By popular demand, here are some of the results from my creativity assignment. Of course, I got permission from all of the artists, some of whom granted me permission to post the art but not their name.

These were posted roughly in order that they handed in the assignment. I will post other pictures as I receive and photograph them.

Lauren Zwolenski:


Anonymous:


Anonymous:


Jeannette Gondringer:


Steph Anderson:


Michelle Wagner:


Michelle Kennedy:

I just finished administering and grading all of my midterms—I had one class on Thursday, and two classes on Friday. I attempt to write midterms that are short. I have written about this topic before

I have two things to add. The first is that I think this gives students a more accurate assessment of their ability. Short exams eliminate the "I knew the material, but I got a bad grade because I ran out of time"-factor. You eliminate an excuse for the students.

The second comment is one of goals: I am generally happy with the length of the exam if the exam has an LD-50 of "10 minutes." By this, I mean that 50% of the students finish with at least ten minutes left in the class period. There is no scientific basis to this, only a sense that I am providing a thorough enough exam without creating much of a time pressure.

I was able to meet my goal in two of my three classes. Interestingly enough, I teach two sections of multivariable calculus and only one class met the LD-50 goal. I am not sure why this is.

Jordan Ellenberg posted a link to a great article about coincidence. The article can speak for itself, although I will add that humans are not very good a probability. This is probably because our focus is too narrow—we can only focus on what is in front of us.

A good example is the "birthday problem" described at the end of the article. I think that most people naturally think "what is the probability that someone has the same birthday as me." This lack of a global view makes people think things are less likely than they are.

I am teaching elementary education majors this semester. We started discussing tessellations, and I decided two things:

1. The walls of our classroom are boring.
   
2. My students need a creative outlet.
   

I decided to ask the students to create a tessellation. Here were the guidelines:

1. Create a tessellation.
   
2. If you do it, you will get full credit. If you don't, you won't get any credit.
   
3. We will decorate the classroom with your assignments (unless you opt out).
   

I wanted to give my students maximum flexibility. They were, frankly, dumbfounded. I was barraged with five minutes of questions. "How big should we make it?" "Can we use color?" "Do we have to draw it, or should we do it on a computer?" "Does it have to be a tessellation of an animal?"

I think that these were all very reasonable questions given our education system—I find that there is not a lot of room for creativity in school (note: the last studio art class I took was in middle school; maybe art classes got better since then). My students were looking for a "catch"— I must have some sort of restriction up my sleeve. I find this a shame. Educators speak about the importance of nurturing creative students, yet we almost always stifle creativity with too many directives. "All assignments must be double-spaced." "Compare Great Expectations to The Joy Luck Club." "Do questions 1,3, and 7 from the textbook."

I am guilty of this, too, but I am trying. For instance, I try to let students choose which questions they want to do on a homework assignment: "Do three of the following five questions" (note: I have not been good at doing this this semester). I hope to incorporate more room for creativity in each of my classes. These will likely be small things at the beginning, such as the tessellation assignment, but I think it is better than doing nothing.

By the time I finished answering their questions, my elementary educations majors seemed to be buzzing about the project; there was almost a tangible excitement in the room.

Matthew Leingang tweeted about a webpage on common errors in college mathematics. This list does a reasonable job cataloguing common mistakes, but one missing error is "inadvertently dividing by zero." For instance, a student might "simplify" the equation "x²=2x" by dividing both sides by x to yield "x=2." This division is allowable as long as x is not zero, so the student is implicitly assuming that x cannot be zero. Unfortunately for the student, he assumes away the solution x=0.

A second problem—I do not know if it should go on this list— is that students everyone seems to be under the assumption that "if I wish it were true, then I'll just assume that it is true without thinking about the evidence." Perhaps students do this when they distribute square roots. This is also common in political discussions, as when Joe Wilson yelled "you lie" at President Obama (the proposal clearly states that illegal immigrants cannot receive health benefits. I am not taking a side on this issue, but am only pointing out a recent example of "if I wish it to be true, it must be true").

It is interesting to see all of the errors enumerated and explained, although I am not sure how useful it is. On the page, the author writes of a professor who warned students that an exam would contain many extraneous solutions, and that he would penalize them heavily if the students included extraneous solutions in their answers. It sounds like this did little to prevent the students making these errors.

When I was a graduate student at the University of Wisconsin, our College Algebra course had a theme called "vital errors."^[*] These were a list of misconceptions that algebra students often make. For example, one vital error is the distribution of powers, such as (x+y)²=x²+y². Our students were given a couple tests on problems that contained these "vital errors." In spite of knowing that every question on these tests would contain one of the 7-8 vital errors they had been warned about, students still did very poorly.

It seems like telling the students "these are the errors that you might make, and so you will be punished heavily if you make them" is not effective; at least, it was not effective at Wisconsin and not for the professor referenced on the page. This begs the question: what is an effective way to get students to avoid making these errors?

I welcome proposals in the comments. My only contribution is that students need to understand arithmetic before they can abstract it to algebra. If I were teaching college algebra now, I would begin with a unit on arithmetic. Only after students felt comfortable with, say, the distributive law with numbers would we move to variables.

Other ideas?

Senator (also former Secretary of Education and president of the University of Tennessee) Lamar Alexander wrote an article on education for Newsweek in last week's issue. The title is "The Three Year Solution."

Here is a brief summary of the article: The "Big Three" auto companies were once great American institutions. They ceased to be great because they failed to adapt to changing conditions. The American university system is currently a great American institution. It could cease to be great by failing to adapt to changing conditions. One way to do this is to provide options for well-prepared students to enter a three, rather than four, year college program.

I value ideas like these. Other ideas that Alexander alludes to: eliminating summer vacation, , re-thinking tenure, and doubling the faculty to allow for year-round utilization of campuses. I have discussed tenure before—I think that it should be eliminated for community colleges and below (where research is not done), definitely kept for Research I universities, and unsure about liberal arts colleges like mine ("Well, that makes me feel better about me, worse about Giles. Kinda shaky about you."). My justification for tenure is tied exclusively to research—I do not want researchers with unpopular ideas to be squelched. Certainly, great progress often occurs as the result of an unpopular idea. Thus, I disagree with Senator Alexander; while I believe that the tenure process may stifle creativity while a professor is working to obtain tenure, eliminating tenure would stifle creativity for a lifetime. After all, one is not likely to explore an odd idea if it would put one's job in danger.

I am intrigued by the Senator's ideas on eliminating summer vacation and holding college year round. There seems to be little reason to retain summer vacation, aside from the fact that we all like it (perhaps, though, this is reason enough). Senator Alexander also proposes that we double the size of the student body, double the faculty, and hold classes year round. This would allow students and faculty to retain their vacations (although it seems like they would not always be in the summer), but better utilize the campus. After all, the campus costs money to run even when students are not present.

One problem with summer classes is that I do not see a large, unfulfilled demand for college education. My institution(s) had fewer students enroll this year than desired. The only way to get more students would be to accept students who are less prepared (which may not be a bad thing).

My impression is that Senator Alexander is largely offering solutions to keep costs down for colleges. Although he did not mention this, eliminating tenure would almost certainly have the effect of lowering costs, since there is a large pool of cheap, recently graduated Ph.Ds who would like jobs (I, of course, would be opposed to this, since I believe that experience is important in education). While running classes year-round would not decrease costs, they very likely would increase revenue. These ideas make sense to me in the context of his article.

Most of the article, though, is about creating a system where well-prepared students could plan on graduating in three years. While this is an interesting idea (I graduated in three years from the University of Minnesota, and I am happy that I did), it eludes me how this relates to his article. He writes that this would save the student money—which it certainly would. However, this would come at the expense of the college—which seems counter to the point of the article.

I wonder if he thought that offering a three-year program would increase revenue, but I am skeptical here. The pool of well-prepared college students is small, and the pool of well-prepared college students who refuse to attend college because it takes exactly one year too long is even smaller. I cannot imagine that there are students who are not attending college for this reason.

I cannot figure out what problem the three-year solution is attempting to solve. I agree that it would be nice for some students (although I fear that too many students would attempt it—I am of the opinion that education is being far too compressed as it is), but I do not see the advantage it offers for colleges. Please let me know if you have ideas.

I am in the middle of midterms. I tend to write three different types of exams: two types of in-class exams, and one time of take-home exam. I will mix the take-home with either type of in-class midterm.

The first type of in-class midterm is a check that students are able to do the basic things from the course. This includes recalling definitions and answering straightforward questions. In a calculus class, I might include a question like "What is the derivative of f(x)=x^2?" The purpose of this in-class exam is to act as an incentive for the students to take time to learn the course material.

The take-home exam has a different purpose. Here, I'll ask questions that require students to think about concepts in novel ways. I often make these open book, open notes, group exams. In a calculus class, I might include a problem like: "Find the equation of a tangent line to f(x)=x^2+1 that goes through the point (4,8)." The purpose of this type of midterm is less to assess the student's knowledge than to help her acquire more. I hope that thinking about these questions leads to a greater understanding of the material.

The second type of in-class midterm is like the take-home, only it is in-class and not a group test (with a couple of exceptions). The main lesson I have learned here is to only give a small number of questions, since each of the questions is fairly involved.

I have given all three types of midterms so far this semester. I tend to always include a component of "learning exam" (rather than "accessing exam"), as my main goal is to help students learn. However, I also need to assign grades, and this is the reason for the pure assessing exams.

I don't feel great about giving the assessing exams. I do not like the idea of making the students demonstrate that they learned the material, largely because I have read psychology results that say this type of "incentive" (a bad grade is a "stick," or a good grade is a "carrot") decreases student learning. I would love to hear of creative ways of having students learn mathematics, assessing what they know, and having the two complement---rather than work against---each other.

Please leave comments, although please offer evidence if you say "students would never learn if I don't give them exams/homework/etc."

Here are two sites that I recently bookmarked: Polymath and Math Overflow. Both are math help sites for the mathematician (graduate student and above).

Most of these questions do not apply to me, but I expect to check them daily (along with the group theory page on the arXiv) as part of a way of keeping up with the mathematical community.

Thank you to Jordan Ellenberg and Tim Gowers for introducing me to these two new websites.

My thesis advisor gave me a great piece of advice: never leave the office without giving your computer something to do.

Since then, I have tried to have a program running overnight whenever possible. This is not the greenest thing to do, but it is productive. Here are two things that I do regularly.

1. I use the computational algebra system GAP in my research, and I try to have a program running whenever I am not at the computer.
   
2. I participate in the Great Internet Mersenne Prime Search (GIMPS) to use my computer to find large primes.
   

Frankly, I haven't been very good about having GAP programs run. However, it is very easy to search for primes--GIMPS has programs that are foolproof.

I read Jordan Ellenberg's excellent Quomodocumque weblog. He referenced a paper by Edward Nelson. This paper made my head hurt.

This paper is not terribly technical, and could be understood by people who are reasonably comfortable with mathematics (no Ph.D. needed). This paper is 12 pages long, so many of you may not want to take the time to read it.

To summarize: Nelson writes about the foundations of mathematics--roughly, verifying that the basics of mathematics is correct (e.g. Does "1+1" equal "2?"). Nelson uses that fact that, as long as we are only considering natural numbers, multiplication is repeated addition, and that exponentiation is repeated multiplication.

The good news: under the basic rules of mathematics, it easy to show that the sum of two natural numbers is a natural number. Similarly, it is easy to show that the product of two natural numbers is a natural number.

The bad news: it is not at all obvious that exponentiation of natural numbers yields a natural number. This creates the unpleasant situation that a number such as 999^123456789 may not be an integer. For all we know, this could be, say, a fraction.

I do not think that we are going to find natural numbers n and m such that n^m is not a natural number, but it seems like we should be able to easily prove this from the very basics. Not all is right with the universe.

One of my colleges just installed a new president. This was accompanied by a 2.5 hour ceremony, complete with some 15 different speakers, academic regalia, strings, brass, choruses, and presentations in at least six different languages.

My question is: what is the benefit to the institution? I am asking this question hoping for a response, although I start from a position of skepticism; I have attended a grand total of two college graduations, neither of which were for any of my three academic degrees. So it could be that I just need an attitude adjustment.

Here are my speculative answers, with lingering questions in parentheses:

1. It serves to introduce the president to the University (But he has been in the community for over 40 years, and his remarks at the ceremony comprised only a small percentage of the ceremony).
   
2. It serves to introduce the president to the larger academic community--many of the attendees were from other schools (See remarks above).
   
3. It serves to create a spirit of cohesion at the University (But not a lot of students were there - I estimated that most people there were alumni, administration, monks, or faculty. Also, it seems like there would have to be regular meetings like these in order to add up to a significant change in student attitude).
   
4. Any excuse to celebrate is enough to justify a celebration (Okay, but why do we only have these ceremonies when we change presidents? Why not have celebrations monthly?)
   
5. It is tradition. (This evades the question--not all traditions are good. We should be able to justify the ceremony with some other reason if we are going to justify such an expense).
   

I would sincerely like to hear a response to my questions. I understand that my point of view is sometimes, well, odd, and I would enjoy understanding how others think about this.

Edit: On Facebook, one of my students pointed out that classes were cut short on the day of the ceremony. This further emphasizes the need to provide evidence that this ceremony is useful, since students pay for it in lost class time.

My goal for today is to discuss large-scale educational goals. I expect this to be a running theme in this weblog, since it is an essential, yet under-recognized, part of education.

This theme will start with one example: standardized testing. This is a polarizing issue in education. One side, which is currently "winning," claims that standardized testing is essential. We cannot know if students learned what they should unless students are given an unbiased exam. Moreover, standardized exams give us information about the teachers and schools; if too many students fail a standardized exam, it is evidence that a teacher and/or school is failing. Largely, standardized tests are the only true way to establish accountability.

The other side claims that standardized testing hurts education. Among other reasons, it is easiest to write a standardized exam about memorized facts; testing higher learning skills is considerably more difficult--and therefore much rarer. This gives us a skewed view of how the students are doing; "no information" would be better than "wrong information." This problem compounds itself when teachers "teach to the test," favoring bite-sized facts to complex problem solving. Furthermore, some standardized exams predict family income better than future grades. This could lead to promising students from poorer backgrounds to be denied access to education. Finally, standardized tests are expensive, and school districts could spend the money better elsewhere.

I did my best to be fair to both sides (I have my own opinion), although my arguments for each is by no means exhaustive.

There is debate about standardized testing in some circles, and arguments like these are thrown back and forth at each other. However, I think a more constructive step would be to delve deeper to determine the education goals and attitudes of both sides. What follows is my attempt to determine what kind of attitudes both sides might have about education.

Pro-standardized testing attitude: Students need to learn what we teach them, and we teach them things that are easy to measure--either a student knows how to add, or she doesn't. Because of this, we need to provide incentive to the students to put in the work to learn. One way of doing this is testing--the student will learn what we teach in order to do well on the exam.

Anti-standardized testing: While facts are important, it is more important for students to develop habits and thought patterns that will make them a successful citizen. Knowing the fifty states is nice, but it is more important that students develop a habit of providing evidence when making assertions (and requiring evidence when hearing assertions).

If I were to have the pro-standardized testing attitude, it would be obvious to me that standardized testing is essential. With the other attitude, it would be clear that standardized testing would be difficult to administer, at best. Because of this, I believe it would be better for the sides to attempt to reach agreement on the educational goals, rather than standardized testing. Even if both sides were to agree on standardized testing, we would have only solved one symptom; the underlying cause of the dispute--different attitudes toward education--would linger and create new disagreements.

I propose that we all identify our educational goals and attitudes before we decide what tools (such as standardized testing) would best meet these goals.

I was a little skeptical that writing a professional weblog would be of any benefit to anyone, but it has actually already paid off for me. As a response to my post on Getting Things Done, two friends/colleagues wrote to me about How to Write a Lot by Paul Silvia. I read this over the weekend, and enjoyed it.

The book roughly says, "To write a lot, schedule time each day to write." This might seem simple, but it can be a little difficult to implement. Before reading the the text, I had scheduled myself time to do math for 70 minutes every other day. So far, this has not paid off a lot. However, I think that I might change that to daily starting next week. When this happens, I hope to see a rise in productivity.

A second thing that I am doing is to apply for a fellowship. This fellowship will release me from one course next year, giving me time to work on a textbook I am writing. I have to imagine that this, combined with a schedule, will help me to "write a lot."

A second benefit of the book is that it spelled out the difference between psychology research (Silvia's field) and mathematics (mine). Silvia wrote that most psychology professors have a backlog of data, and they could produce many papers if they could find time to sort through the data. Basically, psychology professors are not lacking for things to write about. I find this not to be true in mathematics. My struggle is to create original mathematics that people care about. I would love to have the problem of having too many ideas stashed away in my filing cabinet. Because of this, I expect that most of my "writing time" will actually be "thinking time."

My wife teaches math at the public university in the area. I highly recommend marrying an academic, as dinner-time conversations can quickly turn into professional development opportunities.

Our dinner conversation on Tuesday centered around timed exams; that is, exams where you have to do many questions in a relatively short period of time. We debated their merit, and we came up with the following:

1. Timed exams should only be used if they fit your goals and values. A discussion on goals and values will be the topic of an upcoming post; for now, I will just say that they are woefully neglected in education.
   
2. Timed exams really only work if the students are only expected to either recall, or to do a very basic computation repeatedly.
   
3. Timed exams are not appropriate if students are engaged in complex problem solving.
   

We reached these conclusions mainly by acknowledging that brilliant people can sometimes take a long time to figure things out - professors are never expected to start and finish a paper within a week. Deep thinking takes time. Therefore, adding time pressure to an exam can give a faulty assessment of one's understanding (assuming this is the reason why the exam is being given).

On the other hand, there are other times when we do not want our students to think much, and here timed exams could give useful information. Examples of such topics include an elementary student demonstrating that they know their multiplication tables, or a calculus student demonstrating that they know how to quickly compute easy derivatives. In both cases, we want students not to have to think deeply about these questions (it is really difficult to get common denominators when both the concept of adding fractions AND multiplying integers require concentration; the cognitive load is just too much).

As I posted (years) before, I have started to give short midterms. This is because I mainly want to test problem solving and conceptual understanding, and I don't include as much recall and computation (although they have to do computation in order to do other problems). I tend to test recall and computation in different parts of the course.

I would like to welcome myself back to my professional weblog. My semester has begun, and I hope to post regularly to this now (2-3 times per week).

I now have varied responsibilities in my new job, and I have many meetings, responsibilities, and people to keep track of. Because of this, I recently re-implemented David Allen's Getting Things Done methodology. At its core, this system gets you to put all of your obligations into an external system so that you can experience less stress. I am not certain that this helps me do more work, but I am certain that it decreases the amount of stress in my life.

By far my favorite aspect of the system is the "tickler file," which is a file-based calendar where I can leave myself time-sensitive notes. For instance, today would have been my grandfather's birthday, and a note reminding me of these was in my tickler file this morning when I checked it. This reminder helps me to remember to call my grandmother tonight.

More importantly, this note in my tickler file means that I do not have to try to remember this birthday. Loose ends like this are a cause of stress for me, and this system allows me to delegate this worrying to the "Getting Things Done" system.

I decided this semester to make a concerted effort to shorten my examinations. We typically have two hour exams, and this normally translates into 11-12 questions.

However, this creates a great time pressure on some students. I was concerned that exams designed to take 2 hours might not evaluate some students' knowledge of material, as some students think a little more slowly or feel test anxiety. Since I am not concerned with the speed with which a student finishes the exam (we tend to test concepts much more than mechanical skills), I decided to shorten the exam to only 7 questions. After all - there is no rule that says we need to keep students working for the entire duration of the exam.

I was quite pleased with the result. The results were that students did slightly better than usual, but still within a normal range. I felt like I got a good assessment of my students' knowledge, and the students were happier. The students were generally happy with the exam on our midterm evaluations.

There are two other advantages to shorter assessments besides (what I believe to be) more accurate evaluation of student knowledge. The first of which is that I was able to ask one or two harder exam questions. Students have more time to think on any one question, so I can choose a couple questions to be harder and still save time on the exam.

The second advantage is not a pedogagical one, but is pleasant nonetheless: there is less grading.

Ultimately, this stems from my philosophy that it is not important to do math quickly (there are a handful of exceptions to this). Mathematicians are never expected to start and finish a paper within the span of a day, and I don't think that our students should feel such time pressures, either.

One bit of wisdom that I have learned during my time here is the importance of meeting with students at the beginning of the semester. I meet with each of my students one-on-one for ten minutes sometime outside of my normal class and office hour times. These are intended to be social meetings, rather than meetings where we talk about math. This allows me at least one experience interacting with the student with him/her being "the math student" and me being "the math teacher."

A large portion of my job is to help graduate students have a successful teaching experience, and I have been recommending that they meet individually with their students ever since I got here. I have heard several success stories of graduate students "turning the class around" after meeting with the students, and salvaging the semester for everyone after a rocky start. Still, I guess it didn't sink in how effective these meetings are.

The primary reason why I was surprised at how well the meetings work is because I normally teach freshmen. I have seven years of teaching experience, but probably 90% of the students whom I have taught have been freshmen. I have always felt that I could get away without meeting my students (although it is better if I do) because I have developed ways of getting to know students while teaching them.

This semester, I am teaching an upper-level math course: discrete mathematics. I felt that this class got off to a bit of a slow start - things weren't awful, but there was not the normal "atmosphere" in my classroom. In my second week of class, after the semester had settled in, I had my meetings with the students in the discrete math course. I met with them between the Monday lecture and the Wednesday lecture, and Wednesday's lecture was ridiculously better than Monday's - it was like night and day.

I think that there are two reasons for this. First, I felt more comfortable with my students. I had never taught an upper-level course before, and I was not completely comfortable teaching students that I didn't "know." I wasn't sure what their experience was, what their confidence level was, and what their goals were. I learned much of this from my 10 minute meetings, and I am sure that my comfortable was evident during Wednesday's lecture.

The second reason is that I think that the students were more comfortable, too. They found out that I was not a jerk, and at the very least I was willing to give up five hours of my time to meet with the entire class. The students were immediately friendlier toward me, and now everyone enjoys the class more.

We are holding an orientation session for those who are going to be teaching in the calculus program here. Here is a schedule:

10-10:20: Icebreaker
10:20-10:40: Advice from experienced teachers
10:40-11: Goal Setting
11-11:20: First day videos
11:20-12:05: Pop Quiz (issues in teaching)
12:05-12:50: Lunch
12:50-12:55: Informational handout sheet
12:55-1:40: Skits on clarity, response to student work, and classroom management:
1:40-1:50: Feedback

The part that I am most interested in is Goal Setting. We will get together in our courses and try to determine goals for the semester. I find this to be an interesting and essential part of teaching that is too often overlooked.

I define "goals" and "learning objectives" differently. A "goal" is the effect we wish to have on the student. The "learning objective" is roughly the manner in which we choose to accomplish this goal. The best example of this may come from Rocky IV. The American boxer (fighter, not dog) Rocky is scheduled to fight the Soviet boxer Drago. They, of course, train for the bout, but in very different ways. Drago has a state-of-the-art gym full of computers and training equipment; Rocky goes to a logging cabin in the woods. But they share the same "goals;" for instance, they both want to strengthen their quadriceps muscles. However, their trainers give them different "learning objectives:" Rocky's trainer gives him a "learning objective" of pulling the trainer on a dogsled. Drago's trainer gives him a "learning objective" of n number of reps on a machine that mimics the same motion. See the video below:



Even though their goals are the same, they are given different learning objectives to accomplish these goals.

But I digress. We are going to choose goals for our courses. In choosing a goal, I like to try to answer the following questions: Why do we education people at all? There are many possible answers, but I feel like what we do in a math class should support at least one of the answers. This question will appear again in this weblog. In fact, here are a series of questions that I am contemplating:

1. Why do we educate people rather than not?
   
2. Why do we teach mathematics to people rather than some other subject?
   
3. What does it mean to be intelligent?
   

I will likely post on each of these questions separately, and I likely will not provide any answers. All feedback will be welcome.

Hello, and welcome to my professional weblog. I was inspired to do this by my colleague Matthew Leingang, and I hope to keep up with this as well as he does. I expect this weblog to mostly revolve around the teaching part of my career, rather than the research part of my career. I would expect the audience (if any) to be colleagues and students, although not every post will be interesting to both groups.

I think that I am going to try to post one to three times each week. I post more frequently on my personal weblog, and that is because I hope to write here about things that are more interesting than the topics on my personal weblog.

Welcome to the show!