## 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 and as

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

So

And statisticians define Pearson's correlation coefficient r so that

Thus if we set and , then .

The idea is to think not of n ordered pairs (x1, y1), (x2, y2), ..., (xn, yn), 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 and  , the vectors containing all the and respectively.  Instead, the necessary two n-dimensional vectors are the and defined above.

And nicely, the least-squares regression line for the data is y = mx + b, where and .  (Notice that the variance , so m can also be written as  .

One typically derives the least-squares regression line by finding m and b that minimize .  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 .  The vector is the vector of all 1's and the vector   is the orthogonal projection of the vector  onto the space spanned by and .

## 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 , 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 , so .

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

.

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 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

### 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.

## Saturday, September 26, 2009

### The Mod Squad at Pierce

Five members of the Pierce College math department--Cassie Cain, Sheri Lehavi, Brenda Rudin, Zhila Tabatabai, and Kathy Yoshiwara--are trying an experiment to improve student success in elementary algebra.

Five sections of elementary algebra, a total of 200 students, are involved.

The plan is based on the Emporium model (http://www.thencat.org/PlanRes/R2R_Model_Emp.htm ) developed at the Virginia Teach and implemented successfully at a variety of institutions nationwide, and specifically at Foothill College in Los Altos (http://www.foothill.edu/PSME/index.php ).
The course is broken into small pieces, or modules, that students can cover at their own pace (with plenty of guidance and a detailed suggested schedule for successful completion in one semester.)

The modular format is designed for active learning. "Lectures…are replaced with an array of interactive materials and activities that move students from a passive note-taking role to an active-learning orientation." (NCAT) This format should also give students more control over the pace at which they cover the material, and encourage them to take more responsibility for their own learning. If students cannot finish the course in one semester, but do complete at least half the modules, they can opt for an Incomplete and continue the following semester.

Course Structure
The five members of the "Mod Squad" have organized the material into ten modules. A module consists of a pretest, four or five lessons with worksheets for in-class practice, a sample test, and a suggested schedule. There was an initial plan to have homework problems delivered and graded by computer, but that has not been implemented this semester.

Mod Squad members authored the modules, which students can purchase at the Bookstore at cost. Students will not need to purchase a separate textbook.

Study skills are incorporated into lessons as part of the course. The study skills activities are based on the lessons designed by the Mod Squad and already in use in Pierce College Learning Communities iand the ASAP (Algebra Success At Pierce, a 10-unit math + 1 unit study skills + 3-unit Personal Development) algebra course. We hope to be able to add a 1-unit study skills class to the Modular Math classes.

When students finish the assignments in the module, an instructor verifies that the portfolios are complete, and the students are given a ticket allowing them to take the in-class module test. These tests would require mastery at 85%. (The plan to have students take a qualifying sample test on the computer has been postponed.)

Students who do not pass the module test would be required to complete a second set of worksheets (and computer drill problems) before attempting the module test again.

Pierce does not have a testing center to accommodate self-paced courses. Instead, module tests are offered twice weekly, on Tuesdays and Thursdays, with one or more of the modular classrooms used as the test room, and one of the modular team serving as proctor. All students who are eligible to take a module test would report to the test room and be given the appropriate module test. Tests are returned to the respective instructor of record for grading. Students who don't wish to take a test on that day report to one of the remaining classrooms for class as usual.

Delivery
The course is team taught by the five instructors of the Mod Squad. Each instructor is responsible for the grading and record keeping for the students on her roster. However, all modular sections meet at the same time in adjacent rooms, and students may attend class in any of those rooms, depending on which module they are studying.

Class meetings concentrate on group work, practicing the skills on the modular worksheets. Instructors also offer short mini-lectures as needed. At least one classroom keeps to a schedule that allows students to finish the course in one semester, but others work at a pace to accommodate different students. Students working on the same module are formed into groups if they wish. The modular team instructors coordinate daily to plan scheduling and room allocation.

To facilitate active learning, there is a student tutor for each classroom. Tutors help instructors interact with students individually or in small groups and assist with organization and logistics, as students move from one setting to another. "Students need human contact…to assure them that they are on the right learning path. An expanded support system…is critical to persistence, learning, and satisfaction." Tutors are also crucial for checking the steady stream of worksheets and quizzes that provide students with immediate feedback. "Shifting the traditional assessment approach toward continuous assessment is an essential pedagogical strategy. … Low-stakes quizzes motivate students to keep on top of the course material, structure how they study, and encourage them to spend more time on task."

Timeline
It is anticipated that many students will need more than one semester to be able to pass all the modules at the 85% proficiency level required. So students who have completed at least half the modules at the 85% level but do not complete the course in one semester will earn an "incomplete" grade in the course. They can pick up where they left off the following semester.

But in order to encourage reasonable progress, any student in the program who does not complete at least half of the modules in the first semester will earn a grade of F.

The modularized elementary algebra course serves as the elementary algebra component of our departmental MAP (Mediated Algebra Project, mentioned in the previous blog). We can monitor both courses via the common exam given to all Pierce College sections of elementary algebra and the common exam given to all Pierce College sections of intermediate algebra.

More on the MET, our common exams, in the future.

## Friday, September 18, 2009

### Developmental math projects at Pierce

Several members of the Los Angeles Pierce College mathematics department are involved in two extraordinary experiments to improve the school's developmental math program. The Mediated Algebra Project (MAP) aims to improve the success of intermediate algebra students, and the Modular Math Project is trying an alternative method for teaching elementary algebra.

MAP involves:

• reading assignments completed before coming to class (in a textbook authored by Pierce faculty and available to students both online for free, or through the bookstore at the cost of copying and binding),
• online reading and skills questions (using the open source WeBWorK homework delivery and grading system http://webwork.maa.org/moodle/, with problems coded by Pierce faculty),
• online videos (including Pat McKeague's freely available MathTV site (http://www.mathtv.com/ ) and screen capture videos made by Pierce faculty),
• an online Question and Answer student forum using the open source Moodle learning management system,
• Replacement of lectures with in-class activities that explore math concepts, (from an Activities book created by Pierce faculty and sold to the students at cost) within an environmental theme when possible ,
• written homework aligned with the in-class activities, and
• concept "clicker" questions (written by Pierce faculty).

This fall 2009 semester is the first try at the "full" MAP package, with three faculty teaching in the pilot: Kathie Yoder, Kathy Yoshiwara, and myself. The Reading and Activities books have already been class-tested by the principal author, Kathy Yoshiwara. Some of the ancillary MAP materials were created by Roya Furmuly, Sheri Lehavi, Bob Martinez, Jenni Martinez, Brenda Rudin, Ben Smith, Kathie Yoder, and me.

The "Mod Squad", team-teaching a modular self-paced elementary algebra course, consists of Cassie Cain, Sheri Lehavi, Brenda Rudin, Zhila Tabatabai, and Kathy Yoshiwara. More on their efforts later.

Most of the funding to pay for the continuing development of the two programs comes from California's Basic Skills Initiative (http://www.cccbsi.org/ ). Further funding came from a STEM grant at the college and will be supplemented this semester by a Hewlett Foundation (http://www.hewlett.org/ ) grant. The college itself has committed none of its normal budget to the effort.

## Friday, September 11, 2009

### Teacher Prep at Community Colleges

Although only a small proportion of the general public or even the faculty and administrators involved seem to be aware of it, community colleges are in the business of preparing future teachers.

It has been estimated that 40% of U.S. K-12 teachers took math or science at a two-year college (http://www.nsf.gov/pubs/1999/nsf9949/nsf9949.htm ) and that 46% of baccalaureates in science and engineering have attended a two-year school (http://www.nsf.gov/statistics/nsf04302/ ).

The numbers of K-12 teachers who attended two-year colleges only grows if we include courses and degrees in non-STEM disciplines.

At California State University Northridge (CSUN) , which identifies teacher preparation as one of its primary missions, more than half of their students in the multi-subject credential program (for teaching elementary school ) took math courses at a community college. Most of CSUN's math majors are considering a career in teaching, and about two-thirds of CSUN math majors took math at a community college.

Many or most universities require only 3 to 9 units of courses taught by math departments for the students preparing to become elementary school teachers. Some of these units may be taken at a community college, and considering that many students (51% at CSUN) take developmental math classes at a two-year schools, and furthermore that some students (20% at CSUN) require as much as 20 units of remediation, our prospective elementary school teachers are probably taking more math at two-year colleges than at four-year colleges and universities combined.

A number of faculty at two-year colleges take seriously their role in the recruitment and education of future teachers. The Teacher Prep Committee ( http://amatyc.dtcc.edu/ , http://teacherprep.amatyc.org/ ) is one of only eight standing national committees of the American Mathematical Association of Two-Year Colleges. Yet there are two-year colleges where the "Math for Teachers" class either does not exist or is taught primarily by adjunct faculty because the full-time faculty do not have sufficient interest to teach the course, or the institution is not sufficiently motivated to run the course.

## Saturday, September 5, 2009

### A policy against married couples

Los Angeles Pierce College has decided that no married couples can serve together on a hiring committee. This policy was initiated by the vice-president of academic affairs and then endorsed by the Pierce Ethics Committee and the Pierce College Academic Senate.

The motivation behind the ban is putatively to reduce the possibility of Pierce receiving accusations of collusion between a married couple from a candidate who was not offered a position. The policy establishes Pierce's preference to restrict the rights of all productive married couples on the faculty over accepting any risk that a disgruntled candidate with sexist prejudices may file a complaint and thus cause Pierce the inconvenience of having to defend faculty against false accusations.

If a Pierce couple actually does engage in collusion or unfair practices, then the department should not nominate them for a hiring committee, the Vice-president should advise the senate to reject the offending couple, and the academic senate should not approve them. Further, everyone on a hiring committee, including any couple, is supervised by a dean and compliance officer.

There will be no inappropriate collaboration between any married couple if any one of the department, or the VP, or the senate, or the dean, or the compliance officer can do a responsible job, even if no married couple can act responsibly on their own.

Obviously it is simpler for Pierce to ban all couples rather than to take on the responsibility of making a hard decision to deny opportunities a specific couple if it should become appropriate. But a policy banning all married couples from serving together on hiring committees is in direct opposition to the recommendations of the American Association of University Professors (AAUP).

The 1971 position paper “Faculty Appointment and Family Relationship” called for an end to “policies and practices [that] subject faculty members to an automatic decision on a basis wholly unrelated to academic qualifications and limit them unfairly in their opportunity to practice their profession.”

This position paper was prepared initially by the Association’s Committee on Women in the Academic Profession. It was approved by that committee and by Committee A on Academic Freedom and Tenure. The statement was adopted by the Association’s Council in April 1971.

But let us assume that we support Pierce's position that trying to avoid lawsuits is more important that supporting productive faculty. Then we should consider how much better it would be if we also banned any two faculty members sharing religious beliefs from serving together on a hiring committee, because there is a chance that a rejected and disgruntled applicant may complain that the two colluded against him or her. We should also prevent any two faculty who attended the same school together from serving together on a committee. Certainly we cannot allow any two homosexuals to serve together.

Continuing this logic leads us inevitably to restricting the hiring committees to members of the Pierce community who are beyond reproach because of their unquestioned integrity and fairness, such as the Ethics Committee and the Vice-president of Academic Affairs. But the school's philosophy requires that we guard not only against rational complaints, but especially against irrational complaints of collusion. So the appropriate policy would appear to be that all personnel decisions should be decided by lot rather than allowing any human to be involved.

## Friday, August 28, 2009

### Tales of student cheating

At Simon Fraser University in Burnaby, B.C., there is now a grade of FD, considered lower than F, for a students found guilty of egregious cases of academic dishonesty. (http://www.calgaryherald.com/university+adds+grade+worse+than/1890672/story.html)

In contrast, at California community colleges, "...it is not permissible to give a student either a failing grade or an incomplete because a student has cheated on a particular assignment" no matter how egregious the offense (http://www.cccco.edu/Portals/4/Legal/opinions/attachments/07-12.pdf ).

Here are two tales from my campus in the 2008-2009 academic year. In the first, a student paid a ringer $1000 (yes, one thousand US dollars) to sit next to him during a calculus exam, and also paid$100 to each of several other students in class just to sit in seats surrounding the two principals. When the ringer completed the exam (early), a pre-planted cellphone went off. It took the instructor some time to locate the cellphone, which was stashed in a trash can. While the instructor was thus distracted, the ringer handed off his completed exam to the cheater, who then copied answers in his own handwriting . The ringer was not enrolled in the course (someone dropped from the class that day) and simply did not turn in an exam. We learned of the scheme because one of the $100 classmates confessed. In the second case, an online student paid$100 for a ringer to take an elementary algebra quiz. We learned this from the ringer, who forwarded to the math department chair an email from the cheater detailing the instructions for taking the quiz and proposed payment--the ringer was outraged because the cheater "was suppossed [sic] to pay me \$100 for the same which he didnt [sic]."

In neither case could the instructor assign a grade of F. The calculus student passed and the algebra student dropped the course.

And by the way, unless your school is so small that the instructors know all the students on campus, you probably have students on your campus taking classes for others.

Here's how the scam works.

Abe and Bob both sign up for English 1 and Math B, but different sections of each subject. Abe attends and does the work for both English classes and Bob does the same for both math classes. The four instructors involved know the names and faces of all the students in their classes, so they never bother to check IDs. (Professor Yee doesn't know that Abe attends her English class every day answering to the name of Bob, and Professor Zed doesn't know that Bob attends her Math class every day answering to Abe).

The two cheaters get credit for two classes while only needing to master the material for one (and without having to pay any bribe money).

A few years ago we actually had a case of a man taking a class for a woman. Classmates were upset that the ringer was raising the curve, and they informed the instructor that the same person was answering to an entirely different name in other classes. The student who was actually enrolled in the class was Asian, and her instructor did not recognize that her name was a woman's. The only consequences to the students were that they had to speak with a vice-president, and that neither got credit for the class that semester.

## Sunday, August 23, 2009

### Section 508 and the Use of Non-captioned Videos

The ADA compliance officer on my campus says that I am forbidden to provide a link to any of the wonderfully useful videos I find on YouTube or MathTV because of Section 508 of the Rehabilitation Act.

The text of Section 508 can be found at http://www.access-board.gov/sec508/guide/act.htm, and FAQs can be found at http://www.access-board.gov/sec508/faq.htm. The main idea of the statute was to require that information technology resources purchased with public funds be accessible to people with disabilities. And, from the FAQ page cited above, “In general, an information technology system is accessible to people with disabilities if it can be used in a variety of ways that do not depend on a single sense or ability.”

But the subsequent state adoption of the regulation has altered the thrust of the regulation. The popular interpretation is that the regulation forbids the use of any video that is not captioned.

I recognize that captioned videos can be beneficial to many students, not only those with disabilities. And I do plan to include captioning when I create videos. But it is ridiculous that I cannot recommend existing excellent math videos (which are useful even without sound) to my face-to-face classes, not even as an optional resource for which no credit is awarded.

My issue is not about the benefits of having captioned videos, it's about a wrong-headed policy that exceeds the actual statute requirements and forbids using valuable resources.

The mucky-mucks embrace the ban on all non-captioned videos because they want to minimize any chance of a lawsuit of any sort. But they do not consider that the easiest path for an instructor is to avoid making any use of the Web, and the result will be that the students will actually have their learning experience diminished.

I'm trying to find some credible person who can explain what adoption of the statute actually requires. But so far I've only found people who can tell me the policy that their school/district/system has adopted, not anyone who has familiarity with the actual statute.

## Sunday, August 16, 2009

### Using Wolfram|Alpha in a math class

As Maria Andersen convincingly argues in her blog Teaching College Mathematics, the introduction of Wolfram|Alpha will have a significant impact on our math classrooms.

We are not going to make Wolfram|Alpha go away, so we need to learn how to make the best use of it.

Both Maria and Robert Talbert provide interesting examples of what Wolfram|Alpha can do. But how do we take advantage of its power (and weaknesses)?

If we assume that our students will be using Wolfram|Alpha with or without our encouragement, we should at least help them get past some of the things Wolfram|Alpha does that are not appropriate for a given class.

If we ask it to "graph y=3x+5" we get two graphs, the first of which seems to have a negative y-intercept and a positive x-intercept. We can show this to our beginning algebra students, discuss why this graph should surprise us, and, after examining the scale on the horizontal axis, resolve the apparent error.

If we ask it to "solve 2^x = 5" we get an answer that involves complex numbers. We can still show this to our intermediate algebra class, and discuss what appears when we click on the "Show steps" button.