Math Ed Blog from Bruce Yoshiwara

Wednesday, December 23, 2009

Mediated Algebra Project: Success!

Three instructors used the Mediated Algebra Project (MAP) materials during the Fall 2009 semester. Kathie Yoder taught a section that met early afternoon twice weekly , and Kathy Yoshiwara and I taught the two sections that met mid-morning four-days-a-week.

We now have evidence that MAP students learn more than cohorts in other sections of the Pierce College intermediate algebra course: On the departmental Math Exit Test (MET), our three sections all scored at least 2.5 standard errors above the department mean.

But we had significant setbacks during the semester.

We had numerous technical difficulties. Many of the WeBWorK problems I had authored had coding errors and/or needed refinement in wording or formatting. And most of the WeBWorK exercises taken from the national WeBWorK library were poor fits for our project and had to be rewritten or removed from our problem sets during the semester.

Our sets of video tutorials--intended to help with drill and skill exercises--had many gaps in content. And yet we were not given sufficient space on our school's server to store the videos created by our faculty for the MAP. Instead, our IT department arranged that only a subset of those videos would be accessible at any one time.

All three instructors found that the project's classroom activities and clicker questions required more time than was available in a class meeting. Some of the activities, or the clicker questions, or both would go unused in each lesson.

We heard complaints about our WeBWorK assignments, the insufficiency of available videos, and the amount of work we asked the students to do both in and outside of class.

But the students who persisted in MAP averaged much higher on a department-graded common exam than students from the other sections of intermediate algebra.

Tuesday, December 15, 2009

The Correlation Coeffiicent as cosine theta

Mathematicians define the dot product between vectors $\vec{v}= (v_{1}, v_{2}, \, \ldots \, , v_{n})$ and $\vec{w}= (w_{1}, w_{2}, \, \ldots \, , w_{n})$ as

$\vec{v} \cdot \vec{w} = v_{1} w_{1} + v_{2} w_{2} + \, \cdots \, + v_{n} w_{n}$

On the other hand, the alternate geometric definition for the dot product popular with physicists is

$\vec{v} \cdot \vec{w} = \left|\left|{\vec{v}\right|\right| \,\left|\left|{\vec{w}\right|\right| \,\cos \, \theta$

$\cos \, \theta = \frac{\vec{v} \cdot \vec{w}}{\left|\left|{\vec{v}\right|\right| \,\left|\left|{\vec{w}\right|\right|$

And statisticians define Pearson's correlation coefficient r so that

$r = \frac {\sum (x_{i} - \bar{x})(y_{i} - \bar{y}) } {\sqrt{\sum (x_{i} - \bar{x})^2} \sqrt{ \sum (y_{i} - \bar{y})^2}}$

Thus if we set $\vec{v} = (x_1 - \bar{x}, x_2 - \bar{x},\, \ldots \, , x_n - \bar{x})$ and $\vec{w} = (y_1 - \bar{y}, y_2 - \bar{y},\, \ldots \, , y_n - \bar{y})$ , then $r = \cos \,\theta$ .

The idea is to think not of n ordered pairs (x₁, y₁), (x₂, y₂), ..., (x_n, y_n), but rather to think of two vectors in n-dimensional space. When the vectors are pointing in the same direction, the angle between them is zero and the correlation coefficient is cos 0 = 1. When the vectors point in opposite directions, the correlation coefficient is the cosine of a straight angle, r = -1. And when the vectors are orthogonal, the correlation coefficient is the cosine of a right angle, r = 0.

The only tricky part is that the two n-dimensional vectors are not the vectors $\vec{x}$ and $\vec{y}$ , the vectors containing all the $x_{i}$ and $y_{i}$ respectively. Instead, the necessary two n-dimensional vectors are the $\vec{v}$ and $\vec{w}$ defined above.

And nicely, the least-squares regression line for the $(x_i , y_i )$ data is y = mx + b, where $m= r \frac{\left|\left|\vec{w}\right|\right|}{\left|\left|\vec{v}\right|\right| }$ and $b = \bar{y} - m \bar{x}$ . (Notice that the variance $\sigma_{x}^{2} = \frac{\vec{v} \cdot \vec{v}}{n}$ , so m can also be written as $m= r \frac{\sigma_y}{\sigma_x}$ .

One typically derives the least-squares regression line by finding m and b that minimize $\sum (m x_i +b - y_i )^2$ . But one can alternatively use the n-dimensional vector point of view, where the coefficients m and b correspond to the solution of the vector equation $m\vec{x} + b\vec{1} = \hat{y}$ . The vector $\vec{1}= (1, \, 1, \, \ldots \, , \, 1)$ is the vector of all 1's and the vector $\hat{y}$ is the orthogonal projection of the vector $\vec{y}$ onto the space spanned by $\vec{x}$ and $\vec{1}$ .

Monday, December 7, 2009

Generating Pythagorean Triples

The 5 millennia old clay tablet designated Plimpton 322 contains a trig table. The second and third columns represent a leg and hypotenuse of a right triangle with positive integer sides. The rows are arranged in approximately equal steps of angle.

The existence of such a table suggests that the Babylonians were adept at producing Pythagorean triples (integers a, b, and c satisfying a² + b² = c²), a trick which is also useful to many algebra, geometry, and trig teachers attempting to create exercises with nice values.

Every positive Pythagorean triple can be generated by choosing positive integers u and v with u > v and setting a = 2uv, b = u² - v², and c = u² + v² (or by scaling such a triple by a positive integer). We'll derive that fact below. (Pythagorean triples with no common factor are called primitive Pythagorean triples, and all the primitive Pythagorean triples are generated when u and v are relatively prime with exactly one of them being odd.)

It's straightforward to verify that the a, b, and c so defined do form a Pythagorean triple. And conversely, if a, b, and c form a Pythagorean triple, then (a/c, b/c) is a point on the unit circle $x^2+y^2=1$ , so the positive Pythagorean triples can be mapped onto the rational points of the unit circle that lie in the first quadrant.

The line y = 1 + mx will intersect the unit circle at (0,1) and also at a point in the first quadrant when the slope m is between -1 and 0. In fact, we can find the x-coordinate of the second intersection point by solving the equation x² + (1 + mx)² = 1--we find that $x=\frac{-2m}{1+m^2}$ , so $y=1-\frac{2m^2}{1+m^2}$ .

Thus the second intersection point is a rational point if m is rational. Of course the slope between (0,1) and any rational point on the unit circle is rational, so we have a 1-1 correspondence between positive rational points on the unit circle and rational slopes between -1 and 0.

We now assume that m is a rational number between -1 and 0, so we can write m = -v/u, where u and v are positive integers with u > v. Then the second intersection point we found above has the form

$\left(\frac{-2(\frac{-v}{u})}{1+(\frac{-v}{u})^2}, \,\, 1- \frac{2(\frac{-v}{u})^2}{1+(\frac{-v}{u})^2}\right) = \left(\frac{2uv}{u^2+v^2}, \frac{u^2-v^2}{u^2+v^2}\right)$ .

Thus every rational point on the unit circle can be written in this form. In particular, every primitive Pythagorean triple a, b, and c can be expressed as above in terms of u and v.

Thursday, December 3, 2009

Edublog Awards

My nominations for the 2009 Edublog Awards (http://edublogawards.com/) are:

Best resource sharing blog: http://teachingcollegemath.com/

Best teacher blog: http://teachingcollegemath.com/

Best educational tech support blog: http://teachingcollegemath.com/

Many thanks to Maria Andersen for providing a wonderful resource!

Sunday, October 18, 2009

MET: the Math Exit Test at Pierce

At Pierce College there is an MET for elementary algebra and an MET for intermediate algebra. All instructors of those classes are required to have their students participate, but each instructor determines how the MET scores will be weighted in the students' grades.

The MET has both multiple choice and "essay" parts, all submitted on a Scantron form. Instructors volunteer to meet after the exam to team grade the essay questions--the volunteers may but need not be instructors of the relevant courses.

The MET was designed as way to measure the department's success at achieving its stated Student Learning Outcomes (SLOs) in elementary and intermediate algebra. Each instructor is given summaries of his/her students' performance, as well as the summaries across all sections.

The department learns on which problems students overall perform well and on which they perform poorly. Individual instructors can compare their students performance with those of the entire department.

The department chair announced to the department that one intermediate algebra instructor (Kathy Yoshiwara) had far more students in the top 10% than any other instructor. Not officially discussed was the fact that one (anonymous) elementary algebra instructor had an unusually large number of students finishing the semester for a grade, with all scoring below the department MET mean, and providing a class average a few standard deviations below the department MET mean.

Guess which algebra instructor is a favorite among students, the counseling department, and our Special Services faculty and staff?

Saturday, October 10, 2009

My favorite free math stuff

Winplot

Although I have licensed copies of Mathematica, Mathcad, and Maple, my favorite grapher is Winplot. Plots are easy to create and highly customizable. Winplot handles parametric, polar, and implicit 2D graphs and wireframe 3D plots, with numerous other nifty features.

Winplot is one of several clever programs written by Rick Parris of Phillips Exeter Academy. You can download his free programs from http://math.exeter.edu/rparris/.

GeoGebra

Another under-utilized program is Markus Hohenwarter's GeoGebra. If you ever wanted to use (or are using) Geometer's Sketchpad or Cabri, you might want to give this one a try. Like GS or Cabri, GeoGebra allows you to make a geometric construction based on points and/or lines of your choice, then shows you how the constructed object changes as you use the mouse to alter the defining points or lines.

But unlike GS or Cabri, GeoGebra also has an algebra window that records the algebraic representation of the geometric objects. You can either modify an algebraic definition and watch in real-time the change in the figure, or alter the figure and see how parameters change in the algebraic description.

Read more about GeoGebra in articles in the online journal Loci (http://tinyurl.com/yk8w3ks and http://tinyurl.com/yz2yjkj), or download the free program directly from http://www.geogebra.org/cms/ .

Flash Forum

Barbara Kaskosz and Doug Ensley's Flash Forum has lots of clever applets for free download or use online. I particularly like the "Visualizing Regions for Double Integrals" (http://tinyurl.com/ygcfulh) by Barbara and Lewis Pakula. You enter the limits of a double integral (in rectangular or polar coordinates) and the appropriate region is sketched. Or you can ask for a practice problem, and you are given a region for which you need to determine the coordinate system and corresponding limits to define it.

The Flash Forum also has a 3D function plotter, and graphers for surfaces defined parametrically in rectangular, cylindrical, or spherical coordinates (http://tinyurl.com/yhx45lx).

But my favorite applet is "Terminate the Terminator!", (http://tinyurl.com/yzn9m25) a game to introduce radian measure and polar coordinates. It was originally created by my colleague Bob Martinez in Mathcad, but the online version is in Flash.

Sunday, October 4, 2009

Algebra Success at Pierce

Algebra Success at Pierce (ASAP) is a program that allows students to take both elementary algebra and intermediate algebra in one semester.

Whereas our school (and state) typical success rate is around 50% in each of the two courses, Kathie Yoder has had a 70% success rate at getting students through both classes in one semester.

Her students score higher on the department's standardized intermediate algebra exit exam than students in the regular or online intermediate algebra classes.

In addition to having an exceptional teacher, the students in ASAP have several advantages over their peers in other intermediate algebra classes. ASAP students are all enrolled in both elementary algebra and intermediate algebra (5 units each), Personal Development 40 (3 transferable units taught by counseling faculty), and a 1-unit math study skills course. (Yes, the students meet with Kathie for more than 2.5 hours per day, 4 days per week.) The students are not permitted to enroll in other classes during that semester.

In other words, they are immersed in math for the semester.

The course materials are written by Pierce faculty, designed specifically for this course. There is a Supplemental Instruction (SI) leader who holds study sessions outside the assigned class hours.

Pierce has also had students in a Learning Community experiment that had prealgebra, elementary algebra, or intermediate algebra, teamed with the PD 40 class and 1-unit of study skills. Results were not consistently better than for students in ordinary sections of those courses.

We have had previous experiments with SI leaders in algebra classes, but again with no convincing evidence of effectiveness.

This semester we have a second section of ASAP, and the new instructor, Jenni Martinez, reports very encouraging success on the first two exams.