Tuesday, June 7, 2016

"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.

Saturday, February 7, 2015

Strategies to help developmental math students

Nationally about 70% of incoming community college students are placed into developmental (a.k.a. “remedial” or “foundational”) math classes that earn no college degree credit. But only 10% of these students successfully move past developmental math to earn their degrees.

Four broad areas are being addressed to increase student success through developmental mathematics (1) Placement, (2) Pedagogy, (3) Curriculum, and (4) Student attitudes.

Improving Placement
Failing a class is not the only barrier to completion--the length of the developmental math path defeats many students. More developmental math students drop out of college without ever failing a math class than flunk out of math. One strategy to reduce the number of “exit points” is to help students place into as high a math level as reasonable.

For example, CaƱada College uses its Math Jam both as an intensive preparation for the math placement exam and also as a recruitment tool to get more students into STEM fields.

The placement instrument itself, typically a machine-graded standardized test, can be augmented or replaced.  High school GPA, recency of the previous math course, weekly work hours, and total course load could be part of “multiple measures.” Some schools have abandoned placement into developmental math courses, typically offering supplementary resources for students in credit-bearing classes.

Modifying Pedagogy
James Stigler lists three key types of learning opportunities that students need to experience to become flexible learners: productive struggle, explicit connections, and deliberate practice.

Modularized courses can allow students to spend time only on topics they need to study. The Emporium Model relies on software to do the pretest, primary instruction, and mastery testing, with human interaction largely limited to one-on-one tutoring in the computer lab (where students work lessons and take assessments). The University of Illinois uses software for placement and remediation, and California State University Northridge uses software as part of its hybrid lab remediation for students considered “at risk.”

Technology also plays a key role in both MOOCs (Massive Open Online Courses) and the “flipped” classroom. However, the “MO” aspects of MOOCs appear not to improve student success compared with the online developmental math courses that have existed for decades.

Another way to address the attrition between courses in a sequence is to offer “compressed” courses. The students take two courses during one term, but each course meets the standard number of hours per term--students are essentially immersed in math, which comprises most or all of their studies for that term.

Adjusting the Curriculum
The American Mathematical Association of Two-Year Colleges (AMATYC) has a 2014 position paper that states, “Prerequisite courses other than intermediate algebra can adequately prepare students for courses of study that do not lead to calculus.”

There are numerous “pathways” that have been created to allow developmental math students to pass a transferable math course--typically statistics or a quantitative reasoning course--that do not require many topics typically associated with intermediate algebra. The pathways normally reduce the number of developmental math courses required before earning transferable math units.
  • Path2Stats is part of the California Acceleration Project, based on a program developed by Myra Snell at Los Medanos College.
  • Statway and Quantway are projects of the Carnegie Foundation for the Advancement of Teaching.
  • The Dana Center’s Math Pathways include pathways for both STEM and non-STEM students.
  • Mathematical Literacy for College Students (MLCS) and Algebraic Literacy grew out of an AMATYC project. They can serve as alternatives to beginning and intermediate algebra classes for STEM majors, or the MLCS  can serve as prerequisite for a transferable non-STEM math course.

Addressing student attitudes
The “affective domain” includes attitudes, values, beliefs, interests, and motivation.

Carol Dweck’s research indicated that students (from grade school through graduate school) with “growth mindsets” persist and succeed better than peers with “fixed mindsets”. And importantly, students can learn to move from a fixed mindset to a growth mindset.

David Yeager’s research suggests that the performance gap in math--specifically developmental math--suffered by women and other underrepresented groups can be eliminated by specific brief interventions.

City University of New York's Accelerated Study in Associate Programs (ASAP) is an initiative that does not attempt to modify what occurs in the classroom. ASAP stipulates full-time enrollment and provides participants with academic advisement, career services, tutoring, financial supports, specially blocked or linked courses.

Sunday, December 28, 2014

Perfect numbers

"'I'll show you one more thing about perfect numbers," he said..."You can express them as the sum of consecutive natural numbers." Yoko Ogawa, The Housekeeper and the Professor, translated by Stephen Snyder

The examples that immediately follow the statement in the novel show that the perfect numbers 6, 28, and 496 are triangular numbers, that is, can be expressed as a sum of the form \(1+2+3+\cdots+n\).

This got me thinking about perfect numbers. For example,

Theorem: The reciprocals of the divisors of any perfect number sum to 2.
Proof: If the divisors of the perfect number \(N\) are \(1, d_2, d_3, \ldots, d_k\), and \(N\), then the sum of the reciprocals would be
\(1/1 + 1/d_2+ 1/d_3 \cdots + 1/d_k+ 1/N\). If we call this finite sum \(s\), then
\[Ns=N/1 + N/d_2+ N/d_3 \cdots + N/d_k+ N/N\]
\[ = N + d_k +\cdots +d_3+ d_2 + 1\]
\[= N +(\text{proper divisors of N})  = N+N = 2N\] Thus \( s=2\)

Theorem: Every even perfect number has the form \(N=(2^k -1)\cdot 2^{k-1}\), where \(2^k - 1\) is a Mersenne prime.
Proof:  Let \(N\) be an even perfect number. Then \(N\) can be written in the form \(N=2^{k-1} m\), where \(k>1 \)and \(m\) is odd.
Define \(\sigma(n)\) to be the sum of all positive divisors of \(n\).  In particular, \(\sigma(n)=2n\)  whenever \(n\) is perfect.
When \(a\) and \(b\) are relatively prime, \(\sigma(ab)=\sigma(a)\sigma(b)\)  because every divisor of \(ab\) can be uniquely written as the product of a divisor of \(a\) times a divisor of \(b\), so summing the divisors of \(ab\) can be accomplished by first computing \(\sigma(a)\)  and multiplying it by each of the divisors of \(b\), and summing those products.
Now \(\sigma(N)=2N=2^k m\) because \(N\) is perfect. By adding a finite geometric series with common ratio 2, we see that \(\sigma(2^{k-1})=2^k-1\), and we have
\(2^k m=  \sigma(2^{k-1}m) = \sigma(2^{k-1})\sigma(m)=(2^k-1)\sigma(m)\)  

Solving for \(\sigma(m)\), we get \(\sigma(m)=m+ \frac{m}{2^k-1} \)

From the definition of \(\sigma(m)\), \(m+ \frac{m}{2^k-1} \)  must represent the sum of all divisors of  \(m\), and in particular the fraction must be an integer.  But then \(\frac{m}{2^k-1}\) must itself divide \(m\), and as \(m\) clearly divides itself, \(m\) and \(\frac{m}{2^k-1} \)  must be all the divisors of \(m\).  Thus \(m=2^k-1\) must be prime.

Conversely, Theorem: Each Mersenne prime \(2^k -1\) gives the perfect number \(N=(2^k -1)\cdot 2^{k-1}\).
Proof: If \(2^k - 1\) is prime, then the divisors of \(N=(2^k -1)\cdot 2^{k-1}\) are \(1, 2, 2^2, \ldots, 2^{k-1},\) and also the product of any of those powers with the prime \(2^k - 1\). Summing all the proper divisors is the sum of two geometric series, each with common ratio 2. We get
\[\left(1+2+…+2^{k-1}\right) + \left(2^{k}-1\right) \left(1+2+…+2^{k-2}\right)\]
\[ = \left(2^k -1\right) + \left(2^{k}-1\right) \left(2^{k-1}-1\right)\]
\[= \left(2^k -1\right) \left(1+ 2^{k-1}-1\right) = N\]
We can see that every even perfect number is a triangular number, because \(N=(2^k -1)\cdot2^{k-1}\) has the form \(\frac{(n-1)n}{2}\), where \( n = 2^k\).

Euler evidently knew everything about even perfect numbers, but as far as I know, neither he nor anyone else has proven whether or not any odd perfect number exists.

I don't know whether an odd perfect number would need to be a triangular number. But the Professor only asserted that perfect numbers can be expressed as a sum of consecutive natural numbers. And of course any odd number \(O = 2n+1\) --including any odd perfect number-- can be expressed as the sum of two consecutive natural numbers: \(n + (n+1)\).

Friday, November 28, 2014

Inscribed Triangles and the Law of Sines

The Law of Sines follows from the following fact:  

Theorem: Any side of a triangle divided by the sine of the opposite angle gives the diameter of the circumscribing circle.

In other words, \( \frac{a}{\sin \alpha}, \frac{b}{\sin \beta}\), and \(\frac{c}{\sin \gamma} \) are all equal because each represents the same diameter.

The Law of Sines can be proven by using the fact that the area of a triangle is half the product of any two sides and the sine of the included angle. But such a proof gives no hint of the geometric interpretation of \( \frac{a}{\sin \alpha}\).

Here's a geometric argument for the theorem.

Case 1: First, we notice that if we have a right triangle with hypotenuse \(c\), then \(\sin\alpha =\frac{a}{c}\), \(\sin\beta =\frac{b}{c}\), and \(\sin\gamma =\sin 90^{\circ}=1\), so all three ratios  \( \frac{a}{\sin \alpha}, \frac{b}{\sin \beta}\), and \(\frac{c}{\sin \gamma} \)  are equal to \(c\), the hypotenuse. And because an inscribed right angle subtends an arc of \(180^{\circ}\), the hypotenuse coincides with a diameter. Thus the theorem is true for all angles in any right triangle.

Case 2: Now suppose that \(\Delta ABC\) is a triangle with \(\alpha =\angle CAB\) an acute angle. Draw the diameter through \(B\) to the point \(D\), and draw the segment from \(D\) to \(C\).

\(\Delta DBC\) is a right triangle inscribed in the same circle and shares the side of length \(a\) with \(\Delta ABC\). The angle at \(D\) subtends the same arc as the angle at \(A\), so the angles are congruent. By Case 1, \(\frac{a}{\sin\alpha}\) is the diameter of the circumscribing circle. Thus the theorem holds for any acute angle in any triangle.

Case 3: Now suppose that  \(\Delta ABC\) is a triangle with \(\alpha =\angle CAB\) an obtuse angle. Choose \(D\) so it lies on the arc of the circle subtended by \(\angle CAB\). Then \(\Delta DBC\) is inscribed in the same circle as \(\Delta ABC\) and shares the side of length \(a\).

Because \(\angle CAB\) and \(\angle CDB\) subtend arcs that sum to \(360^{\circ}\), they are supplementary angles. In particular, \(\angle CDB\) is acute, so the diameter of the circumscribing circle is \(\frac{a}{\sin(180^{\circ}-\alpha)}\). And because \(\sin(180^{\circ}-\alpha) = \sin\alpha\), we conclude that the diameter of the circumscribing circle is  \( \frac{a}{\sin \alpha}\).

Thus, whether \(\alpha\) is a right angle, acute angle, or obtuse angle, the ratio  \( \frac{a}{\sin \alpha}\) gives the diameter of the circumscribing circle.

Tuesday, September 9, 2014

Higher Education Alignment with the Common Core

The August 29, 2014 letter from California's higher education top administrators  announced that "the a-g requirements for CSU and UC admission, specifically areas ‘b’ (English) and ‘c’ (Mathematics), have been updated to align with the Common Core standards."

How that alignment will look is not specified in the letter.

As of today (9/9/14), the UC Mathematics ("c") subject requirements listed publicly do not show alignment with the Common Core State Standards. Instead, they still show expectations of California standards that existed before the CCSSM. For example, in item 2 of Course requirements, "The content for these courses will usually be drawn from the Common Core State Standards for Mathematics [PDF]. While these standards can be a useful guide, coverage of all items in the standards is not necessary for the specific purpose of meeting the 'c' subject requirement....The ICAS Statement of Competencies in Mathematics can provide guidance in selecting topics that require in-depth study." [Emphasis mine.]

A concern for California community colleges is that the alignment to the CCSSM might become what was proposed by the UC Board of Admissions and Relations with Schools (BOARS) in 2013. In July, BOARS wrote that “… the basic mathematics of the CCSSM can appropriately be used to define the minimal level of mathematical competence that all incoming UC students should demonstrate...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.”

BOARS clarified (December 2013) that “… going forward, all students must complete the basic mathematics defined by the college-ready standards of the Common Core State Standards for Mathematics (CCSSM) prior to enrolling in a UC-transferable college mathematics or statistics course.”

The college-ready standards of the CCSSM are simply all the non-plus standards. As written in the CCSSM“The higher mathematics standards specify the mathematics that all students should study in order to be college and career ready. Additional mathematics that students should learn in preparation for advanced courses, such as calculus, advanced statistics, or discrete mathematics, is indicated by a plus symbol (+). All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students.” [Emphasis mine]

Thus BOARS has twice stated that it expects all UC students to have all the CCSSM non-plus standards as prerequisite to any course that could receive UC credit.

But what undermines BOARS's credibility is its assessment of how the ICAS statement of competencies and the CCSSM content standards compare. In the opening paragraph of the BOARS July letter: "The most recent version of the ICAS mathematical competency statement makes clear the close alignment between it and the CCSSM. Both define the mathematics that all students should study in order to be college ready." [Emphasis mine.]

In actuality, what ICAS considers essential math content for all students is only a small subset of what the CCSSM specify as necessary. The ICAS document lists four sets of possible high school math topics. The first is Part 1: Essential areas of focus for all entering college students. Appendix B of the ICAS document explicitly shows how the CCSSM include not only the math topics of Part 1 but also the math topics of Parts 2, 3, and 4, which are areas of focus for students in quantitative majors or are areas of focus considered desirable but not essential.

Friday, August 29, 2014

Cosine of 72 degrees (and constructing a regular pentagon)

By using (say) DeMoivre's theorem, we have that \( \left( \cos\frac{2 \pi}{5}+ i \sin\frac{2\pi}{5} \right)^5=1\)
Expanding the left side as the fifth power of a binomial, equating the imaginary parts on both sides of the equation, and then replacing  \( \sin^2 72^\circ\) with  \(1- \cos^2 72^\circ \)
 \( i\sin 72^\circ \left(5 \cos^4 72^\circ+10 \cos^2 72^\circ i^2 \sin ^2 72 ^\circ+ i^4 \sin^4 72^\circ \right)=0i\)
 \(16 \cos^4 72^\circ - 12 \cos^2 72^\circ  + 1=0\)
Solving this quadratic in \( \cos^2 72^\circ \), we get
 \[ \cos^2 72^\circ  = \frac{12 \pm  \sqrt{80} } {32}   \]
 \[ \cos^2 72^\circ  = \frac{6 \pm 2 \sqrt{5} } {16}=\frac{\left( \sqrt{5}\pm 1 \right)^2}{4^2}   \]
 \[ \cos 72^\circ  = \pm \frac{\sqrt{5} \pm 1}{4}  \]
where we can choose the correct value of the four possible values by noting that, because 72° is between 45° and 90°,  \( \cos 72^\circ \) must lie between  \(1 / \sqrt{2}\) and 0. Because  \( \cos 72^\circ \) is positive, we choose the "+" before the fraction, and because  \( \cos 72^\circ \) is less than \( 1 / \sqrt{2}\), which in turn is less than \(\frac{\sqrt{5}+1}{4}\), we choose the "-" in the numerator:
\[ \cos 72^\circ = \frac{\sqrt{5}-1}{4} \]

Constructing a regular pentagon

So we can construct \( \cos 72^\circ\). For example, the diagonal of a 1-by-2 rectangle is \(\sqrt{5}\). We could cut off one unit from a segment of length \(\sqrt{5}\), then divide the segment of length \(\sqrt{5}-1\) into four pieces of length \( \frac{\sqrt{5} -1} {4} \). (Or we could construct the appropriate solution to the equation \( 4x^2 +2x -1 = 0 \). See my post on Solving quadratic equations via geometric construction.)

Construct a unit circle centered at O, and construct a radius \(\overline{OA}\).  Construct the point B on \(\overline{OA}\) so that \(\overline{OB}\) has length  \( \cos 72^\circ\). If C is a point on the circle so that \(\overline{BC}\) is perpendicular to \(\overline{OA}\), then \(\angle COA\) is a 72° angle, and both A and C are vertices of a regular pentagon inscribed in the circle.

Tuesday, August 5, 2014

What Math is Needed by All?

The (California version of the) Common Core State Standards in mathematics purport to be what all students need to be college and career ready.

The quantifier "all" in this context indicates that the math content should be the intersection (over all students) of math a student needs to be ready to begin college (or begin a career). Critics of the CCSSM who decry that the standards are not enough to prepare a student for an elite university such as Stanford are missing the point. The intent of the CCSS was never to include the union (over all students) of the math that a student needs to succeed in college. (And if the CCSS could provide all the math and English Language Arts that Stanford students need, then Stanford would not deserve its status as an elite school.)

And what do all students need? In 2013, the National Center on Education and the Economy released a study What Does It Really Mean to Be College and Work Ready?, reporting on both mathematics and English literacy. The report says, "Mastery of Algebra II is widely thought to be a prerequisite for success in college and careers. Our research shows that that is not so... Based on our data, one cannot make the case that high school graduates must be proficient in Algebra II to be ready for college and careers."

California's Intersegmental Committee of the Academic Senates (ICAS) represents the faculty academic senates of the three CA systems of higher education: the University of California (UC), the California State University (CSU), and the California Community College (CCC) system. The ICAS Statement on Competencies in Mathematics Expected of Entering College Students, revised in 2013, describes a number of mathematical topics that are or could be taught in high schools.

The ICAS competency statement describes mathematical subject matter in four categories: Part 1: Essential areas of focus for all entering college students, Part 2: Desirable areas of focus for all entering college students, Part 3: Essential areas of focus for students in quantitative majors, and Part 4: Desirable areas of focus for students in quantitative majors.

The mathematics that the CCSSM describe as what all students need should presumably match with what the ICAS statement describes as "essential" and lists in Part 1. But although the UC Board of Admissions and Relations with Schools (BOARS) states there is "close alignment" between the CCSS and the ICAS statement, the ICAS statement makes clear that there are many CCSS that are not "essential" but rather merely desirable or for only some students. Appendix B of the ICAS statement explicitly shows where Part 2, 3, and 4 areas of math are found in the CCSS (and NCTM standards).

And the Interim Environmental Scan Report to The Common Assessment Initiative Steering Committee has in  Appendix B a Table that shows a number of CCSS that do not occur at all in the ICAS statement.

Here are examples of CCSSM topics that might surprise some community college math faculty, especially those who believe that intermediate algebra as currently taught will be sufficient to cover all the CCSSM.
  • Probability:  sample spaces, independent events, conditional probability, permutations and combinations; analyzing decisions and strategies using probability
  • Statistics: assessing the fit of a function by plotting and analyzing residuals; interpreting the correlation coefficient of a linear model in context; normal distributions, random samples, estimating population parameters, simulations, using probability to make decisions
  • Transformational geometry: congruence defined in terms of rigid motion; similarity defined in terms of dilations and rigid motions
  • Trigonometry: trig ratios, special angles, 6 trig functions of real numbers; modeling periodic phenomena, proof and use of the Pythagorean trig identity \( \cos^2 \theta + \sin^2 \theta = 1 \)