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Topic: Re: Pure Mathematics Courses in High School?
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Jonathan Groves

Posts: 2,068
From: Kaplan University, Argosy University, Florida Institute of Technology
Registered: 8/18/05
Re: Pure Mathematics Courses in High School?
Posted: Apr 23, 2011 1:57 PM
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On 4/22/2011 at 9:56 am, Paul Salomon wrote:

> I teach at a private school in a vibrant math
> department. Over the past couple of years we have
> made curricular changes to allow a place for
> alternative math courses, in both the required and
> elective courses. We're focusing on student choice
> as a guiding force, making the math more personal.


Thanks for your post discussing some of the approaches your school
takes in mathematics education. I take it that this school is
in Brooklyn, NY since Math Forum identifies that as your location.
Is that correct?

Paul Lockhart (yes, another Paul!), the one who "A Mathematician's
Lament," teaches at St. Ann's, also in Brooklyn, NY. Perhaps you
do know him.

Students do need more options in mathematics than what traditional
curricula offer. Not all students are headed towards Calculus
Mountain, yet that is generally the only mathematics path in many
high schools. We also need options like these to help students
to see that there is much more to math than arithmetic and algebra
and recipes. In fact, students should learn to see that mathematics
is a network of ideas that are supposed to flow naturally from each
other, not just a mere collection of disconnected formulas and
recipes to memorize. And certainly students should learn something
about mathematical proofs, but the proofs shouldn't be formal at
the beginning. Most students don't have the foggiest idea of how
to prove something in mathematics, not even informally. For most
of them, something is true merely because the book and their
teachers say so, not because they see that for themselves. What
an utter disaster!

I agree with Paul Lockhart: The beauty behind mathematics is
missing. When the reasoning is cut, leaving just the facts and
procedures to learn, almost all the beauty is gone, leaving behind
meaningless mush. Beauty in mathematics comes from its ideas and
reasoning, not just from the facts. Cantor's ideas on set theory
are beautiful, not just because of the facts he had discovered
but very much as well in how he had discovered and used his ideas
to develop and prove his theory.

> In the required sequence students have two options.
> Differences are usually in approach, but the core
> e content remains the same. In all of the classes,
> however, students are practicing mathematics - asking
> questions, writing up "proofs," defining terms,
> presenting material, conversing formally and
> informally.

Exactly much of the problem with math education for most students:
They don't get to do much of the mathematical development for
themselves, and most of them end up not having the slightest idea
of how mathematics is actually developed and where it comes from.
Heck, most students and adults think that mathematicians are just
human calculators or accountants, that basically their work is
not a whole lot different from what they themselves have learned in
school except maybe that their work is a lot more advanced but
still along the same lines.

The dominant ways of teaching and presenting mathematics presents
canned mathematics, the product itself rather than a process
that produces the product. Many students then get the false notion
that mathematics springs from mathematicians' heads in this canned
form and that we are supposed to learn how to give correct solutions
to problems on the first attempt or at least that our first draft
of a written solution is supposed to be the final version.

Even mathematicians do a lot of mathematics informally to try to
understand it, develop intuition, discover ideas, discover approaches
to a problem, and so on. They themselves work a lot with what
Charles Wells calls the rich view of mathematics:

In fact, for that matter, mathematicians tend to use a lot of
images and metaphors anyway to help them understand ideas.
Yet we do have to be careful with images and metaphors: They are
not perfect, and they can be misleading in certain ways.

Of course, they do not stick to just this rich view all the time:
They use the rigorous view to prove mathematical theorems. They
also use the rigorous view any other time they do not want to let
these images and metaphors to mislead them. And the rigorous
view can tell us that mathematical ideas may mean a lot more than
what many images and metaphors suggest.

> *Algebra 1: Techniques and Applications (pretty
> standard course)
> *Algebra 1: History and Aesthetics (follows
> historical developments)
> *Geometry: System Building (euclidean axiomatic
> course)
> *Geometry: Investigations (more informal and
> experiential)
> *Algebra 2: Analytic Geometry (pretty standard
> course)
> *Algebra 2: Functions and Abstract Algebra (set
> theory and foundations +)
> The last of these is what I have been working hardest
> on. We cover a pretty standard list of topics, but
> we start with sets and relations as a foundation, and
> everything is later put in these terms. We think
> about functions and maps everywhere. Doing this also
> gives me the opportunity to introduce Graph Theory
> and Topology in a semiformal way and ask one or two
> great problems. We can also discuss the axiomatic
> method without necessarily tying ourselves to it.

The axiomatic method definitely lies at the center of mathematical
development, so students should learn something about it. But you
are correct in that students don't need to be tied to it at the

Besides, the axiomatic method is mainly for presenting a rigorous
view of mathematics with rigorous proofs, to check to be sure no
gaps remain, to check to be sure that our reasoning has not been
led astray by misleading images and metaphors, etc. The axiomatic
method is better saved after students have gained a good conceptual
understanding of the mathematical ideas. They are probably better
off learning and exploring and thinking on their own for a while
before attempting more formal proofs and developments.

> Secondly, we offer semester electives, mostly for
> juniors and seniors:
> *The Complex Plane
> *Fractals and Chaos
> *Number Theory
> *Non-Euclidean Geometry
> *Modern Algebra
> *Topics in Topology
> Having exciting course options thrills our students.
> Students are signing up for multiple math courses
> s where previously they had no interest.

It would be good to see some of these course descriptions to get a
better idea of what is discussed in these courses and what depth.
Even at the college level, I have seen a wide variation in depth.
For instance, when I was an undergraduate at Austin Peay State
University, our Abstract Algebra course stopped at what is about
the midway point at Western Kentucky University's Abstract Algebra
course. Thus, when I later took that course at Western Kentucky,
the second half of the course was new to me. For instance, we
did not discuss Sylow's theorems at Austin Peay, but we did at
Western Kentucky.

> 1) Get to the good stuff. Don't worry about
> locking everything in or ensuring complete mastery
> when it comes to advanced, modern topics. Bring in
> your very best, most compelling and gripping problems
> and let the kids get to work on them.
> 2) Use lots of informal narrative to motivate what
> you're doing. It doesn't have to be perfectly
> sequenced so that everything can be laid perfectly on
> top of what came before. Human learning doesn't
> really work like that. Informal precedes formal.
> This is how humans learn; They just make sense of
> f something, relating it to other things, talking it
> out, and later putting it all together, cleaning
> house more and more. Axiomatic process is sometimes
> just editing.

Mastery of modern mathematical ideas takes time and a lot of thinking,
so we can't reasonably expect mastery the first time around. Even
mathematicians need to revisit ideas frequently before they can digest
them thoroughly.

Even in reading math books, we don't necessarily have to understand
a proof thoroughly to help make sense of the proof of the next
theorem. I myself had been confused with some pointers here and
there in a section of reading, but these sticky pointers have not
always prevented me from being able to make pretty good sense of what
followed. Likewise, we can be confused about the proof of a lemma
but still understand the proof of the theorem that uses that lemma.

Sometimes whether statement or idea B requires understanding statement
or idea A to make sense of it may also depend on how the mathematics
is developed. A mathematical theory can often be developed in several
different ways. Likewise, the prerequisites of a proof of a theorem
depends on what proof is presented. Some proofs of the Prime Number
Theorem require understanding some pretty sophisticated complex
analysis, but other proofs do not. By the way, I would be interested
in seeing an elementary proof of this theorem. I have heard some
stories about these elementary proofs but not the actual proofs
themselves, just proofs using complex analysis.

Point 2 here lines up well with my previous comments above.

> 3) Make it personal. Our students feel that their
> choice of math class is a personal statement, a
> reflection of who they are intellectually. By making
> them ask their own questions and write up and present
> their work, we force the students to put themselves
> in the work. On another note, sharing your own
> thoughts and research is very helpful. I talked
> about commutative linear functions under composition,
> and they ate it up.

Personalizing teaching and learning is generally a good idea. We have
different talents, strengths, weaknesses, interests, goals in life,
and so on. Even in mathematics itself, we see a wide variety of
tastes of mathematicians and other professionals who use mathematics
or develop new uses or theories of mathematics. There is no one
right way to approach and use mathematics. And one approach that
some students love, others will find boring or pointless or even
downright hate.

> The trouble has been resources. I have found lots of
> good books, but nothing really really great. I just
> found a really solid and admirable book called
> "Introductory Mathematics" by Seymour Hayden. I
> recommend it highly. I also have two good books
> called "Introduction to Algebraic Structures" and
> "Fundamental Concepts of Algebra," published by
> Dover.
> The truth is, of course, we always need new resources
> and texts. The people who do this kind of thing need
> to work tirelessly to share their best stuff -
> writing books, posting on blogs, tweeting, etc.
> Speaking of which I better get to work on "Algebra
> a 2: A Modern Course." :)
> Thanks for the great discussion!

I don't know many of those kinds of books much myself. Most I know
that discuss modern mathematics are either for mathematicians or
math graduate students or math majors in college or are math appreciation
books for the general public. These latter books are good, of course,
and I have found them fun to read. But they aren't appropriate as a
main book in these kinds of courses because you can't gather from these
books the mathematics ideas in a form that you can use them for actually
doing mathematics beyond what the books discuss. That makes sense anyway
because otherwise the general public would not find those books readable.

And the purpose of those books is not to develop technical competence
but to introduce to the general public a spirit behind mathematics that
they rarely get to see but do not need technical competence to gain a
decent understanding of what that spirit is.

I can think of three books offhand that may be good resources for
you and your fellow teachers:

1. George Andrews' book "Number Theory" published by Dover might be a
good choice to consider. No formal prerequisites are required.

2. Another good one might be Edward Barbeau's "Power Play." I haven't
had the chance to read this book in its entirely, but what I have read so
far is quite enjoyable. From the preface:

"This book focuses on powers of numbers. This is not unduly narrow.
The most notorious problem of all time, the Fermat conjecture that no
positive nth power for n>=3 can be written as the sum of two positive
nth powers, falls into this category. A few of the results recorded
here are quite recent, so that I hope this material will be of interest
to the experienced mathematician as well as the novice.

"Strict amateurs who ignore the exercises and notes will find that a
background of elementary mathematics with basic high school algebra
will carry them through. However, there are other audiences to whom
this monograph is addressed:

(a). Secondary students should read this book with a calculator
handy. Besides checking the results given, they may want to explore
and discover patterns of their own. Advanced secondary students should
try to prove some of the results given. The exercises, keyed to
assertions in the text, will indicate what they can credibly tackle.
The notes will provide further details.

(c). Secondary teachers will be able to construct units that will
supplement the regular curriculum. Many of the arithmetic equations
can be verified through factorization techniques. Searching out and
verifying special relationships among larger numbers will require skills
in numerical analysis and the use of a calculator or computer to a high
degree of efficiency."

The elements (b) and (d) in that list mention college students and
college teachers.

3. Gary Chartrand's "Introductory Graph Theory" published by Dover
has no formal mathematical prerequisites. There are proofs and fun
problems in this book.

Jonathan Groves

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