Wednesday, December 28, 2011

sin(n) is dense in [-1,1]

Numbers of the form sin(n) get arbitrarily close to every real value from -1 to 1.

Consider the integers mod(2pi).   Equivalently, we can consider the points (cos n, sin n) on the unit circle (a.k.a. using the "wrapping function").   Because we have an infinite set of points in a compact set, there must be an accumulation point. 

For any epsilon > 0, we can find distinct integers m and n such that  (cos m, sin m)  and  (cos n, sin n)  are less than epsilon apart on the circle, so  (cos(m-n), sin(m-n))  is within epsilon of (1,0).   Then (by say the Archimedean principle),  every point on the unit circle is within epsilon of some integer multiple of (m-n) wrapped around the unit circle .   Thus the points of the form (cos k, sin k)  form a dense subset of the unit circle.  In particular, numbers of the form sin(k) form a dense subset of [-1,1].

A slightly different argument for density of sin(n) depends on the following lemma.

Lemma:  If x is irrational, then the additive group Z + xZ is dense in R.

If we accept the lemma, then the additive group Z + (2pi)Z is dense in R, so the points (cos t, sin t) for t in Z + (2pi)Z must be dense in the unit circle.  But (cos (n+m2pi), sin (n+m2pi)) = (cos n , sin n), so (cos n , sin n) is dense in the circle.

To prove the lemma, consider the contrapositive and assume that Z + xZ is discrete.  Then Z + xZ must be cyclic:  Z + xZ = dZ for some real number d.  Because Z is contained in dZ, there must be some integer n so that 1 = nd, hence = 1/n is rational.  And then for some m, 1+m/n, so x is rational.

And the converse of the lemma is also true, because of m/n is rational, then Z + xZ =  1/n Z.

Saturday, November 19, 2011

Summing 1/n^2

Here's how Euler evidently first evaluated the sum

We know the MacLaurin series for sin x, and hence

On the other hand, the zeros of this function are the nonzero integer multiples of pi, and writing the series in a factored form, we obtain

Expanding the last (infinite) product of binomials and equating its quadratic coefficient with that of the original MacLaurin series, we obtain

Euler's result follows directly.

Tuesday, June 21, 2011

Productive Persistence, Explicit Connections, and Deliberate Practice

The Carnegie Foundation for the Advancement of Teaching is promoting three key underpinnings to the Statway™ lesson design:  productive persistence, explicit connections, and deliberate practice.

The ideas are intertwined, but of the three, deliberate practice is the hardest to grasp.

Hiebert and Grouws found that mere student engagement was not enough for a program to succeed. The students actually needed to grapple and struggle with their mathematics in all the programs that were successful in achieving high student learning.   That is, all the successful programs fostered productive persistence among the students.

Nor are well-designed, challenging activities necessarily enough for student learning.  Students need to have the connections between what they are doing and desired mathematical learning goals to be made explicit.

But the very phrase "deliberate practice" seems to give the wrong impression.

Most math teachers are familiar with exercise sets consisting essentially of minor variations of the same skill.  Working such exercise sets is not deliberate practice. 

Deliberate practice is not about repetition and drill.  The goal of deliberate practice is not simply to  master a procedure within a fixed context or to acquire automaticity, but rather to broaden the student's understanding of concepts through an organized set of tasks designed to challenge the student to synthesize and extend ideas previously learned.

The Statway™ webinar featuring Jim Stigler and Karen Givvin discussing productive persistence, explicit connections, and deliberate practice was recorded.

Sunday, April 24, 2011

Statway Research About Teaching and Learning

Statway Institute - Jim Stigler - Winter 2011 from Statway on Vimeo.

Statway is one project of the Carnegie Foundation for the Advancement of Teaching seeking to find alternative math pathways to baccalaureate degrees.  There are so many interesting pieces going into Statway that the project promises to provide useful information even to educators who are appalled at the idea of allowing a non-STEM major to earn a BA without passing an intermediate algebra class.

Carnegie posts Statway resources on their site.  Here are a few I find particularly interesting.

  • David Yeager's video discusses productive persistence in students.  This was probably the most talked about presentation at the 2011 Statway mid-year institute.  Skip to 7:00 into the video for data on improved success for community college students (+17 percentage points) by introducing self-regulated learning.  Skip to 10:15 for discussion of Carol Dweck's work on mindsets, or skip to 18:40 for improvement resulting from a single 45-minute psychological intervention (+.3 gpa).  Go to 21:00 for a discussion of stereotype threat, with data on student impact (-39% memory span; -13% on a math test at 24:10) because of the threat, or go to 25:15 to learn how the stereotype threat can be eliminated with two 15-minute interventions. 
  • Jim Stigler's video (above) describes teaching as a cultural activity.  Stigler outlines some of the challenges US educators face to adopt effective practices that are the norm in other countries.  (Hint:  In the typical classroom of the countries that are top-ranked because of high student performance in math and science, the students are expected to struggle with problems that they have not been shown how to solve, and the instructors allow the students to be frustrated for much longer than American teachers could tolerate.) 
  • Jim Stigler, Karen Givven, and Belinda Thompson (all of UCLA) reported to Carnegie on "What Community College Developmental Mathematics Students Understand about Mathematics."  (The report was later the basis of an article of the same title in the MathAMATYC Educator.)

Sunday, March 20, 2011

ICTCM 2011

There were about 750 participants at the International Conference on Technology in Collegiate Mathematics this year in Denver (March 17-20).  The keynoter Theo Gray gave an exciting talk about his vision of what textbooks should be.  He gave snippets of his Elements ebook, which was enough to make me want an iPad.

Lila Roberts gave a great start to the Emerging Technologies strand of presentations.  She proposes widely utilizing browser-independent applets, that is, applets based on HTML5 and javascript rather than using Flash or Java.   A few free resources  mentioned in her talk that I want to explore: for creating and storing slideshows online, for screen capture videos via browser,  MathJax for displaying math notation online, and JSXgraph for dynamic  graphs.

Lila also mentioned WolframAlpha widgets.   You can easily create and embed a Wolfram|Alpha applet in your webpage or Learning Management System (Blackboard, Moodle , WebCT, Angel, etc.) , or simply embed one of the existing widgets from their gallery (as above).

Susan McCourt mentioned embedding videos during her talk about engaging students in discussion boards.  Her YouTube video shows how to  embed a Jing video in a discussion board so that the actual video is on the discussion board, not merely a link to a video.

I was not encouraged by the course redesign sessions I attended.  The strategy appears to limit the curriculum to exercises that computers can grade.  I was in agreement with the speaker when she said that we should automate what is best done by automation, but she lost me when her next statement was that we should never grade homework again.

At another redesign session, the school's goal was to improve the college algebra success rate of their students who pass intermediate algebra.  That goal was reached admirably, but at an expense of lowering the pass rate in intermediate to the level that there did not appear to be any more students able to progress through both classes than before the redesign.

And in the a third redesign session I attended, the speaker confirmed that in Tennessee, intermediate algebra is no longer a developmental course, so that elementary algebra (with systems of equations removed) was now the prerequisite for some college math courses.

I had agreed to man the keyboard for Fred Feldon's Friday morning talk on Wolfram|Alpha.  I arrived early to make sure I could work ok with the provided laptop.  Then Sharon Sledge walked in with an unusual request:  would Fred and I be willing to take over the Wolfram|Alpha workshop that was starting an hour after Fred's talk?  The scheduled speaker cancelled that morning, but the workshop was completely booked.

I think our improvised workshop went reasonably well, but I did need to spend the hour between those sessions editing and uploading some materials I was working on for an AMATYC webinar in May.  

Sunday, January 9, 2011

Joint Mathematics Meetings January 2011

The 2011 Joint Math Meetings started for me with the advisory board meeting of the Mathematical Sciences Digital Library (MathDL). Both the MAA Reviews and the Classroom Capsules are currently free on the MathDL site to both MAA members and non-members.  The Reviews are useful mainly to faculty and libraries making decisions about purchasing books, but the "Classroom Capsules and Notes brings together the best of 114 years of the short classroom materials from the MAA print publications" and should be of use to all math faculty.

The MAA now hosts WeBWorK, an open source (free) web-based homework system originally out of Rochester U.  If your institution is unable or unwilling to host WeBWorK on its own server, the MAA will let you try out WeBWorK with up to 100 students from your school.  (And if you want to try it out with more students, ask anyway, and the MAA will try to find a participating institution that can accommodate you.)

Aaron Wangberg of Winona State U. has created an electronic whiteboard to work within WeBWorK.  His students are required to use the whiteboard to show their work before submitting their answers for the instant correct/incorrect feedback.  Mike Gage says this utility will soon be available to all the WeBWorK authoring community.

Warren Esty of Montana State U:  Theorem  The value of math skills has gone way down.  Corollary  We should focus our teaching toward skills that will add value.  Examples:  Being able to read math and to make the right things "come to mind."  (The idea of the latter example is that information and computation are both cheap, but knowing what is relevant to look up or to compute is harder to come by.)

At the CTiME sponsored panel session on the final day I learned a bit more about MathJAX, which is rapidly becoming the standard tool for putting math notation on the Web.  MathJAX is largely the work of Davide Cervone of Union College.

Much of the excitement about MathJAX is that it provides a much needed piece of the puzzle so that the potential of MathML can be realized.  Although MathML has been the World Wide Web Consortium (W3C) standard for putting math on the Web for 15 years, it has not been widely accepted by browsers.  MathJAX allows the author of a webpage to use TeX or MathML to specify math notation, and MathJAX instructs the browser how to display the math properly, even for browsers that do not know how to display MathML.

Steve Wilson (AMATYC Central Vice-President) told me that he spent a few hours in the exhibits room.  Besides the major textbook publishers and mathematical software providers, the exhibitors at JMM 2011 included, which showcases information and resources for students and faculty about careers, etc.  The ALEKS rep would give me no numbers at all about possible costs, but the WebAssign folks were happy to chat about costs or anything else.

One American Mathematical Society booth freebie was a report on an AMS survey about online homework systems.  By far the most frequently adopted were 1) MyMathLab, 2) WebAssign, and 3) WeBWork.  MML involved the most students (230K), then WeBWorK with over 100,000 and WebAssign with less than 100,000.  There was no effort to rate the relative quality of the different homework systems, but I found it interesting that the study divided the PhD granting institutions into those that were among the top 80 (as ranked by the National Research Council) and those ranking below the top 80.  For example, the response rate among the top 80 was 71%, for the rest of the PhD-granting universities the response rate was 65%.  For the masters-granting universities the response rate was 45%, for bachelor's 30%.  "The two-year college numbers were too small for any meaningful analysis" as "Only 11 responded" of the 30 TYC invited to participate.