Phillips Exeter Academy's math problem sets

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Phillips Exeter Academy's math problem sets

Dave L. Renfro
By accident I stumbled on Phillips Exeter Academy's math problem
sets this afternoon and thought that some here would be interested.
You'll find a very large number of decent problems in these sets.

Go to their math department web page

http://www.exeter.edu/academics/72_6532.aspx

and click on "Teaching Materials" at the left side to get

http://www.exeter.edu/academics/72_6539.aspx

Their math faculty seems quite impressive, and in recent years their
students have been even more impressive in the USAMO and IMO competitions.

Dave L. Renfro
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Re: Phillips Exeter Academy's math problem sets

kirby urner-4


On Thu, Apr 30, 2015 at 4:51 PM, Dave L. Renfro <[hidden email]> wrote:
By accident I stumbled on Phillips Exeter Academy's math problem
sets this afternoon and thought that some here would be interested.
You'll find a very large number of decent problems in these sets.


This is a serious school where faculty sources much of the curriculum.

Phillips / Andover is like that too.

http://www.andover.edu/Pages/default.aspx

Note you can get math credit for this one, the kind of course any serious high school should offer.

Thanks to distance learning, if you're a student trapped in a joke school without something like this, you may still have options:

Introduction to Discrete Mathematics and Programming
MATH-470
(W-S)
Five class periods. This course blends a study of programming (using the Python programming language)
with mathematics relevant to computer science. Students learn how to design simple algorithms and write
and test short programs in Python. The course covers Python syntax and style, as well as data types,
conditional statements, iterations (loops), and recursion. Selected mathematical topics include sets, number
systems, Boolean algebra, counting, and probability. A student in this course is eligible for credit in either
mathematics or computer science. A student who wishes to receive mathematics credit should sign up for
MATH-470 ; a student who wishes to receive computer science credit should sign up for COMP-470
.
Prerequisite:
MATH-330 or permission of the department.

Kirby

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Re: Phillips Exeter Academy's math problem sets

Bishop, Wayne
In reply to this post by Dave L. Renfro
Congratulations on their high standards,
competent instruction, and appropriate
recognition of genuine excellence.  The biggest downside, however, is huge:
http://www.exeter.edu/admissions/109_1370.aspx

Our great educational malfeasance is to not
routinely offer such opportunities – at least
some approximation thereof – to ordinary students
with ordinary socioeconomic status much less to
children from communities of exceptionally low
education level and socioeconomic status.

Wayne

At 02:51 PM 4/30/2015, Dave L. Renfro wrote:

>By accident I stumbled on Phillips Exeter Academy's math problem
>sets this afternoon and thought that some here would be interested.
>You'll find a very large number of decent problems in these sets.
>
>Go to their math department web page
>
>http://www.exeter.edu/academics/72_6532.aspx
>
>and click on "Teaching Materials" at the left side to get
>
>http://www.exeter.edu/academics/72_6539.aspx
>
>Their math faculty seems quite impressive, and in recent years their
>students have been even more impressive in the USAMO and IMO competitions.
>
>Dave L. Renfro

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Re: Phillips Exeter Academy's math problem sets

Carleton Washburne
In reply to this post by Dave L. Renfro
Dave, I couldn't help notice that the page to which you've pointed us at Phillips-Exeter's renowned math department makes mention of TI-89 Titanium calculators, and not in the context of spitting upon their use for some unfathomable reason.

In fact, there's information about technology addressed to the student at the beginning of those problem sets:

"Many of the problems in this book require the use of technology (graphing calculators or computer software) in order to solve them. Moreover, you are encouraged to use technology to explore, and to formulate and test conjectures. Keep the following guidelines in mind: write before you calculate, so that you will have a clear record of what you have done; store intermediate answers in your calculator for later use in your solution; pay attention to the degree of accuracy requested; refer to your calculator’s manual when needed; and be prepared to explain your method to your classmates. Also, if you are asked to “graph y = (2x − 3)/(x + 1)”, for instance, the expectation is that, although you might use your calculator to generate a picture of the curve, you should sketch that picture in your notebook or on the board, with correctly scaled axes."

That's pretty radical stuff (or should I say, "fuzzy"?) for a place with such a good reputation. I'm not sure our resident technology and mathematics experts could or would approve. Since you seem to sit somewhat outside that group, I wondered what your take on all this might be.
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Re: Phillips Exeter Academy's math problem sets

kirby urner-4
In reply to this post by Bishop, Wayne


On Thu, Apr 30, 2015 at 10:45 PM, Wayne Bishop <[hidden email]> wrote:
Congratulations on their high standards, competent instruction, and appropriate recognition of genuine excellence.  The biggest downside, however, is huge:
http://www.exeter.edu/admissions/109_1370.aspx

The question is, why do you have to be an $nnK/yr private school to teach Pythonic Math?
 

Our great educational malfeasance is to not routinely offer such opportunities – at least some approximation thereof – to ordinary students with ordinary socioeconomic status much less to children from communities of exceptionally low education level and socioeconomic status.

Wayne

I completely agree.  It's more than malfeascance though, it's deliberately holding back the 99% so the kids of the 1% can stay privileged.

The US as we know it would fall apart if anything close to real democracy were practiced.  It's not about economics as much as class privilege.

Happy May Day!

Kirby



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Re: Phillips Exeter Academy's math problem sets

Carleton Washburne
In reply to this post by Dave L. Renfro
Kirby Urner wrote:

> On Thu, Apr 30, 2015 at 10:45 PM, Wayne Bishop <
> [hidden email]> wrote:
>
> > Congratulations on their high standards, competent
> instruction, and
> > appropriate recognition of genuine excellence.  The
> biggest downside,
> > however, is huge:
> > http://www.exeter.edu/admissions/109_1370.aspx
> >
>
> The question is, why do you have to be an $nnK/yr
> private school to teach
> Pythonic Math?


Well, of course, no one should have to be, but it's mostly in places like that in which one finds teachers with enough academic freedom to do so and a parent base that understands that going off script is a good thing, not the first sign of the crumbling of America. Or that giving their kids major legs up on the rest of the population is why they pay $nnK/yr and glad as hell to do so.

>
>
> >
> > Our great educational malfeasance is to not
> routinely offer such
> > opportunities – at least some approximation thereof
> – to ordinary students
> > with ordinary socioeconomic status much less to
> children from communities
> > of exceptionally low education level and
> socioeconomic status.
> >
> > Wayne
>
>
> I completely agree.  It's more than malfeascance
> though, it's deliberately
> holding back the 99% so the kids of the 1% can stay
> privileged.
>
> The US as we know it would fall apart if anything
> close to real democracy
> were practiced.  It's not about economics as much as
> class privilege.
>
> Happy May Day!
>
> Kirby

Could you explain to me how you distinguish between economics and class privilege? They seem very intimately linked to me, but perhaps you're breaking it down further in ways that would be useful to consider.

I have to add in closing, however, that Wayne, Haim, Robert, et al., would be screaming bloody murder if you or anyone tried to implement anything vaguely like what goes on in PEA at the local inner city school of extreme poverty and attack it as "fuzzy" and "math avoidance," etc. I'm going to post at length from the math department blurb at PEA in a while and it reads like my dream of great math for high schools. It also reads like nearly everything those cave dwellers of ours rail against every day of the year here when it's proposed or done anywhere but a school of class privilege and wealth. (Keep in mind that such places do on occasion let a less-affluent, less-privileged soul into the school, though generally if that child has some "special talent," like, oh, the ability to play sports of some sort extremely well).

As I read through the PEA's math department problem materials and catalog, I was reminded of places like the Park School of Baltimore, which also does its own math curriculum, one that is also problem-based like that of PEA, and of course St. Ann's in Brooklyn, where Paul Lockhart teaches, and St. Mark's School in Southborough, MA, where James Tanton taught for a decade: all are places that have the academic freedom for faculty and students necessary for truly great things to be tried and done. A bit harder for a Mathematically Correct/HOLD crowd to show up at public meetings to destroy any innovative "too fuzzy" program that is implemented, the "excessive" use of technology, etc., not the least reasons being the quality of the faculty and the sanity and intelligence of the parents.
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Re: Phillips Exeter Academy's math problem sets

Dave L. Renfro
In reply to this post by Dave L. Renfro
Carleton Washburne wrote:

http://mathforum.org/kb/message.jspa?messageID=9761541

> Dave, I couldn't help notice that the page to which you've pointed us at
> Phillips-Exeter's renowned math department makes mention of TI-89 Titanium
> calculators, and not in the context of spitting upon their use for some
> unfathomable reason.
>
> In fact, there's information about technology addressed to the student at
> the beginning of those problem sets:
>
> "Many of the problems in this book require the use of technology (graphing
> calculators or computer software) in order to solve them. Moreover, you are
> encouraged to use technology to explore, and to formulate and test conjectures.
> Keep the following guidelines in mind: write before you calculate, so that
> you will have a clear record of what you have done; store intermediate answers
> in your calculator for later use in your solution; pay attention to the degree
> of accuracy requested; refer to your calculator?s manual when needed; and be
> prepared to explain your method to your classmates. Also, if you are asked to
> "graph y = (2x - 3)/(x + 1)", for instance, the expectation is that, although
> you might use your calculator to generate a picture of the curve, you should
> sketch that picture in your notebook or on the board, with correctly scaled axes."
>
> That's pretty radical stuff (or should I say, "fuzzy"?) for a place with such
> a good reputation. I'm not sure our resident technology and mathematics experts
> could or would approve. Since you seem to sit somewhat outside that group,
> I wondered what your take on all this might be.

I don't have any concerns with calculator usage in these instructions or in
the problems. Most of the problems (at least those for Math 1, which I looked
over before writing this) seem pretty much independent of calculator use,
except maybe for basic arithmetic computations that to me mostly seem designed
to discourage calculator usage (by using "nice numbers"). Also, their advice
seems pretty spot on, and many of the complaints I've heard over the years
about how inappropriate calculator use tends to encourage bad habits seem to be
addressed in their advice.

I'm certainly not adverse to calculator use, only inappropriate calculator use.
See the following problem set of mine from 1 November 1999 (posted in math-teach
a few years ago, but I don't know the post's URL and found it simply by googling
my name and the title of the problem set). This was a collection of problems that
I gave as a review of some of the material we had covered up to that point. We were
getting ready to start Riemann sums, area, and basic integration techniques,
so I thought it would be a good idea to review some of the differentiation material
by spending a couple of days working through these. (If anyone is wondering about
the absence of curve sketching and max/min problems, these were covered at the beginning
of 2nd semester calculus, after a more extensive than usual introduction to integration in
1st semester calculus. Yes, I know this is a little different from usual, but that's
how the topics were covered at that time where I was at.)

Anyway, in these review problems I wanted to (a) review certain skills (so some of
the problems were designed to target those skills, something I've often talked about
in math-teach), (b) I wanted to showcase certain concepts (e.g. finding y' by solving
for y in terms of x first, and by implicit differentiation, and checking that the
results are the same), (c) I wanted something a bit non-trivial to use a calculator
to numerically calculate (see problem 8 on pp. 4-5), (d) I wanted something that
involved a lot of interplay between mathematical reasoning and calculator use
(see E on pp. 5-7; incidentally, I used a handwritten version of "E" for several
years in the late 1990s teaching high school at LSMSA), and (e) ... probably other
things that don't jump out at me right now in looking at the problems.

http://mathforum.org/kb/servlet/JiveServlet/download/206-1874348-6544585-537994/cal_1W2.pdf

Incidentally, the graph on p. 4 is messed up (I didn't make the .pdf file),
but if anyone is interested, I can (using the new computer I bought in July 2013)
supply a corrected version.

Dave L. Renfro
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Re: Phillips Exeter Academy's math problem sets

Richard Strausz
In reply to this post by Dave L. Renfro
One further bit of info  related to PEA, they host a quite famous summer institute for math and science teachers around the world.

https://www.exeter.edu/summer_programs/7325.aspx

I was one of the instructors  there in the early 2000s, and despite my later reputation for math avoidance I was invited back to teach the following two summers as well.

Richard
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Re: Phillips Exeter Academy's math problem sets

Dave L. Renfro
In reply to this post by Dave L. Renfro
Wayne Bishop wrote:

http://mathforum.org/kb/message.jspa?messageID=9761536

> Congratulations on their high standards,
> competent instruction, and appropriate
> recognition of genuine excellence. The biggest downside, however, is huge:
> http://www.exeter.edu/admissions/109_1370.aspx
>
> Our great educational malfeasance is to not
> routinely offer such opportunities –at least
> some approximation thereof –to ordinary students
> with ordinary socioeconomic status much less to
> children from communities of exceptionally low
> education level and socioeconomic status.

Yes, obviously given my upbringing that was at the
front of my mind yesterday when I posted this, but it
was nearly the end of the day for me and I decided to
skip getting into it at that time.

However, unlike in years past when one might not even
be aware of the existence of such schools, now students
(and their teachers) can access much of the material
that students from these very privileged backgrounds
(probably well into the upper echelons of the top 1%)
get to see. Of course, they're not going to have access
to the Academy's teachers' individual attention nor to
the Academy's many excellent students to bounce ideas
off of, but it's way better than the situation 40 some
years ago when I was in high school. The main problem
I see is one that I've mentioned before, namely that
too much of a good thing can be bad in the sense that
motivated students from less privileged backgrounds can
easily drown in all the choices open to them, especially
if they don't have anyone around them who can help guide
them in the choices.

Dave L. Renfro
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Re: Phillips Exeter Academy's math problem sets

Bishop, Wayne
In reply to this post by kirby urner-4
I could not disagree more.  This is not class warfare; it is a ridiculously growth of our "professional" educational schools.

Wayne

At 06:53 AM 5/1/2015, kirby urner wrote:


On Thu, Apr 30, 2015 at 10:45 PM, Wayne Bishop <[hidden email]> wrote:
Congratulations on their high standards, competent instruction, and appropriate recognition of genuine excellence.  The biggest downside, however, is huge:
http://www.exeter.edu/admissions/109_1370.aspx


The question is, why do you have to be an $nnK/yr private school to teach Pythonic Math?
 

Our great educational malfeasance is to not routinely offer such opportunities – at least some approximation thereof – to ordinary stustudents with ordinary socioeconomic status much less to children from communities of exceptionally low education level and socioeconomic status.

Wayne


I completely agree.  It's more than malfeascance though, it's deliberately holding back the 99% so the kids of the 1% can stay privileged.

The US as we know it would fall apart if anything close to real democracy were practiced.  It's not about economics as much as class privilege.

Happy May Day!

Kirby


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Re: Phillips Exeter Academy's math problem sets

kirby urner-4


On Sat, May 2, 2015 at 9:16 AM, Wayne Bishop <[hidden email]> wrote:
I could not disagree more.  This is not class warfare; it is a ridiculously growth of our "professional" educational schools.

Wayne



These are not mutually exclusive positions. 

It's to the benefit of the 1% to have the military do the serious education with immediate applications, leaving the downtrodden to the incompetent.

Then, in a few elite schools, allow the cultivation of excellence, such that entitlements may be passed down, nobility to heirs.

That's the shape of class warfare in the USA:  scare 'em about the scary world out there and make 'em do the cowardly thing. 

Ben Franklin understood this.  They're cowards, and lie a lot.

Once you understand what makes Americans tick, governance is not that hard.

Kirby



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Re: Phillips Exeter Academy's math problem sets

Timotha Trigg
In reply to this post by Bishop, Wayne
[Wayne] Our great educational malfeasance is to not routinely offer such opportunities – at least some approximation thereof – to ordinary students with ordinary socioeconomic status much less to children from communities of exceptionally low education level and socioeconomic status.

But don't you think this


and this


are pretty much equivalent to the PEA questions?

-Timotha

On Sat, May 2, 2015 at 12:16 PM, Wayne Bishop <[hidden email]> wrote:
I could not disagree more.  This is not class warfare; it is a ridiculously growth of our "professional" educational schools.

Wayne

At 06:53 AM 5/1/2015, kirby urner wrote:


On Thu, Apr 30, 2015 at 10:45 PM, Wayne Bishop <[hidden email]> wrote:
Congratulations on their high standards, competent instruction, and appropriate recognition of genuine excellence.  The biggest downside, however, is huge:
http://www.exeter.edu/admissions/109_1370.aspx


The question is, why do you have to be an $nnK/yr private school to teach Pythonic Math?
 

Our great educational malfeasance is to not routinely offer such opportunities – at least some approximation thereof – to ordinary stustudents with ordinary socioeconomic status much less to children from communities of exceptionally low education level and socioeconomic status.

Wayne



I completely agree.  It's more than malfeascance though, it's deliberately holding back the 99% so the kids of the 1% can stay privileged.

The US as we know it would fall apart if anything close to real democracy were practiced.  It's not about economics as much as class privilege.

Happy May Day!

Kirby



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Re: Phillips Exeter Academy's math problem sets

Bishop, Wayne
My bad.  Thanks for the correction,

Wayne

At 01:49 PM 5/2/2015, Jake W wrote:
[Wayne] Our great educational malfeasance is to not routinely offer such opportunities – at least some approximation thereof – to ordinary students with ordinary socioeconomic status much lesss to children from communities of exceptionally low education level and socioeconomic status.

But don't you think this

http://tinyurl.com/mr299er

and this

http://tinyurl.com/m5ulnbu

are pretty much equivalent to the PEA questions?

-Timotha

On Sat, May 2, 2015 at 12:16 PM, Wayne Bishop <[hidden email]> wrote:
I could not disagree more.  This is not class warfare; it is a ridiculously growth of our "professional" educational schools.

Wayne

At 06:53 AM 5/1/2015, kirby urner wrote:


On Thu, Apr 30, 2015 at 10:45 PM, Wayne Bishop <[hidden email]> wrote:
Congratulations on their high standards, competent instruction, and appropriate recognition of genuine excellence.  The biggest downside, however, is huge:
http://www.exeter.edu/admissions/109_1370.aspx


The question is, why do you have to be an $nnK/yr private school to teach Pythonic Math?
 
Our great educational malfeasance is to not routinely offer such opportunities – at least some approximation thereof – to or ordinary stustudents with ordinary socioeconomic status much less to children from communities of exceptionally low education level and socioeconomic status.
Wayne


I completely agree.  It's more than malfeascance though, it's deliberately holding back the 99% so the kids of the 1% can stay privileged.

The US as we know it would fall apart if anything close to real democracy were practiced.  It's not about economics as much as class privilege.

Happy May Day!

Kirby

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Re: Phillips Exeter Academy's math problem sets

Bishop, Wayne
For those unfamiliar with dy/Dan's mentor at Stanford:
http://www.piedmont.k12.ca.us/learn/blog/2015/01/18/jo-boaler-stanford-university-professor-advocating-for-change-in-u-s-math-classrooms/
Don't miss her prestigious position at Stanford, Professor of Mathematics.

Whether or not her claim that, "All students can learn higher mathematics.", is accurate depends mostly on one's understanding of what "higher mathematics" means, but her history certainly tells us that is not what some of us mean.  Her famous "Railside", reported in Education Week (and made her nationally famous) that, "41 percent of the Railside students had taken calculus by the end of 12th grade."  Unreported was that none of those students took the College Boards exam AP Calculus AB nor that, if any of them had taken it, their score would have been 1, the score you get for signing up and showing up for the exam.  Many of those who then went to a CSU campus did not pass the CSU ELM (system-wide Elementary Level Mathematics test) so had to start with remedial non-credit math.
  http://www.edweek.org/ew/articles/2005/02/16/23math.h24.html?querystring=viadero&levelId=2300

Wayne

At 06:31 AM 5/4/2015, Wayne Bishop wrote:
My bad.  Thanks for the correction,

Wayne

At 01:49 PM 5/2/2015, Jake W wrote:
[Wayne] Our great educational malfeasance is to not routinely offer such opportunities – at least some approximation thereof – to ordinary students with ordinary socioeconomic status much lesss to children from communities of exceptionally low education level and socioeconomic status.

But don't you think this

http://tinyurl.com/mr299er

and this

http://tinyurl.com/m5ulnbu

are pretty much equivalent to the PEA questions?

-Timotha

On Sat, May 2, 2015 at 12:16 PM, Wayne Bishop <[hidden email]> wrote:
I could not disagree more.  This is not class warfare; it is a ridiculously growth of our "professional" educational schools.
Wayne
At 06:53 AM 5/1/2015, kirby urner wrote:


On Thu, Apr 30, 2015 at 10:45 PM, Wayne Bishop <[hidden email]> wrote:
Congratulations on their high standards, competent instruction, and appropriate recognition of genuine excellence.  The biggest downside, however, is huge:
http://www.exeter.edu/admissions/109_1370.aspx


The question is, why do you have to be an $nnK/yr private school to teach Pythonic Math?
 
Our great educational malfeasance is to not routinely offer such opportunities – at least some approximation thereof – to or ordinary stustudents with ordinary socioeconomic status much less to children from communities of exceptionally low education level and socioeconomic status.
Wayne


I completely agree.  It's more than malfeascance though, it's deliberately holding back the 99% so the kids of the 1% can stay privileged.
The US as we know it would fall apart if anything close to real democracy were practiced.  It's not about economics as much as class privilege.
Happy May Day!
Kirby
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Re: Phillips Exeter Academy's math problem sets

Dave L. Renfro
In reply to this post by Dave L. Renfro
Dave L. Renfro wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=9761775

> Incidentally, the graph on p. 4 is messed up (I didn't make the .pdf file),
> but if anyone is interested, I can (using the new computer I bought in July 2013)
> supply a corrected version.

[See the rest of my post (URL above) for more context.]

O-K, I reformatted the handout a little this weekend (better page
breaks and graphs), and the reformatted version is attached to this post.

Incidentally, in my original handouts (not just this one) I often used
the A4-paper style setting and put in page breaks in order to fit a tiny
bit more on some pages. Also, I often structured things so that there was
either one page or an even number of pages, and in the case of an even
number of pages I made photocopies so that both sides of the paper were
used. However, many of the .pdf files of my stuff I've posted have wound
up looking "ugly" because they were not appropriately reformatted prior
to conversion to .pdf files.

Dave L. Renfro

cal-1W2.pdf (171K) Download Attachment
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Re: Phillips Exeter Academy's math problem sets

Richard Strausz
In reply to this post by Dave L. Renfro
Wayne, it be satisfying to see how much of the PEA math department's philosophy and policies matches what you have long preached:
=====

The goal of the Mathematics Department is that all of our students understand and appreciate the mathematics they are studying; that they can read it, write it, explore it and communicate it with confidence; and that they will be able to use mathematics as they need to in their lives.

We believe that problem solving (investigating, conjecturing, predicting, analyzing, and verifying), followed by a well-reasoned presentation of results, is central to the process of learning mathematics, and that this learning happens most effectively in a cooperative, student-centered classroom.

We see the following tenets as fundamental to our curriculum:

* that algebra is important as a modeling and problem-solving tool,with sufficient emphasis placed on technical facility to allow conceptual understanding;
* that geometry in two and three dimensions be integrated across topics at all levels and include coordinate and transformational approaches;
* that the study of vectors, matrices, counting, data analysis and other topics from discrete mathematics be woven into core courses;
* that computer-based and calculator-based activities be part of our courses;
* that all topics be explored visually, symbolically and verbally;
* that developing problem-solving strategies depends on an accumulated body of knowledge...
============
Richard

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Re: Phillips Exeter Academy's math problem sets

Bishop, Wayne
In reply to this post by Dave L. Renfro
Healthy exercises, no doubt, but I particularly liked the easiest one
stuck in among the various permutations, y = (pi)^(pi).

Wayne

At 09:03 AM 5/4/2015, Dave L. Renfro wrote:

>Dave L. Renfro wrote (in part):
>
>http://mathforum.org/kb/message.jspa?messageID=9761775
>
> > Incidentally, the graph on p. 4 is messed up (I didn't make the .pdf file),
> > but if anyone is interested, I can (using the new computer I
> bought in July 2013)
> > supply a corrected version.
>
>[See the rest of my post (URL above) for more context.]
>
>O-K, I reformatted the handout a little this weekend (better page
>breaks and graphs), and the reformatted version is attached to this post.
>
>Incidentally, in my original handouts (not just this one) I often used
>the A4-paper style setting and put in page breaks in order to fit a tiny
>bit more on some pages. Also, I often structured things so that there was
>either one page or an even number of pages, and in the case of an even
>number of pages I made photocopies so that both sides of the paper were
>used. However, many of the .pdf files of my stuff I've posted have wound
>up looking "ugly" because they were not appropriately reformatted prior
>to conversion to .pdf files.
>
>Dave L. Renfro
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Re: Phillips Exeter Academy's math problem sets

Dave L. Renfro
In reply to this post by Dave L. Renfro
Wayne Bishop wrote:

http://mathforum.org/kb/message.jspa?messageID=9763902

> Healthy exercises, no doubt, but I particularly liked the easiest one
> stuck in among the various permutations, y = (pi)^(pi).

I liked to put stuff like this on tests, but I always went over it
in class (or put it on home/class work like this) and warned them
something like it would probably be on the test. This way there's
no "gotcha" reaction. In fact, students tended to like it because
I made its point value worth the same as the others in its section
of problems.

Of course, there's always the question (very reasonable I think)
of why the n*x^(n-1) rule fails or why the (a^x)(ln a) rule fails,
which makes for a useful teaching moment. I would put both rules
(in general form) on the board and then we'd look at what is going on.

Letting u = u(x) be some general function of x, the two differentiation
rules are

d/dx of u^n equals n * u^(n-1) * u'

and

d/dx of a^u equals a^u * (ln a) * u'

Using the first rule, with n = pi and u(x) = pi, we get
d/dx of pi^pi equal to 0 because u' = d(pi)/dx = 0.

Using the second rule, with a = pi and u(x) = pi, we get
d/dx of pi^pi equal to 0 because u' = d(pi)/dx = 0.

Dave L. Renfro
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Re: Phillips Exeter Academy's math problem sets

Bishop, Wayne
In reply to this post by Richard Strausz
Let us start with the last in your list:
* that developing problem-solving strategies
depends on an accumulated body of knowledge...
I could not agree more.

The rest of the list (actually, that one
included) is entirely consistent with the
"integrated math" of the high-performing schools
all over the world often identified as Maths 1,
2, 3, and 4 (with 4 – often the 4 or even 3 and 4
– only for those headed for the university ready
to successfully pursue math-based "STEM"
disciplines.  If you look at their materials,
presentation in development of concepts is very
similar to the good old days in the US
(absolutely NOTHING like our so-called
"Integrated Math").  The difference is once
taught and supposedly learned, it is assumed
known and used thereafter – students'
responsibility if you can imagine.  I
enthusiastically support the approach and the
collective student performance results it
generates. Try it sometime instead of preaching
math-avoidance of the dy/Dan variety.  You might find it effective, too.

Wayne

At 09:57 AM 5/4/2015, Richard Strausz wrote:

>Wayne, it be satisfying to see how much of the
>PEA math department's philosophy and policies
>matches what you have long preached:
>=====
>
>The goal of the Mathematics Department is that
>all of our students understand and appreciate
>the mathematics they are studying; that they can
>read it, write it, explore it and communicate it
>with confidence; and that they will be able to
>use mathematics as they need to in their lives.
>
>We believe that problem solving (investigating,
>conjecturing, predicting, analyzing, and
>verifying), followed by a well-reasoned
>presentation of results, is central to the
>process of learning mathematics, and that this
>learning happens most effectively in a cooperative, student-centered classroom.
>
>We see the following tenets as fundamental to our curriculum:
>
>* that algebra is important as a modeling and
>problem-solving tool,with sufficient emphasis
>placed on technical facility to allow conceptual understanding;
>* that geometry in two and three dimensions be
>integrated across topics at all levels and
>include coordinate and transformational approaches;
>* that the study of vectors, matrices, counting,
>data analysis and other topics from discrete
>mathematics be woven into core courses;
>* that computer-based and calculator-based activities be part of our courses;
>* that all topics be explored visually, symbolically and verbally;
>* that developing problem-solving strategies
>depends on an accumulated body of knowledge...
>============
>Richard
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Re: Phillips Exeter Academy's math problem sets

Richard Strausz
In reply to this post by Dave L. Renfro
> Let us start with the last in your list:
> * that developing problem-solving strategies
> depends on an accumulated body of knowledge...
> I could not agree more.
>
> The rest of the list (actually, that one
> included) is entirely consistent with the
> "integrated math" of the high-performing schools
> all over the world often identified as Maths 1,
> 2, 3, and 4 (with 4 – often the 4 or even 3 and 4
> – only for those headed for the university ready
> to successfully pursue math-based "STEM"
> disciplines.  If you look at their materials,
> presentation in development of concepts is very
> similar to the good old days in the US
> (absolutely NOTHING like our so-called
> "Integrated Math").  The difference is once
> taught and supposedly learned, it is assumed
> known and used thereafter – students'
> responsibility if you can imagine.  I
> enthusiastically support the approach and the
> collective student performance results it
> generates. Try it sometime instead of preaching
> math-avoidance of the dy/Dan variety.  You might find
> it effective, too.
>
> Wayne
>
Snide remarks aside, you and I agree with what you chose to reprint. *However*, things you chose *not* to print from the PEA philosophy are things you have advocated strongly against in all your years on Math Teach. They include:

...learning happens most effectively in a cooperative, student-centered classroom.
...data analysis and other topics from discrete
mathematics (should) be woven into core courses;
...that computer-based and calculator-based activities be part of our courses

If we can get you to sign off on these, we have a historic agreement!

Richard
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