Sunday, January 19, 2014

JMM 2014

Over 6400 mathematicians descended upon Baltimore January 15-18 for the 2014 Joint Mathematics Meetings. Sessions included current research in math, discussions on pedagogy, content, collaborations across institutions, social events, and more.

The first session on Wednesday 15 January was the MAA Minority Chairs committee meeting at 7:00 am, although there were actually some short courses, workshops, AMS council meetings and MAA Board of Governors meeting on the preceding Monday and Tuesday. And there were dozens of contributed paper sessions throughout the morning and the rest of the day.

The JMM unveiled the theme of the 2014 Math Awareness Month (April 2014): Mathematics, Magic, & Mystery ( On each day of April 2014 a new square of the Activity Calendar goes live, giving access to mathematical puzzles and magic. (Once opened, the resources are to be kept available for as long as the AMS exists.)

JMM2014 also included a panel session launching TPSE Math: Transforming Post-Secondary Education in Mathematics (@tpsem, The project is sponsored by the Carnegie Foundation of New York and the Alfred P. Sloan Foundation.

One of the sessions on the last day was The Public Face of Mathematics  The panel was organized by mathemagician Art Benjamin ( and included "Math Guy" Keith Devlin (, NY Times columnist Steven Strogatz (, mathbabe Cathy O'Neill (, freelance journalist Tom Siegfried (, and US Congressman Jerry McNerny ( 

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 \(\sqrt{s(s-a)(s-b)(s-c)}\).

Thursday, December 20, 2012

Alternative Pathways vs Common Core State Standards

A primary goal of the Common Core State Standards (CCSS) is to provide a curriculum to ensure that all high school graduates are college and career ready. The CCSS  math topics through grade 11 include not only all of the topics of the traditional U.S. Algebra 1-Geometry-Algebra 2 sequence, but also topics typically taught in courses named trigonometry and statistics.

Alternative pathways provide a means for non-STEM (i.e., non- Science, Technology, Engineering, and Math) students to transfer from a two-year college to a four-year institution and earn a bachelor's degree without needing to show mastery of traditional intermediate algebra topics. The promotion of alternative pathways challenges the premise that the CCSS for math are needed for all students to be college ready.

The common goal of both alternative pathways and the CCSS is to improve U.S. education.

 "Core Principles for Transforming Remedial Education: A Joint Statement" from the Charles A. Dana Center, Complete College America, Inc., Education Commission of the States, and Jobs for the Future, calls for revamping the two-year college remediation structure.  The paper lists seven Core Principals for 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.

"...Principle 2. The content in required gateway courses should align with a student’s academic program of study — particularly in math.

"Gateway courses provide a foundation for a program of study, and students should expect that the skills they develop in gateway courses are relevant to their chosen program. On many campuses, remedial education is constructed as single curricular pathways into gateway math or English courses.

"The curricular pathways often include content that is not essential for students to be successful in their chosen program of study. Consequently, many students are tripped up in their pursuit of a credential while studying content that they do not need. Institutions need to focus on getting students into the right math and the right English.

"This issue is of particular concern in mathematics, which is generally considered the most significant barrier to college success for remedial education students. At many campuses, remedial math is geared toward student preparation for college algebra. However for many programs of study, college algebra should not be a required gateway course when a course in statistics or quantitative literacy would be more appropriate….

"...One final note: Postsecondary leaders must work closely with K–12, adult basic education, and other training systems to reduce the need for remediation before students enroll in their institutions.  Postsecondary institutions should leverage the Common Core State Standards by working with K–12 schools to improve the skills of their students before they graduate from high school. Early assessment of students in high school, using existing placement exams and eventually the Common Core college and career readiness assessments, which lead to customized academic skill development during the senior year, should be a priority for states. Similar strategies should be employed in adult basic education and English as a second language programs."

Monday, December 17, 2012

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,

Thursday, November 22, 2012

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.