Friday, July 26, 2013

More on Alternative Pathways and transferability in California

California's adoption of the Common Core State Standards in Mathematics (CCSSM) helps to shape the expectations of universities regarding the mathematical background of their incoming students.

The July 2013 statement ( from the University of California's Boards of Admissions & Relations with Schools (BOARS) comments that most California Community Colleges (CCCs) continue to use "traditional Intermediate Algebra (i.e., Intermediate Algebras as defined prior to CCSSM implementation)" as prerequisite to a transferable mathematics course.

The BOARS statement continues, "Specifying that transferable courses must have at least Intermediate Algebra as a prerequisite is not fully consistent with the use of the basic mathematics of the CCSSM as a measure of college readiness...Requiring that all prospective transfer students pass the current version of Intermediate Algebra would be asking more of them than UC will ask of students entering as freshmen who have completed CCSSM-aligned high school math courses. As such, BOARS expects that the Transferable Course Agreement Guidelines will be rewritten to clarify that the prerequisite mathematics for transferable courses should align with the college-ready content standards of the CCSSM."

Meanwhile, the Academic Senate of California Community Colleges (ASCCC) has endorsed the CCSSM, but has no formal position on alternative pathways.  A Fall 2012 resolution to support innovations to improve success in under-prepared non-STEM pathways was referred to the executive committee.  However, former ASCCC president Ian Walton did publish in the ASCCC Rostrum an opinion ( that "The wide range of conversations demonstrates that a strong case can be made for the exploration and implementation of alternative preparations for transfer level math courses that differ from the content of the traditional intermediate algebra course."

Sunday, July 21, 2013

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.

Thursday, March 28, 2013

Heron's formula for the area of a triangle

The angle bisectors of the triangle meet at the center of the inscribed circle of radius r.  If we let \(2\alpha=A\), \(2\beta = B\), and \(2\gamma=C\), we have \(\alpha+\beta+\gamma=\frac{\pi}{2}\). 

Let x be the distance from the vertex at A to points of tangency, y the distance from B, and z the distance from C.  Then then lengths of the triangle sides opposite A, B, and C are respectively \(a=y+z\), \(b=x+z\), and \(c=x+y\).

Thus if we name the semiperimeter sthen \(s=x+y+z\), \(x=s-a\), \(y=s-b\), and \(z=s-c\).

\(\tan \alpha =\frac{r}{x}\), \(\tan \beta =\frac{r}{y}\), and \(\tan \gamma =\frac{r}{z}\).  Because \(\gamma\) and \( (\alpha+\beta )\) are complementary angles, we obtain

\[ \tan\left( \frac{\pi}{2}  - (\alpha+\beta) \right)   = \frac{r}{z} \]
\[ \tan\left( \alpha+\beta \right)   = \frac{z}{r} \]
\[ \frac{ \tan \alpha+\tan\beta}{1-\tan\alpha \tan\beta}   = \frac{z}{r} \]
\[r \left(\tan \alpha+\tan\beta  \right)= z (1-\tan\alpha\tan\beta) \]
\[r \left( \frac{r}{x}+\frac{r}{y} \right) = z \left( 1 - \frac{r}{x}\frac{r}{y} \right) \]
\[ r^2 y + r^2 z = xyz - r^2 z \]
\[ r^2 ( x+y+z) = xyz \]
\[ r^2 s = xyz \]

The radii at the points of tangency and the angle bisectors form 3 pairs of congruent triangles.  The area of \(\Delta ABC\) is \(xr+yr+zr= r(x+y+z)\), so area \(=rs\), and \( (\text{area})^2=r^2s^2\).  Using results we have above, we obtain
\[ (\text{area})^2 = s\cdot xyz = s(s-a)(s-b)(s-c)\]
so the area is \(\sqrt{s(s-a)(s-b)(s-c)}\).

Wednesday, January 16, 2013

Contradictory mandates to community colleges

A goal of the Common Core State Standards (CCSS) is to prepare students to be college and career ready.  That goal is also part of the mission of community colleges.  There has been considerable discussion regarding how  the CCSS might affect students' chances for getting into college, but scant discussion about how community colleges fit into the implementation of the CCSS.

Community Colleges might be assumed to have a distorted view of what it means to be college or career ready.  After all, they typically use the word "college" when naming themselves, yet eligibility to become a community college student does not require any minimum GPA nor any minimum score or ranking in any test.  It is sometimes said that the University of California serves the top 12.5% of California high school graduates, the California State University system the top 33.3%, and the California Community Colleges serve the top 100%.  But this is too limiting--a high school diploma is not a requirement for enrollment at any California Community College.

So at community colleges, we may worry less about "college ready" but rather focus on "transfer ready".  UCLA and CSUN are my school's two nearest public universities, and both report that our transfer students perform slightly better than their native students.  So there is evidence that community colleges are not grossly underestimating what is needed to be transfer ready.

California Community Colleges are presented with two conflicting mandates .   Community colleges  are encouraged 1) to align with K-12 standards for college and career readiness (according to the California Community College  Student Success Task Force, and 2)  to provide alternative pathways to transfer (according to the Carnegie Foundation for the Advancement of Teaching, the Charles A. Dana Center, etc.).

Explicitly, a consortium of the Charles A. Dana Center, Complete College America, Inc., Education Commission of the States, and Jobs for the Future, asks community colleges to provide a  "fundamentally new approach for ensuring that all students are ready for and can successfully complete college-level work that leads to a postsecondary credential of value.

"...The content in required gateway courses should align with a student’s academic program of study — particularly in math... Institutions need to focus on getting students into the right math and the right English." (from "Core Principles for Transforming Remedial Education: A Joint Statement" :

The researcher at my institution estimated that 75% of our students interested in transfer are in disciplines that require no mathematics beyond an introductory statistics class to earn a baccalaureate degree at CSUN.  Evidently there are many students who can earn baccalaureate degrees without taking  single course from the mathematics department of any 4-year school.

The California Community College Success Network (, the Carnegie Foundation, the Dana Center, and the Student Success Task Force all recommend removing  curricular requirements  that act as barriers rather than aids to program completion.  The SSTF report contends, "Improved student support structures and better alignment of curriculum with student needs will increase success rates in transfer, basic skills, and career technical/workforce programs." [emphasis added]

The existing and proposed curricula of alternative pathways for non-STEM students omit many topics of intermediate algebra.

On the other hand, neither the University of California nor the California State University accepts a math or statistics course to meet  math transfer requirements unless that course has intermediate algebra as a prerequisite. 

If aligning with the CCSS implies that "intermediate algebra" should mean CCSS Algebra 2 (which includes circular trig and some inferential statistics), then the math requirements for transfer math courses will increase significantly.

And because intermediate algebra is the California Community College minimum math requirement for an associate's degree, the requirement for an AA degree will also increase simultaneously.

It is impossible for community colleges to remove unnecessary but currently required topics (for transfer to non-STEM disciplines) while simultaneously not merely maintaining but augmenting that list of required topics for all students.