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 
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 This is a serious school where faculty sources much of the curriculum. 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 MATH470 (WS) 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 MATH470 ; a student who wishes to receive computer science credit should sign up for COMP470 . Prerequisite: MATH330 or permission of the department. Kirby 
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 
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 PhillipsExeter's renowned math department makes mention of TI89 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. 
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: The question is, why do you have to be an $nnK/yr private school to teach Pythonic Math?
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 
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 lessaffluent, lessprivileged 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 problembased 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. 
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 > PhillipsExeter's renowned math department makes mention of TI89 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 mathteach 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 mathteach), (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 nontrivial to use a calculator to numerically calculate (see problem 8 on pp. 45), (d) I wanted something that involved a lot of interplay between mathematical reasoning and calculator use (see E on pp. 57; 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/20618743486544585537994/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 
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 
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 
In reply to this post by kirby urner4
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: 
On Sat, May 2, 2015 at 9:16 AM, Wayne Bishop <[hidden email]> wrote:
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 
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:

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. 
For those unfamiliar with dy/Dan's mentor at Stanford:
http://www.piedmont.k12.ca.us/learn/blog/2015/01/18/joboalerstanforduniversityprofessoradvocatingforchangeinusmathclassrooms/ 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 (systemwide Elementary Level Mathematics test) so had to start with remedial noncredit 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, 
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.] OK, 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 A4paper 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 cal1W2.pdf (171K) Download Attachment 
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 wellreasoned presentation of results, is central to the process of learning mathematics, and that this learning happens most effectively in a cooperative, studentcentered classroom. We see the following tenets as fundamental to our curriculum: * that algebra is important as a modeling and problemsolving 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 computerbased and calculatorbased activities be part of our courses; * that all topics be explored visually, symbolically and verbally; * that developing problemsolving strategies depends on an accumulated body of knowledge... ============ Richard 
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.] > >OK, 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 A4paper 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 
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^(n1) 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^(n1) * 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 
In reply to this post by Richard Strausz
Let us start with the last in your list:
* that developing problemsolving 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 highperforming 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 mathbased "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 socalled "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 mathavoidance 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 wellreasoned >presentation of results, is central to the >process of learning mathematics, and that this >learning happens most effectively in a cooperative, studentcentered classroom. > >We see the following tenets as fundamental to our curriculum: > >* that algebra is important as a modeling and >problemsolving 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 computerbased and calculatorbased activities be part of our courses; >* that all topics be explored visually, symbolically and verbally; >* that developing problemsolving strategies >depends on an accumulated body of knowledge... >============ >Richard 
In reply to this post by Dave L. Renfro
> Let us start with the last in your list:
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:
> * that developing problemsolving 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 highperforming 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 mathbased "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 socalled > "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 > mathavoidance of the dy/Dan variety. You might find > it effective, too. > > Wayne > ...learning happens most effectively in a cooperative, studentcentered classroom. ...data analysis and other topics from discrete mathematics (should) be woven into core courses; ...that computerbased and calculatorbased 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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