I don't understand this

I just finished grading my Math 097 exams this afternoon. There were two parts to the exam. The first part was a 125 minute in-class exam with one page of notes. This is the standard sort of in class final that you would expect from a math class. The second part was a two day take-home final, where students could work together in groups, talk to tutors, and do whatever it takes for them to figure out what's going on (except asking me for help). It's also exactly the same test as the in-class exam.

The average for the in-class: 42.5
The average for the take-home: 55.5

13 point improvement... That's not too bad. However, the exam was out of 94 points. That means that even if the students were allowed almost unlimited resources, they were still only able to get about 60% credit. I find this very disappointing. I also don't understand why the students couldn't pull off at least a solid finish. This is even more confusing to me because the take-home part was worth a full 13% of their final grade!

Sadly, this result on the final exam means that over half the class failed. There were a couple of bright spots. I had a couple students who worked very hard and came away with a well-deserved B.

Coming to the end of the semester

Even though there's another week left in the semester, I've reached the point where I have no more teaching to do. At least, no more planned teaching of new material. I have one more class this evening to teach, but we're just doing review. I've got a couple hours right now with no specific plans, so I'm going to spend the time rambling away with my thoughts on how this semester has gone in order to prepare myself for next semester. I'm pretty sure some of this stuff I have mentioned before, but it's worth reconsidering again.

General things

I'm not going to be using WebCampus for my homework postings next year. Doing it this year was too much of a hassle without enough benefits to make it worth my time and energy. I think by hosting it on a normal webpage will make my life much easier. (Hopefully, I can figure out how to get proper FTP access from home to make this part even more flexible and accessible.) Along the same lines, I need to get better posting quiz solutions and that sort of thing.

I let myself get lazy with my bookkeeping and grading, which was a little bit frustrating just because I don't like to have stuff like that hanging over my head. I also found myself about halfway through the semester not having my quizzes ready until just before class. I think if I get myself a little more organized (the point above with the homework posting frustrations comes into play here), I should be able to make that happen a little more regularly (and efficiently).

I made a homework cover sheet that I think I will have my students use. The point of it is to increase the impetus on my students to be organized. It has questions on the front that I want students to answer, which will force them to think a little bit about what they did with their homework instead of just turning it in. I also gave them a space to ask questions. I didn't leave myself any room to make comments in return, so maybe I'll change that.

I want my students to check their answers in the back of the book. I don't feel that grading their homework is actually something that is particularly benficial for them. I think they need to learn how to use the tools they have to assess themselves. Of couse, I need to explain this process to them on the first day of class.

Math 097

I feel that I need to change my approach to this class completely. I almost want to break it into two different sections: One hour lecture and one hour lab. That might even need to be broken down into two half hour lectures and two half hour labs. I want to keep the daily quizzes, though I may change them into weekly quizzes. There's something about giving them a problem to do and forcing them to show what they know that has seemed to help things stick in the minds of some of my students.

I think the first 4 weeks of the class are the most critical as the foundational algebra must be set up by then for any of the rest of the stuff to have a real chance of making sense. But this doesn't just mean the mechanics of algebra, but a sense that the algebra is actually connected to real life (as a way of representing real values). I've got some ideas that I'm working out right now, but nothing solid yet. It has to do with starting the class using just numbers and then introducing variables *after* they have a sense of seeing patterns and *after* they have a notion of what it means to generalize a pattern.

I am also going to be less ambitious about the number of topics I cover in that class. I need to think a little more carefully about it, but I've got a preliminary schedule that feels like it's only about 2/3 the speed as I had this semester. I would like to follow the "teach less, teach better" mentality this time around and see if I can cut back to the core essentials.

There are things I want to do, but I don't know of any textbooks out there that do it. Maybe I'll write my own someday...

Math 124

There is a move right now for us to get rid of this class. The entire math department wants to do it. We've got a proposal in the works to make it happen.

The problem is that the course has turned into a terminal course, meaning that most students who take this will have this as their last math experience. Unfortunately, the content of the class doesn't really climax very well as a mathematical exeperience, and just leaves students with the feeling that math is a bunch of nonsense that doesn't go anywhere or have any real application. That's all very disappointing because it continues to perpetuate the sense of how hard and weird math is.

As far as how I might change how I teach this class, I'm really not sure. The difficulty is that this class has less time than the other one, so I can't break it up into lots of little pieces and have time for things. However, I still like the idea of breaking it into a lecture/lab combo, but I need to spend time working out how much information I can put in front of them in a 20 minute span and have them make sense of it.

Math 283

This was my fun class all semester long. I hope my number theory class next semester goes just as well if not better than this one. I don't really have too much to say about this one just because it was the one that I could relax the most in (even though it was the most difficult content) and everything that happened in that class just seemed to be very enjoyable.

I scared my student...

I had was I guess I would consider to be my first big mistake as a professor, although I'm not sure if it really was a mistake or if i'm allowed to fault the student for this. I had a student who I knew was having a tough time with the math and I believed that it was mostly a result of her unwillingness to think about things before she just started randomly guessing what she should do. For example, we were looking at equations of circles and we were trying to write them in "standard form." She knew that a square had to go *somewhere*, but she had no idea where and just sort of stuck on the end. This is bad news from a mathematical perspective because it violates many of the precepts, in particular that every step has logic behind it and that nothing is purely random or haphazard.

With this in the background, she made a post on Webcampus asking me to post solutions to an earlier quiz over the weekend before the midterm. I explained to her that I would be unable to post it until at least Monday (after getting back to campus), but also directed her to the book and other quizzes to use as a guide. I also warned her against the misuse of solutions, namely trying to just memorize them for a test and making sure that she thought through her problems.

She responded by accusing me of criticizing her publicly (her post was public and my response was public) and felt insulted enough to drop the class before even taking the exam.

I had a chance to talk with Sandip about this incident (chair of the Physical Sciences), and it was a constructive conversation. I'm not sure if I agree with his approach, namely that I should treat every student as if they were the most fragile person in the world, but I do know that there can be a bit of a rough edge around me because when I make an observation of the type described above, I tend to believe that I stand on a fairly solid analysis.

I don't know whether I should I feel responsible for a student misinterpreting something like that as a criticism. There's nothing in my reading of what I wrote that I believe is talking down at her or insulting her. I also don't know how else I can communicate the teaching without giving the teaching. This is something that I'll just have to feel my way through in the future, I guess.

A chance to refelct on the first half of my first semester

I'm taking some time right now to think about the first 8 weeks of the semester. I'm doing this now, right before the big onslaught of graind that is to come next week. There are certainly a number of things I want to do differently next semester.

#1 - Homework: I'm going to try something new next semester. I'm going to make part of the homework for my algebra classes simply copying the examples out of the book. This is first to force them to read the examples and work through them. Secondly, I want to give them the opportunity to circle parts of the examples that doesn't make sense to them that they would like clearified. I also want to force them to compare the examples to their own work that follows. Hopefully, this will help them get into a good routine of using the book effectively instead of just a source of homework problems. Thirdly, I think giving students a full week to do homework is simply an excuse for them to put it off. Only a couple students ever start the homework ahead of time like I ask, and so I need to force the issue a little with them. Finally, I need to assign problems from previous sections with much more regularity than I have been. This will hopefully reinforce the necessity of not forgeting past concepts.

#2 - Grading: I made the mistake of making homework out of too many points. It really only needs to be worth 3 or 4 points. I had a student ask me why she lost 5 points out of 20 when I only marked one error on her homework. My response was that her error demonstrated that she didn't understand a particular concept very well. She argued that she did lots of other problems correctly, and I agreed with her that she did. However, the point is that she failed to grasp one of four or five concepts for that section (which she agreed didn't make sense), and that her grade reflected that reality. However, there's a psychological difference between 15/20 and 3/4 that she had a hard time getting over.

#3 - Quizzes: I might do the quizzes at the beginning of class and schedule class start time 10 minutes after the start of the class. It's a much more regular routine that way. I might also cut it to one quiz per week.

#4 - I want to buy "The Inner Game of Tennis" and read it so that I can potentially send some of that message to the students on the first day of class. Many of the students have already defeated themselves by coming in with an attitude of failure. I've heard this sports psychology book is very good at discussing how players lose before they even begin. I think I'll jump to Amazon right now to do that because I'm done posting.

I'm glad I'm not a researcher/teacher

I'm finding it very hard to find the time to put my thoughts together in this blog. I meant for it to be a habit of professional reflection, and I guess by "habit" I meant it to be somewhat regular. But reality sinks in and there's just not all that much time. I really should be making more of an effort to carve out time for this, but since I'm already starting to fall behind with my grading (gone all weekend at a wedding), I can expect the remaining semester to continue to be hectic.

Changes that would allow me more time:

1. Different homework grading system
2. Quizzes weekly instead of daily
3. Not having to write lecture notes from scratch (future semesters)

Presentation matters

Next semester, I'm going to start off by placing a much higher emphasis on presentation to my algebra classes. My experiences this quarter indicate that this is actually a foundational part of their math education. It has a number of benefits:

1. It sets the standard of reading and following directions carefully
2. It forces students to think about what they're doing and exposes their lack of thought
3. It makes the papers significantly easier to grade
4. It outlines a style of thinking that encourages logical and organized thought
5. It prepares them for more difficult and more complicated algebraic maneuvers down the line

I've made some adjustments and most of my Math 097 students are on board with the presentation (and I hope they're seeing their current material of adding/subtracting/multiplying polynomials as a clear extension of these things). Jason is considering writing up a document about these algebraic presentation things, and I might help him out if I can.

My first stumbling block

I'm now just about two weeks into the semester, and I've hit that first bump that I've been anticipating. What is the problem? It's complicated, and I think it's the same problem that everyone who teaches developmental/college algebra runs into. The students don't get it. Simply stated, the students are underperforming by a wide margin. Why? I'm not entirely sure. I have some conjectures:

1. Bad teachers: I can probably blame other teachers for teaching sloppy, lazy math to students and probably be right. But that doesn't really help anyone.


2. Bad habits: Consider the following musical analogy -- Students of music who are classically trained from the beginning learn their fundamentals right away and develop the good habits early. Students who sort of pick it up as they go along develop bad habits because they simply do not know better; it's a matter of making it work however it does. Classically trained musicians who stick with the program all come out consistently good, with some very high caliber exceptions. The self-trained musicians are generally not so good, but those with natural talent and musical instinct still come out of it playing exceptionally well. Those musicians who are self-trained and are not doing so well have a difficult time when put into a formalized classical setting because their bad habits prevent them from doing better. It takes considerably more effort to break bad habits than to form new ones from scratch.


3. Bad self-assessment: I asked my students on the first day of class to rate their mathematical ability. Most students rated themselves in the 5-8 range out of 10. However, the work that I see puts them in the 3-6 range or even a little lower. Why do they think they're better than they are? Probably because they don't have a full vision of what the range of mathematical talent is. Most of them have probably only been in classes where they were average or above average. They have probably only seen students as good as a 7, and so they see their talent as about 50%-80% of that (3.5-5.6) and so they are working on an broken scale.


4. Bad expectations: I think they just don't know what is expected of them. Because they don't know what's expected, how can they reach that goal? I think this is where I need to begin. I don't know how just yet, but I'm working on it.

More thoughts about grades

Conventionally, a grade of a C means "average". But have you ever thought about what "average" means? In order to have an average, you must have some population over which you compute this average. What is the average for any given class? Is it the classroom itself? I would argue not, because that implies that even for a classroom full of students who fully understand the material, a certain percentage must receive a D or F. Is it average over the entire population? No, that's not it either, for then almost every student in math will be a C or better student. Is it the historical average knowledge of students in the past who have taken this class? One might make that a theoretical argument, but in practice I have no idea how students did last year.

In my syllabi for this semester, I have given the following description of the letter grades:

A = Highly proficient
B = Proficient
C = competent
D = Minimally competent
F = Not competent

This may seem like a minor distinction to some, but I think it provides an appropriate framework for a lot of pedagogical claims. For example, "Everybody can pass this class." Under the "average" system, there must necessarily be a certain subset of the population that cannot pass. This, of course, does not say that everyone *WILL* pass.

Students as Clients

Link to the article

I just finished reading an extremely insightful article that was referenced in "Enhancing Scholarly Work on Teaching and Learning." The article is titled "Students as Clients in a Professional/Client Relationship" by Jeffrey J. Bailey.

I'm quite pleased that this article exists, as it reflects some of my own perspectives on teaching that I already have and enhancees it by increasing the depth of those views. What is even more exciting is that this article exists in a journal for management education, not a journal for mathematics education. One of the main points of "Enhancing Scholarly Work" is that cross-disciplinary reading in education is both valuable and necessary because there are education issues that extend beyond the boundaries of a particular subject.

There were a few poignant quotes:

The enhanced role of the professional and client in the client metaphor (compared to a sales clerk/customer metaphor) embodies additional rights, responsibilities, and expectations for both professors and students. The client rightfully has expectations that the professional operates within accepted standards and ethical guidelines and will fulfill responsibilities associated with being a member of the profession.

This statement expresses something that I've already felt and even codified in my "Standard Syllabus" in the contract that I have at the end. I feel less odd about the contractual nature of that document now.

As Franz notes, the attainment of physical fitness cannot be given to the client but must be accomplished by the client. A trainer can, however, show a client what to do, encourage him or her, and provide accountability. Similarly, students need to realize the importance of their active involvement in learning. The students must work at learning just as the trainee must exercise (work) to obtain physical fitness.

This helps to put the classroom experience in the right perspective from the students' side. If you go to the gym but don't exercise, do you expect to derive any benefit from the experience?

The professor/student relationship has dimensions to it that parallel the accounting firm/client relationship. If a student is not satisfied with a grade, he or she does not get it changed simply because of the dissatisfaction. Just as the audited client cannot say, “I’m not satisfied with your audit so change some numbers here to make me satisfied,” so too the student is generally bound by the grade the professor has assigned.

I wish I had this article five years ago, when I started as a TA. It would have given me a much clearer explanation to students as to why it is unproductive to argue for a better grade when one it is clearly not deserved.

New Reading Material

I can't quite figure out where I came across this title, but I've been reading "Enhancing Scholarly Work on Teaching and Learning" by Maryellen Weimer and I must say that I'm pretty impressed with what I've seen so far. The primary discussion at hand is how to view the growing body of work that addresses teaching from a scholarly perspective. Without going into a history that will be full of errors, let me simply say that this is an area that is still struggling to find its identity and figure out what standards it should set for itself.

But as I've been reading through this book, I've also been getting my hands on the exemplary articles that she suggests, such as "Confidence in the classroom: Ten maxims for new teachers" and "Helping students understand grades," both of which are interesting reads, and I might have more to say about them later (after I have had some more time to think about them).

"Opposites" of numbers

I'm taking a little break from writing lecture notes to complain about things that textbooks do that are really stupid. I'm going through the Math 097 textbook, and I have reached the third or fourth time where I just shake my head at the textbook and hope that somehow either the textbook or the author will feel my frustration and it will be changed in the next edition. I suppose I could write a letter to the publisher or the authors, but I'm not going to do that now.

Why does it matter? I have an expectation that the students will read the textbook. As a consequence of this expectation, I am reading the textbook as well to make sure what they read actually makes sense.

As an aside, in what classes do students pay $100+ for texts that they have no expectation of reading? As far as I know, it's only the math classes.

Anyway, something that really bothered me was the discussion of negative numbers. They talk about 1 and -1 being "opposites". This random terminology that math people would never use reminds me of some terminology I ran into in high school with the CPM program. Apparently, somewhere along the line someone felt that a "common denominator" should be called a "fraction buster". If you don't believe me, just Google "fraction buster." Anyway, I suppose the idea is that when solving an equation involving fractions , you can "bust" the fraction by multiplying by the right number... or something like that. Besides the absurdity of nonstandard terminology, it also means that when you add fractions with different denominators that you would have to introduce the "common denominator" as a separate concept, even though it's the exact same thing.

But back to "opposites". The standard terminology for this is "additive inverse" for the simple reason that the relationship between them is additive, namely that if you add them together and you get zero (4 + (-4) = 0), and zero is the special number that doesn't change the result if you add it to something else (10 + 0 = 10). Then there's a link between this idea and the "multiplicative inverse" of a number. The relationship between a number and it's multiplicative inverse is that when you multiply them together you get one (4 * (1/4) = 1), and one is the special number that doesn't change the result if you multiply it by something else (10 * 1 = 10).

It just got worse as I pushed onward into the following section. Here is a quote from the text:

A symbol like -x, which has a variable, should be read "the opposite of x" or "the additive inverse of x" and not negative x," since to do so suggests that -x represents a negative number.

Furthermore,

As you read mathematics, it is important to verbalize correctly the words and symbols to yourself. Consistently reading the expression -x as "the opposite of x" is a good step in this direction.

If you talk to anyone who does math (this includes chemists, engineers, physicists, and many others), and pointed to -x and told them to read it, they would all say "negative x." So while it would make sense to a student brought up in this way, to the rest of the world, it's nonsense.

But it gets worse. Consider the following sentence "Multiply x by the opposite of x." If given that sentence outside of the context of this subject, I would consider "opposite" to be related to "multiplication" and end up with x * (1/x) instead of x * (-x). The book will likely introduce a different word, "reciprocal" for 1/x. Unfortunately, all this does is introduce more words for students to memorize, and those words are not particularly informative to the nature of the relationship. Why is "opposite" additive and "reciprocal" multiplicative? It just is. Memorize it. (At least "reciprocal" is a common terminology to people who use math...)

I think that's enough of a rant. I need to get back to writing up lecture notes. I won't get into the "Rules for Addition of Real Numbers" table right now...

The data is what it is...

Every now and then, you think something should be pretty obviously true, but it turns out to be false... or at least not entirely true. Consider the following supposition:

Students who take their remedial math class and first college-level math classes in consecutive semesters are more likely to pass the college-level class than those who take a break between the two.

On the face of it, it's seems arguably plausible and pretty simple. And when you look at the numbers, it turns out to be true. From the data available, we found that 52% of students pass when there is a break in between and 62% of students pass when there isn't. This is a 10% difference and a relative 19% increase.

However, when you take a closer look at the data, a surprising trend arises. For the weaker students (B- to C- students), the pass rate is approximately the same regardless of whether they took the classes sequentially! The data set got a bit small for this part of the analysis, with less than 100 students, but the signs point to something lurking underneath the surface that has not yet been identified.

The mystery to understand why this is the case begins...

Looking ahead

I went to my first meeting yesterday. The discussion addressed something that I was not really aware of as an "outsider" to the department. One of the major problems of the camus is that students are not exiting with as high a level of basic mathematical competency as they would like. I didn't say much during the meeting, but there were a few things which seemed very reasonable approaches to the problem

• No gaps between math classes: Many students (for whatever reason) choose to take several semesters off between their math classes. Intuitively, this seems entirely counter-productive because math, like any other skill/knowledge, gets lost with lack of use. It seems unclear whether it is possible to mandate students to take the courses in consecutive semesters, but it is something to be pushed strongly.
  


• Tracking student data: Apparently, the Nevada System of Higher Education is using a very very old program to track their students (an upgrade is in the process, but it's a number of years off before completion). This means that while there is lots of anecdotal information to support various positions, it is hard to produce the hard evidence required to back it up. Apparently, some of the faculty figured out how to access the information and get the data they want out of it, so there's an effort to get this information and process it so see if the data supports the claims.
  


• Online class limitations: Apparently, there is an effort to make a lot of classes available online. There's even an effort to have entirely online degrees. I have some misgivings about online degrees (quality control, making sure that the name on the application actually corresponds to the person doing the work), but I can see the value of it (in principle). However, the degrees that some people are trying to push include things like physics, biology, and chemistry, which is absolutely absurd. These lab component of the physical sciences is such a large part of the degree that I cannot imagine that the degree would be given much credibility if it were earned entirely online!
  

I guess if I had thought about it for a while that I could have surmised that there would be conversations of this sort in a growing campus. But actually sitting in on such a meeting and hearing the discussion really drives home the feeling of urgency to "get it right" and to provide a quality product for the students, and to put students in the best possible position to succeed. (Of course, success rests on their shoulders, not ours -- if they don't put in the work, they shouldn't earn the degree.)

First class.... sort of.

I filled in for Jason yesterday in his precalculus class. I didn't do much lecturing, but instead spent the whole time answering questions. As a result, it felt a whole lot like teaching section all over again.

I didn't do anything different from teaching sections back at UCSD. I asked them what question from the homework they had problems with, and then I proceeded to explain the concept and solution to them. However, I did notice a distinctive "desire" to learn, mostly from the students who seemed to be non-traditional (I originally typed "older", but I feel that it might be received as derogative to some).

There was one student in particular who struggled with something involving absolute values, but could not find the right question to ask to solve her difficulty. I had to ask her a couple times to try to rephrase her question because I couldn't figure out what was tripping her up. But she politely and patiently persisted in trying to wrap her mind around it until she was finally able to put it together. I've seen that sort of thing happen before at UCSD, but I also remember the student simply say "never mind" after struggling with it for a couple minutes.

I was a bit amused by her response after she put it all together: "Thanks for humoring me." That's a new one. I hope to see more of this from my evening classes this Fall.

Learning Styles

I recently got my login information for the Blackboard Learning System at NSC and have been playing around with the different links that I found there. One of the links is the Index of Learn Styles Questionnaire. I thought it would be interesting to take the quiz and see what the results say about me.

Results for: AW

      ACT                              X                    REF
          11  9   7   5   3   1   1   3   5   7   9   11
                             <-- -->

     SEN                      X                            INT
          11  9   7   5   3   1   1   3   5   7   9   11
                             <-- -->

     VIS          X                                        VRB
          11  9   7   5   3   1   1   3   5   7   9   11
                             <-- -->

     SEQ      X                                            GLO
          11  9   7   5   3   1   1   3   5   7   9   11
                             <-- -->

The test took only about 5 minutes to complete. It's 44 questions long and I probably spent about 5-7 seconds thinking about each one. How do you read the results? You can read about the four categories on the Learning Styles and Strategies page.

I'm not very surprised by the first one. I tend to learn well in all sorts of settings and I don't really believe I'm strongly biased one way or the other. The second is also a balance that doesn't surprise me, either.

The last two are clearly dramatically skewed. I had initially thought that I would be balanced between visual and verbal, since I tend to be able to hold a lot of auditory information in my head, but after some reflecting I found that the questionnaire is probably right. Memorizing information is not the same as learning. So while I can hold audio in short term memory, it's not information that is being processed in any way. When students ask me questions in section, I tend to have to write down what they say and stare at it before I can see what they're asking.

Seeing that I'm a sequential learner is unsurprising. I've always been better at the "nuts and bolts" operations than trying to see the everything at once. I think this is part of why students tend to like my teaching style: I often break things down into small steps (or sometimes decision trees).

I'm greatly amused by this sort of thing. Perhaps this is an indication that I should really start looking into Math Ed research or at least SoTL research as a scholarly pursuit.

Spring 2007 Math 20D Evaluation Results

If you've looked at my poker blog, then you would know that I like record-keeping and data. I've compiled the results from my TA Evaulations and put them in a this document for future reference.

What must I get on the final to earn an A?

I bumped into a professor in the math department the other day on the way to the math department party and during the course of the conversation he shared the following little anecdote that I think is worth sharing:

Why is it that students ask, "How well must I do on the final to earn ...?" If you them that they must get 80% on the final to earn a B, are they somehow more capable of getting an 80%?

Non-rhetorically, why do students ask this absurd question? I don't know, but I think it's all an illusion and I dislike it. Instead of the "do the best you can" mentality I hope for students to have, the exam is seen as a game of grades. Yes, the assessment leads to a grade, but the grade is the afterthought to the educational value of the content being taught. The right question is "What can I learn?" not "What must I do?"

Thinking about this question has made me contemplate the type of response I should have to students who ask that question. I have gone through a few which I've listed here along with some commentary:

• "Your best chance of getting the best possible grade is to get a perfect score." - I'm not happy with this because it brings the grade into focus, and not the act of learning.
• "Just do the best you can." - While this is better, I don't think it demonstrates to the students that I care. Instead, this sort of sidesteps the question and doesn't give a helpful response (of course they will do the best they can).
• "Does it matter?" - It's getting closer. A question puts the ball back in their court to try to justify why they care. I'm not quite happy with it because I know the answer. It matters because they care about their grade.
  
• "Would it change anything if I told you?" - This is the best answer I have so far. It puts the responsibility on them to look at their question in a different light. I'm not denying the importance that they have put on their grade (as misplaced as I believe it is), but I am leading them to realize that this question is ultimately unhelpful to them.
  

I won't have to face this question for about 5-6 months, so I'll probably put these thoughts on the back-burner for now.

Evaluations

I've tried to do TA evaluations for every single class. I've tried a few different types, but the majority of them were this form from the UCSD Center for Teaching Development. I think it's a pretty good form, but now that I'm starting to think even more about teaching than I had before, I think it's a mistake to use that form over and over again. It is a good place to start, but I think that evaluations need to be customized to the goals of the class.

After all, in any class you teach, you set out with specific goals in mind. The evaluation should ask whether those goals were accomplished. As I read through the form that I had given to my students over and over again, I felt that I should try to make some adjustments to reflect my personal teaching goals. I wish I had thought to do this more than a couple days before I needed to have them filled out, but I still think I have a decent first level modification. You can download it here.

I haven't actually read their responses because I haven't graded their exams yet. This is something that I do more for myself than for them. As a matter of avoiding the appearance of impropriety, I don't want to give students the chance to perceive any undue influence on their grades as a result of their comments. I doubt this makes any difference in their responses, but I know for a fact that I can wait a week to find out what they said.

I'm just going to go from top to bottom and describe why I'm interested in their responses to various questions.

I borrowed the first couple questions from the CTD form, but I think they're wasted questions. How often students attend lecture is a reflection of the professor and not me. Also, by the end of the quarter, the students who are completing the evaluation are almost uniformly those who least frequently.

1) Aaron is easy to understand (both in speaking and writing).

There seems to be a recurring theme when people find out that I'm going to teach math. "At least you speak English." For some reason, it seems like lots of people have had bad experiences with TAs who struggle with the English language. However, my purpose for asking this is because I know I have a tendency to move really quickly through material. I am consciously aware when I start thinking faster than I can talk and when I start stumbling over my words. I also know my handwriting isn't always neat. It's generally legible, but I still want to keep track of these things as I continue to teach so that I don't become lazy about it. In retrospect, this should have been a 1a) and 1b) type question to isolate the the two parts.

2) Aaron has encouraged student participation.

There's a subtlety here that I think most students will catch. The question is whether I encourage student participation, not how successful I am at it. I've had some classes that are basically unresponsive no matter how I try. However, I think that students can see when you try to reach out to them and when they fail to respond.

3) Aaron has made the class more understandable.

This is an important marker for me. Students can often tell the difference between understanding something (or at least being in an illusion of understanding) and when they are going through rote motions.

4) Aaron has taught me how to think about the problems instead of just giving me the answers.

Most of my section time is filled with giving away answers. 90% of the time is spent doing some part of a homework problem. I want to know if students feel like I'm teaching them instead of just doing homework on my own. A more interesting question might be the following: I learn to think through problems as Aaron presents them instead of just copying the solution from the board.

5) Aaron has been helpful and accessible during office hours and via email.

This is a pretty standard availability question that could have been broken into two parts like #1.

6) Aaron managed section efficiently and effectively.

I believe that this is an area of strength in my teaching. I am very time conscious and I try hard not to go beyond the 50 minute section time. I also try to get to as many questions as possible in the allotted time. In my mind, this is more of a personal check-up to make sure that my perception doesn't deviate too far from the students'.

7) If given the chance, I would choose Aaron as my TA for my next math class.

This is the big picture question. Have I done well enough for them to want me to do it again? This allows the students to boil everything down and think through what they think is important and tell me if I'm meeting those educational needs. When it occurred to me to ask this question, it made me wonder why it wasn't on any of the other forms from the CTD.

I neglected to ask a couple things on this form:

• Aaron treated the students respectfully and ethically. (Taken from my Standard Syllabus)
• What could Aaron have done better? (An open-ended question to allow students space to comment on whatever they want)

I like how the self-designed evaluation feels. I can see some risk in asking questions that simply seek to affirm myself as a teacher, so I do need to find ways to ask questions that can help students to verbalize things they would like to see changed. This is more difficult because it's asking them to see what's missing. I think with more than a couple days' notice, I might be able to come up with questions that do that. The next evaluation won't happen until December, so I hope I don't forget. (That's why I have this blog!)

The Standard Syllabus

Every class deserves its own syllabus, but a lot of the content in a syllabus remains static from one class to the next. I spent much of this afternoon reflecting on the common content and pulling it together into a single document. I call it the "Standard Syllabus" and you can find it by following this link.

There are three components to the syllabus.

The first is a set of pithy comments that address the basic philosophical approach to teaching and assessment.

The second piece is a short essay to help students think about how they actually learn math. Just as problems are broken into two types on the first page (computation and concepts), their main structure for learning is also broken into these two components. Homework builds up the computational skill and classroom time will be spent trying to build up the conceptual framework. I think what I have now is an acceptable form, but it may be changed later.

The third piece is a layout of the expectations that the students should have of me and what they should expect from themselves (it's also what I expect of them). I think it's vital to the students' education that they actually own their own education. They must be responsible for it and I've made that explicit in having the space for them to sign their name. I admit it feels a little contrived, but I will gladly exchange that for being able to make this point absolutely clear to them.

Daily Quizzes

As I've been thinking about my Math 097 class in the Fall, I seem to have broken the class into two components:

1. The development of computational proficiency
2. The understanding of mathematical reasoning

On the first point, I believe it is a reasonable task to have the students complete a 5-10 minute daily quiz to emphasize the importance of being able to compute things correctly in a reasonable amount of time. My immediate guess is that giving a student 7-10 seconds to complete a one or two step algebra problem is a perfectly reasonable amount of time to give them. This means that a 5 minute test will be at least 30 problems long. Does that seem reasonable? I think it does.

I would give this quiz at the end of the class so that they can leave when they finish and not have to sit around and wait. Also, doing it at the end of class instead of before a break means that students who compute more slowly do not have their break time penalized.

I can also see how this can also be an instructive tool. For example:

• 85 + 74 = ???
• (80 + 5) + (70 + 4) = ???
• (80 + 70) + (5 + 4) = ???

It should not be hard to imagine doing this for the distributive property of multiplication over addition and for common errors involving fractions.

By giving problems that are suggestively sequential, I can introduce various aspects of arithmetic that will become relevant to their future algebraic manipulations. It could also be used as a starting point for a discussion for the next class period. I'll definitely have to take a closer look at the structure of the textbook to see how effectively such a scheme could be woven into the material.

The Campus Interview

This is another post preparing for the job panel. I want to focus this one on the campus interviews. I had two interviews, and they were quite different from each other for a number of reasons.

After trying a couple times to organize my thoughts, I think the best way for me to do this is to set it up like a Q&A session. The topics are so varied that it seems that this is probably the most efficient way to do it.

Q: Where did you interview?
A: I interviewed at Simpson University (Christian college in Redding, CA) and Nevada State College (Henderson, NV).

Q: Was there anything distinct about the Christian college?
A: It was structurally very similar to the other interview. The market is less competitive (because there are fewer qualified applicants -- where "qualified" includes agreeing to a statement of faith). They also asked me some theological questions during the interview that I would not have been asked at a secular college. But otherwise, they were essentially the same.

Q: Where does the campus interview fall in the process?
A: Most of the time, you have a phone interview, first. If they like you enough after that then they'll fly you out for a campus interview. The campus interview seems to be the last step before they make their decision. However, I was called by CSU Fresno to schedule a campus interview without having a phone interview, so this is not universal.

Q: What is the structure of the interview?
A: The details will depend on the campus, but there seem to be a number of common components. They will send you an itinerary with all of this information:

• Teaching sample: The teaching sample for both interviews was very different. In one, it was a one hour talk where I was able to teach whatever I wanted and the other one was only 20 minutes long and they told me exactly what they wanted from me.
• The interviewing panel ("Bad Cop"): This is the part where they sit you down at a table with 4-5 other faculty and they ask you a bunch of questions.
• Meeting with Human Resources ("Good Cop"): This is where they tell you about your benefits (if you should get employed). These people have very little to do with whether you get employed, so there's nothing to fear here.
• Meeting with the Dean: This is the "vision-building" part of the interview, where you hear about the goals and the direction of the department.
  
• Campus Tour: It's exactly what you think it is. They take you around to various parts of campus and point out things to you.
• Meal: Most campuses will take you out to lunch or dinner, depending on the time and length of the interview.

Q: How long did it last?
A: One of them lasted all day (9 AM start, and we finished dinner at about 8 PM, but I did have a one hour break before dinner) and the other was just a morning (they picked up from the hotel at 7 AM and we finished before noon).

Q: What was the interview part like?
A: I just remember lots of questions. Some of the things they asked me were my views on how I would teach a certain type of class, what I think is important about teaching, and things like that. If you spend lots of time preparing your teaching statement, you'll likely have asked yourself at least a few of the questions they will ask you. If you can remember, have a cup or bottle of water with you for this part. It serves as a chance to take a quick rest, plus you'll going to be doing lots of talking.

Q: Will there be reimbursements?
A: Yes. You should keep all your receipts for everything on your trip, but you may not need them all (you may get a per diem meal stipend). There will be a form to fill out and send back to them when you're done. If you drive, keep track of the mileage and gas expenses. If you fly, you should be able to include airport parking or shuttle service.

Q: How did you prepare the teaching sample?
A: First, figure out who your audience will be. The full hour talk was given to a number theory class, so I picked a stand-alone number theory topic (continued fractions). The other talk had a designated topic, but was to be given to the interviewing committee plus a few students in the program. Since the school has some emphasis on preparing teachers, I assumed there would be some math education majors in the audience. They won't have as broad of a math background, but they are going to be interested in the presentation. So I planned to ask a few extra questions to the audience than I normally would. Everything else is just a matter of teaching. Be yourself and teach like you would normally teach.

Q: Any bits of random advice?
A: Sure...

• Take full advantage of bathroom breaks. Matthew Horton, a former UCSD math grad, gave me this advice when I happened to cross paths with him at the Meetings in New Orleans. They are watching you all the time and constantly building an impression of you, and you can start to feel a bit worn down by it. If there is time for it, take a minute in the bathroom to sit and relax, even if you don't really have business there.
• Make sure you wear comfortable shoes. You don't know how much walking they will have you do.
• Greet people by name if you can remember, otherwise ask them for their name again when you part company.
• This may be somewhat controversial, but don't overdress. I had a chance to talk with the students at one of my interviews, and I asked them about the other people who interviewed. One of their comments was that the other guy wore a suit, and that made him seem very formal. I wore khaki pants with a shirt and tie to both interviews. But then again, I got a job at a school where there were lots of southern Californians. I don't know if it would have gone over as well if I had interviewed at a school in the Northeast.
  

My Experience on the Teaching Market

This is the third of what will end up being either four or five posts in preparing myself for the "Finding Jobs in Academia" panel (tomorrow). This one is just going to be a simple timeline of events as I applied for jobs. I might come back later to fill in some details.

• Spring Quarter Before Applying for jobs - Started to ask questions
• What are the mechanics of applying for jobs?
• Deadlines: The first applications are due in early November
• Letters of recommendation: Give your letter-writers a full month to write the letter and remind them often. Letters are put on file the faculty support person. Just tell them where to send them and it is done for you at the department's expense.
  
• Teaching statement: See my other post about this one.
  
• Research statement: This is less important for a teaching job, but it must still be passable. Your advisor would be more helpful than me.
  
• What sort of job do I want? (This was easy for me -- I would much rather teach than do research, and if I could avoid mandatory research it would be even better.)
  
• Where am I willing to go? What sorts of situations may influence my decisions? (If I had a choice between a teaching job at a 4-year university in the middle of Arkansas or a teaching job at a community college in California, which one would I take? What if the California job paid less? Other issues to consider include family and dating relationships.)
• Summer Before Applying for Jobs
• Started writing drafts of my teaching statement
• Devoted extra time to my dissertation work so that I would be able to focus on the job application process.
• Continued to reflect on the questions above -- I think it's important to be reflecting on what you're doing (or about to do) at all times. It helps to gain perspective and insight.
• Worked on an informational webpage for prospective employers. I wanted a place where I would have more room to provide information without cluttering up the application itself. I don't know whether this had any benefit, but I feel like it didn't hurt anything.
• Late September - Early October (Beginning of the Fall Quarter)
• Secured four letter writers (I only needed three for most of my applications)
  
• Wrote up a generic cover letter to serve as a starting point for writing cover letters for specific schools
• Looked online for job postings
  
• AMS-EIMS
• Mathjobs
• The Chronicle of Higher Education
• Organized the information in a spreadsheet
• Researched the institutions
  
• Signed up for the Employment Center at the AMS-MAA National Conference (A resource I ended up not using very much... but that's another story)
  
• Late October - Early November (Middle of the Fall Quarter)
  
• Wrote Cover Letters
• Sent out my first batch of applications
• Updated the spreadsheet of job postings
  
• Late November
• Wrote more Cover Letters
• Sent out two more batches of applications
• Followed up on the first batch of applications
• Updated the spreadsheet of job postings
  
• December
• Followed up on the second and third batches of applications
• Wait and pray...
• Early January
• Attended the Meetings
• Semi-participated in the Employment Center
• Attended Scholarship of Teaching and Learning Talks
• Attended MAA mini-course in directing undergraduate research
  
• Got a two phone interviews
• Got a few rejection letters
  
• Late January
• Got a lot of rejection letters
• Prepared for on-campus interviews
• Mostly preparing two teaching talks
• Some time spent researching the institutions
• Early February
• Went for on campus interviews (I'll try to do this in a separate post)
  
• Started to schedule another on campus interview
  
• Late February
• Received an offer from my first choice and took it
• Canceled the third campus interview

I have to admit that I was a bit lucky to get my first choice and to get it relatively early. The interviewing process can go on for a couple months, and new job postings (especially smaller schools) continue to pop up in January and possibly even into February.

Before You Apply...

This post is another set of reflections regarding the job application process in preparation for my participation on the "Finding Jobs in Academia" panel. The focus on this one is addressing the things that you can do to prepare yourself for a good teaching job (and increasing your marketability in such a position) before you start applying to places.

Beginning (as always) with the basic principles, if you are applying for a teaching position then you must show that teaching is important to you. The impression that I have at this time is that places that are looking for teachers are looking for very strong teachers, and that the market is full of them, and the market is getting stronger as more emphasis is being placed on university teaching. Therefore, you must be enter the market as strong teacher and you must be committed to (or at least appear to be committed to) becoming a better teacher if you want to have a good chance at getting one of these jobs. The focus of this entire post is to present things that can demonstrate that you have an active interest in teaching.

First, attend conferences and look for teaching-related talks and mini-courses. Unfortunately, I did not do this until it was too late. I think if I had a less substantial teaching background, I might have needed some extra boosts to my vita to increase my chances of getting a job. I attended the New Orleans Conference this January and attended several talks in the "Scholarship of Teaching and Learning," which is an excellent way to discover and develop your perspective on teaching which is useful when writing your teaching statement. You should also look at the math education talks to try to attend one that discusses pedagogy. Also, UCSD sponsors the MDTP conference every year, which is a short afternoon conference on campus. Guershon Harel has given the main talk in the last couple years and seems likely to continue, and he has some wonderful insights into how students think about math.

Second, look to the Center for Teaching Development for workshops and other opportunities to increase your teaching ability. There are lots of resources there that remain untapped by most graduate students (to their loss). I didn't really pay much attention to the email advertisements sent out by the CTD until after I got involved with them through being a Summer Graduate Teaching Fellow. They have a Preparing Profession Faculty Program which gives you a much better insight into what employers will be looking for you to do. There's also a TA Development Program that enhances your skills as a TA. All of this can go into your vita.

Third, look for chances to teach. I mentioned that I was a Summer Graduate Teaching Fellow, which is a program sponsored through the Center for Teaching Development by the
Department of Academic Affairs. This gave me some experience with writing a syllabus and some of the other administrative duties of a professor that you do not get as a graduate student. Also, check with the Senior TA to see if there are opportunities to teach during the academic year. You can also get a job at one of the many local community colleges. Finally, spend time as a tutor. If you are a freelance tutor, you may not be able to document it as well to make an impressive mark on your vita (but it won't hurt). On the other hand, if you are part of the OASIS program, your experiences are likely to be viewed with higher legitimacy (this last sentence is somewhat speculative, but I believe it's very likely to be true).

Fourth, actively evaluate your own teaching. One of the stated goals of this blog was "to build up a good habit of professional reflection to become better at what I'm being paid to do." You don't necessarily have to do an online journal, but you should have some sort of chronicle of your teaching experience (this is called a "teaching portfolio"). You can ask the Senior TA to help you by observing your sections or helping you find someone who can (alternatively, the CTD has a similar program). Be aware that some colleges will ask you to submit a teaching portfolio as part your application. It will be quite apparent to those colleges if you put one together in a couple weeks as opposed to one that has been in development for well over a year.

Again, the emphasis of this post is to help point out ways that you can demonstrate that you have an active interest in teaching. These are just some ideas. I'm sure there are more out there that I've missed.

Writing a Good Teaching Statement

I was asked by the department if I would be willing to be part of a panel addressing the issue of "Finding Jobs in Academia." In particular, since I got a teaching job they are interested in me giving my perspective (as limited as it may be) on getting a teaching job. I'll have about 10 minutes to talk openly about my experience (same with the other panel members), then the floor will be open for questions. In preparation for the questions, I'm going to spend some time thinking about various parts of the application/interview process and try to make my thoughts more concise.

I've decided to start with the teaching statement because this is probably the most distinctive part of the application. All of the other pieces (CV, research statement, cover letter, transcripts, ...) have a very standard form and do not allow much freedom for unique expression. The freedom you have in writing your teaching statement gives you a chance to stand out from everyone else. However, this same freedom means you don't have boundaries to let you know when you've gotten off track.

Before I even talk about the teaching statement itself, I should point out some of the guiding principles I used when forming my teaching statement:

1. Be yourself - I find that I don't express myself as well when I try to write in a very formal manner. It is both more comfortable and more effective for me to write as if I were speaking. This causes some sentences to run a little too long sometimes, or perhaps the word choice may be somewhat awkward, but those things will get sorted out during the proof-reading process.
2. Balance specific details with pedagogical positions - I think part of a good teaching statement is telling a good story about how you teach. I think it's helpful to talk about specific interactions with students students, topics, or incidents because it highlights something unique to you. However, experiences alone are not sufficient. You should also spend some time to discuss why the story matters. For example, a story that demonstrates good rapport with students is helpful because it shows that the students a more comfortable environment to ask questions. Or alternatively, even though the students were disappointed that you didn't show all the details, you wanted to emphasize the main ideas and deemphasize the algebra.
3. Be honest - I think it's relatively easy to see through people who aren't being honest about themselves. If you are applying for a teaching position and there's something about your teaching statement seems off, you're not likely to get an interview.
   

I was thinking about my teaching statement during the Spring quarter before I applied for jobs (most applications are due in November). Part of this was due to my participation as a Summer Graduate Teaching Fellow, and part of it was due to the fact that I like to think about teaching math more than I like to think about math itself. However, none of my thoughts were written down until sometime during the summer.

As part of preparing to write a teaching statement, I searched the web for advice. There are lots of pages that offer such advice, but some of it is contradictory. For example,

• Do not read any other teaching statements before you write your own. This will prevent you from expressing yourself in a unique way.
• Read lots of other teaching statements for inspiration.

In the end, I started off by jotting down a whole bunch of notes on things that I could potentially talk about and different ways to present myself, then read other teaching statements to see if there were any other good thoughts that I missed. This way, I used both pieces of advice. Do whatever makes you happy. I don't think it matters much either way.

The process of formulating thoughts for your teaching statement begins by asking questions. In fact, it's probably fair to say that a teaching statement answers the question,"What does 'teaching' mean to you?" Of course, such a vague question doesn't offer much guidance, so here are some other questions to prompt your thinking:

• What does it mean for students to "think mathematically"? How do you encourage students to "think mathematically"?
• What is your view on the student-TA relationship? (How should you think of your students/how should they think of you?) What do you do to develop this type of relationship?
• What is your biggest teaching mistake? What did you learn from this mistake?
• What is your biggest pedagogical complaint about being a TA at UCSD? (This could be about the course content or a professor's teaching style... just don't name names.) What pedagogical principles did it break?
  

Notice that these questions come in pairs. One is designed to get you to think about teaching as an abstract process and the other is designed to reflect some real-life experience (again, balancing specific situations with the underlying philosophy). If you only answer one, it's an incomplete thought with respect to your teaching statement.

You can also find questions on webpages that offer advice (see below) and formulate your own by reading comments on your teaching evaluations.

You're not going to fit all of your thoughts on teaching into your teaching statement. So after you've written up a few different responses to these questions (or other ones that you find), read them through carefully and try to determine which one is most representative of your thoughts towards teaching and which one presents you in the strongest light (whatever that means -- this is a personal interpretation). Feel free to give drafts to a few people who know you to see if they think it accurately reflects who you are. Remember that this is the only chance in your application that you have to let your personality show.

Once you figure out what works best with you, the hard work is done. Now it's time for the tedious part: refining your statement. This is where your too-long sentences get hacked up and your word choice is scrutinized for clarity. Ask some friends to proofread your statement. Consider their comments carefully, make some adjustments, and then ask some more friends to proofread it. Do this until you get sick of it. Do it once more, then you're done.

---

There are lots of links for more advice on writing teaching statements. Here are a few:

• http://www.ams.org/notices/200609/fea-lewis.pdf
• http://www.ams.org/employment/academic-job-search.html
• http://www.math.lsu.edu/gradfiles/TeachingStatement_Oxley.pdf
• http://galois.math.ucdavis.edu/UsefulGradInfo/HelpfulAdvice/ProfDev/Coppolla_TeachingStatement.pdf

I also want to recommend reading the book "What the Best College Teachers Do" by Ken Bain. There are lots of ideas that you can incorporate into your teaching, which will translate into ideas you can incorporate into your teaching statement. (Don't put the cart before the horse!)

Reworking the Grading System

Chapter 7 of What the Best College Teachers Do talks about evaluating students. While I can't recall any specific passage or thought from the book that got me thinking this way, it did prompt me to think more about grading systems.

What does it mean for a student to have a problem "80% correct"? If I cannot answer this question, then it makes no sense for me have some conclusion that sounds like "therefore, 80% is an B- in my class." This is one of the reasons I am thinking of grading problems out of 5 points (see the last part of this post). I want to have a grading system that is simple enough to be consistent, but diverse enough that there are enough strata to have an accurate gauge on students in the class.

When I look at student scores right now, I see a spreadsheet that only has the final scores on tests. That means in a hypothetical 4 problem exam, I could not tell the difference between an 80-80-80-80 student and a 100-100-100-20 student. I would feel that the 80 student is probably a stronger student than then 100 student because the 80 student has shown that he has an understanding of all the topics, but the 100 student has not developed the same breadth of knowledge. This is also an indication of me not really knowing what it means to be 80% correct. Alternatively, it could also be that the 100 student just made some small error at the beginning of the last problem and that ruined the rest of his otherwise perfect work.

My current thought is data intensive, but sounds like something I might be willing to do with three small classes. It might be beneficial to keep a record of all the grades on the individual problems (homework and exams). This has multiple benefits:

1. I would be able to see overall trends in the class. If there are specific sections that are giving students more difficulty than others, it will be immediately apparent and give me a chance to cover that material more carefully.
2. It can get students away from thinking about their understanding in terms of percentages. I could assign grades based on the relative numbers of points they got on their problems, and not the sum of their scores (this would also require me to use a grading system of 4-5 points for every problem so that it is a consistent measure).
   
3. It may also provide good feedback for students if this was presented to them in a reasonable manner. It would have to be organized in a nice way and not just a list of numbers.
4. It will also provide historical data to recognize (for example) that most students really struggle with this particular topic, this problem is extra tricky, or other observations that could potentially slip through unnoticed.

As always, the devil is in the details because I have no idea at this point how such a system could be presented to a class in a way that would make sense to them. I need to think about this a little bit more.

First Day of Class Quiz

I've been toying around with an idea to overcome some of the hurdles discussed in this post. In particular, I want to change students' perception of what math is really about and the negative perception that they either are not good at math or cannot become good at math (obviously, I'm not talking about all students, since some don't have this problem).

My idea is to open up the class with a quiz. It's not so much a quiz as a self-assessment. I want to get them to start thinking differently from the beginning. My mind is geared right now towards my Elementary and Intermediate Algebra class (Math 097, Section 3). Here are some of the questions I want to ask (besides name, email, and those things):

1. Why are you taking this class?
2. What are the necessary skills required to be good at math?
   
3. What do you hope to gain from taking this class?
   
4. On a scale of 1-5 with 5 being "highly proficient", rate your mathematical ability.

I want to do a last day of class something where I give this back to them and have them evaluate the class. This is still a work in progress.

Stubborn Students

Here is a brief passage from Ken Bain's What the Best College Teachers Do (Chapter 2, opening pages):

In the early 1980s, two physicists at Arizona State University wanted to know whether a typical introductory physics course... changed the way the students thought about motion...

Did the couse change student thinking? Not really... They had memorized formulae and learned to plug the right numbers inoto them, but they did not change their basic conceptions...

[The professors] wanted to probe this disturbing result a little further. They conducted individual interviews with some of the people who continued to reject Newton's perspectives to see if they could dissuade them from their misguided assumptions. During those interviews, they asked the students questions about some elementary motion problems, questions that required them to rely on their theories about motion to predict what would happen in a simple physics experiment. The students made their projections, and then the researchers performed the experiment in front of them so they could see whether they got it right. Obviously, those who relied on inadequate theories about motion had faulty predictions. At that point, the physicists asked the students to explain the discrepancy between their ideas and the experiment.

What they heard astonished them: many of the students still refused to give up their mistaken ideas about motion. Instead, they argued that the experiment they had just witnessed did not exactly apply to the law of motion in question; it was a special case, or it didn't quite fit the mistaken theory or law that they held as true. "As a rule," [the professors] wrote, "students held firm to mistaken beliefs even when confronted with phenomena that contradicted those beliefs." If the researchers pointed out a contradiction or the students recognized one, "they tended at first not to question their own beliefs, but to argue that the observed instance was governed by some other law or principle and the principle they were using applied to a slightly different case." The students performed all kinds of mental gymnastics to avoid confronting and revising the fundamental underlying principles that guided their understanding of the physical universe. Perhaps most disturbing, some of these students had received high grades in the class.

I earned a B.A. in physics as an undergrad at UCSB. (I didn't earn a B.S. because I didn't take the physics labs; they were too time consuming and I wasn't intending to continue in physics.) I was fortunate enough to be in a specialized program (College of Creative Studies) where we had a very good teacher for our lower-division physics classes. He taught us in ways that forced us to think about how we thought about the subject by assigning very difficult problem sets. What was difficult about them? It wasn't only computationally difficult, but he asked us to interpret the meaning of the results. This forced us to reconcile our results with reality and expanded our ability to think physically.

Here are a couple examples of things he would do:

1. In multi-body problems, we would see what would happen to the results as one of the masses becomes infinitely big or small. Did our new results match the expectations? Why or why not?
2. Some problems resulted in two solutions. Did they both have a physical interpretation?

I would like to find ways to do this with math. I don't know how it can be done, but I expect that it can be done. The details will have to wait for a specific class, or even a specific topic or problem, because I think it's impossible to talk about real life in the abstract. It can only be done in the context of things that are actually happening.

The book continues to discuss the views of the successful teachers with respect to the development of knowledge. I'll give the bullet points here and have you buy/read the book for yourself if you want to find out more:

1. Knowledge is constructed, not received
2. Mental models change slowly
3. Questions are crucial
4. Caring is crucial

The biggest hurdle for the students who are not mathematically inclined (most of them) is that they come in with the presumption of what math is and that they aren't very good at it. They don't really focus on the processes, but the end result. Why? This is how they are trained to think about math in school from the beginning. (Read this post for more on this topic.) The biggest hurdle for me as a teacher is trying to turn away the negative light on the subject. I have some ideas, but I won't know if they will work until I get the chance to try.

The importance of thinking

I was up late talking with a friend the other night. He lived with me last year, but spent this year in China teaching English and computer skills in a rural town somewhere. We had a nice conversation about a number of topics, and one in particular was relevant to this blog.

It turns out from our collective experience that most undergraduate students don't know how to think about math at all (ourselves included). This doesn't mean that they are incompetent or stupid, it's just that they have not ever developed the skills of critical thinking and self-reflection.

Before talking about math, let me point out a specific example in real life. Over by the Mandeville center, down where the art people have a space to do their thing, there used to be a painting on the ground (someone painted over it). It was a picture of what I've always supposed was Mary and baby Jesus (woman, child, halo). Next to the figure was painted "Do not push your beliefs on others" (or something like that). I am not sure whether the person who painted that appreciates the irony. As a personal philosophy ("I will not try to force you to believe what I believe"), it works just fine. However, it cannot possibly be given as an instruction to someone else in an internally self-consistent manner ("You should not impose your beliefs on me" -- by making this statement, the speaker is trying to impose HIS beliefs on the listener, thus doing what he says one should not do). A lack of critical analysis leads to nonsense.

The same internal inconsistency exists with bland, naive relativistic statements such as "all religions are the same" (different religions have different claims to truth, and they are obviously incompatible) and "you should accept everybody" (the person doesn't accept you because you don't accept someone, but then that person is himself not accepting of everyone). Most of the people I know who talk that way haven't ever really spent time thinking and reflecting on their system of beliefs -- it's much like how Christians who never study the Bible come to false conclusions about how God "should" behave (really, how God "does" behave).

But what does this have to do with math? I'll grant that philosophical reasoning is a bit stickier than mathematical reasoning. At least with mathematical reasoning, we (the math community) have a set of mostly agreed upon fundamental beliefs (axioms) upon which we build mathematical structures. There also isn't a whole lot of room for "personal interpretation" when it comes down to the formalism of mathematics (there is actually room for personal interpretation when it comes to trying to understand math... but that's a different story).

Where was I? Oh yeah, what does this have to do with math? A lack of critical analysis leads to nonsense. The nature of mathematics for a lot of students is that you find some numbers, or a formula, and plug stuff in, and move some terms around, and get some answer. It doesn't matter where the problem came from and it doesn't matter how silly the answer is. It doesn't matter that it's a word problem computing the terminal velocity of a falling object, and that the final answer they got is "t = 10 minutes." The problems have no intrinsic meaning to the students, so their answers often have no intrinsic meaning.

Here are a few examples of relatively simple mathematical processes that a lot of people know they're supposed to do, but don't actually know why it's the right thing to do (other than they were told to do it that way):

1. Why do we "carry the 1" in addition? (Even more fundamentally, why do we start on the right side when we add? -- This one isn't about being right or wrong)
   
2. Long division: Divide, subtract, multiply, bring down, repeat... What does this process actually do?
   
3. x^n * x^m = x^(n+m) -- If you're not familiar with typing math, x^n is x to the nth power.
   

The lack of understanding of these operations is the result of how math is taught. My memory of learning math in the California public school system goes something like this:

• Kindergarten - Learn to count
• 1st Grade - Learn to add and subtract
• 2nd Grade - Learn to add and subtract in multiple columns
  
• 3rd Grade - Learn to multiply, introduction to fractions
  
• 4th Grade - Learn to multiply in columns and do long division
  
• 5th Grade - I don't even remember what I was supposed to learn...
  

The point here is that math was taught as a process and a bunch of rules that you memorize and reproduce. There were some parts where math's intrinsic value was highlighted. For example, I can remember "clock arithmetic", which was a low-level introduction to modular arithmetic, but it still played out like another set of rules to memorize.

I was fortunate that my school and my teachers were willing to put in extra effort for the bright students. There was a weekly thing in the library for the more proficient math students (I remember learning about exponential growth there by the chessboard and grains of wheat problem).

My fourth grade teacher borrowed books from the library to keep me occupied while she taught arithmetic to the rest of the class (she got me books on finance, but since I didn't have a good conception of money at the time, the real value didn't sink in until much later... for example, if I understood no-interest government student aid loans and all that stuff, I would have realized that when I started my undergraduate studies, I could have taken out a maximal student loan and earned 5% interest in a CD somewhere and make free money off the government. Even at 3.5% APY on a $10,000 student loan, because I was in school for 9 years and the loan would be without interest would have given me an extra $5500 for free. Then taking the $5500 and continuing to earn 5% APY on it from age 27 to age 65 is an extra $35000 in retirement funds for doing essentially nothing. Of course, I probably could have qualified for more at the start, and probably could have gotten better than 5% return on average (I think the average market growth rate is around 8%), but I think this makes my point about the gap between being able to compute things and being able to critically analyze what the math is saying.

Who is to blame for the inability of students to think critically about their math? It's not the fault of any individual, but it's a faulty system. Teachers teach math this way because this is basically how they learned it. Students learn math this way because they have no other model for learning math. The students grow up, and this way of thinking is never corrected, so it propagates itself to the next generation.

Fortunately, I do believe change is in the works. There are a number of educators who are looking into the failing mathematics system, and while I'm not involved there, I do hope they find true solutions and not false ones like having teachers teach to standardized testing... that just reinforces the same problems.

MDTP San Diego Conference 2007

MDTP stands for "Mathematics Diagnostic Testing Project". I attended the conference without really knowing (or caring) what MDTP actually does. I went because I was interested in learning about teaching and expanding my ideas about teaching.

The first talk I attended was given by Jeff Rabin (UCSD). I've met him a couple times on an inter-personal level and I took his course mathematical methods course my first year here. He opened by introducing the following problem:

Give an example of two triangles that have 5 congruent parts but are not congruent triangles.

He gave everyone about 10 minutes to work on it. I think I got it at around 7-8 minutes. More important than actually obtaining the answer was the intellectual process that one takes to get from the question to the answer. I can write down an answer and (if you're mathematically inclined) it will take mere seconds for you to conclude that it is a valid example. However, the point was to notice the thought patterns involved in working out a problem like this. There were two important features in play:

1. I did not know the answer ahead of time. I only knew that an answer existed. (Alternatively, the question could be phrased "If possible, give an example..." This would create an interesting situation in a classroom setting with more time for interactions.)
2. I did not know the specific tools required in order to obtain the result. However, the fact that we were talking about incongruent triangles gave enough of a hint as to where to start.

This is the essence of problem-solving. Finding and justifying answers to questions whose answers and method of solution were not previously known to the problem solver. I actually don't remember what else he had to say, but this was already useful information to me. I would like to keep this in mind as I plan problem sets for students in my classes. Of course, there is also the necessity for having exercises (straight-forward computations, questions where the method of solution is actually known in advance -- perhaps given in lecture at some point).

The plenary session was given by Guershon Harel. I met with him once when I was thinking about post-grad school jobs and ways to perhaps transition into math education instead of researching in pure math. He talked was titled "Thinking in Terms of Ways of Thinking." He talked specifically about DNR-based instruction (Click here to read a short paper on it), I would like to find some way of having math students (especially those not going into math) to evaluate themselves and how they think about things. I do like to emphasize that there are different skills in mathematics, which is something I talked about in the Teaching Statement I used when I was applying for jobs. My ideas were not nearly as refined and technical as his, which is perfectly fine by me.

"I don't know" is a good answer

In the summer of 2006, I had the opportunity to teach Math 10A (calculus for the non-technical students) through a fellowship from UCSD's Center for Teaching Development. You can see the syllabus here. I think the format is a little bit clunky, but I'm not very fluent with HTML and making the boxes that resize themselves properly was a big deal for me.

This post is going to highlight the "About Tests" section, which I've copied below:

Tests measure your ability to demonstrate your understanding of the course material.

I have an unusual stance when it comes to exams. You can earn up to 20% credit for admitting that you don't know what you're doing instead of haphazardly guessing at what you should be doing. I want to discourage the "shotgun" method of test taking; that is, I don't think you deserve credit for writing down a bunch of stuff and hoping that some part of it resembles something that might come close to the right answer. The tests attempt to measure how well you understand the material, not your ability to spew information on your paper. This does not apply to multiple choice questions. On the tests, there will a box to mark if you want to take the credit.

Similarly, you will earn credit on your exams for having a good presentation. While the answer is important, it is also important that you are able to demonstrate how you got to that answer. Math reads left to right, top to bottom, just like in English. (It helps to practice good presentation by doing this on your homeworks!) IfI you have questions about the clarity of your presentation, you are welcome to stop by during office hours and I will help you out.

The 20% credit for not randomly guessing was an idea I came up with as a graduate student while I was lamenting the terrible scribbles students left on their paper when they clearly had no idea what was going on. It frustrated me enough to make me want to give negative points. Of course, that's not an option. I don't think it's good to penalize students in that way.

But instead, I think it's appropriate to award students for academic honesty and integrity by giving them the chance to say "I don't know." In real life, I think "I don't know" is a perfectly legitimate answer, and is often the best one when it's true. Too many times I have seen people (myself included) get trapped in difficult situations because they didn't want to admit that they were not qualified to give an answer on the basis of lack of knowledge or experience.

There are a number of positive aspects to this idea:

1. As mentioned above, it rewards students who are able to give academically honest answers.
2. It encourages students to evaluate the quality of their work, something which seems to be conspicuously absent, especially among students who are less mathematically inclined.
3. It prevents students from being penalized inequitably for that one topic that they never quite understood that happened to be the one that showed up on the test.
4. It makes grading those problems much faster.
   

As I tried to implement this, I discovered a few problems which I will hopefully be able to address and clean up with some more experience:

1. The grading must be done in such a way that 20% is a meaningful enough amount to make it worth while for the students to consider it as an option. Many students felt that their wild guessing would get them more points. I can see a two possible solutions. I can increase the value from 20% to 40%, or I can change the grading so that it is harder to earn 20%. I'll have to experiment and see what happens.
2. Many students don't know how to interact with this option. Their entire academic lives, they have been taught *NOT* to leave questions blank and to always guess something because "perhaps you'll get partial credit." I think students need to be retrained to use this system to their advantage.
3. The grading must be consistent from problem to problem. It cannot be difficult to earn 20% on one problem, then a piece of cake to earn 20% on another one. I think this can be resolved by making all problems worth 5 points. With a narrower grading system, there is less room for fudging around with -1 for this mistake and -2 for that mistake. What is the difference between 14/20 and 15/20 on a particular problem, anyway?
   

I know my son can't read...

Here's a story from one of my high school teachers. I think it demonstrates how some people have a poor sense of what it means to get an education.

Every student must pass a semester of civics in order to graduate from high school. Because everyone must pass this class, it's not particularly hard. Not everyone will get an A in it, but everyone who works at it should be able to get out with a passing grade. There was a particular student who was failing this class. The teacher called up the student's mother to talk to her about what's going on, and to encourage her to encourage him to put in the effort so that he can pass.

Initially, the mother tried to do some negotiating with the teacher, but the teacher would not budge. He refused (on principle) to give a student who was clearly failing the class a passing grade. After a while, the mother got exasperated with the teacher's position and said, "I know my son can't read, but I want him to have a high school diploma."

To me, this is very sad. It's likely that the mother does not have much of an education herself based on this comment. It's likely that her son does not appreciate the education he has been getting because he still can't read, even though he's in high school. (That he got so far in the first place is a sad commentary on the state of education.)

A degree will only have meaning if it allows one to differentiate between those who are qualified and those who are not. It's a system that does draw a very clear line between the "haves" and "have nots," and some people don't like that. I'm not ignorant to the fact that social conditions have an effect on students and the levels of education they are able to attain. I'm in favor of outreach programs for low income students and other things to help them to navigate the educational system (especially higher education). However, "help" should never be turned into
a "giveaway." The students must still demonstrate that they have the knowledge and the skills appropriate for the degree. Otherwise, all you do is treat a symptom without providing any real help to cure the problem.

The Cheating International Student

This is one of the things I don't look forward to about teaching. There are students who put themselves in bad situations and then look to you to bail them out. Here's a story that happened recently in a class I was TAing.

Before I begin, I will say that I don't know the student personally (I don't even know his name) and I don't think there's really any information being put forth here that will compromise any sort of privacy or anything.

This student was suspected of cheating on the first exam. He didn't cheat during the exam, but he went home, changed some answers, then brought it back under the pretense of a regrade. The other TA felt as if something was funny, and made a photocopy of his second exam. Sure enough, he came back for regrades again, and was caught red-handed.

I would be completely oblivious to this if the student had not come to the professor's office to beg (literally) for mercy. I happened to be there to help him grade the final exam. He was an international student and kept asking for "forgiveness" (which is an entirely separate matter). He was clearly distraught by the prospect of being forced to leave and took a physical posture of submission by being on his knees. He cried a lot and kept repeating "forgive me, professor."

I found out later that this student was already on academic probation for poor grades, and that he was not doing well in his other classes. Failing this class would mean an automatic dismissal, but so would being caught cheating. The personal side of his story was that his parents worked very hard to send him out here, and there was an implied sense of shame if he were to return home because he was kicked out of school.

What do you do in this situation and what are the guiding principles?

As a Christian, my entire system of beliefs is based on grace and the delicate balance of mercy and justice:

Micah 6:8 -
He has showed you, O man, what is good.
And what does the LORD require of you?
To act justly and to love mercy
and to walk humbly with your God.

How do you act justly while simultaneously demonstrating a love for mercy in this situation? I talked with the other TA for a little bit about this student. He felt bad for the student and didn't want to come down hard on him, because people make bad choices and make mistakes, and they should not be held against them forever (being kicked out of school probably means he's going back to his own country, and he may never get the chance to come back). That sounds good because it sounds like mercy.

But what of justice? This student has already shown that he was borderline because he's already on academic probation. He was clearly caught cheating on an exam (the type of cheating that requires him to lie to the face of his TA). He did the crime, he doesn't come in with a clean record, and it is appropriate for him to receive some sort of punishment for this. That sounds good, because it fits exactly with how you're supposed to respond to a student caught cheating: Report it to the appropriate academic council and let the system do its thing.

The thing that got me the most about this whole thing was that the student never seemed repentant for what he did. All the time that he was saying "forgive me, professor" he was also trying to negotiate some sort of deal. He wanted to receive a D instead of an F (even though we had not even graded his exam, scoring 0 on the second midterm was almost certainly going to make him fail the class). He wanted to do something to avoid getting kicked out. To me, this weighs very heavily on the scales towards taking the hard line of justice.

Grace is undeserved favor, but mercy can depend on actions. People who "turn their lives around" are more deserving of mercy than those who continue to do wrong. In his approach to the professor, this student showed that he was not concerned with the actions that led to his situation, but the consequences of his actions. He was not interested on being on the side of truth, merely avoiding punishment.

In my mind, this student has already failed himself out of school. He didn't have just one bad quarter, or just one class where he was not doing well. He was on the road to not succeeding in school. Even if the professor gave him a D in this class, the student would not be much better off than he was before. He would still be just a borderline student, he is likely to cheat again (he's also likely to have cheated before). Allowing him to get away without recognizing the magnitude of his situation is not likely to benefit him at all. Perhaps he will finish and "earn" a degree. Or perhaps he'll fail out in the next quarter, or the next year. It's hard to say. If you send him away, he may understand that his actions have consequences, and then proceed to earn a degree in his own country (with integrity).

This reminds me of a story from one of my high school teachers, which I will post separately. Click this link to the story.