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 WeUseMath.org, 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.

Gaps in developmental math students' knowledge

The Pierce College Modular Math 115 project for a modular, self-paced, mastery-based elementary algebra course is in its third semester.

We've made adjustments each semester.  This semester we've used the open source online homework system WeBWorK so that students can verify that they have the correct answers on all the drill questions, so we no longer collect those homework sets on a daily basis.

This semester we are acknowledging how inadequately we have met the self-paced aspect of the plan.  The four sections of Modular Math 115 all meet at the same hour so that we can physically relocate students to the classroom that is progressing at the pace most appropriate for them.   In order to accommodate students who cannot master our elementary algebra materials within one semester, we are moving the slower students (electronically) to a course in the district database that covers only the first half of a two-semester elementary algebra course.

Among the 43 students in my classroom, none had mastered even two (of the nine total) units by the end of the twelfth week of fifteen weeks in the semester.

The students suffer all the mathematical gaps we have come to expect, such as inability to distinguish the concepts of area and perimeter.  But I was surprised by how many of the students have only a very shallow understanding of subtraction (and of course of multiplication and division).

They are all capable of computing 5 - 3.  And they can easily answer, "If you had 5 pencils and I took away 3, how many pencils would you still have?"

That particular word problem involves the simplest model for subtraction, "take away".  But my students have trouble with the more sophisticated, "If your pencil box holds 5 pencils and you already have 3, how many more pencils do you need to fill the pencil box?" 

The students do not automatically recognize their task as computing a difference.  Instead, they solve such a problem by counting up from 3, and so they have even more trouble with "If you had some pencils and then I gave you three more so that you had a total of 5 pencils, how many pencils did you have at the start?"

And they have more difficulty with a comparison question, "If you have 3 pencils and I have 5, how many more pencils do I have than you?"

Some of the students seems unfamiliar with the idea of multiplication as repeated addition--they know some multiplication facts but do not recognize that one can compute 3+3+3+3 by multiplying 4*3. 

Even if we assume that our students have access to technology to carry out computations and symbolic manipulations, some of these students do not recognize what calculation or manipulation is useful when given a context outside of pure mathematical computation.

A short wish list for online homework systems

I take it for granted that electronic homework systems cannot effectively grade any math problem that requires students to write coherent sentences.

All the electronic homework systems allow the possibility of students submitting answers that won't be machine graded.  This capability greatly increases the variety of types of questions that can appear in an electronic exercise set.  There is an issue of how students enter math notation and figures (hey, just let them  photograph their handwritten answers with their cellphones and upload the jpg file), but the principal reason that I'm reluctant to include such problems is the fear that grading online will be cumbersome.

There are several ways to make life easier for the instructor faced with grading a single "essay" problem from a large set of students.  First, the interface for viewing individual responses should be intuitive and effortless.  Don't make us click on a link to open one student's response and then have to close that file before opening the next. 

It would be better to have "zoomable" thumbnails of each student's answers spread across the screen, with mouse flicks or dragging to scroll.  Should the instructor have the foresight to provide a grading rubric for the problem, that rubric should be visible (or at least available) to the student when working the problem.

In many cases, the availability of the rubric could reduce the instructor's need for to make copious comments.  For further convenience, the instructor should have a few editable paste buffers holding common comments (like "You need the product rule" or "This is not an equation").

WeBWorK as an answer key

The principal author (Kathy Yoshiwara) of the project materials being used at Pierce believes that our  students misuse the answer keys found at the back of math textbooks.  She believes students need to struggle at times for an answer, rather than always be able simply to find the answers in the book (and to work backwards from there).

On the other hand, we recognize that students can benefit from the  reassurance of knowing that they've successfully solved a math exercise, or from the knowledge that their first efforts were in error.  As part of our student success projects for developmental math, we have been hiring student tutors to check off that students have correct answers before the students are allowed to submit their portfolios.

But we'd prefer that the tutors' time be spent in actually working with the students.  So we have begun to code answers to drill-type exercises into the open source online grading system WeBWorK.  Students will type in their answers online at home or in a computer lab, and will get immediate feedback as to whether or not their answers are correct.

We will still grade by hand the questions that require complete sentences as answers, and we will still  check the student work on the drill problems when we collect portfolios.  But students will now have a means of checking the accuracy of their answers before coming to the classroom and without needing to consult our tutors.

WeBWorK

I've been using the open source homework system WeBWorK for the past few semesters.  The first couple of times, my classes were hosted by the University of Rochester (thanks to Michael Gage) and now by xyzhomework.com (thanks to Patrick McKeague).

The bank of problems in the WeBWorK National Problem Library is thin on the types of problems we want to use at Pierce College in our intermediate algebra class, so I've been authoring most of the problems.  The fact that all the authoring tools are made available to instructors is one of the nice features of WeBWorK, probably second only to the fact that it's free.

The downside is that without the huge work force associated with the big publisher-owned homework systems, the product is not so instructor-friendly.  We don't get all the bells and whistles that we might expect if we've only see the popular MyMathLab from Pearson.

I only just learned how to change a student's score on a graded assignment.  It goes like this:

• Go to the Classlist Editor (the first link under Instructor Tools in the Main Menu).
• In the row for the student of interest, click on the "Assigned Sets" value (in the fourth column--it has a form like  "m/n").
• Click on the name of the appropriate set (in the "Edit set for..." column).
• Find the problem of interest, and adjust the value of "status" (typically change "0" to "1") to give credit.

On the other hand, WeBWorK is (like the commercial product WebAssign) publisher-independent.  You could use it with a textbook from any publisher you use, or even (as with our program) if you are using materials that do not belong to any publisher.

The MAA is assuming the responsibility of maintaining WeBWorK (http://webwork.maa.org/moodle/) from its original home at the University of Rochester.