Simpson's Rule on Cubics

Simpson's rule approximates a definite integral

by replacing the integrand f with the quadratic  function that agrees with f at the endpoints and midpoint of each sub-interval.  (For comparison, the Left- and Right-Hand Riemann sums each replace f with a constant function, the Trapezoid and Midpoint rules replace f with the linear function respectively agreeing with f at the endpoints of the interval or agreeing with both f and f' at the midpoint of the interval.)

It is remarkable that Simpson's rule gives the exact values of definite integrals not only for any quadratic but also for any cubic polynomial, using only one sub-interval.

This can be algebraically verified by using the change of variable x = a + (b - a)t and verifying that Simpson's rule with one sub- interval gives the exact value for

Here is a more geometric argument.

Let f be a cubic polynomial, and let q be the quadratic function satisfying f(a) = q(a), f(b) = q(b), and f((a+b)/2) = q((a+b)/2).

Then the error in using Simpson's rule for approximating

is

where E is the cubic polynomial defined by E(x) = q(x) - f(x).

Because  E(a) = E(b) = E((a+b)/2) = 0, the inflection point in the graph of E occurs at x = (a+b)/2.  Cubic polynomials are symmetric about their inflections points, so the region lying between the curve and the x-axis on one side of the inflection point is congruent to the region between the curve and the x-axis on the others side of the inflection point. 

Hence

That is, the approximation has no error.

CCSS and Community College Math Programs

We may need a complete redesign of the developmental math program in US two-year colleges.

My campus currently uses a placement test (Mathematics Diagnostic Test Project) to determine if students are ready for transfer level courses (math for elementary school teachers, stats, trig, precalculus, calculus)  or what remedial course (arithmetic, prealgebra, elementary algebra, intermediate algebra) they should take.

But the Common Core State Standards for mathematics will have high school students studying mathematics organized in a fashion that does not align with our existing math courses.

California is one of the 45 states that have formally adopted the CCSS for mathematics, and I am on a recently appointed state committee whose charge is to align California’s math standards (a.k.a. the California Framework) with the CCSS.

One of the main reasons that I applied to be on the Mathematics Curriculum Framework and Evaluation Criteria Committee (MCFCC) was to better familiarize myself with what is to be taught in California's K-12 schools.  (Another reason was to lose myself in abbreviations:  SBE for State Board of Education, CDE for California Department of Education, IQC for Instructional Quality Commission, the body that forwarded my name to the SBE for approval to serve on the MFCC to align the CF with the CCSS.)

The CCSS specify a consensus of what math is required for students to be college or career ready.  The standards are grouped into six conceptual categories:  Number and Quantity, Algebra, Functions, Modeling, Geometry, and Statistics and Probability.  (There are separately eight standards for mathematical practice that go across all grade levels.)

The CCSS differ significantly from what is typically required for graduation in most American high schools today.  For example, the treatment of statistics and probability includes not only descriptive statistics but also conditional probability, inference, decisions based on probability, and rules of  probability. 

The CCSS include not only right-triangle trigonometry but also trig functions of a real variable, to be used in modeling periodic behavior.  Thus trig spans the geometry, algebra, and function categories.

The CCSS gives math standards for high school without specifying courses or order of topics.  But evidently the introduction of functions includes an emphasis on (linear and) exponential functions with domains restricted to a subset of the integers--sequences are explicitly studied as functions.

California community colleges do not require a high school diploma for admission.  A student who masters the first CCSS high school math course will already have compared exponential functions with linear functions and solved equations both algebraically and graphically. The student will have had explicit instruction on descriptive statistics.  The student may have worked with constructions and transformations in the plane and proved simple geometric theorems algebraically but not yet worked with polynomials (and specifically not with quadratic functions or quadratic equations).

How will our placement system advise this student?

One of the recommendations  of California's Student SuccessTask Force is for better alignment between high school and college curricula.  With the CCSS adopted across states, it looks as if most community colleges will need to make adjustments to their way of placing and educating their math students.

MathFest 2012 and Common Core State Standards in Math

Andrew Hacker’s article "Is Algebra Necessary?" in the New York Times was a hot topic last week and mentioned by several presenters at the 2012 MathFest session "What Mathematics Should Every Citizen Know?".  The panelists, Bil lMcCallum, Lynn Steen, Hyman Bass, Joseph Malkevitch, and co-organizer Sol Garfunkel, were actually reacting to the Core Curriculum State Standards in mathematics.

Mathematicians and math educators agree that we are not  currently doing the best job of teaching algebra.  But unlike Hacker, the math community believes the appropriate strategy is to improve algebra instruction, not to abandon it to all but an elite few pupils.

On the other hand, the speakers on the panel, although quite civil with each other, clearly had disagreements about the best strategy to improve math education in the US.

McCallum, who was the lead mathematician in the development of the CCSS, emphasized the benefits of having commonality across states.  Having a set of standards that could be adopted by 45 of the 50 states (so far) required compromises, but the benefits accrue not only to pupils and teachers in our mobile society, but to all who do business with textbook publishers who currently provide materials for the multitude of different curricula.

Steen gave some numbers showing the dismal success of preparing US students for STEM, but argued that we should improve rather than remove algebra from the curriculum.  He favors a modeling-based approach and avoidance of common assessments.  When asked how to accomplish his recommendations, he cheerfully remarked that he doesn't need to worry about that now that he's retired.

Bass focused on pedagogy rather than curriculum as the key to improving math education.  Student learning is increased when the instructor employs appropriate classroom strategies.

Malkevitch promotes widening the curriculum.  He argued that we need to show many ways that mathematics impinges on daily lives.  He gave combinatorial graphs and fair choice algorithms as examples of mathematical topics that are new and accessible to very young children.

Garfunkel believes that the entire K-12 mathematics curriculum should be centered on modeling. He echoed Malkevitch's suggestions that the US curriculum needs to be widened, and said that Bill Schmidt had paid an advertising agency to create the phrase "a mile wide and an inch deep" that is used to characterize the US K-12 curriculum following the disappointing ranking of the US high school students in the Third International Mathematics and Science Study.

Reviewing Grant Proposals

I recently served on a panel reading NSF grant proposals.  We were admonished not only to respect the confidentiality of the proposal contents but also not to divulge the dates we did the reviews nor the title or nature of the grant types being solicited.

I've reviewed NSF grant proposals on about a half dozen occasions, and I've enjoyed each of my experiences.   There are typically 4 or 5 members to a committee, a few dozen committees representing the different STEM disciplines, and each committee is assigned about a dozen proposals to review.  NSF tries to create panels with a diversity of geographical regions and of institution type  (research universities, state universities, liberal arts colleges, and sometimes even two-year colleges). NSF brings in both faculty and administrators.

Panelists register online using the NSF program FastLane long before they see any of the proposals. FastLane facilitates many aspects of the review process, such as tracking panelist coordinates and other information for  travel arrangements, giving access to proposals, recording panelist reviews, and arranging letters to home institutions acknowledging service.

Panelists upload to Fastlane their personal evaluations of each proposal before going to DC and physically meeting as a panel.  Project descriptions are restricted to 15 pages, but with all the additional documentation of references, biographical sketches, budget, etc., the entire proposal usually exceeds 50 pages.  So it is also common that when the panels finally meet in DC on the first morning, not all the panelists have successfully prepared reviews of all the proposals.

The panels meet essentially all of the first day discussing each proposal.  Panelists are logged on the FastLane program during discussion.  This allows each panelist the ability to see the reviews of all the panelists for each proposal.

NSF assigns each panelist to be "scribe" for two or three proposals, and the scribe's task is to record the discussion about the proposal.  NSF does not require consensus, but it is common that the discussions persuade some panelists to change their initial evaluations.  Most panelists work into the night preparing the summaries required by their duties as scribe and resubmitting their personal proposal reviews.

The second morning is spent largely in approving the scribe summaries.  Again the panelists are logged on FastLane, and they are not allowed to leave until each summary has been approved.  NSF reconvenes the panels by discipline late in the morning for a debriefing before sending the panelists back home.

The debriefing is one of the highlights of the whole experience, because each panel typically describes its two favorite proposals.  So we hear the highlights, and it's exciting to learn of innovative ideas that are being pursued.

MS Input Panel and Equation Writer

Microsoft's Equation Writer converts handwritten mathematical expressions to typeset expressions which can be inserted into MSWord documents.  I played with the Equation Writer in January 2010 and found it amusing but not usable.

Now Microsoft's Math Input Panel, which I found under Accessories in Windows 7, is definitely an improvement.

I first tried writing the same equation that had given the Equation Writer so much trouble.

Although my handwriting has not improved, the Math Input Panel recognizes my writing much better than the Equation Writer had, provides a more pleasingly formatted result, and is easy to edit.

The "Select and Correct" allows you to lasso a part of your expression and gives you a list of likely interpretations.  Sadly, the program's best guesses for my cursive "dx"were "m," omega,"max," "cos," "log," "def," or the infinity symbol.  But it was easy to erase my connected "dx" and replace it with two separate letters, which were properly interpreted.

To get the expression into a Word document, I needed to have both the Word document and  the Input Panel visible, I clicked on the point of insertion in the Word document, then clicked on "Insert" in the Input Panel.

What might make this worthwhile is that once you insert the expression into a MSWord document, you can use the Equation Tools.

For example, suppose you've input "sin x" using either the Math Input Panel or typed directly into the Equation editor (by first simultaneously pressing the Alt and = keys).  Now when you right click on the expression, you get a menu of possible actions.

If you click on Integrate on x:

It's interesting that the constant of integration is included in the result.  What's also interesting is that if you type a 1 immediately after the C, right-click to get the menu of options, then select Graph -> Plot in 2D, you get an insertable graph that can first be reformatted and/or display an animation based on varying the constant of integration.

You can also go back and right click on your antiderivative, integrate on x, and type a 2 immediately after your new constant of integration.  Now when you ask for a 2D graph, you can run an animation based on either of the two constants.

And yes, you can draw 3D graphs in your Word document. Press the Alt key and the = key simultaneously, type in an expression in two variables (such as sin x + cos y), right click and select Graph -> Plot in 3D.

You can drag the image to a preferred orientation before inserting it into your Word document.

You can also use the "Calculate" option to solve some simple equations or to evaluate/simplify algebraic expressions.

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.