Painters and Pure Mathematicians

The Atlantic posted an article this week with the title Here’s How Little Math Americans Actually Use at Work. The article is a good summary of what it is about.

This article annoyed me for four reasons. First, I do not think that the data in the article support its conclusion that people do not use much math at work. It cites that 94% of people use “any math,” which alone makes the title seem ridiculous (would they be happy with 96%? 99%? Would it have to be 100%?). The have a better point that only 22% of workers use any mathematics beyond arithmetic and fractions. But is this number actually low? Would at least one out of every five workers use American history at work? Science? French? Phy Ed? The only school subject that I can think of that would be higher is “English,” since many workers have to write at work. A better, less-provocative-and-more-accurate headline would be “Here’s How Much Math Americans Actually Use at Work.”

Next, I am annoyed because I feel that math teachers are largely the cause of this. As a community, we have put a lot of effort into teaching students that they should care about mathematics because it is useful. While this is true, we would have done a much better job motivating students if we had spent the same amount of energy switching to more effective pedagogies. And I can see why people like the author of the above article might be concerned: students were promised that mathematics would be useful, and then they feel let down/lied to/vindicated when 78% of the workforce only uses at most elementary school mathematics in their jobs (Edit: Thanks to Kate Owens for catching an arithmetic mistake here).

I am also annoyed at the double-standard. I have written about this before. But it still bothers me that mathematics is held to a different standard than other school subjects precisely because it is so useful, but then people (like the author of the Atlantic article) suggest that we over-emphasize mathematics because it is not useful enough. As I stated in the previous paragraph, I think that this is largely the fault of the mathematics community.

Finally, I am annoyed as a pure mathematician that my subject is being perverted. A quote from a recent This American Life (Episode 493: “Picture Show”) sums up my feelings beautifully. The show talks about how art is often traded, held, and re-traded as a commodity like wheat or corn. One artist found her works traded in this market and reflects:

 ”Painters really paint because there is sort of like this beautiful magic moment in it, you know.  And after you are constantly making stuff all of the time, and people are buying stuff, and then they are flipping paintings, and it is all about money—it’s like you, you just crave for that magic moment again.  It becomes corrupted if you let it.”

Replacing “painters” with “pure mathematicians” leads to an accurate description of how I felt when I read the Atlantic article. I do mathematics because of the magic moment. The article seems as ridiculous to me as if someone wrote an article suggesting we should consider eliminating art classes because very few people have to paint the walls of their office as part of their job.

My goals worked; I said “no.”

I previously listed some goals/guiding principles for my career. Here they are:

• I want to continue to improve my teaching.
• I want to do research in finite group theory.
• I want to help K-12 teachers improve their teaching. This is mostly done by teaching pre-service courses, but this could be met in other ways.
• I want to help close the gender gap in mathematics.

I have two bits of news related to these:

First, I think that I can now provide more detail for the second goal. Part of the goal will be to focus on undergraduate research. By the end of the week, I will have been to four thesis defenses in ten days (I was an advisor for two of them and a reader for the other two). I thoroughly enjoy this process, and it is something that benefits the college and the students. I think that “undergraduate research” might be the focus of my upcoming sabbatical.

Second, these goals are working. I previously wrote, “This means that my default answer should be ‘no’ to any other requests of my time” (aside from these goals), and I used this list to figure out that I should say “no” to a request. I was given the opportunity to volunteer for a service job last week. There are several reasons why I wanted to volunteer for the job, but it did not meet the criteria on this list (and I already had enough other service jobs to fulfill the requirements for my job). So I said “no” (in the sense that I did not volunteer).

I am pleased that I will be able to spend more time next year on my career goals.

RIP Jing Pro

I am very sad to report that my favorite screencasting software, Jing Pro, has been discontinued as of February 28, 2013 (The non-Pro version of Jing is still availble, though).

Jing Pro was my favorite tool for making quick and dirty screencasts. I liked that it was easy to use, uploaded to YouTube with a click of a button, and had no advertising (it cost me $15 per year to upload to YouTube for the last two features; the regular version of Jing has advertising and does not upload to YouTube).

(Edit: Andy Rundquist let me know that I was mistaken about Jing having advertising).

But I have found an alternative that I like as much: Screencast-O-Matic (note: I have not been paid by the makers of Jing nor Screencast-O-Matic; I am just a happy customer). Here is what I like about Screencast-O-Matic:

1. It is easy to use.
2. It is easy to upload videos to YouTube.
3. There is no download required! (an improvement over Jing)
4. There are ads in the free version (a banner advertising Screencast-O-Matic at the bottom of the screen), but not in the premium version (like Jing Pro, this costs $15 per year).

One difference between Jing/Jing Pro and Screencast-O-Matic: the former has a 5 minute limit, and the latter has a 15 minute limit.

The main drawback: Screencast-O-Matic seems to take much longer to upload to YouTube than Jing/Jing Pro does—probably by a factor of 4.

I will eventually get a premium version of Screencast-O-Matic, but I will not have enough screencasts to do in the near future to justify getting it now.

One final note: I tried to use Camtasia. Camtasia would be great if you want screencasts that are not quick and/or not dirty. It was way too much for me, and I did not want to pay the price of really long upload times for higher quality video.

An example of why lecturing does not work very well

We just started discussing confidence intervals in probability and statistics. As expected, students had a difficult time with it.

As usual, they read the section, answered some questions online, and came to class. In class, we worked on clicker questions. The first was basically:

Q: The 95% confidence interval for the population mean is [x,y]. Based on this interval:

1. There is a 95% chance that is in this interval.
2. 95% of the observations are in this interval.
3. This method of creating intervals works 95% of the time.

This is a tricky idea, but the third choice is the best answer of the three. In my second class, only 2 out 26 students got it correct. This was to be expected, though, since it is a tricky subject.

So I basically gave a 15-20 minute lecture as to why the third one was correct and the first two were wrong. Actually, it is more accurate to say that I repeated a six minute lecture three times about how to think about this.

We had two more clicker questions related to confidence intervals, and then I gave them the following question (perhaps you recognize it):

Q: The 95% confidence interval for the population mean is [x,y]. Based on this interval:

1. There is a 95% chance that is in this interval.
2. 95% of the observations are in this interval.
3. This method of creating intervals works 95% of the time.

The class was completely split into thirds as to which of the three answers was correct (to be fair, the question was only isomorphic to the first question, not equal).

I re-gave my two more variations of my six minute lecture explaining how to think of each of the three choices.

Then I re-gave the question, only with the following choices:

1. There is a 95% chance that is in the interval.
2. The probability that is in the interval is 0.95.
3. 95% of the observations are in this interval.
4. Exactly two of these answers are correct.
5. Each of the first three answers are correct.
6. None of the above answers are correct.

The correct answer is “None of the above,” of course. Three of the 26 students got it correct, even though I had literally just told them why the first three choices were wrong two minutes prior to voting.

This means one of two things. Either

1. Either learning is incredibly complex, and lecturing is not a good tool to help people understand, or
2. I suck at lecturing.

To be fair, Peer Instruction was not working, either. But it is surprising to me that Peer Instruction works as well as it does, and it is surprising to me that lectures fails as miserably as it does. The confidence interval lesson is a good reminder of the latter.

The point is not that my students are dumb—they are not. Nor is it that they are bad students—they are not. The point is that learning is difficult (especially with tricky ideas like “confidence intervals”), and one must be sensitive to this fact.

Students are Gaming Your System

There was an amusing story recently about some clever students who found a way to get an A on the final without doing any work.

The professor’s policy was that all exam scores are re-normed so that the highest score on an exam becomes the new “100%” (I have a serious issue with his statement that this system is the “most predictable and consistent way” of comparing students’ work to their peers, since I think that students should judged on the basis of their knowledge of the material and NOT by comparing them to their peers. But that is a topic for a different post). His students recently got together and decided that none of them would take the final. Thus, the highest grade was 0%, and everyone got an A.

The professor did give everyone an A on the final exam, but later said: “I have changed my grading scheme to include ‘everybody has 0 points means that everybody gets 0 percent, and I also added a clause stating that I reserve the right to give everybody 0 percent if I get the impression that the students are trying to ‘game’ the system again.”

Here is the thing: students are always trying to game the system; this is because they are largely rational people. Moreover, professors mostly want students to game the system.

For example: I know of many professors who count homework as maybe 5-10% of a student’s final grade. The most common reason I hear from giving a low, but non-zero, weight to homework is “I want to make the students do the homework.” Translated: “I want my students to ‘game’ the system by doing the homework, whether they learn from it or not.” This is also true of attendance policies, participation policies, or really anything that has points attached to it.

What we (thoughtful people, at least) are really interested in is “student learning.” This is difficult to measure, so we use a proxy—”points”—to measure it. But then we fall subject to Campbell’s Law, where we confuse the proxy with the real thing.

Thus, the phrase “I reserve the right to give everybody 0 percent if I get the impression that the students are trying to ‘game’ the system again” really means
“I reserve the right to give everybody 0 percent if I get the impression that the students are trying to ‘game’ the system again in a way that I do not approve of.”

Moreover, the professor is not being clear about what are allowable ways of gaming and what aren’t. This conjures the memory of the famous case of cheating in Central Florida. The students found a test bank available online and studied from it. To me, this sounds like a completely reasonable way to study, and—to the best of my knowledge—this was not explicitly prohibited by the instructor.

The Central Florida was asking the students to ‘game’ his system by performing well on the exam. He was not clear about the allowable ways to ‘game’ the system, but he expected them to know what was allowable and what was not. This seems very unreasonable to me (there are some things that I think that we mostly agree on. It is not allowable to ‘game’ the system by writing down exactly what your neighbor wrote down, for instance. But I don’t think the Central Florida example is such a culturally agreed upon situation).

This is one of the many reasons why I switched to Standards-Based Grading: the proxies are at the very least less familiar, and most likely better associated with our ultimate goal of “student learning.” My proxies are not points, but rather “the number of times you demonstrated that you can do a particular type of problem to me.” It is tough to ‘game’ this system in a way that I am not in favor of, since most of the ‘gaming’ involves learning something well enough to convince me that you understand it.

It is still possible to game the system, though. For instance, students can demonstrate understanding through quizzes, and they can game the system by copying down their neighbors’ answers. But most of the examples of gaming SBG that I can think of falls into the “everyone knows that you are not supposed to do that type of gaming” category.

But the main point is: let’s not pretend that we don’t want students to game the system in certain ways. Let’s remember that the system is not what is important, and we must not lose sight of the reason why there is a system in the first place.

Career Goals

I have been thinking recently about how I would like my career to be. I am trying to narrow my focus so that I can figure out what is important (and then say “no” to things that get in the way of the important thing). Then I had a serendipitous conversation with my wife last night. She was making a list of “values” for herself. This list will help her make decisions about what she should and should not do in the future.

I realized that this is exactly what I have been considering recently, only I have been thinking about my “career values” rather than “life values” (this is not a perfect match—the word “goals” applies much more to my example than hers).

So here is what I currently have for a list of values. These are the things that I want to focus on for my career, and I will work to minimize things that do not belong on this list. This is also only a draft—I will be preparing my tenure file this summer, and I will do some deeper thinking about this then.

1. I want to continue to improve my teaching. This is the most obvious one, but it should be included.
2. I want to do research in finite group theory. This is one of the two values that has the least direction. So far, I have kind of been all over the place in my research, and not in a good way. The only theme I have seen so far is “I seem to be interested in maximal subgroups.”
3. I want to help K-12 teachers improve their teaching. This is mostly done by teaching pre-service courses, but this could be met in other ways.
4. I want to help close the gender gap in mathematics. Kate Owens has been inspiring me on this lately, and it is the value that I have the very least direction on. I have also only begun to think about it. At some point, I am going to have to sit down and read a bunch of the literature.

This is my list for now. This means that my default answer should be “no” to any other requests of my time (in theory, at least). For instance, I had considered eventually running for Chair of the Faculty Senate, but faculty governance is much less important to me than these four things. So I will stop actively thinking about this, although I would be happy to be a senator again in the Faculty Senate (which requires a lot less time—I have to do some service, after all).

Has anyone else thought of their careers like this? Any advice?

When to start flipping

Joshua Bowman tweeted the following question:

Has anyone tried “flipping” a class they hadn’t taught before? #mathchat #flippedclassroom

— Joshua Bowman (@Thalesdisciple) January 31, 2013

The underlying question is: should your first flipped class be a class you have taught before, or should it be a new class?

The argument for the former seems clear to me: it is smart to reduce the number of moving parts. If you have the content and assessments down, you can focus more on the pedagogy.

But I probably lean the other way: I think that it might be better to first flip a class you have not taught before. The reason: you don’t have the safety net of a pre-prepared lecture to fall back on, so you are forced to solely think about the class from a flipped perspective.

Of course, this might just be because of my personal experience. My first attempt at flipping a class was in linear algebra, which I had taught twice before. I had the students watch some Khan Academy videos and do problems out of the textbook before class, and we worked on problems during class.

The problem was that students would ask me questions in class, and I could immediately turn to all of my pet examples (which I had not reviewed beforehand) that I developed the two previous semesters. So the first third of the semester was as much a straight lecture as a flipped classroom. Once I realized this was happening, I rebooted the class to be a better version of a flipped classroom (but you never want to be forced to reboot anything).

Other people may not have this trouble, but I did. But it worked out: I taught real analysis—which I had not taught before—and the flipped classroom went well. There are a variety of reasons for this, but it helped that I did not already have a lecture-mindset for that class.

Anyway, here is my advice for anyone considering flipping a classroom:

1. Start with a class you haven’t taught before.
2. Use Peer Instruction (PI). Not only will it provide you with a great framework for your in-class work, but many people do it so you can borrow/steal a lot material. Best yet: even if you completely screw up the class, you will still be no worse off than a brilliantly-done lecture.
3. Choose a class textbook that is readable for the students. Have the students read it before each class.
4. Have some sort of mechanism for collecting the students’ questions prior to each class. Classroom management systems like Moodle/Blackboard/etc work, you could set up a class blog on wordpress.com and have them use the “comments” for their questions, or you could just use email.
5. Get someone else’s PI “clicker questions” to use a foundation for your course.
6. To prepare for a class, read through the section and create several clicker questions of your own before reading the clicker questions you stole from someone else (this is to get practice, but also to focus on what you think is important about the section). After you have written some of your own, merge them with the reference questions you got from someone else. This can be done well before the class actually meets.
7. The morning before the class, look through your students’ questions. Pick the appropriate clicker questions from your reserve that will best answer their questions, writing new ones if needed (this is optional, especially if you have an 8 am class). Be sure to keep some questions on the most important topics, though, since students sometimes do not ask questions on this.
8. Go to class, ask the questions, and have fun.

Notice that I did NOT recommend “creating videos.” I think that this is a nice thing to do for the students, but it is a lot of work. Students can definitely learn from a reasonable textbook.

As for “clickers,” I use TurningPoint, but only because that is what my campus decided on. Several people use iClicker, and Learning Catalytics is supposed to be awesome if you are sure that everyone has a device (and you have some money to spend). But do not discount low-tech solutions, either: I believe Andy Rundquist prefers colored notecards to electronic clickers (students raise a red notecard for option a, green for b, etc).

I am a big fan of the flipped classroom for most college-level classrooms. Please contact me if you are interested in getting started.

As always, please feel free to critique anything that I have said in the comments.

(photo “Flip” by flickr user SierraBlair, Creative Commons License)

Grading for Probability and Statistics

Here is what I came up with for grading my probability and statistics course. First, I came up with standards my students should know:

“Interpreting” standards (these correspond to expectations for a student who will earn a C for the course.

1. Means, Medians, and Such
2. Standard Deviation
3. z-scores
4. Correlation vs. Causation and Study Types
5. Linear Regression and Correlation
6. Simple Probability
7. Confidence Intervals
8. p-values
9. Statistical Significance

“Creating” standards (these correspond to a “B” grade):

1. Means, Medians, and Standard Deviations
2. Probability
3. Probability
4. Probability
5. Confidence Intervals
6. z-scores, t-scores, and p-values
7. z-scores, t-scores, and p-values

(I repeat some standards to give them higher weight).

Finally, I have “Advanced” standards (these correspond to an “A” grade):

1. Sign Test
2. Chi-Square Test

Here is how the grading works: students take quizzes. Each quiz question is tied to a standard. Here are examples of some quiz questions:

(Interpreting: Means, Medians, and Such) Suppose the mean salary at a company is $50,000 with a standard deviation of $8,000, and the median salary is $42,000. Suppose everyone gets a raise of $3,000. What is the best answer to the following question: what is the new mean salary at the company?

(Interpreting: Standard Deviation) Pick four whole numbers from 1, . . . , 9 such that the standard deviation is as large as possible (you are allowed to repeat numbers).

(Creating: Means, Medians, and Standard Deviations) Find the mean, median, and standard
deviation of the data set below. It must be clear how you arrived at the answer (i.e. reading the answer off of the calculator is not sufficient). Here are the numbers: 48, 51, 37, 23, 49.

Advanced standard questions will look similar to Creating questions.

At the end of the semester, for each standard, I count how many questions the students gets completely correct in each standard. If the number is at least 3 (for Creating and Advanced) or at least 4 (for Interpreting), the student is said to have “completed” that standard (the student may opt to stop doing those quiz questions once the student has “completed” the standard).

If a student has “completed” every standard within the Interpreting standards, we say the student has “completed” the Interpreting standards. Similarly with Creating and Advanced.

Here are the grading guidelines (an “AB” is our grade that is between an A and a B):

-A student gets at least a C for a semester grade if and only if the student “completes” the Interpreting standards and gets at least a CD on the final exam.
-A student gets at least a B for the semester grade if and only if the student “completes” the Interpreting and Creating standards and gets at least a BC on the final exam.
-A student gets an A for the semester grade if and only if the student “completes” all of the standards, gets at least an AB on the final exam, and completes a project.

The project will be to do some experiment or observational study that uses a z-test, t-test, chi-square test, or sign test. It can be on any topic they want, and they can choose to collect data or use existing data. The students will have a poster presentation at my school’s Scholarship and Creativity Day.

I would appreciate any feedback that you have, although we are 1.5 weeks into the semester, so I am unlikely to incorporate it.

Peter Elbow is awesome

I am busy preparing for classes, but I want to post something here so that I can find it later: Peter Elbow writes about “minimal grading,” which is essentially the wheel that I am reinventing. Enjoy the article.

(hat tip to Angela Vierling-Claassen, who tweeted the article)

Grading Epiphany (or, Bret was completely wrong about everything)

I am teaching probability and statistics, a course for first-year students, for the first time this spring. I have been struggling with how to grade the students.

This course is unusual in that there is only a little mathematics in the course (we throw in all of the probability that we can, but it is still mainly statistics). This requires that I think like a statistician, which is new and somewhat painful.

It also makes designing a course more difficult. I have the basics of what I want to do, but—as mentioned above—how to grade the students is the most difficult part. I want to stick with an SBG approach, but I was not sure how to set up the standards.

In calculus last semester (and other courses in previous semesters), I had a general format of: “if you can do the basic skills from the course, you will get a C. To get a grade higher than C, you must demonstrated some conceptual understanding.”

I realized as I was brushing my teeth last night that this is completely and utterly backwards.

I want my C students to understand the concepts of the course, but not necessarily be able to do the computations and symbolic manipulations. My B students should, in addition to understanding the concepts, be able to do many of the computations and symbolic manipulations. My A students should, in addition to understanding the concepts, be able to do all of the computations AND demonstrate that they can do some self-guided work.

Here is my rationale for requiring understanding the concepts to get a C:

1. I am convinced that the concepts are easier in most college-level mathematics courses—students are better at drawing tangent lines on graphs of functions than they are at finding the equations of tangent lines.
2. The students who need the calculation and symbolic manipulation skills are the ones who are going to continue taking more mathematics (and related) courses. I am guessing that C students are less likely to continue taking these courses.
3. Computers can now do much of the calculation and symbolic manipulation, although the user has to understand the concepts to correctly enter the information.
4. Most importantly, the concepts are the most important part of the course! I want to explicitly encourage students to focus on the concepts—I don’t want, say, a calculus student to be able to get an A in the course by only having good algebra skills (a colleague yesterday complained to me about such students; I view this as a flaw in the grading system).

I am not happy about having this completely backwards, and I feel bad for my previous students. But I am happy that I now understand what I want.

The tough part is designing assessments that isolate concepts. But that is part of my job, and I find it fun to come up with such questions.

How can this be improved? Am I wrong about this?

(image is “LED Light Bulb” by flickr user Wade Brooks, Creative Commons License)