Tuesday, July 29, 2014

Schizophrenic Common Core Supporter


Back in 2012 Sol Garfunkel wrote "I feel like a schizophrenic. I truly think that the Common Core State Standards for Mathematics (CCSSM) are a disaster...So why do I feel like a schizophrenic? Because I am at the same time working to make the implementation of the CCSSM be as effective as possible!"

As mathematician Keith Devlin has emphasized, the heart of the CCSSM is the set of 8  standards of Mathematical Practice:

  • MP1. Make sense of problems and persevere in solving them.
  • MP2. Reason abstractly and quantitatively.
  • MP3. Construct viable arguments and critique the reasoning of others.
  • MP4. Model with mathematics.
  • MP5. Use appropriate tools strategically.
  • MP6. Attend to precision.
  • MP7. Look for and make use of structure.
  • MP8. Look for and express regularity in repeated reasoning.


It would be hard to imagine that any mathematician or math educator would not applaud these standards. And these standards, the key to the CCSSM and presented at the start of each set of grade level standards, are rarely if ever mentioned in the attacks on the CCSSM.

Much of the resistance to the CCSS is political: the democratic President of the United States has endorsed the CCSS, so there is automatic opposition from the Tea Party, Republicans, and Libertarians, who argue that the CCSS is a federal program. But although President Obama is giving incentives for states to adopt the CCSS, the standards are the result of 48 state governors and secretaries of education agreeing to cooperate to create educational standards that would be consistent across state lines.

The resistance from the classroom teachers is understandable because they will be held accountable to how their students will do on the CCSS standardized testing. But the standardized testing that will be used is not part of the CCSS but rather is being created by SBAC or PARCC, consortia created to write CCSS assessments. That is, although the news media report teacher opposition to the CCSS, the teachers' actual objection is to the assessments and how they will be used.

The widely seen mocking and vilification of CCSS lessons by the public also confuse the CCSS with methods for testing students for mathematical proficiency. The CCSS explicitly require that students master the standard algorithms that critics mistakenly say are "real math" and missing from the CCSS. But significantly, the CCSS also require (MP1) that students can make sense of the mathematical tasks they are performing.

I think the CCSSM grossly overshoot the mark when trying to specify the math that all students need to be college and career ready. But like Sol Garfunkel, I think we should simultaneously embrace the CCSS and work to improve them.

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. If the assumption is false, the paper is advocating moves to change higher education to accommodate inappropriate standards, changes that could harm students and impede their paths to college degrees.

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 (http://www.mathaware.org/). 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, http://www.tpsemath.org/). 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 http://blogs.ams.org/jmm2014/2014/01/18/the-public-face-of-mathematics/.  The panel was organized by mathemagician Art Benjamin (http://www.math.hmc.edu/~benjamin/) and included "Math Guy" Keith Devlin (http://www.stanford.edu/~kdevlin/), NY Times columnist Steven Strogatz (http://www.stevenstrogatz.com/), mathbabe Cathy O'Neill (http://mathbabe.org/), freelance journalist Tom Siegfried (http://www.sciencenoise.org/), and US Congressman Jerry McNerny (http://mcnerney.house.gov/). 



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 (http://senate.universityofcalifornia.edu/committees/boars/BOARSStatementonMathforAllStudentsJuly2013.pdf) 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 (http://asccc.org/content/alternatives-traditional-intermediate-algebra) 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 \(\sqrt{s(s-a)(s-b)(s-c)}\).