Wednesday, July 23, 2014

Common Core Goes to College

The New America Foundation’s position paper by Lindsey Tepe gives recommendations for how higher education can support the Common Core State Standards. However, this paper and related articles in the Chronicle  and Hechinger Report  miss the most important way for higher education to support the CCSS, namely, to work to repair or ameliorate the existing flaws in the CCSS. 

An implicit assumption in Tepe's paper is that the CCSS have successfully captured what all students need to be college and career ready--A move to change higher education to accommodate inappropriate standards could harm students and impede their paths to college degrees.

But the CCSS have missed the mark at what is necessary for all students to succeed in college.

Many of the non-plus CCSS are currently introduced to students in credit-bearing courses of baccalaureate granting institutions. That is, the CCSS overshoots what is needed to be ready for college and includes topics that are part of what some college students need to learn while in college.

The intent of the CCSS was to help get students college (and career) ready. It is an abuse of the CCSS to use those standards as an opportunity for colleges and universities to raise admissions and/or degree requirements, and that abuse will work against the goal of giving more students the opportunity to earn college degrees.

Thursday, July 17, 2014

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.

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."