Thursday, March 28, 2013

Heron's formula for the area of a triangle

The angle bisectors of the triangle meet at the center of the inscribed circle of radius r.  If we let $$2\alpha=A$$, $$2\beta = B$$, and $$2\gamma=C$$, we have $$\alpha+\beta+\gamma=\frac{\pi}{2}$$.

Let x be the distance from the vertex at A to points of tangency, y the distance from B, and z the distance from C.  Then then lengths of the triangle sides opposite A, B, and C are respectively $$a=y+z$$, $$b=x+z$$, and $$c=x+y$$.

Thus if we name the semiperimeter sthen $$s=x+y+z$$, $$x=s-a$$, $$y=s-b$$, and $$z=s-c$$.

$$\tan \alpha =\frac{r}{x}$$, $$\tan \beta =\frac{r}{y}$$, and $$\tan \gamma =\frac{r}{z}$$.  Because $$\gamma$$ and $$(\alpha+\beta )$$ are complementary angles, we obtain

$\tan\left( \frac{\pi}{2} - (\alpha+\beta) \right) = \frac{r}{z}$
$\tan\left( \alpha+\beta \right) = \frac{z}{r}$
$\frac{ \tan \alpha+\tan\beta}{1-\tan\alpha \tan\beta} = \frac{z}{r}$
$r \left(\tan \alpha+\tan\beta \right)= z (1-\tan\alpha\tan\beta)$
$r \left( \frac{r}{x}+\frac{r}{y} \right) = z \left( 1 - \frac{r}{x}\frac{r}{y} \right)$
$r^2 y + r^2 z = xyz - r^2 z$
$r^2 ( x+y+z) = xyz$
$r^2 s = xyz$

The radii at the points of tangency and the angle bisectors form 3 pairs of congruent triangles.  The area of $$\Delta ABC$$ is $$xr+yr+zr= r(x+y+z)$$, so area $$=rs$$, and $$(\text{area})^2=r^2s^2$$.  Using results we have above, we obtain
$(\text{area})^2 = s\cdot xyz = s(s-a)(s-b)(s-c)$
so the area is $$\sqrt{s(s-a)(s-b)(s-c)}$$.

Thursday, December 20, 2012

Alternative Pathways vs Common Core State Standards

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

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

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

"Core Principles for Transforming Remedial Education: A Joint Statement" from the Charles A. Dana Center, Complete College America, Inc., Education Commission of the States, and Jobs for the Future, calls for revamping the two-year college remediation structure.  The paper lists seven Core Principals for a "fundamentally new approach for ensuring that all students are ready for and can successfully complete college-level work that leads to a postsecondary credential of value.

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

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

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

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

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

Monday, December 17, 2012

Common Core State Standards Algebra

One issue of concern for the California K-12 educators is that California currently requires students to pass Algebra 1 in order to earn a high school diploma. The Common Core State Standards (CCSS) version of Algebra 1 includes topics not traditionally associated with Algebra 1, for instance, exponential functions and some statistics.

Unless new legislation addresses this change in content, the adoption of the CCSS automatically raises the California high school graduation requirement.

A related issue more directly linked to California Community Colleges (CCCs) is that the CCSS has created a higher level Algebra 2. If community college intermediate algebra is to align with high school Algebra 2, then we will be raising our math requirement for the AA degree and for the prerequisite for transfer level math.

And the California Community College Student Success Task Force calls for better alignment:
"Aligning K-12 and community colleges standards for college and career readiness is a long-term goal that will require a significant investment of time and energy that the Task Force believes will pay off by streamlining student transition to college and reducing the academic deficiencies of entering students...

"Recommendation 1.1
"Community Colleges will collaborate with K-12 education to jointly develop new common standards for college and career readiness that are aligned with high school exit standards.

"The Task Force recommends that the community college system closely collaborate with the SBE and Superintendent of Public Instruction to define standards for college and career readiness as California implements the K-12 Common Core State Standards and engages with the national SMARTER Balanced Assessment Consortium to determine the appropriate means for measuring these standards. Doing so would reduce the number of students needing remediation, help ensure that students who graduate from high school meeting 12th grade-level standards are ready for college-level work, and encourage more students to achieve those standards by clearly defining college and career expectations."
I don't know who speaks for CCCs in the collaboration with the State Board of Education and Superintendent of Public Instruction.  But I do think it likely that one strategy to bring better alignment will be to use the Smarter Balanced assessments at grade 11 as placement instruments at the community colleges.  The other consortium creating CCSS assessments, PARCC, already has agreement among its adopting states to use its assessments for college placement.  (See, for example, http://bit.ly/QYVjUF.)

Thursday, November 22, 2012

Alternative pathways

It seems that student-success discussion at two-year colleges is shifting from "course redesign" to "alternative pathways" for the math required to transfer.

The topics overlap considerably, because the idea of alternative pathways normally involves modifying course prerequisites in format and/or content.

Some community colleges are exploring alternative pathways via multiple versions of intermediate algebra.  For example, several campuses have a "pre-stats" course which prepares students for the regular statistics course, but the pre-stats course does not cover all of intermediate algebra (and may not have elementary algebra as a prerequisite).

The CarnegieFoundation for the Advancement of Teaching, the Dana Center at UT Austin, and 3CSN (California Community College Success Network) are three groups promoting the development of alternative pathways through dev math for non-STEM majors. The California State University system has agreed to accept Statway™ (Carnegie's two-semester course beginning at the elementary algebra level and ending with a transferable statistics credits) for meeting the Area B4 (math/quantitative  reasoning ) requirement for transfer.

But  last month the CSU emailed California Community College articulation officers the following:

"When the CSU reviews community college courses proposed to satisfy Area B4, we look for a prerequisite of intermediate algebra. We’re aware that many community colleges are experimenting with alternative prerequisites to their approved B4 courses, in an effort to improve student persistence. Some of these alternatives take away topics traditionally included in intermediate algebra; others substitute a different course altogether.

"Please take this email as a reminder that only courses with a full prerequisite of intermediate algebra, as traditionally understood, will continue to qualify for CSU Area B4.

"The CSU has made a recent exception for the Statway™ curriculum, under controlled and very limited circumstances, so we can evaluate whether other approaches will satisfactorily develop student proficiency in quantitative reasoning. In the meantime, we count on the articulation community to uphold the current standard."

That email seems to cast doubt on the future of alternative pathways.  But in the meanwhile, the CSU appears to be fine with the strategy proposed by Palomar College.  Palomar is not changing the intermediate algebra prerequisite for statistics, but evidently students who pass the alternative pre-stats course will be allowed to waive the intermediate algebra prerequisite.

Thursday, November 15, 2012

Student struggle is a good thing!

NPR recently interviewed UCLA researcher Jim Stigler about the differences between how the US and other cultures view student struggle.

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.

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.

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:

1. Make sense of problems and persevere in solving them.

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.

Sunday, October 14, 2012

Mathematical Practices

The Common Core State Standards list Standards for Mathematical Practice at each grade level.  These practices are

Mathematical Practices
1.     Make sense of problems and persevere in solving them.
2.     Reason abstractly and quantitatively.
3.     Construct viable arguments and critique the reasoning of others.
4.     Model with mathematics.
5.     Use appropriate tools strategically.
6.     Attend to precision.
7.     Look for and make use of structure.
8.     Look for and express regularity in repeated reasoning.

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.

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).

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 10x to represent the value (in cents) of x dimes.

But not all word problems involve mathematical modeling.  It is not 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.

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 .

I personally need further explanation of Mathematical Practice #8.  Here is how it's first described in the CCSS:
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 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 attending to the details. They continually evaluate the reasonableness of their intermediate results.

The phrase "repeated reasoning" presumably refers to the "R" in Guershon Harel'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.

Friday, October 12, 2012

Supporting community college faculty across the disciplines

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.

The workshop was hosted by Carleton College and its Science Education Research Council.  SERC (http://serc.careltoncollege.edu) has amassed an impressive collection of resources across multiple disciplines including geoscience (the first discipline), chemistry, economics, mathematics, physics, psychology,  and more.

The Pedagogy in Action page (http://serc.carleton.edu/sp/) 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.

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.

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.

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.