tag:blogger.com,1999:blog-58348863887522155252017-02-08T20:46:56.451-08:00Math Ed Blog from Bruce YoshiwaraA retired community college math professorBruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.comBlogger63125tag:blogger.com,1999:blog-5834886388752215525.post-14882796184301960012016-12-03T15:31:00.002-08:002016-12-05T14:07:22.618-08:00Learning Styles and Optimizing Learning"<a href="https://www.psychologicalscience.org/journals/pspi/PSPI_9_3.pdf" target="_blank">Learning Styles: Concepts and evidence</a>," <i>Psychological Science in the Public Interest</i>, V. 9 No 3, December 2008, by Harold Pashler, Mark McDaniel, Doug Rohrer, and Robert Bjork examined whether or not there was scientific evidence to support the "learning-styles" practice of matching the format of instruction to the learning style of the individual learner.<br /><br />They authors described what evidence would be appropriate to validate the learning-styles practice. For distinct learning styles A and B, learners with learning style A should learn better with an instruction/intervention with learning style format A than with instruction/intervention with learning style format B, and vice versa: learners with learning style B should learn better with instruction/intervention of format B than format A.<br /><br />The authors were actually more generous in what they considered acceptable evidence. But simply finding that learners A did better with intervention A than with intervention B was not sufficient--it was also necessary to show that learners B did worse with intervention A than with B.<br /><br />And at that time (December 2008) the authors found no evidence base to justify incorporating learning-style practices. And in 2016, there still has been no adequate evidence to justify the use of learning-styles practices.<br /><br />Among professionals who research learning, there is consensus that it is not effective to teach towards the learning style of the individual student. Yet across the K-16 spectrum there are ardent adherents to the learning-styles practice.<br /><br />From the opening of Philip M. Newton's "<a href="http://journal.frontiersin.org/article/10.3389/fpsyg.2015.01908/full" target="_blank">The Learning Styles Myth is Thriving in Higher Education</a>," <i>Frontiers in Psychology</i>, 15 December 2015, "The existence of ‘Learning Styles’ is a common ‘neuromyth’, and their use in all forms of education has been thoroughly and repeatedly discredited in the research literature. However, anecdotal evidence suggests that their use remains widespread."<br /><br />Both Newton and Pashler et al. acknowledge that people can readily identify with a preferred learning style and that there is validity to the identification. But there still is no evidence that learning is enhanced by identifying a student's learning style and providing instruction geared towards that style.<br /><br />The promotion of the learning-styles practices is not an innocuous indulgence. Not only does the emphasis on learning styles take time and resources away from proven effective interventions, the emphasis on learning styles can potentially reinforce a fixed mindset and steer students away from certain leaning challenges and academic paths.Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-82145302663730473232016-06-07T17:11:00.000-07:002016-06-07T17:30:50.101-07:00"Unjustified use of Algebra 2"<div style="margin-bottom: 1.35em;">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.<br /><br />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:<br /><blockquote class="tr_bq">“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.”</blockquote>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:<br /><blockquote class="tr_bq">“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.” </blockquote>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.<br /><br />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.<br /><br />The U.S. DOE evidently intends to hold another such meeting in 3 or 4 months to check on what progress has been made.<br /><div><br /></div></div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-10476560286504651282015-02-07T13:39:00.000-08:002016-06-07T17:29:25.132-07:00Strategies to help developmental math students<div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">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.</span></div><b id="docs-internal-guid-9d225714-65ea-c8cc-d59e-d23e8d85687d" style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Four broad areas are being addressed to increase student success through developmental mathematics (1) Placement, (2) Pedagogy, (3) Curriculum, and (4) Student attitudes.</span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: bold; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Improving Placement </span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">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.</span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">For example, Cañada College uses its <a href="http://www.canadacollege.edu/STEMcenter/mathjam.php" target="_blank">Math Jam</a> both as an intensive preparation for the math placement exam and also as a recruitment tool to get more students into STEM fields.</span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The placement instrument itself, typically a machine-graded standardized test, can be augmented or replaced. <a href="http://ccrc.tc.columbia.edu/publications/predicting-success-placement-tests-transcripts.html" target="_blank">High school GPA</a>, 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.</span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: bold; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Modifying Pedagogy</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">James Stigler lists <a href="http://www.carnegiefoundation.org/blog/creating-opportunities-students-become-flexible-experts/" target="_blank">three key types of learning opportunities</a> that students need to experience to become flexible learners: productive struggle, explicit connections, and </span><span style="font-family: "arial"; font-size: 15px; line-height: 17.25px; white-space: pre-wrap;">deliberate practice</span><span style="font-family: "arial"; font-size: 15px; line-height: 1.15; white-space: pre-wrap;">.</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><br /></span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Modularized courses can allow students to spend time only on topics they need to study. The <a href="http://www.washingtonpost.com/local/education/at-virginia-tech-computers-help-solve-a-math-class-problem/2012/04/22/gIQAmAOmaT_story.html?tid=pm_pop" target="_blank">Emporium Model</a> 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<a href="http://www.math.illinois.edu/ALEKS/" target="_blank"> University of Illinois uses software</a> for placement and remediation, and <a href="http://nextgenlearning.org/grantee/california-state-university-northridge" target="_blank">California State University Northridge</a> uses software as part of its hybrid lab remediation for students considered “at risk.”</span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Technology also plays a key role in both <a href="http://chronicle.com/article/San-Jose-State-U-Puts-MOOC/140459/" target="_blank">MOOCs</a> (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.</span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Another way to address the attrition between courses in a sequence is to offer “compressed” courses. The students take <a href="http://207.62.63.167/offices/counseling_center/learncomm.asp" target="_blank">two courses during one term</a>, 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.</span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: bold; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Adjusting the Curriculum</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The American Mathematical Association of Two-Year Colleges (AMATYC) has a 2014 <a href="http://www.amatyc.org/?page=PositionInterAlg" target="_blank">position paper</a> that states, “Prerequisite courses other than intermediate algebra can adequately prepare students for courses of study that do not lead to calculus.”</span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">There are numerous “<a href="http://www.learningworksca.org/wp-content/uploads/2013/10/LWBrief_ChangingEquations_WEB.pdf" target="_blank">pathways</a>” 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.</span></div><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><a href="http://cap.3csn.org/developing-pilots/pre-statistics-courses/,%20http://3csn.org/2014/04/28/just-released-rp-group-evaluation-of-16-cap-colleges/" target="_blank">Path2Stats</a> is part of the California Acceleration Project, based on a program developed by Myra Snell at Los Medanos College.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.carnegiefoundation.org/developmental-math" target="_blank">Statway and Quantway</a> are projects of the Carnegie Foundation for the Advancement of Teaching.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The Dana Center’s <a href="http://www.utdanacenter.org/higher-education/new-mathways-project/" target="_blank">Math Pathways</a> include pathways for both STEM and non-STEM students.</span></div></li><li dir="ltr" style="background-color: transparent; color: black; font-family: Arial; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.devmathrevival.net/?page_id=8" target="_blank">Mathematical Literacy for College Students</a> (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.</span></div></li></ul><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: bold; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Addressing student attitudes</span></div><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The “affective domain” includes attitudes, values, beliefs, interests, and motivation. </span></div><b style="font-weight: normal;"><br /></b><br /><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;"><span style="background-color: transparent; color: black; font-family: "arial"; font-size: 15px; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.npr.org/templates/story/story.php?storyId=7406521" target="_blank">Carol Dweck</a>’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.</span></div><br /><span style="font-family: "arial"; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.carnegiefoundation.org/blog/creating-classroom-culture-student-success/" target="_blank">David Yeager</a>’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.</span><br /><span style="font-family: "arial"; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"><br /></span><span style="font-family: "arial"; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.mdrc.org/sites/default/files/doubling_graduation_rates_fr.pdf" target="_blank">City University of New York's Accelerated Study in Associate Programs</a> (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. </span><span style="font-family: "arial"; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"></span><br /><div><span style="font-family: "arial"; font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"><br /></span></div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-70910864199985287062014-12-28T17:21:00.000-08:002015-11-18T01:01:21.549-08:00Perfect numbers<script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>"'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, <i><a href="http://www.ams.org/notices/201005/rtx100500635p.pdf" target="_blank">The Housekeeper and the Professor</a></i>, translated by Stephen Snyder<br /><br />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 \(1+2+3+\cdots+n\).<br /><br />This got me thinking about perfect numbers. For example,<br /><br /><b>Theorem</b>: The reciprocals of the divisors of any perfect number sum to 2.<br /><b>Proof</b>: 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<br />\(1/1 + 1/d_2+ 1/d_3 \cdots + 1/d_k+ 1/N\). If we call this finite sum \(s\), then<br />\[Ns=N/1 + N/d_2+ N/d_3 \cdots + N/d_k+ N/N\]<br />\[ = N + d_k +\cdots +d_3+ d_2 + 1\]<br />\[= N +(\text{proper divisors of N}) = N+N = 2N\] Thus \( s=2\)<br /><br /><b>Theorem</b>: Every even perfect number has the form \(N=(2^k -1)\cdot 2^{k-1}\), where \(2^k - 1\) is a Mersenne prime.<br /><b>Proof</b>: 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.<br />Define \(\sigma(n)\) to be the sum of all positive divisors of \(n\). In particular, \(\sigma(n)=2n\) whenever \(n\) is perfect. <br />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.<br /><div>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</div><div>\(2^k m= \sigma(2^{k-1}m) = \sigma(2^{k-1})\sigma(m)=(2^k-1)\sigma(m)\) </div><br />Solving for \(\sigma(m)\), we get \(\sigma(m)=m+ \frac{m}{2^k-1} \)<br /><br />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.<br /><br />Conversely, <b>Theorem</b>: Each Mersenne prime \(2^k -1\) gives the perfect number \(N=(2^k -1)\cdot 2^{k-1}\).<br /><b>Proof</b>: 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<br />\[\left(1+2+…+2^{k-1}\right) + \left(2^{k}-1\right) \left(1+2+…+2^{k-2}\right)\]<br />\[ = \left(2^k -1\right) + \left(2^{k}-1\right) \left(2^{k-1}-1\right)\]<br />\[= \left(2^k -1\right) \left(1+ 2^{k-1}-1\right) = N\]<br />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 \( n = 2^k\).<br /><br />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.<br /><br />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: \(n + (n+1)\).Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-27254248230167604442014-11-28T11:46:00.000-08:002014-11-28T15:32:57.678-08:00Inscribed Triangles and the Law of Sines<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-RAsShkCReoI/VHgSy0YKCpI/AAAAAAAAM0E/nyLiY2VFZK0/s1600/Inscribed-triangle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-RAsShkCReoI/VHgSy0YKCpI/AAAAAAAAM0E/nyLiY2VFZK0/s1600/Inscribed-triangle.png" /></a></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The Law of Sines follows from the following fact: </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><b>Theorem</b>: Any side of a triangle divided by the sine of the opposite angle gives the diameter of the circumscribing circle.<br /><br />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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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 \( \frac{a}{\sin \alpha}\).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Here's a geometric argument for the theorem.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><i>Case 1</i>: 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.</div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-pK0a6Uu6dsM/VHgOix_zVHI/AAAAAAAAMz4/WnFRaXbHQg8/s1600/Inscribed-Right-Triangle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-pK0a6Uu6dsM/VHgOix_zVHI/AAAAAAAAMz4/WnFRaXbHQg8/s1600/Inscribed-Right-Triangle.png" /></a></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><i>Case 2</i>: 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 \(C\).</div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-QGwJAiZ7kEE/VHgbWUsgUII/AAAAAAAAM0U/nifrI_0tNL4/s1600/Inscribed-acute.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-QGwJAiZ7kEE/VHgbWUsgUII/AAAAAAAAM0U/nifrI_0tNL4/s1600/Inscribed-acute.png" /></a></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">\(\Delta DBC\) 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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><i>Case 3</i>: 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 \(a\).<br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-kfqxY3i25Dw/VHjJ79SaMCI/AAAAAAAAM0w/EH8ubtPXJRo/s1600/inscribed-obtuse.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-kfqxY3i25Dw/VHjJ79SaMCI/AAAAAAAAM0w/EH8ubtPXJRo/s1600/inscribed-obtuse.png" /></a></div><br />Because \(\angle CAB\) and <span style="font-size: 11pt;">\(\angle CDB\)</span><span style="font-size: 11pt;"> subtend arcs that sum to \(360^{\circ}\), they are supplementary angles. In particular, </span><span style="font-size: 11pt;">\(\angle CDB\)</span><span style="font-size: 11pt;"> is acute, so the diameter of the circumscribing circle is \(\frac{a}{\sin(180^{\circ}-\alpha)}\). And because \(\sin(180^{\circ}-\alpha) = \sin\alpha\)</span><span style="font-size: 11pt;">, we conclude that the diameter of the circumscribing circle is </span><span style="font-size: 11pt;"> </span><span style="font-size: 11pt;">\( \frac{a}{\sin \alpha}\).</span><br /><span style="font-size: 11pt;"><br /></span><span style="font-size: 11pt;">Thus, whether \(\alpha\) is a right angle, acute angle, or obtuse angle, the ratio </span><span style="font-size: 11pt;"> </span><span style="font-size: 11pt;">\( \frac{a}{\sin \alpha}\) gives the diameter of the circumscribing circle.</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-88807878995368442672014-09-09T17:00:00.001-07:002014-09-20T11:53:49.462-07:00Higher Education Alignment with the Common Core<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The <a href="http://www.cde.ca.gov/be/pn/nr/yr14sberel04att.asp" target="_blank">August 29, 2014 letter</a> 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."</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">How that alignment will look is not specified in the letter. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">As of today (9/9/14), the <a href="http://ucop.edu/agguide/a-g-requirements/c-mathematics/" target="_blank">UC Mathematics ("c") subject requirements listed publicly</a> 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, <span style="font-weight: bold;">coverage of all items in the standards is not necessary </span>for the specific purpose of meeting the 'c' subject requirement....<span style="font-weight: bold;">The ICAS Statement of Competencies in Mathematics</span> can provide guidance in selecting topics that require in-depth study." [Emphasis mine.]</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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. <a href="http://senate.universityofcalifornia.edu/committees/boars/BOARSStatementonMathforAllStudentsJuly2013.pdf" target="_blank">In July, BOARS wrote</a> 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.” </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><a href="http://senate.universityofcalifornia.edu/committees/boars/BOARSStatementonBasicMath.pdf" target="_blank">BOARS clarified (December 2013)</a> 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.”</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The college-ready standards of the CCSSM are simply all the non-plus standards. As written in the <a href="http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf" target="_blank">CCSSM</a>, <span style="font-size: 11pt;">“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 (+). </span><span style="font-size: 11pt; font-weight: bold;">All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students</span><span style="font-size: 11pt;">.” [Emphasis mine]</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">But what undermines BOARS's credibility is its assessment of how the <a href="http://icas-ca.org/competencies-in-mathematics" target="_blank">ICAS statement of competencies</a> 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 <span style="font-weight: bold;">close alignment</span> between it and the CCSSM. Both define the mathematics that all students should study in order to be college ready." [Emphasis mine.]</div><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In actuality, what ICAS considers essential math content for all students is only a small subset of what the CCSSM specify as necessary. The<a href="http://icas-ca.org/Websites/icasca/images/ICAS-Statement-Math-Competencies-2013.pdf" target="_blank"> ICAS document</a> 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.</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-46130901739355684742014-08-29T18:47:00.000-07:002014-08-31T16:36:36.469-07:00Cosine of 72 degrees (and constructing a regular pentagon)<script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> By using (say) DeMoivre's theorem, we have that \( \left( \cos\frac{2 \pi}{5}+ i \sin\frac{2\pi}{5} \right)^5=1\)<br />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 \(1- \cos^2 72^\circ \)<br /><div style="text-align: center;"> \( i\sin 72^\circ \left(5 \cos^4 72^\circ+10 \cos^2 72^\circ i^2 \sin ^2 72 ^\circ+ i^4 \sin^4 72^\circ \right)=0i\)</div><div style="text-align: center;"> \(16 \cos^4 72^\circ - 12 \cos^2 72^\circ + 1=0\)</div>Solving this quadratic in \( \cos^2 72^\circ \), we get<br /><div style="text-align: left;"> \[ \cos^2 72^\circ = \frac{12 \pm \sqrt{80} } {32} \]</div><div style="text-align: left;">so</div><div style="text-align: left;"> \[ \cos^2 72^\circ = \frac{6 \pm 2 \sqrt{5} } {16}=\frac{\left( \sqrt{5}\pm 1 \right)^2}{4^2} \]</div><div style="text-align: left;"> \[ \cos 72^\circ = \pm \frac{\sqrt{5} \pm 1}{4} \]</div><div>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:</div>\[ \cos 72^\circ = \frac{\sqrt{5}-1}{4} \]<br /><h2>Constructing a regular pentagon</h2>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 <a href="http://byoshiwara.blogspot.com/2010/01/solving-quadratic-equations-via.html" target="_blank">Solving quadratic equations via geometric construction</a>.)<br /><br />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.<br /><br />Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com1tag:blogger.com,1999:blog-5834886388752215525.post-74375181508581357532014-08-05T20:25:00.002-07:002014-08-08T20:14:03.244-07:00What Math is Needed by All?<div class="separator" style="clear: both; text-align: center;"><script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script><a href="http://www.ncee.org/wp-content/uploads/2013/05/cover.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://www.ncee.org/wp-content/uploads/2013/05/cover.png" height="320" width="236" /></a></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The (California version of the) <a href="http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf" target="_blank">Common Core State Standards in mathematics</a> purport to be what all students need to be college and career ready.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">And what do all students need? In 2013, the National Center on Education and the Economy released a study <span style="font-style: italic;"><a href="http://www.ncee.org/college-and-work-ready/" target="_blank">What Does It Really Mean to Be College and Work Ready?</a></span>, 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."</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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 <span style="font-style: italic;"><a href="http://icas-ca.org/competencies-in-mathematics" target="_blank">Statement on Competencies in Mathematics Expected of Entering College Students</a></span>, revised in 2013, describes a number of mathematical topics that are or could be taught in high schools. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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) <a href="http://senate.universityofcalifornia.edu/committees/boars/BOARSStatementonMathforAllStudentsJuly2013.pdf" target="_blank">states</a> 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).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">And the <a href="https://www.bakersfieldcollege.edu/sites/bakersfieldcollege.edu/files/Environmental%20Scan%20Report%20to%20CAISC%206.14.pdf" target="_blank">Interim Environmental Scan Report to The Common Assessment Initiative Steering Committee</a> has in Appendix B a Table <span style="font-size: 11pt;">that shows a number of CCSS that do not occur at all in the ICAS statement.</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"></div><ul><li><span style="font-size: 11pt;">Probability:</span><span style="font-size: 11pt;"> </span><span style="font-size: 11pt;">sample spaces, independent events, conditional probability, permutations and combinations; analyzing decisions and strategies using probability</span></li><li><span style="font-size: 11pt;">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</span></li><li><span style="font-size: 11pt;">Transformational geometry: congruence defined in terms of rigid motion; similarity defined in terms of dilations and rigid motions</span></li><li><span style="font-size: 11pt;">Trigonometry: trig ratios, special angles, 6 trig functions of real numbers; modeling periodic phenomena, proof and use of the Pythagorean trig identity \( \cos^2 \theta + \sin^2 \theta = 1 \) </span></li></ul><br />Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-19053061809308648912014-07-29T14:21:00.001-07:002014-08-05T19:01:02.620-07:00Schizophrenic Common Core Supporter<div class="separator" style="clear: both; text-align: center;"><a href="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQbllbI0rVvC-pmH1mDYCJKwHyH64nCYmcOydfT-8R3b6d0yk7T" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQbllbI0rVvC-pmH1mDYCJKwHyH64nCYmcOydfT-8R3b6d0yk7T" /></a></div><br />Back in 2012 <a href="http://www.ams.org/journals/notices/201207/rtx120700909p.pdf" target="_blank">Sol Garfunkel wrote</a> "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!"<br /><br />As mathematician <a href="http://www.huffingtonpost.com/dr-keith-devlin/common-core-math-standards_b_5369939.html" target="_blank">Keith Devlin has emphasized</a>, the heart of the CCSSM is the set of <a href="http://www.corestandards.org/Math/Practice/" target="_blank">8 standards of Mathematical Practice</a>:<br /><br /><ul><li>MP1. Make sense of problems and persevere in solving them.</li><li>MP2. Reason abstractly and quantitatively.</li><li>MP3. Construct viable arguments and critique the reasoning of others.</li><li>MP4. Model with mathematics.</li><li>MP5. Use appropriate tools strategically.</li><li>MP6. Attend to precision.</li><li>MP7. Look for and make use of structure.</li><li>MP8. Look for and express regularity in repeated reasoning.</li></ul><br /><br />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.<br /><br />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.<br /><br />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 <a href="http://www.smarterbalanced.org/" target="_blank">SBAC</a> or <a href="http://www.parcconline.org/" target="_blank">PARCC</a>, 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.<br /><br />The widely seen <a href="http://christopherdanielson.wordpress.com/2014/04/06/5-reasons-not-to-share-that-common-core-worksheet-on-facebook/" target="_blank">mocking and vilification of CCSS lessons by the public</a> 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.<br /><br />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.<br /><div><br /></div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-91830374058108188422014-07-23T13:11:00.000-07:002014-09-20T12:06:02.544-07:00Common Core Goes to College<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The New America Foundation’s <a href="http://www.newamerica.net/sites/newamerica.net/files/policydocs/CCGTC_7_18_2pm.pdf" target="_blank">position paper by Lindsey Tepe</a> gives recommendations for how higher education can support the Common Core State Standards. However, this paper and related articles in the <a href="http://chronicle.com/article/Colleges-Must-Help-Further-the/147843/" target="_blank">Chronicle </a> and <a href="http://hechingerreport.org/content/report-higher-ed-behind-common-core_16785/" target="_blank">Hechinger Report</a> 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. </div><div class="separator" style="clear: both; text-align: center;"><a href="http://www.newamerica.net/sites/newamerica.net/files/policydocs/CCGTC_7_18_2pm.pdf"><img alt="cover of position paper" border="0" src="http://www.newamerica.net/sites/newamerica.net/files/imagecache/standard_node_image/articles/images/commoncore.PNG" title="Position paper" /></a></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The CCSS have missed the mark at what is necessary for all students to succeed in college.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><span style="font-family: Calibri; font-size: 11pt;">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.</span>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-26223435088755382082014-07-17T10:31:00.000-07:002014-07-17T10:31:31.103-07:00Math Initiatives for Student SuccessThe LearningWorks paper <i><a href="http://www.learningworksca.org/wp-content/uploads/2013/10/LWBrief_ChangingEquations_WEB.pdf" target="_blank">Changing Equations: How Community Colleges Are Re-thinking College Readiness in Math</a></i>, 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.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://content.screencast.com/users/yoshiwbw/folders/Jing/media/26cc6051-e8ee-491e-971d-85151de2f125/MathReadinessReformMenu.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://content.screencast.com/users/yoshiwbw/folders/Jing/media/26cc6051-e8ee-491e-971d-85151de2f125/MathReadinessReformMenu.png" height="235" width="640" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal">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.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">The initial data on alternative pathways, some cited in <i>Changing Equations</i>, 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. </div><div class="MsoNormal"><br /></div><div class="MsoNormal">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 <i>Changing Equations</i>, 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.</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-29033789149494418762014-01-19T15:00:00.002-08:002014-01-19T15:00:48.885-08:00JMM 2014<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The JMM unveiled the theme of the 2014 Math Awareness Month (April 2014): Mathematics, Magic, & Mystery (<a href="http://www.mathaware.org/">http://www.mathaware.org/</a>). 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.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">JMM2014 also included a panel session launching TPSE Math: Transforming Post-Secondary Education in Mathematics (@tpsem, <a href="http://www.tpsemath.org/">http://www.tpsemath.org/</a>). The project is sponsored by the Carnegie Foundation of New York and the Alfred P. Sloan Foundation.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://blogs.ams.org/jmm2014/files/2014/01/unnamed1-300x300.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://blogs.ams.org/jmm2014/files/2014/01/unnamed1-300x300.jpg" /></a></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">One of the sessions on the last day was The Public Face of Mathematics <a href="http://blogs.ams.org/jmm2014/2014/01/18/the-public-face-of-mathematics/">http://blogs.ams.org/jmm2014/2014/01/18/the-public-face-of-mathematics/</a>. <span style="font-size: 11pt;">The panel was organized by mathemagician Art Benjamin (</span><a href="http://www.math.hmc.edu/~benjamin/" style="font-size: 11pt;">http://www.math.hmc.edu/~benjamin/</a><span style="font-size: 11pt;">) and included "Math Guy" Keith Devlin (</span><a href="http://www.stanford.edu/~kdevlin/" style="font-size: 11pt;">http://www.stanford.edu/~kdevlin/</a><span style="font-size: 11pt;">), NY Times columnist Steven Strogatz (</span><a href="http://www.stevenstrogatz.com/" style="font-size: 11pt;">http://www.stevenstrogatz.com/</a><span style="font-size: 11pt;">), mathbabe Cathy O'Neill (</span><a href="http://mathbabe.org/" style="font-size: 11pt;">http://mathbabe.org/</a><span style="font-size: 11pt;">), freelance journalist Tom Siegfried (</span><a href="http://www.sciencenoise.org/" style="font-size: 11pt;">http://www.sciencenoise.org/</a><span style="font-size: 11pt;">), and US Congressman Jerry McNerny (</span><a href="http://mcnerney.house.gov/" style="font-size: 11pt;">http://mcnerney.house.gov/</a><span style="font-size: 11pt;">). </span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-54431140011752426652013-07-26T09:32:00.000-07:002013-07-26T11:04:08.732-07:00More on Alternative Pathways and transferability in California<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The July 2013 statement (<a href="http://senate.universityofcalifornia.edu/committees/boars/BOARSStatementonMathforAllStudentsJuly2013.pdf" target="_blank">http://senate.universityofcalifornia.edu/committees/boars/BOARSStatementonMathforAllStudentsJuly2013.pdf</a>) 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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The BOARS statement continues, "<span style="color: blue;">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.</span>"</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Meanwhile, the Academic Senate of California Community Colleges (ASCCC) has endorsed the CCSSM, but has no formal position on alternative pathways. A Fall 2012 <a href="http://asccc.org/resolutions/support-innovations-improve-underprepared-non-stem-student-success-mathematics" target="_blank">resolution to support innovations to improve success in under-prepared non-STEM pathways</a> was referred to the executive committee. However, former ASCCC president Ian Walton did publish in the ASCCC Rostrum an opinion (<a href="http://asccc.org/content/alternatives-traditional-intermediate-algebra" target="_blank">http://asccc.org/content/alternatives-traditional-intermediate-algebra</a>) that "<span style="color: blue;">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.</span>"</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-25956467214184599092013-07-21T15:44:00.001-07:002013-07-22T09:17:21.517-07:00Alternative Pathways and transferability in California<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">California is home to the Carnegie Foundation for the Advancement of Teaching, the current force behind two pathway projects: <a href="http://www.carnegiefoundation.org/developmental-math" target="_blank">Statway and Quantway</a>. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; margin: 0in;"><span style="font-size: 11.0pt;">An underlying assumption behind alternative pathways is that mathematics requirements for degrees and/or certificates should vary according to discipline. California's </span><a href="http://www.californiacommunitycolleges.cccco.edu/Portals/0/Executive/StudentSuccessTaskForce/SSTF_Final_Report_1-17-12_Print.pdf" target="_blank"><span style="font-size: 11.0pt;">Student Success Task Force report</span></a><span style="font-size: 11.0pt;"> contends, "Improved student support structures and better </span><span style="color: #333333; font-size: 11.25pt; font-style: italic;">alignment of curriculum with student needs </span><span style="font-size: 15px;">[Emphasis added] </span><span style="font-size: 11pt;">will increase success rates in transfer, basic skills, and career technical/workforce programs." The National Center on Education and the Economy 2013 report, "</span><a href="http://www.ncee.org/college-and-work-ready/" target="_blank"><span style="font-size: 11.0pt;">What Does It Really Mean to Be College and Work Ready?</span></a><span style="font-size: 11pt;">" 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."</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="margin: 0in;"><span style="font-family: Calibri; font-size: 11.0pt;">The Carnegie Foundation, The Charles A. </span><a href="http://www.utdanacenter.org/mathways/" target="_blank"><span style="font-family: "Trebuchet MS"; font-size: 9.75pt;">Dana Center</span></a><span style="font-family: Calibri; font-size: 11.0pt;"> at U.T. Austin, and the California Community College Success Network (</span><a href="http://cap.3csn.org/why-acceleration/" target="_blank"><span style="font-family: "Trebuchet MS"; font-size: 9.75pt;">3CSN</span></a><span style="font-family: Calibri; font-size: 11.0pt;">) 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. </span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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). </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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:</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin-left: .375in; margin: 0in;"><span style="color: blue;">"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."</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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:</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin-left: .375in; margin: 0in;"><span style="color: blue;">"Please take this email as a reminder that only courses with a full prerequisite of intermediate algebra, as <a href="http://en.wikibooks.org/wiki/Intermediate_Algebra" target="_blank">traditionally understood</a>, will continue to qualify for CSU Area B4 [math/quantitative reasoning requirement to transfer].</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin-left: .375in; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin-left: .375in; margin: 0in;"><span style="color: blue;">"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."</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">But Ken O'Donnell sent <a href="http://cap.3csn.org/2013/04/12/csu-clarifies-intermediate-algebra-policy/" target="_blank">an April 2013 email</a> 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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><br /><div style="font-family: Calibri; font-size: 11.0pt; margin-left: .375in; margin: 0in;"><span style="color: blue;">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.<span style="font-family: Tahoma; font-size: 8pt;"> </span></span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Thus both the UC and the CSU are tacitly giving CCCs the go-ahead to develop alternative pathways.</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-79027065671549193542013-03-28T16:33:00.001-07:002014-08-08T20:21:10.138-07:00Heron's formula for the area of a triangle<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-wCGEJAunkWY/U-WTfCKwE3I/AAAAAAAAMXE/Kbt-Jn0aprg/s1600/heron.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-wCGEJAunkWY/U-WTfCKwE3I/AAAAAAAAMXE/Kbt-Jn0aprg/s1600/heron.png" height="151" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The angle bisectors of the triangle meet at the center of the inscribed circle of radius <i>r</i>. If we let \(2\alpha=A\), \(2\beta = B\), and \(2\gamma=C\), we have \(\alpha+\beta+\gamma=\frac{\pi}{2}\). </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Let <i>x </i>be the distance from the vertex at <i>A</i> to points of tangency, <i>y</i> the distance from <i>B</i>, and <i>z </i>the distance from <i>C</i>. Then then lengths of the triangle sides opposite <i>A</i>, <i>B</i>, and <i>C </i>are respectively \(a=y+z\), \(b=x+z\), and \(c=x+y\).<br /><br />Thus if we name the semiperimeter <i>s</i>, <span style="font-size: 11pt;">then </span><span style="font-size: 11pt;">\(s=x+y+z\), </span><span style="font-size: 11pt;">\(x=s-a\), \(y=s-b\), and \(z=s-c\)</span><span style="font-size: 11pt;">.</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="margin: 0in;"><div style="font-family: Calibri; font-size: 11pt;">\(\tan \alpha =\frac{r}{x}\), <span style="font-size: 11pt;">\(\tan \beta =\frac{r}{y}\)</span><span style="font-size: 11pt;">, and </span><span style="font-size: 11pt;">\(\tan \gamma =\frac{r}{z}\)</span><span style="font-size: 11pt;">. Because \(\gamma\) and \( (\alpha+\beta )\) are complementary angles, we obtain</span></div><div style="font-family: Calibri; font-size: 11pt;"><span style="font-size: 11pt;"><br /></span><span style="font-size: 11pt;"><br /></span><span style="font-size: 11pt;">\[ \tan\left( \frac{\pi}{2} - (\alpha+\beta) \right) = \frac{r}{z} \]</span><br /><span style="font-size: 11pt;">\[ \tan\left( \alpha+\beta \right) = \frac{z}{r} \]</span></div><div style="font-family: Calibri; font-size: 11pt;">\[ \frac{ \tan \alpha+\tan\beta}{1-\tan\alpha \tan\beta} = \frac{z}{r} \]</div><div style="font-family: Calibri; font-size: 11pt;">\[r \left(\tan \alpha+\tan\beta \right)= z (1-\tan\alpha\tan\beta) \]</div><div style="font-family: Calibri; font-size: 11pt;">\[r \left( \frac{r}{x}+\frac{r}{y} \right) = z \left( 1 - \frac{r}{x}\frac{r}{y} \right) \]</div><div style="font-family: Calibri; font-size: 11pt;">\[ r^2 y + r^2 z = xyz - r^2 z \]</div><div style="font-family: Calibri; font-size: 11pt;">\[ r^2 ( x+y+z) = xyz \]</div><div style="font-family: Calibri; font-size: 11pt;">\[ r^2 s = xyz \]</div><div style="font-family: Calibri; font-size: 11pt;"><br /></div><div style="font-family: Calibri; font-size: 11pt;"><span style="font-size: 11pt;"><br /></span></div></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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<br />\[ (\text{area})^2 = s\cdot xyz = s(s-a)(s-b)(s-c)\]<br />so the area is \(\sqrt{s(s-a)(s-b)(s-c)}\).</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-11978664566343003222013-01-16T21:13:00.001-08:002014-08-08T20:21:58.065-07:00Contradictory mandates to community colleges<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">A goal of the Common Core State Standards (CCSS) is to prepare students to be college and career ready.<span style="mso-spacerun: yes;"> </span>That goal is also part of the mission of community colleges.<span style="mso-spacerun: yes;"> </span>There has been considerable discussion regarding how<span style="mso-spacerun: yes;"> </span>the CCSS might affect students' chances for getting into college, but scant discussion about how community colleges fit into the implementation of the CCSS. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Community Colleges might be assumed to have a distorted view of what it means to be college or career ready.<span style="mso-spacerun: yes;"> </span>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.<span style="mso-spacerun: yes;"> </span>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%.<span style="mso-spacerun: yes;"> </span>But this is too limiting--a high school diploma is not a requirement for enrollment at any California Community College.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">So at community colleges, we may worry less about "college ready" but rather focus on "transfer ready".<span style="mso-spacerun: yes;"> </span>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.<span style="mso-spacerun: yes;"> </span>So there is evidence that community colleges are not grossly underestimating what is needed to be transfer ready.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">California Community Colleges are presented with two conflicting mandates .<span style="mso-spacerun: yes;"> </span>Community colleges<span style="mso-spacerun: yes;"> </span>are encouraged 1) to align with K-12 standards for college and career readiness (according to the California Community College<span style="mso-spacerun: yes;"> </span>Student Success Task Force<span style="mso-spacerun: yes;"> </span><a href="http://bit.ly/xOC5aK" target="_blank">http://bit.ly/xOC5aK</a>), and 2)<span style="mso-spacerun: yes;"> </span>to provide alternative pathways to transfer (according to the Carnegie Foundation for the Advancement of Teaching<span style="mso-spacerun: yes;"> </span><a href="http://bit.ly/y1EZhX" target="_blank">http://bit.ly/y1EZhX</a>, the Charles A. Dana Center<span style="mso-spacerun: yes;"> </span><a href="http://www.utdanacenter.org/mathways/" target="_blank">http://www.utdanacenter.org/mathways/</a>, etc.).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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<span style="mso-spacerun: yes;"> </span>"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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">"...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" : <a href="http://bit.ly/TPqqCp" target="_blank">http://bit.ly/TPqqCp</a>)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.<span style="mso-spacerun: yes;"> </span>Evidently there are many students who can earn baccalaureate degrees without taking<span style="mso-spacerun: yes;"> </span>single course from the mathematics department of any 4-year school.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The California Community College Success Network (<a href="http://3csn.org/" target="_blank">3CSN.org</a>), the Carnegie Foundation, the Dana Center, and the Student Success Task Force all recommend removing<span style="mso-spacerun: yes;"> </span>curricular requirements<span style="mso-spacerun: yes;"> </span>that act as barriers rather than aids to program completion.<span style="mso-spacerun: yes;"> </span>The SSTF report contends, "Improved student support structures and better <span style="font-style: italic;">alignment of curriculum with student needs </span>will increase success rates in transfer, basic skills, and career technical/workforce programs." [emphasis added] </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The existing and proposed curricula of alternative pathways for non-STEM students omit many topics of intermediate algebra.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">On the other hand, neither the University of California nor the California State University accepts a math or statistics course to meet<span style="mso-spacerun: yes;"> </span>math transfer requirements unless that course has intermediate algebra as a prerequisite.<span style="mso-spacerun: yes;"> </span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">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.</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com2tag:blogger.com,1999:blog-5834886388752215525.post-28825859624885957342012-12-20T13:33:00.001-08:002012-12-20T14:46:56.549-08:00Alternative Pathways vs Common Core State Standards<br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">A primary goal of the <a href="http://byoshiwara.blogspot.com/2012/12/common-core-state-standards-algebra.html" target="_blank">Common Core State Standards</a> (<a href="http://www.corestandards.org/" target="_blank">CCSS</a>) 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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><a href="http://byoshiwara.blogspot.com/2012/11/alternative-pathways.html" target="_blank"><span id="goog_616871354"></span>Alternative pathways<span id="goog_616871355"></span></a>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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The common goal of both alternative pathways and the CCSS is to improve U.S. education.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"> "<a href="http://www.jff.org/sites/default/files/RemediationJointStatement-121312update.pdf" target="_blank">Core Principles for Transforming Remedial Education: A Joint Statement</a>" 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.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">"...Principle 2. The content in required gateway courses should align with a student’s academic program of study — particularly in math.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">"Gateway courses provide a foundation for a program of study, and students should expect that the <span style="font-size: 11pt;">skills they develop in gateway courses are relevant to their chosen program. On many campuses, </span><span style="font-size: 11pt;">remedial education is constructed as single curricular pathways into gateway math or English courses.</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">"The curricular pathways often include content that is not essential for students to be successful in their <span style="font-size: 11pt;">chosen program of study. Consequently, many students are tripped up in their pursuit of a credential </span><span style="font-size: 11pt;">while studying content that they do not need. Institutions need to focus on getting students into the </span><span style="font-size: 11pt;">right math and the right English.</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">"This issue is of particular concern in mathematics, which is generally considered the most significant <span style="font-size: 11pt;">barrier to college success for remedial education students. At many campuses, remedial math is geared </span><span style="font-size: 11pt;">toward student preparation for college algebra. However for many programs of study, college algebra </span><span style="font-size: 11pt;">should not be a required gateway course when a course in statistics or quantitative literacy would be </span><span style="font-size: 11pt;">more appropriate….</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">"...One final note: Postsecondary leaders must work closely with K–12, adult basic education, and <span style="font-size: 11pt;">other training systems to reduce the need for remediation before students enroll in their institutions. </span><span style="font-size: 11pt;">Postsecondary institutions should leverage the Common Core State Standards by working with K–12 </span><span style="font-size: 11pt;">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 </span><span style="font-size: 11pt;">career readiness assessments, which lead to customized academic skill development during the senior </span><span style="font-size: 11pt;">year, should be a priority for states. Similar strategies should be employed in adult basic education and </span><span style="font-size: 11pt;">English as a second language programs."</span></div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com1tag:blogger.com,1999:blog-5834886388752215525.post-87685354686002687772012-12-17T15:48:00.001-08:002013-01-11T07:10:50.293-08:00Common Core State Standards AlgebraOne 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.<br /><br />Unless new legislation addresses this change in content, the adoption of the CCSS automatically raises the California high school graduation requirement.<br /><br />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.<br /><br />And the <a href="http://www.californiacommunitycolleges.cccco.edu/Portals/0/StudentSuccessTaskForce/SSTF_FinalReport_Web_010312.pdf" target="_blank">California Community College Student Success Task Force</a> calls for better alignment:<br /><blockquote>"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...<br /><br />"<b>Recommendation 1.1</b><br />"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.<br /><br />"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 <a href="http://www.corestandards.org/" target="_blank">K-12 Common Core State Standards</a> and engages with the national <a href="http://www.smarterbalanced.org/" target="_blank">SMARTER Balanced Assessment Consortium</a> 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."</blockquote>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, <a href="http://www.parcconline.org/" target="_blank">PARCC</a>, already has agreement among its adopting states to use its assessments for college placement. (See, for example, <a href="http://bit.ly/QYVjUF">http://bit.ly/QYVjUF</a>.)<br /><div><br /></div><br />Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-20963135250533960262012-11-22T15:50:00.000-08:002013-07-20T11:47:05.564-07:00Alternative pathways<br />It seems that student-success discussion at two-year colleges is shifting from "course redesign" to "alternative pathways" for the math required to transfer.<br /><br />The topics overlap considerably, because the idea of alternative pathways normally involves modifying course prerequisites in format and/or content.<br /><br />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).<span style="font-size: 11pt;"> </span><br /><br />The <a href="http://www.carnegiefoundation.org/developmental-math" style="color: #0068cf; cursor: pointer;" target="_blank">Carnegie Foundation for the Advancement of Teaching</a>, the <a href="http://www.utdanacenter.org/mathways/" style="color: #0068cf; cursor: pointer;" target="_blank">Dana Center</a> at UT Austin, and <a href="http://cap.3csn.org/why-acceleration/" style="color: #0068cf; cursor: pointer;" target="_blank">3CSN</a> (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.<br /><br />But last month the CSU emailed California Community College articulation officers the following:<br /><br />"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.<br /><br /> "Please take this email as a reminder that only courses with a full prerequisite of intermediate algebra, as <a href="http://en.wikibooks.org/wiki/Intermediate_Algebra" style="color: #0068cf; cursor: pointer;" target="_blank">traditionally understood</a>, will continue to qualify for CSU Area B4.<br /><br /> "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."<br /><br />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. Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-11172341367433839462012-11-15T16:22:00.000-08:002012-11-15T17:01:15.141-08:00Student struggle is a good thing!<br /><embed allowfullscreen="true" base="http://www.npr.org" height="386" src="http://www.npr.org/v2/?i=164793058&m=164940191&t=audio" type="application/x-shockwave-flash" width="400" wmode="opaque"></embed> <br />NPR recently interviewed UCLA researcher Jim Stigler about the differences between how the US and other cultures view student struggle.<br /><br />In the US, we typically attribute academic success to intelligence, and often give praise by admiring how smart someone is. In many east Asian cultures, success is attributed to continued effort, and children are praised for their persistence to overcome obstacles.<br /><br />A possible consequence is that US children who do not have immediate success at a task will abandon the effort--their intelligence was evidently insufficient. And US education authorities view student struggle as an indicator that something is wrong--the term "struggling student" is used to designate a student who requires some intervention, rather than to describe a student experiencing an essential stage of deep understanding.<br /><br />Asian cultures often embrace student struggle as a key indicator of future success. And it actually should be embraced by educators following the Common Core State Standards for Mathematics, which has as its first standard of Mathematical Practices:<br /><br /><ol><li>Make sense of problems and persevere in solving them.</li></ol><br />Praising intelligence rather than effort also reinforces a fixed mindset, which can limit a person's successes, whereas praising effort promotes the development of a growth mindset. Carol Dweck has fascinating data on how mindsets affect learning and how mindsets can be changed.<br /><br /> <iframe allowfullscreen="allowfullscreen" frameborder="0" height="315" src="http://www.youtube.com/embed/TTXrV0_3UjY" width="560"></iframe><br /><br /><br />Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-89873459846412041442012-10-14T11:40:00.002-07:002013-01-12T13:05:00.122-08:00Mathematical Practices<script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> <div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The <a href="http://www.corestandards.org/Math" target="_blank">Common Core State Standards</a> list <a href="http://www.corestandards.org/Math/Practice" target="_blank">Standards for Mathematical Practice</a> at each grade level. These practices are</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="color: #632423; font-family: Arial; font-size: 11.0pt; font-weight: bold; margin: 0in;">Mathematical Practices</div><div style="font-family: Arial; font-size: 9.0pt; margin: 0in;">1. Make sense of problems and persevere in solving them.</div><div style="font-family: Arial; font-size: 9.0pt; margin: 0in;">2. Reason abstractly and quantitatively.</div><div style="font-family: Arial; font-size: 9.0pt; margin: 0in;">3. Construct viable arguments and critique the reasoning of others.</div><div style="font-family: Arial; font-size: 9.0pt; margin: 0in;">4. Model with mathematics.</div><div style="font-family: Arial; font-size: 9.0pt; margin: 0in;">5. Use appropriate tools strategically.</div><div style="font-family: Arial; font-size: 9.0pt; margin: 0in;">6. Attend to precision.</div><div style="font-family: Arial; font-size: 9.0pt; margin: 0in;">7. Look for and make use of structure.</div><div style="font-family: Arial; font-size: 9.0pt; margin: 0in;">8. Look for and express regularity in repeated reasoning.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">These same eight expectations are listed in the descriptions for every grade level and for every advanced (a.k.a. high school) course. But carrying out the Mathematical Practices will look different at different grade levels.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The practices that may require the most clarification are probably #4 (Model with mathematics), #7 (Look for and make use of structure) and #8 (Look for and express regularity in repeated reasoning).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Modeling with mathematics typically involves using and perhaps even creating mathematical objects (such as algebraic expressions, equations, inequalities, graphs, etc.) to capture key aspects of a situation to be explored. A kindergartner might use 2+3 to represent the number of people involved if two people are joined by 3 more; a sixth grader might describe a relationship between the numbers of tables to chairs in a room by the ratio 1:4; an Algebra I student might use the expression 10<span style="font-style: italic;">x</span>to represent the value (in cents) of <span style="font-style: italic;">x</span>dimes.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">But not all word problems involve mathematical modeling. It is <i>not </i>mathematical modeling to use a contrived algebraic expression such as a quadratic expression obtained by curve-fitting bi-variate data without any plausible a priori reason for believing that the two variables should be related quadratically.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Mathematical Practice #7, seeking and using structure, is key to both pure and applied mathematics. A first-grader begins to recognize that an addition fact such as 2+3=5 carries with it a family of related arithmetic facts, e.g., 3+2=5, 5-2=3, 5-3=2, etc. ; a seventh-grader can see that because a+0.05a = 1.05a, increasing a quantity by 5% is equivalent to scaling the quantity by 1.05; a geometry student recognizes and introduces structure by adding an auxiliary line to a geometric diagram. Mathematical Practice #7 is definitely not about memorizing or plugging into formulas--both practices, when applied inappropriately, can allow students to ignore the underlying structure .</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I personally need further explanation of Mathematical Practice #8. Here is how it's first described in the <a href="http://www.corestandards.org/Math/Practice" target="_blank">CCSS</a>:</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><i>Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line </i><i style="font-size: 11pt;">through (1, 2) with slope 3, middle school students might abstract the equation \( \frac {y – 2}{x – 1} = 3\) . Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while </i><i style="font-size: 11pt;">attending to the details. They continually evaluate the reasonableness of their intermediate results.</i></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The phrase "repeated reasoning" presumably refers to the "R" in <a href="http://www.math.ucsd.edu/~harel/" target="_blank">Guershon Harel</a>'s DNR. Some of Harel's work is listed in the mathematics CCSS references, but there do not appear to be any direct attributions cited.</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-46164072616960426782012-10-12T21:16:00.001-07:002012-10-12T21:17:33.641-07:00Supporting community college faculty across the disciplines <br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I recently spent 48 hours in Northfield, MN (home of Malt-O-Meal) to work with educators from different disciplines and different organizations trying to find ways to increase two-year college faculty awareness of and participation in professional development opportunities.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"></div><div style="font-size: 11pt; margin: 0in;">The workshop was hosted by Carleton College and its Science Education Research Council. SERC (<a href="http://serc.careltoncollege.edu/">http://serc.careltoncollege.edu</a>) has amassed an impressive collection of resources across multiple disciplines including geoscience (the first discipline), chemistry, economics, mathematics, physics, psychology, and more. </div><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The Pedagogy in Action page (<a href="http://serc.carleton.edu/sp/">http://serc.carleton.edu/sp/</a>) has links for Teaching Methods, Activities, and Research on Learning. SERC is continually seeking to improve its website to become a one-stop launching point for finding discipline-specific lesson plans, research-based pedagogical strategies, student projects, career information--essentially anything of interest to an educator seeking to improve student learning.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">SERC has also been learning how to run effective workshops. We were given pre-workshop assignments to upload essays into designated spaces on the SERC website that were visible to the other participants but not to the rest of the world. And during the workshop we were constantly moving from whole group to small group activities, mixing tasks from cross-discipline to the discipline-specific. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Each group would choose a recorder, who wirelessly entered directly into the SERC system. The others in the small working group could see the notes on their own computers during their discussion, and the notes were available to the whole group during the "share out" session. Working across disciplines allowed us to learn of challenges and strategies that gave us fresh perspectives for our discipline-specific discussions. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The real-time recording of discussions means that our notes won't be accidentally lost among papers or luggage during our journeys home. Eventually the notes from our workshop will be organized, polished, and made publicly accessible on the SERC site.</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-52167853908855595532012-09-28T19:49:00.000-07:002012-09-29T13:47:49.738-07:00Simpson's Rule on Cubics<br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Simpson's rule approximates a definite integral<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=$\int_{a}^{b}%20f(x)%20dx$" style="font-size: 11pt; margin-left: 1em; margin-right: 1em;" target="_blank"><img alt="$\int_{a}^{b} f(x) dx$" src="http://latex.codecogs.com/gif.latex?$\int_{a}^{b}%20f(x)%20dx$" title="$\int_{a}^{b} f(x) dx$" /></a></div><br /><span style="font-size: 11pt;">by replacing the integrand </span><i style="font-size: 11pt;">f </i><span style="font-size: 11pt;">with</span><span style="font-size: 11pt;"> </span><span style="font-size: 11pt;">the quadratic function that agrees with </span><i style="font-size: 11pt;">f </i><span style="font-size: 11pt;">at the endpoints and midpoint of each sub-interval. (For comparison, the Left- and Right-Hand Riemann sums each replace </span><i style="font-size: 11pt;">f </i><span style="font-size: 11pt;">with a constant function, the Trapezoid and Midpoint rules replace </span><i style="font-size: 11pt;">f </i><span style="font-size: 11pt;">with the linear function respectively agreeing with <i>f</i> at the endpoints of the interval or agreeing with both </span><i style="font-size: 11pt;">f </i><span style="font-size: 11pt;">and </span><i style="font-size: 11pt;">f</i><span style="font-size: 11pt;">' at the midpoint of the interval.)</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">It is remarkable that Simpson's rule gives the exact values of definite integrals not only for any quadratic but also for any cubic polynomial, using only one sub-interval.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">This can be algebraically verified by using the change of variable <i>x = a + </i>(<i>b - a</i>)<i>t</i> and verifying that Simpson's rule with one sub- interval gives the exact value for<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=$\int_{0}^{1}%20x^3%20dx$" target="_blank"><img src="http://latex.codecogs.com/gif.latex?$\int_{0}^{1} x^3 dx$" title="$\int_{0}^{1} x^3 dx$" /></a></div></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br />Here is a more geometric argument.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Let <i>f </i>be a cubic polynomial, and let <i>q </i>be the quadratic function satisfying <i>f</i>(<i>a</i>) = <i>q</i>(<i>a</i>), <i>f</i>(<i>b</i>) = <i>q</i>(<i>b</i>), and <i>f</i>((<i>a</i>+<i>b</i>)/2) = <i>q</i>((<i>a</i>+<i>b</i>)/2).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then the error in using Simpson's rule for approximating<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=$\int_{a}^{b}%20f(x)%20dx$" style="font-size: 11pt; margin-left: 1em; margin-right: 1em;" target="_blank"><img alt="$\int_{a}^{b} f(x) dx$" src="http://latex.codecogs.com/gif.latex?$\int_{a}^{b}%20f(x)%20dx$" title="$\int_{a}^{b} f(x) dx$" /></a></div><br />is<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=$\int_{a}^{b}%20f(x)%20dx$" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img alt="$\int_{a}^{b} f(x) dx$" src="http://latex.codecogs.com/gif.latex?$\int_{a}^{b} E(x) dx$" title="$\int_{a}^{b} f(x) dx$" /></a></div><br />where <i>E</i> is the cubic polynomial defined by <i>E</i>(<i>x</i>) = <i>q</i>(<i>x</i>) - <i>f</i>(<i>x</i>).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Because <i>E</i>(<i>a</i>) = <i>E</i>(<i>b</i>) = <i>E</i>((<i>a</i>+<i>b</i>)/2) = 0, the inflection point in the graph of <i>E</i> occurs at <i>x </i>= (<i>a</i>+<i>b</i>)/2. Cubic polynomials are symmetric about their inflections points, so the region <span style="font-size: 11pt;">lying between the curve and the x-axis </span><span style="font-size: 11pt;">on one side of the inflection point </span><span style="font-size: 11pt;">is congruent to the region </span><span style="font-size: 11pt;">between</span><span style="font-size: 11pt;"> the curve and the x-axis </span><span style="font-size: 11pt;">on the others side of the inflection point</span><span style="font-size: 11pt;">. </span><br /><span style="font-size: 11pt;"><br /></span><span style="font-size: 11pt;">Hence</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=$\int_{a}^{b}%20E(x)%20dx=0$" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img src="http://latex.codecogs.com/gif.latex?$\int_{a}^{b} E(x) dx=0$" title="$\int_{a}^{b} E(x) dx=0$" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">That is, the approximation has no error.</div><span style="font-size: 11pt;"></span></div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0tag:blogger.com,1999:blog-5834886388752215525.post-77552423487496162472012-08-07T17:02:00.000-07:002012-10-11T17:26:27.221-07:00CCSS and Community College Math Programs<br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">We may need a complete redesign of the developmental math program in US two-year colleges.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">My campus currently uses a placement test (<a href="http://www.calstate.edu/capp/projects/proj-mdtp.shtml" target="_blank">Mathematics Diagnostic Test Project</a>) to determine if students are ready for transfer level courses (math for elementary school teachers, stats, trig, precalculus, calculus) or what remedial course (arithmetic, prealgebra, elementary algebra, intermediate algebra) they should take.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">But the <a href="http://www.corestandards.org/about-the-standards/key-points-in-mathematics" target="_blank">Common Core State Standards for mathematics</a> will have high school students studying mathematics organized in a fashion that does not align with our existing math courses.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">California is one of the 45 states that have formally adopted the CCSS for mathematics, and I am on a recently<a href="http://www.cde.ca.gov/ci/ma/cf/mathfwcfccmembers2013.asp" target="_blank"> appointed state committee</a> whose charge is to align California’s math standards (a.k.a. the California Framework) with the CCSS.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">One of the main reasons that I applied to be on the <a href="http://www.cde.ca.gov/ci/ma/cf/mathcfcclatestinfo.asp" target="_blank">Mathematics Curriculum Framework and Evaluation Criteria Committee</a> (MCFCC) was to better familiarize myself with what is to be taught in California's K-12 schools. (Another reason was to lose myself in abbreviations: SBE for State Board of Education, CDE for California Department of Education, IQC for Instructional Quality Commission, the body that forwarded my name to the SBE for approval to serve on the MFCC to align the CF with the CCSS.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The CCSS specify a consensus of what math is required for students to be college or career ready. The standards are grouped into six conceptual categories: Number and Quantity, Algebra, Functions, Modeling, Geometry, and Statistics and Probability. (There are separately eight standards for mathematical practice that go across all grade levels.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The CCSS differ significantly from what is typically required for graduation in most American high schools today. For example, the treatment of statistics and probability includes not only descriptive statistics but also conditional probability, inference, decisions based on probability, and rules of probability. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The CCSS include not only right-triangle trigonometry but also trig functions of a real variable, to be used in modeling periodic behavior. Thus trig spans the geometry, algebra, and function categories.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The CCSS gives math standards for high school without specifying courses or order of topics. But evidently the introduction of functions includes an emphasis on (linear and) exponential functions with domains restricted to a subset of the integers--sequences are explicitly studied as functions.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">California community colleges do not require a high school diploma for admission. A student who masters the first CCSS high school math course will already have compared exponential functions with linear functions and solved equations both algebraically and graphically. The student will have had explicit instruction on descriptive statistics. The student may have worked with constructions and transformations in the plane and proved simple geometric theorems algebraically but not yet worked with polynomials (and specifically not with quadratic functions or quadratic equations).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">How will our placement system advise this student? </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">One of the <a href="http://www.californiacommunitycolleges.cccco.edu/Portals/0/Executive/StudentSuccessTaskForce/SSTF_Final_Report_1-17-12_Print.pdf" target="_blank">recommendations </a>of California's <a href="http://www.californiacommunitycolleges.cccco.edu/PolicyInAction/StudentSuccessTaskForce.aspx" target="_blank">Student SuccessTask Force</a> is for better alignment between high school and college curricula. With the CCSS adopted across states, it looks as if most community colleges will need to make adjustments to their way of placing and educating their math students.</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com3tag:blogger.com,1999:blog-5834886388752215525.post-42114642725900792522012-08-06T17:35:00.000-07:002012-09-01T12:34:13.547-07:00MathFest 2012 and Common Core State Standards in Math<span style="font-family: Calibri; font-size: 11pt;">Andrew Hacker’s article </span><a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?emc=eta1" style="font-family: Calibri; font-size: 11pt;" target="_blank">"Is Algebra Necessary?</a><span style="font-family: Calibri; font-size: 11pt;">" in the New York Times was a hot topic last week and mentioned by several presenters at the </span><a href="http://www.maa.org/mathfest/" style="font-family: Calibri; font-size: 11pt;" target="_blank">2012 MathFest</a><span style="font-family: Calibri; font-size: 11pt;"> session "What Mathematics Should Every Citizen Know?". The panelists, </span><a href="http://commoncoretools.me/2012/07/29/new-forum/" style="font-family: Calibri; font-size: 11pt;" target="_blank">Bil lMcCallum</a><span style="font-family: Calibri; font-size: 11pt;">, </span><a href="http://en.wikipedia.org/wiki/Lynn_Steen" style="font-family: Calibri; font-size: 11pt;" target="_blank">Lynn Steen</a><span style="font-family: Calibri; font-size: 11pt;">, </span><a href="http://en.wikipedia.org/wiki/Hyman_Bass" style="font-family: Calibri; font-size: 11pt;" target="_blank">Hyman Bass</a><span style="font-family: Calibri; font-size: 11pt;">, </span><a href="http://www.york.cuny.edu/~malk/" style="font-family: Calibri; font-size: 11pt;" target="_blank">Joseph Malkevitch</a><span style="font-family: Calibri; font-size: 11pt;">, and co-organizer </span><a href="http://en.wikipedia.org/wiki/Sol_Garfunkel" style="font-family: Calibri; font-size: 11pt;" target="_blank">Sol Garfunkel</a><span style="font-family: Calibri; font-size: 11pt;">, were actually reacting to the </span><a href="http://www.corestandards.org/the-standards/mathematics/" style="font-family: Calibri; font-size: 11pt;" target="_blank">Core Curriculum State Standards in mathematics</a><span style="font-family: Calibri; font-size: 11pt;">.</span><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Mathematicians and math educators agree that we are not currently doing the best job of teaching algebra. But unlike Hacker, the math community believes the appropriate strategy is to improve algebra instruction, not to abandon it to all but an elite few pupils.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">On the other hand, the speakers on the panel, although quite civil with each other, clearly had disagreements about the best strategy to improve math education in the US.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">McCallum, who was the lead mathematician in the development of the <a href="http://illustrativemathematics.org/" target="_blank">CCSS</a>, emphasized the benefits of having commonality across states. Having a set of standards that could be adopted by 45 of the 50 states (so far) required compromises, but the benefits accrue not only to pupils and teachers in our mobile society, but to all who do business with textbook publishers who currently provide materials for the multitude of different curricula.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Steen gave some numbers showing the dismal success of preparing US students for STEM, but argued that we should improve rather than remove algebra from the curriculum. He favors a modeling-based approach and avoidance of common assessments. When asked how to accomplish his recommendations, he cheerfully remarked that he doesn't need to worry about that now that he's retired.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Bass focused on pedagogy rather than curriculum as the key to improving math education. Student learning is increased when the instructor employs appropriate classroom strategies.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Malkevitch promotes widening the curriculum. He argued that we need to show many ways that mathematics impinges on daily lives. He gave combinatorial graphs and fair choice algorithms as examples of mathematical topics that are new and accessible to very young children.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Garfunkel believes that the entire K-12 mathematics curriculum should be centered on modeling. He echoed Malkevitch's suggestions that the US curriculum needs to be widened, and said that Bill Schmidt had paid an advertising agency to create the phrase "a mile wide and an inch deep" that is used to characterize the US K-12 curriculum following the disappointing ranking of the US high school students in the Third International Mathematics and Science Study.</div>Bruce Yoshiwarahttp://www.blogger.com/profile/11735941402933510186noreply@blogger.com0