to understand value, fuel consumption, making things (that deck, how
many boards, cost, etc...), navigation.

They took us through a lot of those cases. The first time I applied this was
powers of 2 for computing. Learn the first 16 bits worth, and that helps with
all sorts of things in computing, same as powers of ten and common easy to
compute things do in most other areas of life.

So, what I did was make it a car game. Figure out how tall things might be, or
prices, whatever comes up. Practice doing it helps to actualize the skill and
once it's done, they will apply in in ways they find useful.

------
Dowwie
"we need to provide our children with better cognitive maps of the subjects we
want them to learn"

This style of learning works for _everyone_ \-- including children. There is a
growing community of people sharing educational, animated gifs on Reddit that
recognizes the value of visual learning.

Popular subreddits include:

\- /r/educationalgifs \- /r/mechanical_gifs/ \- /r/GifRecipes/

~~~
wodenokoto
I'm not sure that is what he means with a "cognitive map". I believe, he means
that children need a better understanding of what is possible to do with, and
what the point is with math.

~~~
wanderingstan
Agree. For me it was games.

While other kids complained about algebra, I recognized it as exactly what I
had been doing with my little Apple II games, basically fumbling on my own.
Now each math lesson was a chance to make my games even more interesting.

My first graphing calculator had a simple example of 3D point rotation. It was
like learning a magical incantation! (This is pre-internet.) My friend and I
were so excited at the possibilities and immediately wrote a primitive ray-
tracer.

The point is: by connecting it to games, we were literally _hungry_ to learn
more math.

~~~
PeCaN
The other benefit is a lot of math intuitively _makes sense_ in the context of
using it for games. I started learning linear algebra when I was ~14 or so
because I was trying to make a 3D game and had a specific idea of how the
camera should operate—quaternions, once I got past the sci-fi name, just made
sense in the context of rotating the camera around. That, in turn, made
complex numbers a piece of cake when I got to them in school.

------
nothis
That old topic. Ironically, I find people who work with math ( _chosen_ to
work with math, that is) to probably be quite unqualified to truly understand
the problem. He opens by saying that learning avoiding mistakes before any
bigger concepts is a major root of this but then again, people _do_ make
mistakes in math (even math professors) and it's a field where any mistake can
completely invalidate the work. Also most people learn math for practical uses
rather than as a science or field of research which also gives a different
perspective (for most people, it's more important in life to not drop a zero
when calculating a bill than it is to understand, say, a Homomorphism).

Math simply is hard. It goes against all our good-enough, subconscious brain
functions we use every day to make basic "bigger or smaller" decisions. The
problems mathematicians call "interesting" are counter-intuitive pretty much
by definition and now imagine someone thinking like that writing a math book
for children. The language used is vast and was built over literally
_millennia_ of mathematicians appending their own little modifiers above,
below, and all around other mathematical symbols, resulting in monstrosities
like the function for the Fourier Transformation which can actually be
described quite intuitively geometrically.

Mathematicians want "simplicity" and "beauty" but their definition of either
is absurdly different from any that would take into account basic human
intuition. That's the problem. And there barely are any mathematicians that
would admit that this could ever be a bad thing.

~~~
anon4
I'm not sure how you would define the Fourier Transform geometrically without
needing 4 spatial dimensions, at which point intuitive geometric reasoning is
impossible.

~~~
lordCarbonFiber
Far from impossible, it's even reasonably intutitive. Wikipedia has an
excellent graphic
[https://en.wikipedia.org/wiki/File:Fourier_transform_time_an...](https://en.wikipedia.org/wiki/File:Fourier_transform_time_and_frequency_domains_%28small%29.gif).

Then for people for which that's too abstract, I find this graphic helps a lot
in explaining.
[http://static.nautil.us/1635_42a3964579017f3cb42b26605b9ae8e...](http://static.nautil.us/1635_42a3964579017f3cb42b26605b9ae8ef.png)

You picked a really bad example for the unintutiveness of mathematics.

~~~
BurritoAlPastor
Speaking in my capacity as a person who doesn't understand Fourier transforms,
I'm afraid I have to say that your "intuitive" graphics are anything but. I
can't even make sense of your second graphic.

Just a data point for you.

~~~
lordCarbonFiber
Fair enough, it works a lot better as a gif. Out of curiosity, does this clear
things up, or just as confusing?

[https://upload.wikimedia.org/wikipedia/commons/1/1a/Fourier_...](https://upload.wikimedia.org/wikipedia/commons/1/1a/Fourier_series_square_wave_circles_animation.gif)

~~~
BurritoAlPastor
Oooh. Yes, actually; I think I have an idea about what's going on (and why
they're so damn useful) now.

Thank you!

------
vdnkh
"Why do some children find Math hard to learn? I suspect that this is often
caused by starting with the practice and drill of a bunch of skills called
Arithmetic—and instead of promoting inventiveness, we focus on preventing
mistakes. I suspect that this negative emphasis leads many children not only
to dislike Arithmetic, but also later to become averse to everything else that
smells of technology. It might even lead to a long-term distaste for the use
of symbolic representations."

Substitute "Arithmetic" for "Interview Problems"

~~~
j2kun
To the contrary, don't most interview problems _overemphasize_ inventiveness?

------
socrates1998
It's hard to know exactly what would work for ALL students.

Honestly, for me, the issue is "one size fits all" that the vast majority of
our students get.

I learned math just fine the way it was taught. Sure, it was boring at times,
and I was mathematically inclined, but it worked for me.

For others, it clearly doesn't work.

We have to ask ourselves, what exactly do we want people to know who don't
want to learn math. What is the "bottom line" that we need a citizen to know?

Algebra 2? Trig? Some combination of Algebra and basic math finance concepts?

That's the main question. And it's very difficult to answer for ALL students.

I don't know what the answer is, but adopting one method for all students
isn't the answer.

~~~
SobolevSpaces
Back when I was in high school, they used to break my class into to groups:
those who study German and those who study English. I was in English class. I
don't see why this won't work for mathematics. Break students into two groups:
ones that never question anything and those who can't move forward until their
why questions are answered in a satisfactory manner. The first group can study
precalculus, computational analysis, matrix methods....basically anything that
can be done by rote. The second group could be introduced to proof writing,
beginnings of set/category theory, abstract algebra/topology, statistical
inference...

------
aout
There is something quite interesting about the form of the article. While Mr
Minsky talks about teaching using the "deck of cards method" and why it might
not be the best approach if the child wants to advance to more complex
subjects he also wants us to learn about his experiments (which are quite
good!).

Having read the article I find it well written but WHAT THE HELL?! Why doesn't
he use pictures? simple diagrams? "sexy" stuff?

It seems that I'm having the same problem as the child is having: the form
does suck and it discourages me from reading a boatload of articles like this
one. If the presentation of the subject was better I could read 10 more and
maybe, finally, take an interest in "teaching methods" that would lead me to
being curious and accept any "bad form" of content :)

~~~
sergiosgc
He doesn't use pictures because neither his target audience nor the subject
matter require pictures.

His target audience is composed of people who are interested in mathematics
teaching (parents, teachers and such). He expects (correctly in my opinion)
that these have handled and can handle long texts.

The subject matter is an opinion piece without much underlying data, so
graphical representation is not required.

The writing style must be adapted to circumstances. Increases in picture
density do not monotonically increase text appeal.

~~~
aout
I do agree with you. Preaching the converted doesn't require that much effort.

Now to nuance a bit this over-simplistic sentence I just written: do you think
I am interested in mathematics teaching? I did read the article anyway so you
might be right but I felt a bit overwhelmed by all this "jargon" when an
example would have been so simple to understand.

After all, Mr Minsky was a teacher, certainly a mighty good one :)

------
fmap
This is a wonderful series of articles and I find myself nodding along with
most of it. However, these lines really made me cringe:

    
    
      A child was sent to me for tutoring because of failing a geometry class, and gave this excuse: " I must have been absent on the day when they explained how to prove a theorem." 
    
      No wonder this child was confused—and seemed both amazed and relieved when I explained that there was no standard way to make proofs—and that “you have to figure it out for yourself”.  One could say that this child simply wasn’t told the rules of the game he was asked to play.  However, this is a very peculiar case in which the ‘rule’ is that there are no rules! (In fact, automatic theorem-provers do exist, but I would not recommend their use.)
    

I think interactive theorem provers would go a long way towards making
children understand symbolic reasoning. The way these programs work is that
you have feedback available at every step of a proof. Children can learn basic
causal relationships by looking at the world around them. By visualizing the
basic relationships between logical formulae you can similarly learn to inuit
the effects of reasoning steps. Interactive theorem provers provide the
visualization.

The game has rules, and you can learn them.

~~~
qb45
> I think interactive theorem provers would go a long way towards making
> children understand symbolic reasoning.

I can agree with that, but symbolic reasoning isn't everything. As shown by
Gödel, you can't have symbolic system capable of proving every true fact about
natural arithmetic without also proving some false "facts" about natural
arithmetic.

The ultimate game doesn't have rules. On some level, math is just few pieces
of meat speculating about abstract notions originating from and tied to their
physical experience. Or something like that :)

------
libeclipse
He makes a really good point about traditional teaching methods. What's the
point with the obsession over 'the correct method'? If the answer is correct,
and the method in question can reliably reach the correct answer, in my
opinion, the correct method is that one, and it differs from person to person.

~~~
adrianN
The correct method might be the method that scales better. Counting on your
fingers for example is a pretty decent method to add until you reach bigger
numbers. If you don't use the more complicated methods, you'll never improved
past some point.

~~~
pas
Then ask harder questions that show the value of the harder (more complicated,
less intuitive) methods.

But at the same time, show the value of those answers, because on the face of
it, knowing how much is 342x520+92 is not really valuable. Even knowing how
much is any number plus any number is not that interesting for basically 90%
of the population.

~~~
JadeNB
> Then ask harder questions that show the value of the harder (more
> complicated, less intuitive) methods.

It's not clear to me that there's an obvious right answer here, but I do
believe that most obvious answers are, to some degree, wrong. Namely, saying
"just ask harder questions" doesn't take into account that then the student is
trying to solve hard questions _and_ learn a new method at the same time.

Teaching a more general method on easy problems gives time to become
acclimatised to the method, while also offering sanity checks (by easier,
faster, and / or approximate methods) to make sure that it's not going wrong.
On the other hand, as you say, this approach seems unmotivated, and bores or
frustrates students—there's no obvious right answer! I think that all that can
be said is that one must find the appropriate balance for each individual
student, or class.

------
nkhodyunya
I think early math education suffers from the lack of awareness about how
important is exactness in math. People come to math from real world, where
they don't mind to be rigorous about the meaning of every word. Most newcomers
tend to skip words they don't find interesting and then penalty seems to come
from nowhere. And education even encourages such error-prone behaviour, by
giving wrong exercises and poor material.

Math isn't hard, it just doesn't work for those who don't know how to use it.
And knowing how to use it means understanding what exactly you are doing. Many
students don't bother about what does it mean to do when asked to solve an
equation.

~~~
dboreham
This resonates very strongly with my experience helping my two kids (11,13)
with their math work. Their teachers have not always been as helpful as they
might in this respect, for example marking test solutions as entirely
incorrect when the student has made a stupid arithmetic mistake in the working
but otherwise demonstrated a proper understanding of how to solve the problem.
I'm not arguing that incorrect answers should be rewarded, but it does seem to
demotivate the student when they are punished for a mistake performing a task
that no adult would need to perform (we use computers and calculators for our
arithmetic, and have done since I was my kids age in the 1970s).

------
ColinDabritz
I love these insights, they have strong echoes of "A Mathematicians Lament" by
Paul Lockhart, another excellent essay on the topic.

[https://www.maa.org/external_archive/devlin/LockhartsLament....](https://www.maa.org/external_archive/devlin/LockhartsLament.pdf)
[PDF]

------
theoh
It has apparently been shown, years ago and on a limited sample, that avoiding
arithmetic until 7th grade (in favour of general discursive education) can
produce good results in terms of the ease and success of eventual learning of
arithmetic:
[http://www.inference.phy.cam.ac.uk/sanjoy/benezet/](http://www.inference.phy.cam.ac.uk/sanjoy/benezet/)

------
gambiting
" This child imagined ‘Math’ to be a continuous string of mechanical tasks—an
unending prospect of practice and drill. It was hard to convince him that
there would not be any more tables in subsequent years"

Well, I enjoy maths, but for me my entire mathematical education was always
about memorizing tables. It was made worse by the fact that during Polish
exams you cannot use advanced calculators(only ones that can do
addition/multiplication) so for the final exams at the age of 19 I had to
remember loads of different formulas, trigonometric values for common angles
etc etc.

I'm not saying he was wrong, but in a lot of educational systems maths is 90%
about remembering stuff.

~~~
qb45
> I had to remember loads of different formulas, trigonometric values for
> common angles etc

Real math is about being able to derive this stuff when woken up at 5AM.

What you get in school is drilling kids to _appear_ as if they understood math
so that adults who don't remember this stuff anymore feel intimidated and
leave with impression that their children are learning some rocket science and
the "education system" deserves their money.

No teacher literally reasons like that and no one will tell you that, but
that's exactly what they do when they are overwhelmed by a horde of barely
interested kids who somehow have to pass the final exams. The other part of
this insanity being politicians who set up those exams.

~~~
sago
It's cargo cult mathematics.

People who understand math can do X, Y, and Z.

Therefore we should make sure children can do X, Y, and Z.

Then they'll understand math too!

------
toddan
Not everyone can learn math. I my self had brilliant and great math tutors
that my parents hired for me. I event went to special private math classes
with brilliant math teachers and i tried to the best of my abilities to get
good at it and pass the advanced math courses in highschool. Still i suck
greatly at math and i always will.

I am very grateful for all the help my parents provided for me without it i
would not pass even the first math classes in highschool.

~~~
j2kun
Barring an extremely small number of cases, this is not true by any means. The
cognitive skills required to learn math are the same as the cognitive skills
required to learn language. If you're interested in learning more about this,
check out Keith Devlin's "The Math Gene" [1]

[1]: [http://www.amazon.com/The-Math-Gene-Mathematical-
Thinking/dp...](http://www.amazon.com/The-Math-Gene-Mathematical-
Thinking/dp/0465016197)

~~~
toddan
If it was as simple to read one book to be good at math I would be an engineer
a long time ago with all the tuition and help i got thru my years in school.

~~~
j2kun
You misunderstand me. The book I referenced does not try to teach you math.
The book I referenced surveys scientific studies on the question of "whether
everyone can learn math." There is overwhelming evidence presented in this
book that, in fact, everyone can learn math.

------
SobolevSpaces
Please, never mind my English it's awkward sometimes.

Math could be taught qualitatively.

First, one can give their students ten or so hairy problems where the goal is
to write down what the problem is asking in the most explicit and gory
details.

Then one can present another batch of hairy looking problems and ask the
students what definitions/lemmas/theorems/corollaries look appropriate to
apply to the problems and why?

Another batch of convoluted problems can be offered where the goal is to
simply formulate one's questions regarding the problems in the most precise
way one can. Asking the right, penetrating questions solves more than half the
problem.

Ask the students to rewrite all the definitions/lemmas/theorems/corollaries
presented in class in their own words.

Present students with a text full of unproven statements (most grad level math
texts). The exercise here is to identify as many of these statements as
possible. The bravest students are encouraged to supply proofs.

Give students a bunch of required problems peppered with generalizations and
extensions of required concepts(typically reserved for big boy students) and
ask them why they are required to solve these problems. What exactly to be
gained out of solving every specific problem?

After all that students can tackle the required set of problems written up for
them by the Department Of Serious Business. At this point math will come naked
and explicit in front of one's eyes.

None of this is going to happen in classrooms, though. Students will have to
learn these things on their own as per the traditions.

------
kyled
It's simple. If you're not passionate for a topic you're not going to care for
it.

I hated math when I wasn't passionate about it. In school a lot of math
classes were about route memorization, not going through or writing beautiful
proofs.

Why don't they teach more abstract math in school? I love studying number
theory and category theory.

------
tdaltonc
I really like the educational policy recommendation in Where Mathematics Comes
From [0]. Even the most abstract mathematics is an analogy with our intuitive
embodied experience.

One striking example: Training children to individually articulate their
fingers makes them better at processing arabic numerals. [1]

[0]
[https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From](https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From)
[1] [https://www.researchgate.net/profile/Marie-
Pascale_Noel/publ...](https://www.researchgate.net/profile/Marie-
Pascale_Noel/publication/5464114_Does_finger_training_increase_young_children%27s_numerical_performance/links/0c96051a8abd238141000000.pdf)

------
graycat
Solution: For the students, answer "Why should I learn this?" and, then, give
good intuitive explanations for basically how it all works from algebra,
geometry, trigonometry, and calculus to probability, statistics, functional
analysis, and more. For the intuition, do include pictures.

------
peter303
My father made it memorization a fun quiz game. Maybe a couple times a week
he'd ask whats "5 times 7" and so on. Or how to spell a word. There was no
penalty for getting something wrong. You just wanted to learn the right answer
to please your father and yourself.

~~~
ashark
> Maybe a couple times a week he'd ask whats "5 times 7" and so on.

Ah, the Henry Jones Sr. approach to parenting! I'm a fan.

Young Indy: I have to show you something!

Professor Henry Jones: It can wait. Count to twenty.

Young Indy: No, Dad, I-!

Professor Henry Jones: Junior!

Young Indy: One, two, three, f...

Professor Henry Jones: In Greek.

Young Indy: [rolls his eyes] Ana, theo, thea...

(from IMDB)

------
OliverJones
This is interesting. I volunteer at a homework-help clinic for public housing
kiddos. They've heard "the answer is right but you got there the wrong way" a
lot. It's frustrating to work with the kids, because they say "if I do it that
way, the teacher will say I'm wrong."

And, I basically don't have a clue about how they're being taught to do this
stuff. Any sane person turns 2 x 15 into 2 x 16 - 2 or 10 + 10 + 5 + 5.

But this Common Core arithmetic is not quite sane. The intention is to teach
reasoning skills, but the reality is that some curriculum-writer's idea of
reasoning skills has become codified. Grumble.

You'll be missed, Dr. Minsky. So long, and thanks for all your teaching.

------
EvanPlaice
Mathematics is hard to learn because it's a 'solved' problem.

Everything is taught as a convention through brutally boring rote memorization
and repetitive mechanical operation.

Unfortunately, the people creating the training materials are usually so far
removed from a basic level of understanding, they're blind to their own expert
bias.

What we need sre training materials that bridge basic to advanced
understanding theough practical application. Except practical application is
hard to measure so it'll never be used. In a traditional teaching environment.

------
maker1138
I found his note at the end most interesting and mirrors my observations
since. Governmental standards are making our children math-illiterate. There
should be a myriad of different schools with different specialties. There
should be schools for the learning impaired as well as the gifted. If every
school was private it would allow experimentation with every form of teaching
and promote innovation and effective methods to be discovered and thrive. One
size fits all education works as well for education as it does for clothes.

------
aroman
For an actual implementation of the "students need maps" idea, check out
expii: [https://www.expii.com](https://www.expii.com)

Click on "Topic map" in the header — it is a real domain-expert-created topic
map of many areas of math and science.

It's a free, interactive, crowd-sourced learning platform that does exactly
this, especially for math — the founder is lead Coach of the USA International
Mathematical Olympiad team, and a math professor at CMU.

f/d: I interned there last summer.

------
melling
What makes anything hard?

It's difficult to make initial progress and you see no path, with little
rewards along the way, to get good at it. What's the reward at the end?

Math doesn't have a path and there's no "cool" prize at the end, for most
people.

Computer programming has a path with lots of little rewards and "cool" prizes
at the end. So, you get lots of self-taught programmers.

Create an interesting path for math with lots of "rewards" along the way and
many more people would be interested.

------
cygnus_a
Math is hard for the same reasons that anything is hard. It is literally a
mental strain on your brain.

I agree with the author in that a teacher's enthusiasm and perspective, and
'rigidity' in school can compound difficulties.

But as a physicist, the most difficult part of learning math, for me, is
overcoming my own mental barriers. I need patience and focus to comprehend
even the most simple sentences. If I don't want to learn/strain in the moment,
then I won't.

------
unabridged
Imagine if learning english required getting to college before you could even
get past spelling. And that your teachers who majored in english, have only
constructed a sentence a few times in a 400 level class and have never read
more than a paragraph. How do you even begin to talk about literature? The
payoff is so far away and unrelated to the subject taught in school that 99%
of the people in charge have no idea what is going on.

------
princeb
Mathematics is hard to learn because the way classes and performance testing
is done, there is often only one answer and until you get to the answer the
question is not complete. the end state is binary - either the student was
successful in it, or the student was not.

On the other hand, in a subject like history a student can make good ground
towards an answer by laying down fundamental facts and then arguing and
reasoning back and forth. Even with minimal facts the student may use some
approximate recollection of events and still be able to build an argument. The
answer may not be complete, and it may not even be good, but the final outcome
has the appearance of being complete, and is often somewhat far from the
starting point of nothing.

Humanities subjects use assessments that provide intermediate rewards for a
student. Mathematics offer no such respite. That is what I believe makes it
intimidating. Even worse, some questions have intermediate solutions look
extremely hairy, leading the student to believe that he has gone further away
from the solution. In such cases, the intermediate step has punished rather
than reward!

Very often suggestions for improving mathematics pedagogy is to make use of
"open-ended" questions that admit multiple solutions (as Marvin did here). I
believe most of us who make these suggestions understand the effect that
traditional maths questions have on a learner. These suggested questions are
just like those in history or literature that allow students to have a decent
go at it and still make progress.

I don't know if I truly believe in the theory (hypothesis) that memorization
makes it extremely boring. Until historical accounts are utilized in some
argument (during a performance assessment), these are just empty facts to be
memorized. And given the wide range of questions that can be asked in a
history exam, the effort involved in memory work in history class seem to be
more onerous and less rewarding. Yet I have seen students who are gifted in
remembering in detail historical events, capable of building logical
structures and narrative flow, and they struggle with maths. So clearly they
must have committed a decent amount of effort to memorization - but not in
maths but other subjects instead.

In any case, at the higher levels, rote work in mathematics is extremely
important: it is very difficult to pursue an advanced class in probability
theory without the basic tools picked up in real analysis, for example (that
means all your theorems relating to sequences and continuity and functional
analysis and so on). That usually involves some amount of "practice", thinking
and solving problem sets before moving on to more advanced topics.

------
kelvin0
My high-school math teacher (Older man) had his favorite pets (girls) sit in
front, and would not even address the rest of the class. Questions and
inquiries were completely ignored. The only reason I was able to get ahead is
due to my interest in programming my own games, and thus learning the relevant
math (trigonometry at the time).

~~~
JoeAltmaier
Sounds like how women are treated in most classes (ignored, not addressed)

------
crb002
If you want to change this, fund my story problems book.
[https://experiment.com/u/Uwxi7A](https://experiment.com/u/Uwxi7A)

Algebra and Functional Programming taught together in Middle School is a huge
win.

------
tschiera
Anybody know of any schools/programs that introduce math to children in some
of the ways that Minsky suggests? (In case it isn't obvious, I'm not a
programmer (nor do I work with math)...but I am a parent.)

~~~
pmalynin
Why not do it yourself? I think that fostering a particular topic by the
parents should be a first step.

~~~
tschiera
I agree completely. But I also think it is helpful to understand how others
who may have focused on this issue more than I have to date are approaching
this. I see a fair amount of discussion regarding how we can do a better job
of introducing math to children, but far less on what we should actually do.
Should I just start playing around with LOGO or Scratch myself?

~~~
cjp222
When my kids were younger, the curriculum that their school used was more
about the aha!s and connections than about rote recipes. I volunteered in
one's first grade math class where they spent 20 minutes of the 45 minute math
period learning to draw puppies and kitties so they could answer the question,
"Sue has 15 puppies and kitties. How many of each does she have?" The goal was
to get the kids to come up with every combination from 0 puppies and 15
kitties through 15 puppies and 0 kitties, and first graders are still mostly
concrete thinkers and need those pictures.

At this level, getting 3+2=5 was MUCH less important than being able to
explain in writing that 3 apples and two bananas added up to a bunch of fruits
in words. 4, 6, 8, 5, whatever. Alas, they never got 3+2=5 hardwired in this
curriculum. But they were better artists for it!

And my kids did _not_ learn math, because the curriculum was more about
writing than math and explaining reasoning (I called it "math for people who
don't like math and would rather write about it than do it"). So in elementary
school, I taught my kids math using Singapore Math curriculum and let the
school do its thing so it was balanced, and I volunteered weekly in math
during the elementary school years so I was on top of what they were learning
and how.

In addition, we always did math at home in informal ways without being
explicit about it, from counting lug nuts on cars when they were toddlers
through guessing the color of the next car to go by the stop sign on our walks
to estimating which box of cereal was cheapest without a calculator to noting
that the arrival rate at a traffic signal wasn't random in one direction
because the only way cars could get to it was by going through another signal.
Heck, even counting in binary using the 3 lights above an airplane row on
visits to the grandparents - Look, with three lights I can only count to 8!
And now with high schoolers and college kids, math jokes like log(fu) = log(f)
+ log(u) are the height of hilarity, or at least lighten the mood at exam
time.

Point is - there is "studying math" (with whatever curriculum and rubrics) and
there is "playing with math." As a parent, especially with younger kids, I
found that it was great to establish both. Math isn't a "thing", it's part of
life.

~~~
Jemmeh
On the total flip side I remember when studying math in high school most of
the other students HATED the math questions that required writing. It was like
by that point they preferred the memorization. Maybe because that is what they
were used to. Having to think about how to actually apply math was difficult
for most of them.

------
pakled_engineer
Roger Penrose writes about how fractions are incorrectly taught in school in
the preface of "Road to Reality" always the pie visual is used as an
approximation which later confuses students.

~~~
tanker
I remember asking my 4th grade teacher what the difference between fractions
and division. She clearly didn't understand what I was asking. That was the
moment I started seeking my own answers and stopped trusting my teachers
blindly.

------
lugus35
DragonBox: Secretly teach algebra to your children

[https://news.ycombinator.com/item?id=9469364](https://news.ycombinator.com/item?id=9469364)

------
pinkrooftop
Spelling too is also memorization before higher concepts like grammar or essay
construction. Perhaps its perceived as easier than math because it has a
natural cognitive map.

~~~
jimmaswell
I think you generally understand grammar before you can spell, if not very
rigorously

------
xyzzy4
Math is extremely interesting when you apply it to real world things, such as
finance or physics. It's hard to stay focused on math topics that are purely
abstract.

------
o_nate
_Is it possible that when John Smith moved from Apple to Microsoft, this
raised the average IQ of both companies?_

Ouch... I guess Minsky was a Mac guy.

------
bluedino
It's not that it's hard to learn, it's just not easy. You can't take shortcuts
(you can at first but it bites you later), you must do things in the correct
order, you must be exact. There are lots of little things to learn and
remember.

I think some peoples brains are just wired to 'get' math just as some people
'get' programming. And some people just 'get' art. Certain people just 'see'
certain things and you can't really teach that.

------
artursapek
> Fascinated by electronics and science, the young Mr. Minsky attended the
> Ethical Culture School in Manhattan, a progressive private school from which
> J. Robert Oppenheimer, who oversaw the creation of the first atomic bomb,
> had graduated.

Crazy to read this sentence while I am halfway through Cat's Cradle.

