Hi, I'm tutoring a really bright kid in junior high .... I was
wondering whether someone knows of an introduction to group theory that
is intuitive enough that he can get a feeling for essential concepts
without too much formalism....

Hi, I'm tutoring a really bright kid in junior high, just trying to turn
him on to branches of math so he can see what's out there. He likes
quirky number theory stuff, he loved Conway's Book of Numbers, and he
loved the book "Knots and Surfaces" by Farmer and Stanford, which we
went through in a blitz - he did almost all of the exercises. I was
wondering whether someone knows of an introduction to group theory that
is intuitive enough that he can get a feeling for essential concepts
without too much formalism. I feel like something that had a good
presentation of why there are only a few groups of size smaller than
some limit, and what those groups are, for example, could capture his
imagination. (He likes to know what "all the possibilities are" in
various contexts.) The texts I know are too terse for a kid that age,
who will have time (and patience) to do the formal stuff later, but I
think it would be good for him to discover as soon as possible that he
loves math in a broad way - if indeed he does love math in a broad way.
He's fine on simple set stuff and linear algebra.

Thanks all -Chaz

Hi, I'm tutoring a really bright kid in junior high, just trying to turn
him on to branches of math so he can see what's out there. He likes
quirky number theory stuff, he loved Conway's Book of Numbers, and he
loved the book "Knots and Surfaces" by Farmer and Stanford, which we
went through in a blitz - he did almost all of the exercises. I was
wondering whether someone knows of an introduction to group theory that
is intuitive enough that he can get a feeling for essential concepts
without too much formalism. I feel like something that had a good
presentation of why there are only a few groups of size smaller than
some limit, and what those groups are, for example, could capture his
imagination. (He likes to know what "all the possibilities are" in
various contexts.) The texts I know are too terse for a kid that age,
who will have time (and patience) to do the formal stuff later, but I
think it would be good for him to discover as soon as possible that he
loves math in a broad way - if indeed he does love math in a broad way.
He's fine on simple set stuff and linear algebra.

In article <brqtq6$nka$1@news.fas.harvard.edu>,
chaz2422 <chaz@dont-email-me.com> wrote:
Hi, I'm tutoring a really bright kid in junior high, just trying to turn
him on to branches of math so he can see what's out there. He likes
quirky number theory stuff, he loved Conway's Book of Numbers, and he
loved the book "Knots and Surfaces" by Farmer and Stanford, which we
went through in a blitz - he did almost all of the exercises. I was
wondering whether someone knows of an introduction to group theory that
is intuitive enough that he can get a feeling for essential concepts
without too much formalism. I feel like something that had a good
presentation of why there are only a few groups of size smaller than
some limit, and what those groups are, for example, could capture his
imagination. (He likes to know what "all the possibilities are" in
various contexts.) The texts I know are too terse for a kid that age,
who will have time (and patience) to do the formal stuff later, but I
think it would be good for him to discover as soon as possible that he
loves math in a broad way - if indeed he does love math in a broad way.
He's fine on simple set stuff and linear algebra.

Are you sure that he needs an intuitive approach to
group theory? One can consider examples after seeing
formalism, and that is the time to do it.

Why do you think that texts are too terse? Let him
do the formal stuff as soon as he can, and then he
can PROPERLY apply it later. Bright, and especially
gifted, do not benefit from having to do it clumsily
the first time, and then get the simple conceptions
later. In fact, I do not think anyone benefits.

As for age, my son was auditing graduate courses
when in junior high. He did logic and algebra
before starting first grade.

Knot theory is more complex than formal group theory.

As for age, my son was auditing graduate courses
when in junior high. He did logic and algebra
before starting first grade.

Herman Rubin <hrubin@odds.stat.purdue.edu> wrote:
In article <brqtq6$nka$1@news.fas.harvard.edu>,
chaz2422 <chaz@dont-email-me.com> wrote:
Hi, I'm tutoring a really bright kid in junior high, just trying to turn
him on to branches of math so he can see what's out there. He likes
quirky number theory stuff, he loved Conway's Book of Numbers, and he
loved the book "Knots and Surfaces" by Farmer and Stanford, which we
went through in a blitz - he did almost all of the exercises. I was
wondering whether someone knows of an introduction to group theory that
is intuitive enough that he can get a feeling for essential concepts
without too much formalism. I feel like something that had a good
presentation of why there are only a few groups of size smaller than
some limit, and what those groups are, for example, could capture his
imagination. (He likes to know what "all the possibilities are" in
various contexts.) The texts I know are too terse for a kid that age,
who will have time (and patience) to do the formal stuff later, but I
think it would be good for him to discover as soon as possible that he
loves math in a broad way - if indeed he does love math in a broad way.
He's fine on simple set stuff and linear algebra.

Are you sure that he needs an intuitive approach to
group theory? One can consider examples after seeing
formalism, and that is the time to do it.

It's an interesting question. I'm not trying to start his formal
foundational mathematical training here, I'm trying to get him excited
about things and have him feel confident about his intuition, which is
excellent. At the moment this is going well because he *wants* to do it.
Excessive formalism turns him off at the moment, and I think that's fine
for now. If it was someone's kid whose parents were paying me to turn
into the best mathematician I could, I would agree with you (and I'd also
suggest the parents send him to someone else, very soon, rather than me).
But this is just a friendly session that so far has been fun for him, and
for now I'm keeping it that way.

Why do you think that texts are too terse? Let him
do the formal stuff as soon as he can, and then he
can PROPERLY apply it later. Bright, and especially
gifted, do not benefit from having to do it clumsily
the first time, and then get the simple conceptions
later. In fact, I do not think anyone benefits.

Informal and intuitive do not necessarily imply clumsy!

As for age, my son was auditing graduate courses
when in junior high. He did logic and algebra
before starting first grade.

That's great, though also slightly scary. If it can be done while
maintaining a well-rounded social personality, all the better.

Knot theory is more complex than formal group theory.

Definitely. But knot theory is 'sexier' for a kid, so even though it's
more complex, it would be easy for a bright kid to have a big appetite for
it. Come on, very few kids could be expected to learn what the
requirements of a group are with as much relish as mastering the very
accessible proof that every map is 5-colourable, for example. There's a
'cool' factor there that it would be great to find in an intro to group
theory.

Thanks for your advice though Herman, it's appreciated.
Thanks very much also to the other participants in the thread for their
suggestions! I will check out all of those books.

Chaz

As for age, my son was auditing graduate courses
when in junior high. He did logic and algebra
before starting first grade.

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

Herman Rubin <hrubin@odds.stat.purdue.edu> wrote:
In article <brqtq6$nka$1@news.fas.harvard.edu>,
chaz2422 <chaz@dont-email-me.com> wrote:
Hi, I'm tutoring a really bright kid in junior high, just trying to turn
him on to branches of math so he can see what's out there. He likes
quirky number theory stuff, he loved Conway's Book of Numbers, and he
loved the book "Knots and Surfaces" by Farmer and Stanford, which we
went through in a blitz - he did almost all of the exercises. I was
wondering whether someone knows of an introduction to group theory that
is intuitive enough that he can get a feeling for essential concepts
without too much formalism. I feel like something that had a good
presentation of why there are only a few groups of size smaller than
some limit, and what those groups are, for example, could capture his
imagination. (He likes to know what "all the possibilities are" in
various contexts.) The texts I know are too terse for a kid that age,
who will have time (and patience) to do the formal stuff later, but I
think it would be good for him to discover as soon as possible that he
loves math in a broad way - if indeed he does love math in a broad way.
He's fine on simple set stuff and linear algebra.

Are you sure that he needs an intuitive approach to
group theory? One can consider examples after seeing
formalism, and that is the time to do it.

It's an interesting question. I'm not trying to start his formal
foundational mathematical training here, I'm trying to get him excited
about things and have him feel confident about his intuition, which is
excellent. At the moment this is going well because he *wants* to do it.
Excessive formalism turns him off at the moment, and I think that's fine
for now. If it was someone's kid whose parents were paying me to turn
into the best mathematician I could, I would agree with you (and I'd also
suggest the parents send him to someone else, very soon, rather than me).
But this is just a friendly session that so far has been fun for him, and
for now I'm keeping it that way.

Why do you think that texts are too terse? Let him
do the formal stuff as soon as he can, and then he
can PROPERLY apply it later. Bright, and especially
gifted, do not benefit from having to do it clumsily
the first time, and then get the simple conceptions
later. In fact, I do not think anyone benefits.

Informal and intuitive do not necessarily imply clumsy!

As for age, my son was auditing graduate courses
when in junior high. He did logic and algebra
before starting first grade.

That's great, though also slightly scary. If it can be done while
maintaining a well-rounded social personality, all the better.

Knot theory is more complex than formal group theory.

Definitely. But knot theory is 'sexier' for a kid, so even though it's
more complex, it would be easy for a bright kid to have a big appetite for
it. Come on, very few kids could be expected to learn what the
requirements of a group are with as much relish as mastering the very
accessible proof that every map is 5-colourable, for example. There's a
'cool' factor there that it would be great to find in an intro to group
theory.