
Math from Three to Seven: Chapters One and Three [pdf] - Mz
http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf
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cubano
I believe the most important learning device used is the fact that he is
sitting patiently with the kids and giving them his full attention, sending
the meta-message to them that "this is important stuff and you will be
rewarded with an adult's undivided interest if you work with this".

All children (and most adults) crave attention, and the giving and taking of
it is perhaps the most powerful tool that a parent and/or teacher has in the
arsenal.

I've always thought a lot of parents do it totally wrong...they give the
misbehaving kid lots of (albeit negative) attention while the quiet, well-
behaved one they ignore.

My parenting style was the exact opposite...I always stroked good behaviours
and tried my best to ignore bad ones.

~~~
johnloeber
This is an exceptionally good point. I've read a lot about early child
development, and I've never come across this before. It sounds very plausible
though: children seek attention, so reward good behavior with attention. Of
course.

The same goes for schools, of course. The misbehavers earn the attention of
both teachers and students alike, and if they do it enough, even that of their
parents and the administrators. The promising students, on the other hand, are
often expected to do their work in near-monastic solitude: usually, the only
reason for special attention is if a student is performing weakly.

This is necessarily the case given that a teacher having to divide their
attention between twenty students and the current priority structure of our
schools. Regardless, that's a clear area for improvement.

~~~
cubano
_This is necessarily the case given that a teacher having to divide their
attention between twenty students and the current priority structure of our
schools._

But is it? Don't we all remember that "one teacher" who seemed to give us that
little extra attention or push when we needed it the most?

Good teachers (at least my good ones) all seem to understand the attention-
tool intuitively, but it just became glaringly obvious to me raising my three
children and seeing young children behave badly to obviously get their parents
attention, and many parents obliging them without thinking.

The rather odd part, for me, was, when mentioning this to parents in social
groups, I was often looked at like I was nuts...like the idea of bothering a
well-behaving kids with attention was beyond the realm of reason.

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Animats
(Chapter Two is left as an exercise for the student?)

The comment “What an idiot I was,” I thought. “That was just an axiom, it is
called commutativity. One doesn’t prove axioms.” is interesting. What's chosen
as an axiom, and why, is an advanced question. Unless you get into foundations
of mathematics, that question is seldom addressed. It's way beyond most pre-
college math teachers. It's the sort of question that occurs to smart kids,
but there's no easy answer you can give them. The usual answers are
theological, and boil down to "shut up, kid". Here's a discussion on Stack
Exchange of that subject: [http://math.stackexchange.com/questions/127158/in-
what-sense...](http://math.stackexchange.com/questions/127158/in-what-sense-
are-math-axioms-true)

(If you get into automatic theorem proving, you have to address such issues
head-on. Adding an inconsistent axiom can create a contradiction and break the
system. This leads to constructive mathematics, Russell and Whitehead, Boyer
and Moore, and an incredible amount of grinding just to get the basics of
arithmetic and number theory locked down solid. In constructive mathematics,
commutativity of integer addition is a provable theorem, not an axiom.

I once spent time developing and machine-proving a constructive theory of
arrays, without the "axioms" of set theory. The "axioms" of arrays are in fact
provable as theorems using constructive methods. It took a lot of automated
case analysis, but I was able to come up with a set of theorems which the
Boyer-Moore prover could prove in sequence to get to the usual rules for
arrays. Some mathematicians who looked at that result didn't like seeing so
much grinding needed to prove things that seemed fundamental. This was in the
1980s; today's mathematicians would not be bothered by a need for mechanized
case analysis.)

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otoburb
Absorbing tales, most likely for those of us with young children in the
aforementioned age bracket. First time I'd read about Jean Piaget too. Was
reading through @tokenadult's[1] site earlier and didn't see much mention of
Piaget, so another good reference to add to the list.

This is the probably the most insightful part of the AMS narrative:

>OK, if it is so hard to teach kids the notion of a number, what am I trying
to do? What is the point of my lessons? I said it many times and I am going to
say it again: the meaning of the lessons is the lessons themselves. Because
they are fun. Because it’s fun to ask questions and look for the answers. It’s
a way of life.

[1]
[https://news.ycombinator.com/user?id=tokenadult](https://news.ycombinator.com/user?id=tokenadult)

~~~
Mz
More direct link to his homeschooling site (because his profile is quite
lengthy and filled with links):
[http://learninfreedom.org/](http://learninfreedom.org/)

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abecedarius
Summary and table of contents at [http://www.ams.org/bookstore-
getitem/item=MCL-5](http://www.ams.org/bookstore-getitem/item=MCL-5)

~~~
Mz
Since posting the excerpt, I have learned you can find the entire book here:
[http://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf](http://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf)

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oldbuzzard
Alexander Zvonkin's book is worthwhile both for math pedagogy and Soviet
insight... the OP doesn't link the full book... either
[http://www.ams.org/bookstore-
getitem/item=MCL-5](http://www.ams.org/bookstore-getitem/item=MCL-5) or
[http://www.amazon.com/Math-Three-Seven-Mathematical-
Preschoo...](http://www.amazon.com/Math-Three-Seven-Mathematical-
Preschoolers/dp/082186873X) is a great example of a preschool Math Circle.

If math pedagogy is your main interest any of the MSRI Math Circle Library
books are worthwhile. This includes "Circle in a Box" which is a Math Circle
starter kit freely available here
[http://www.mathcircles.org/GettingStartedForNewOrganizers_Wh...](http://www.mathcircles.org/GettingStartedForNewOrganizers_WhatIsAMathCircle_CircleInABox)

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indrax
On coins and buttons: Perhaps the children are intuitively comparing the areas
of the convex hulls, a thing humans are good at, is an interesting
mathematical problem, and could be the basis of interesting alternative
mathematics.

You could have a whole set of sessions with children exploring different
arrangements of coins and noting that no matter how many you add within the
hull, you don't get any 'more coin' (altering the plurality may help adults
understand this problem.) If you have some button[s] and much more coin[s],
can you add just one coin so that you have more coin than button? How far away
do you need to add it?

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ww2
There is a big gap between chapter 1 and chapter 3, which makes me doubt this
book is very unrealistic. In chapter 1, it argues kids cannot follow logic, it
is not trivial for them to understand the counting with rearrangement; in
chapter 3, they suddenly can count c(5,2), and they even can understand this
number by using symbols URR.., which requires an algebraic mind.

don't be fooled by the title of the book. The content is beyond 7 year old
kids in the US.

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JadeNB
How did you find this sample? Is there any list of sample chapters available
from all AMS books, or do you just have to look at catalogue pages and hope?

~~~
Mz
If you go to the AMS bookstore and click on a title, there is often a link
near the top of the page called "preview materials." I have no idea if that is
for every single book or most or a few. The small sample I checked all had a
preview available.

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cphuntington97
Do we really need to work so hard to stimulate curiosity in people? Aren't
people inherently very curious?

~~~
JScarry
I think his point is that we need to work hard to avoid discouraging
curiosity. Often it is replaced with rote memorization or knowing things
because an authority figure says it is so.

~~~
japhyr
This is particularly true in math, as it is presented in many schools.

~~~
agumonkey
Math is a weird topic. It can be completely wrong when teachers don't
understand maths, and even with teachers that do convey the goal and added
values of math, it's still off to me. The algorithms are too complex and
learned without any real clues (you basically program pupils as biological
Turing Machines / Arithmetic Units.. very sad).

