JMM 2008

The Joint Math Meetings for 2008 (San Diego) just finished, and so it's time to start sorting through my notes to figure out what they mean.

Class related:

• Minute Paper - I've seen this used and discussed many times, but maybe I'll actually give it a try. The basic idea is to give the student one minute at the end of class to write down what he thinks were the main points for the day. This forces the student to reflect on the day's work before it gets lost.
• Algebra for Dummies - This book exists (as well as others that present themselves in the same way), but the question is what they try to do to make the math more understandable (and whether it works). Part of me thinks that anyone motivated enough to buy such a book will be the type of person willing to put in the work to learn, whereas not all of the students in my classes will be like that. I tend to believe that the personal motivation plays a huge role in education. However, it might be worth my time to look at that book to see what it says.
• Math Labs - It would be nice if I could get students to do self-directed labs (like science labs) by giving them a handout with some instructions to follow and some mathematical things to compute. The problem for pulling this off right now is that I have no idea what topics would be good to pursue in this way. One example for a higher level class (like number theory) is the "McNugget Problem" (boxes of 6, 9, and 20 -- for what n can you get exactly n nuggets?)
• Technology - I've always kept my distance from using technology in the classroom because the students won't be able to use them on tests or anything. However, I can see some time-related shortcuts with graphing where it would be nice to be able to generate several graphs quickly and have students make observations to get the ideas behind the graphs before we actually go through the details. I need to find an internet resource that will allow this (Sage?)
• Pretests - I need to give my Algebra students a pretest so that they can get a better measure of "progress" throughout the quarter.
• Handouts - I haven't made much use of handouts, but maybe I should go back to that. I made these my first year or two as a TA and the response was strongly positive.
• Hiding grades - I went to talk where someone did a study on student improvement when you didn't tell them their grades, but only made comments. I don't know if this works at the level of developmental algebra because the students may not be mature enough (mathematically speaking) to make sense of it.
• Grading - I had an interesting thought about how I can grade my students. I still don't really like the idea of percentage grading. So perhaps I can make competence grading by passing a series of "Levels." For example, a level 1 arithmetic computation would be something like 45 + 24. Then a level 2 arithmetic computation would be something like 6 - 4 * 5^2 (introducing PEMDAS). Then a level 3 arithmetic computation would be something like 5 * 2^2 / 4 + 3 * 8 (a complicated string of PEMDAS where the only way they would get it right is if they knew how to completely break it down). There would be similar levels for other ideas, such as solving linear equations, graphing, and so forth. Then their final grade would be a measure of how many topics for which they were able to show a high enough level of competence. This still needs to be worked out in greater detail.
• Colors - Maybe I can use black/red for positive negative numbers at the beginning of class to highlight the difference between the minus sign as a binary operation (5 - 2) and as a unitary(?) operation (-2)
• Spoken/written mathematics - This wasn't from the conference, but it was something I've been thinking about. I think I need to make my students write "five minus two" and "negative two" because their words and their written math often say different things (and sometimes the ideas in their heads are different from both of those!).
  

Other ideas:

• Placement Exams - I want to look over the placement exams to see what they are testing and how they are graded.
• NSHS Math Students - I don't know who the bright high school math students are, but we should probably be actively looking for them and encouraging them in some way. We could try to get students to take the AHSME or something like that.
• Other math students in the area - Can we make a presentation to math clubs and that sort of thing at other schools? Will this be a helpful endeavor for advertising? (It would help if we had a math major to offer them!)
• Putnam - There was an interesting-looking book titled "Putnam and Beyond" that I might want to buy
• Minicourse - I went to a minicourse on Departmental Self Reviews. At this point, I'm going to treat it as background information for me to have as I start looking forward into where the department is going. But there is one thing that I thought would be helpful, which is to get a list of who taught which classes for the past few years to see what the teaching distribution has been and learn more about our part time instructors.
• Articles - "The Way We Think" (Fanconnier? and ??), "Where Mathematics Comes From" (Lakoff and Nunez)
  

Miscellaneous:

• Webpage - I need to update my CV and webpage
• LaTex - "More Math to LaTeX" looked like a good reference. Also, I want to see if I can learn how to make hyperlinks in LaTeX.
• Mathematics of Poker Class - Given the current budget situation, this probably won't happen. But it would be fun if it did.

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.

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.

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.)

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.

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!)

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.

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!)

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.

"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.