"Unjustified use of Algebra 2"

The U.S. Department of Education organized a meeting (“California Math Convening: Gateways to Access – May 31, 2016”) to discuss California's use of Algebra 2 (a.k.a. Intermediate Algebra) in higher education. The meeting was held at the chancellor's office of the California State University (CSU) system. The participants included representatives from the CSU, the University of California, the California Community Colleges, K-12 educators, and educational policy organizations.

The meeting was the DOE's response to a September 30, 2015 letter from Christopher Edley, Jr., to the Catherine Lhamon, Assistant Secretary of Civil Rights, U.S. Department of Education. The letter begins with:

“I write to request that your office investigate the educationally unjustified use of Algebra 2 as a gateway course by all three segments of California’s higher education system: the University of California system; the California State University system; and the California Community College system. There is evidence to suggest that, in varying ways, these institutions have adopted policies and practices that impose a disparate impact on protected groups in violation not only of the equal protection clause of the California State Constitution, but also in violation of federal regulations implementing Title VI of the Civil Rights Act of 1964.”

The letter cites the success of Statway, a project of the Carnegie Foundation for the Advancement of Teaching, as evidence that Intermediate Algebra is not actually necessary for success in completing math requirements for baccalaureate degrees in some majors. The letter concludes with:

“If there are villains here, they are the indifference and inertia that confirm and perpetuate unequal educational opportunity. I believe this discrimination is, for the most part, without animus. Regardless, the injury is real.” 

At the meeting, Christopher Edley Jr. explained that neither intent nor a history of practice would be considered relevant when determining if there is a violation of the Civil Rights Act. The presence of both Catherine Lhamon and also the Under Secretary U.S. DOE, Ted Mitchell, made abundantly evident that the DOE wants California's higher education community to recognize and address the issue.

Another speaker was William McCallum, mathematician with numerous distinctions including being one of the three lead writers of the Common Core State Standards in Mathematics (CCSSM). Bill explained that because College Algebra was the de facto mathematics requirement in U.S. baccalaureate granting institutions at the time of writing the CCSSM, the document needed to include the math that would lead to College Algebra, namely Algebra 2. He commented that it is  inappropriate for colleges or universities to cite the CCSSM to define what is currently needed to be college ready--it makes no sense to argue against modifying college math requirements based on the content of the CCSSM, as the CCSSM were created trying to reflect what the earlier college math requirements had been.

The U.S. DOE evidently intends to hold another such meeting in 3 or 4 months to check on what progress has been made.

Math Initiatives for Student Success

The LearningWorks paper Changing Equations: How Community Colleges Are Re-thinking College Readiness in Math, written by Pamela Burdman, is a nice summary of current initiatives attempting to help capable students negotiate developmental math needs to succeed in transfer-level mathematics.

Much of the paper discusses the strategy of alternative pathways. In this strategy, students pass a course that is identical to, or has the same content and rigor of, accepted transfer math courses, but instead of first passing an intermediate algebra course, the students take a math course designed specifically to prepare them for the transfer course—that preparatory course omits some standard topics of intermediate algebra which are not necessary to succeed in the transfer math course.

The initial data on alternative pathways, some cited in Changing Equations, show that a much higher percentage of students initially placed in a developmental math course can pass a transfer level math course following an alternative pathway than by following the traditional chain of prerequisites. 

But both the University of California and the California State University systems require that intermediate algebra be a prerequisite for any transferable course. Keeping the intermediate algebra prerequisite based on the data that have shown success in intermediate algebra is a predictor of college success is, as pointed out in Changing Equations, following the error of confusing correlation with causation, and in fact the widespread practice of requiring success in intermediate algebra (a.k.a. Algebra 2) as a admissions requirement virtually guarantees the high correlation that has been often noted.

Alternative Pathways and transferability in California

California is home to the Carnegie Foundation for the Advancement of Teaching, the current force behind two pathway projects:  Statway and Quantway. 

An underlying assumption behind alternative pathways is that mathematics requirements for degrees and/or certificates should vary according to discipline. California's Student Success Task Force report contends, "Improved student support structures and better alignment of curriculum with student needs [Emphasis added] will increase success rates in transfer, basic skills, and career technical/workforce programs." The National Center on Education and the Economy 2013 report, "What Does It Really Mean to Be College and Work Ready?" states, "But our research...shows that students do not need to be proficient in most of the topics typically associated with Algebra II and much of Geometry to be successful in most programs offered by the community colleges."

The Carnegie Foundation, The Charles A. Dana Center at U.T. Austin, and the California Community College Success Network (3CSN) all promote alternative pathways to allow students in non-STEM disciplines an option of completing a university-transferable mathematics course without requiring the students to demonstrate completion of an intermediate algebra course.

The two California university systems, the University of California (UC) and the California State University (CSU) have been cautious in embracing the idea of alternative pathways in California Community Colleges (CCCs).

One pathway strategy is to provide students with an alternative  prerequisite to an existing transferable statistics class.  The alternative prerequisite does not have all traditional intermediate algebra topics and does not have elementary algebra as prerequisite. And in response to this strategy, Nancy Purcille of the UC Office of the President sent a March 7, 2013 email to CCC articulation officers:

"The prerequisite for UC-transferable math courses continues to be intermediate algebra or equivalent.  No attempt at this time will be made by UC to define specific content/courses that may be deemed “valid” alternate prerequisites.  When submitting a course for TCA review, if CCC faculty propose a prerequisite that they judge to be the equivalent of intermediate algebra, then UCOP articulation analysts will treat the prerequisite as such and evaluate the course outline as usual.  UC will not be evaluating the prerequisites listed – unless it is jointly requested by the CCC and UC faculty."

This position appears to respect the tenet that the community college should be able to decide the appropriate developmental math required to prepare its students for the articulated transfer-level math course.

The CSU provided a different position to accommodate alternative pathways.   Ken O'Donnell of the CSU Office of the Chancellor sent a November 2, 2012 email to CCC articulation officers that appeared to be discouraging alternative pathways:

"Please take this email as a reminder that only courses with a full prerequisite of intermediate algebra, as traditionally understood, will continue to qualify for CSU Area B4 [math/quantitative reasoning requirement to transfer].

"The CSU has made a recent exception for the Statway curriculum, under controlled and very limited circumstances, so we can evaluate whether other approaches will satisfactorily develop student proficiency in quantitative reasoning.  In the meantime, we count on the articulation community to uphold the current standard."

But Ken O'Donnell sent an April 2013 email acknowledging without objection the strategy of keeping the intermediate algebra the official prerequisite for the transfer math course but facilitating CCC student challenges to that prerequisite.

The CSU Chancellor’s General Education Advisory Committee has looked into this use of the prerequisite challenge process, and determined that it has no grounds to comment.  How community colleges meet curricular requirements that are below baccalaureate level is up to the colleges, and not up to the receiving transfer institutions.  In other words, community colleges may participate in initiatives like Acceleration in Context and the California Acceleration Project without jeopardizing articulation, because the transferable B4 course is unchanged; only the intermediate algebra prerequisite is challenged. 

Thus both the UC and the CSU are tacitly giving CCCs the go-ahead to develop alternative pathways.

Common Core State Standards Algebra

One issue of concern for the California K-12 educators is that California currently requires students to pass Algebra 1 in order to earn a high school diploma. The Common Core State Standards (CCSS) version of Algebra 1 includes topics not traditionally associated with Algebra 1, for instance, exponential functions and some statistics.

Unless new legislation addresses this change in content, the adoption of the CCSS automatically raises the California high school graduation requirement.

A related issue more directly linked to California Community Colleges (CCCs) is that the CCSS has created a higher level Algebra 2. If community college intermediate algebra is to align with high school Algebra 2, then we will be raising our math requirement for the AA degree and for the prerequisite for transfer level math.

And the California Community College Student Success Task Force calls for better alignment:

"Aligning K-12 and community colleges standards for college and career readiness is a long-term goal that will require a significant investment of time and energy that the Task Force believes will pay off by streamlining student transition to college and reducing the academic deficiencies of entering students...

"Recommendation 1.1
"Community Colleges will collaborate with K-12 education to jointly develop new common standards for college and career readiness that are aligned with high school exit standards.

"The Task Force recommends that the community college system closely collaborate with the SBE and Superintendent of Public Instruction to define standards for college and career readiness as California implements the K-12 Common Core State Standards and engages with the national SMARTER Balanced Assessment Consortium to determine the appropriate means for measuring these standards. Doing so would reduce the number of students needing remediation, help ensure that students who graduate from high school meeting 12th grade-level standards are ready for college-level work, and encourage more students to achieve those standards by clearly defining college and career expectations."

I don't know who speaks for CCCs in the collaboration with the State Board of Education and Superintendent of Public Instruction.  But I do think it likely that one strategy to bring better alignment will be to use the Smarter Balanced assessments at grade 11 as placement instruments at the community colleges.  The other consortium creating CCSS assessments, PARCC, already has agreement among its adopting states to use its assessments for college placement.  (See, for example, http://bit.ly/QYVjUF.)

Alternative pathways

It seems that student-success discussion at two-year colleges is shifting from "course redesign" to "alternative pathways" for the math required to transfer.

The topics overlap considerably, because the idea of alternative pathways normally involves modifying course prerequisites in format and/or content.

Some community colleges are exploring alternative pathways via multiple versions of intermediate algebra.  For example, several campuses have a "pre-stats" course which prepares students for the regular statistics course, but the pre-stats course does not cover all of intermediate algebra (and may not have elementary algebra as a prerequisite). 

The Carnegie Foundation for the Advancement of Teaching, the Dana Center at UT Austin, and 3CSN (California Community College Success Network) are three groups promoting the development of alternative pathways through dev math for non-STEM majors. The California State University system has agreed to accept Statway™ (Carnegie's two-semester course beginning at the elementary algebra level and ending with a transferable statistics credits) for meeting the Area B4 (math/quantitative  reasoning ) requirement for transfer.

But  last month the CSU emailed California Community College articulation officers the following:

"When the CSU reviews community college courses proposed to satisfy Area B4, we look for a prerequisite of intermediate algebra. We’re aware that many community colleges are experimenting with alternative prerequisites to their approved B4 courses, in an effort to improve student persistence. Some of these alternatives take away topics traditionally included in intermediate algebra; others substitute a different course altogether.

 "Please take this email as a reminder that only courses with a full prerequisite of intermediate algebra, as traditionally understood, will continue to qualify for CSU Area B4.

 "The CSU has made a recent exception for the Statway™ curriculum, under controlled and very limited circumstances, so we can evaluate whether other approaches will satisfactorily develop student proficiency in quantitative reasoning. In the meantime, we count on the articulation community to uphold the current standard."

That email seems to cast doubt on the future of alternative pathways.  But in the meanwhile, the CSU appears to be fine with the strategy proposed by Palomar College.  Palomar is not changing the intermediate algebra prerequisite for statistics, but evidently students who pass the alternative pre-stats course will be allowed to waive the intermediate algebra prerequisite.

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.

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.

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.

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: 

280slides.com for creating and storing slideshows online, screen-o-matic.com 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.  

A joyful conspiracy

Uri Treisman's Joyful Conspiracy from CarnegieViews on Vimeo.

The Carnegie Foundation for the Advancement of Teaching is organizing a “joyful conspiracy” to help community colleges provide pathways to success for students who initially are placed in developmental mathematics courses.  The Statway will bring non-STEM students from the level of elementary algebra up to and through a transfer-level statistics course in one year.

The Statway 2010 Summer Institute brought teams from 19 community college campuses to the Stanford University campus July 25-30 to meet, share with, and learn from each other and from Carnegie Foundation leaders and consultants.  

We practiced the protocol for presenting, critiquing, and giving feedback on the lessons we will be piloting in the coming year.  Each lesson will involve students working on a rich task with clearly defined learning goals.  A key assumption of Statway is that statistics can provide a context for students to learn to think and reason quantitatively.  The necessary algebraic skills will be embedded within the lesson, rather than holding center stage.

Another core part of the instructional experience is that having students struggle with problems is desirable.  This student engagement, even when students do not discover or invent the necessary mathematics on their own, can be crucial to preparing the students for making sense of the central topic of the lesson.

Statway: A pathway from developmental math through statistics

Los Angeles Pierce College has been invited to be one of sixteen community colleges to participate in the Carnegie Foundation's Statway project.

The key goal of the project is to provide a pathway for developmental math students to progress successfully from elementary algebra to completion of a transferable statistics course, all in one year.

The Statway project has already collaborated with AMS, ASA, MAA, AMATYC, NADE, NACME (National Action Council for Minorities in Engineering), and CAUSE (Consortium for the Advancement of Undergraduate Statistics Education).  Selected faculty from the professional mathematics societies make up the Carnegie Committee on Statistics Learning Outcomes, which has been working on identifying the core concepts, topics, and learning outcomes for transfer-level statistics.  The CCSLO is also identifying the developmental math learning outcomes needed to prepare students for learning statistics.

But in addition to redesigning the content and pathway to statistics, Statway will incorporate a student engagement component--roughly survival skills for a college student.

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.

Solving quadratic equations via geometric construction

We can solve a quadratic equation of the form x² - sx + p = 0, , using the standard construction tools of compass and straightedge.  The method has been attributed to critic Thomas Carlyle.

Construct the circle in the Cartesian plane with center and passing through A(0,1).  By symmetry, the circle also passes through and .

Because the circle has center   and passes through A(0,1), the equation of the circle is

This reduces to

x² - sx + y² - (p + 1)y + p = 0

So the x-intercepts of the circle are the solutions to  x² - sx + p = 0.

Alternate justification:

The segment joining the x-intercepts has a length , hence x₁ + x₂ = s.

intercepts the arc AX₁X₂, and  intercepts the opposite arc, hence the two angles are supplementary.  But is also supplementary with , so is congruent to , which in turn impies that  .  Hence

 ,

which implies that , so x₁ x₂ = p.

Thus (x -  x₁)(x - x₂) = x² - sx + p, and the solutions to x² - sx + p = 0 are x₁ and x₂ .

Mediated Algebra Project: Success!

Three instructors used the Mediated Algebra Project (MAP) materials during the Fall 2009 semester.  Kathie Yoder taught a section that met early afternoon twice weekly , and Kathy Yoshiwara and I taught the two sections that met mid-morning four-days-a-week.

We now have evidence that MAP students learn more than cohorts in other sections of the Pierce College intermediate algebra course:  On the departmental Math Exit Test (MET), our three sections all scored at least 2.5 standard errors above the department mean.

But we had significant setbacks during the semester.

We had numerous technical difficulties.  Many of the WeBWorK problems I had authored had coding errors and/or needed refinement in wording or formatting.  And most of the WeBWorK exercises taken from the national WeBWorK library were poor fits for our project and had to be rewritten or removed from our problem sets during the semester.

Our sets of video tutorials--intended to help with drill and skill exercises--had many gaps in content.  And yet we were not given sufficient space on our school's server to store the videos created  by our faculty for the MAP.  Instead, our IT department arranged that only a subset of those videos would be accessible at any one time.

All three instructors found that the project's classroom activities and clicker questions required more time than was available in a class meeting.  Some of the activities, or the clicker questions, or both would go unused in each lesson.

We heard complaints about our WeBWorK assignments, the insufficiency of available videos, and the amount of work we asked the students to do both in and outside of class.

But the students who persisted in MAP averaged much higher on a department-graded common exam than students from the other sections of intermediate algebra.

The Correlation Coeffiicent as cosine theta

Mathematicians define the dot product between vectors and as

On the other hand, the alternate geometric definition for the dot product popular with physicists is

So

And statisticians define Pearson's correlation coefficient r so that

Thus if we set and , then .

The idea is to think not of n ordered pairs (x₁, y₁), (x₂, y₂), ..., (x_n, y_n), but rather to think of two vectors in n-dimensional space. When the vectors are pointing in the same direction, the angle between them is zero and the correlation coefficient is cos 0 = 1. When the vectors point in opposite directions, the correlation coefficient is the cosine of a straight angle, r = -1. And when the vectors are orthogonal, the correlation coefficient is the cosine of a right angle, r = 0.

The only tricky part is that the two n-dimensional vectors are not the vectors and  , the vectors containing all the and respectively.  Instead, the necessary two n-dimensional vectors are the and defined above.

And nicely, the least-squares regression line for the data is y = mx + b, where and .  (Notice that the variance , so m can also be written as  .


One typically derives the least-squares regression line by finding m and b that minimize .  But one can alternatively use the n-dimensional vector point of view, where the coefficients m and b correspond to the solution of the vector equation .  The vector is the vector of all 1's and the vector   is the orthogonal projection of the vector  onto the space spanned by and .