Math Pathways: Designing for Success

On Friday April 27, 2012, Los Angeles Pierce College hosted a conference to share ideas about curricular and institutional redesign efforts for mathematics at two-year colleges.  A central theme was to improve the rate that  students are able to achieve degrees, certificates, and transfers to four-year institutions--essentially the "to and through" goal embraced by the Statway and Quantway projects of the Carnegie Foundation for the Advancement of Teaching and by the New Mathways projects of the Charles A. Dana Center.

A substantial majority of community college students who take a placement exam place into remedial education courses.  And perhaps only one in five of those who place in remedial math ever succeed in passing a college level math course.

Julie Phelps of Valencia Community College FL was the keynote speaker. She provided a national perspective on the the scope of the problem of students languishing in developmental math classes and discussed some of the initiatives throughout the U.S. that are trying to address the issue.

The break-out sessions at the conference were grouped into three themes:  Before Algebra, STEM Pathways, and Non-STEM Pathways.  At each break-out, math faculty panelists from local community colleges (Pasadena City College, College of the Canyons, and Pierce) discussed strategies being implemented at their campuses.

The conference was sponsored by the California Community Colleges Success Network (3CSN) under the leadership of Deborah Harrington and organized by dean Crystal Kiekel of Pierce College. The 3CSN.org website will host slideshows for not only the keynote presentation from Julie Phelps, but also from the break-out presenters Linda Hintzman, Charlie Hogue, and Roger Yang  of Pasadena City College; Kathy Kubo and Matt Teachout of College of the Canyons; Bob Martinez, Jenni Martinez, Ben Smith, Kathie Yoder, and Kathy Yoshiwara of Los Angeles Pierce College.

ICTCM 2012

The 24th International Conference on Technology in Collegiate Mathematics (March 23-24, 2012) in Orlando, FL, had about 1000 participants.  Keynoter Conrad  Wolfram ("Stop Teaching Calculating, Start Teaching Math") told the audience that the way to fix math education is to adopt computer-based math.

This is the message in his TED video.  

Wolfram likened the teaching of paper-and-pencil computations to ancient Greek.  It's great that some people want to study such things, but these topics should not be part of a core education.  Because people (other than math teachers) in the real world who need mathematics do their computations with computers, we should not be teaching computations but teaching instead how to ask the correct questions, how to translate the questions into mathematical syntax, and how to interpret the results of computer calculations into a solution in the real world.

He also gave the analogy that composition is to English as programming is to mathematics.  We should be teaching programming in our math classes, but using a higher level language such as (coincidentally) Mathematica.

I was personally involved in two ICTCM sessions.  I co-presented with Julie Phelps and Andre Freeman on a talk about the Statway and Quantway projects of the Carnegie Foundation for the Advancement of Teaching.  Andre did the lion's share, describing the homework (a.k.a "out of classroom experience") system MyStatway (based on Carnegie Mellon's OLI statistics course) and spreadsheet simulations that are part of the Statway package of resources.

I also gave a solo session: "Knowledge Exchange Networks and MathDL".

The sessions highlighting NCAT Emporium models for course redesign continue not to impress me.  At least one two-year college campus can claim that the students show great success (not only in developmental math courses but also) in the transfer math courses following an Emporium model developmental math course.  But the (unstated) caveat is that all the classes are taught in the Emporium model, which means that the student assessments in the transfer math course are all graded by the computer, specifically by MyMathLab.  (I do not believe that MyMathLab or any other current computer-graded system can reasonably score questions that ask for interpretations or explanations in complete sentences, but I  believe that we should expect our college students to be able to answer such questions.)

I liked Valencia Community College's idea of a "Math 24/7 Tutorial Website" (Jody DeVoe, Cathy Ferrer, and Jennifer Lawhon).  25 VCC math faculty created hundreds of videos (via Smartpens, flip cameras, Jing, etc.) and then created a webpage of links.

I'll also want to think more about Sarah Mabrouk's one-way use of Twitter--students follow her (class-specific Twitter account), she does not follow any students--to increase student engagement.

But I never made it to any of the theme parks.

Joint Mathematics Meetings 2012

About 7000 mathematicians gathered in Boston for the 2012 Joint Mathematics Meeting January 4-7, 2012.

My first event was the Mathematics Digital Library advisory board meeting.  One of MathDL director Lang Moore's items was that the MathDL's  Course Communities  now include Developmental Math. These resources are from the Knowledge Exchange Networks spearheaded by Tom Carey of U. Toronto, San Diego State U., and the Carnegie Foundation for the Advancement of Teaching.

Haynes Miller (MIT) reported on  MIT's online resource for faculty collaboration.   First developed as an in-house tool to help facilitate writing across the curriculum, the resource is now being opened up for faculty collaborations throughout the US.

The CRAFTY (Curriculum Renewal Across the First Two Years) committee sponsored a contributed papers session on preparing students for calculus, and the College Board/MAA Committee on Mutual Concerns sponsored a panel session  on promoting student success in calculus.  Alison Ahlgren of U. Illinois and Marilyn Carlson of Arizona State University painted vastly different pictures.

Alison believes the UI has found a working solution using ALEKS (Assessent and LEarning in Knowledge Spaces) as both an assessment and placement tool.  All UI students receive ALEKS assessment, and ALEKS placement scores are strictly enforced--even students who pass the UI precalculus course must earn the appropriate ALEKS score to be eligible to enroll in calculus.

Alison has data from many thousands of students on what ALEKS tests.  Analysis of what topics were or were not mastered by successful vs unsuccessful students informs UI about what precalculus topics should receive greater of less emphasis.  For example, UI students all appear to have mastered polynomials, but the great majority of pre-calculus students have little mastery of exponential functions and logarithms.

In contrast, Marilyn argues that ALEKS and other currently popular assessment systems do not measure conceptual understanding, and hence student success with ALEKS does not necessarily facilitate mastery of what we really want our students to learn.  Marilyn led the development of ASU's Precalculus Concept Assessment.

The Committee for Two-Year Colleges sponsored a panel session on math for non-STEM students.  Panelists Bernadine Chuck Fong and Jane Muhich of the Carnegie Foundation spoke about the philosophical and research-based underpinnings of the Statway™ and Quantway™ projects, while Larry Gray of U. Minnesota spoke on his reaction to the actual Statway™ lessons which he has reviewed.  The audience included Mary Parker of Austin Community College and Katherine Yoshiwara of Los Angeles Pierce College, both faculty teaching  Statway™ this year, and both speaking in favor of the project but indicating that there is still much work to be done.

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 d = 1/n is rational.  And then for some m, 1+x = m/n, so x is rational.

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

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.

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.

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.)