Spring 2009 Reflections

I see that I didn't make any comments about Fall 2008. The only thing I would have to say there is that I decided that I didn't really like the Math 093 textbook.

Math 093 - I'm probably about 30-40% done with a textbook/workbook for this class. My primary concern is that the content is not well organized in the current textbook, and that it does not emphasize the points that will help students succeed in Math 097. One of the frustrating things about this textbook is that I feel like I'm giving the students something other than what they really need, and consequently propagating a systemic problem with mathematics education out here. I hope to have this textbook completed in the next four weeks so that I can get it to print with some help from the bookstore.

Math 330 - Linear algebra is a notoriously difficult class because it's very easy to mess up in one of two extremes: Too much computation without enough theory or too much theory without enough computation. I don't know how well I did on this balance. The students came to the class underprepared as a majority of the students are ARL students who don't necessarily have a good mathematics background. In particular, the students really lacked several basic notions of proof and had some problems with basic manipulations at the beginning of class that I wasn't expecting to see.

Math 466 - Numerical analysis is a class that I never took as a student, so I was sort of making it up as I went. The book was pretty good, and I put the emphasis more on the computational methods than the theory. The problem (again) is that the students' background is not as strong as it could be. Their calculus and general proof-writing backgrounds were limited, so the emphasis on methods made sense to me. I found the material to be somewhat interesting, though the derivations were somewhat tedious.

Math 093 Boot Camp

I have made it my goal to put together a functional textbook for Math 093 and have it ready for the Fall 2009 semester. I'm quite disappointed with our current book. It's poorly organized and not well presented. It also does not have nearly as many problems as there should be. I ran a draft outline past Russel today, and he likes what I've put together. Now I need to think through the layout and come to a decision about how I'm going to teach presentation through examples.

The plan is to start by developing a workbook. I'm not going to spend a lot of time working on writing up explanations right now because that's not as important as laying out the problems in the right order, having the right number of them, and other such issues.

If I get far enough in the next month, I might even be able to pilot some of the worksheets in my Math 093 classes next semester. The first two chapters (arithmetic and fractions) are likely going to be the most difficult ones to put together.

To do for/before Spring 2009

1) Presentation matters for Math 093
2) Update images for Interactive Technologies
3) Start work on n-hedral group paper
4) Start outlining a new Math 093 text
5) Timed arithmetic spreadsheet
6) Update homework page template to instruct students to look down if I didn't copy the next assignment to the top
7) Update webpage information

JD Smith Middle School

I had the opportunity over the last month to work with some math teachers at JD Smith Middle School. This sort of fell on my lap at the last minute, but it seemed like a good chance for me to try something new and expand my perspective on education in the region.

During the past few weeks, I visited classrooms and made observations of the individual teachers. I then compiled a short list of topics that I felt were relevant to department as a whole, and then presented my findings to them. I also wrote up a short summary of my thoughts on the individual teachers.

My first point was about the careful use of spoken and written mathematics in the classroom. The teachers were using "solve" as a universal instruction for "do what you're supposed to do." This is fine when the students already know what to do, but the ones who don't know get confused because this one word means many different things in different contexts. I suggested to them that there is a clear distinction between solve, compute, simplify, and other word they might use in the future like factor, reduce, expand, and so forth.

The second topic was to look at arithmetic. Arithmetic is really an exercise in bookkeeping, not anything deep or profound (except the fact that we *can* do bookkeeping, which is truly deep and profound). So I talked about doing addition as "big-to-small" instead of "small-to-big" and explained why it works better once students understand the basics of the place value system.

The third topics was more exploratory, which was the topic of fractions. This is probably the most difficult topic because of the breadth of pictures we use to discuss them. Unfortunately, these pictures are not all compatible with each other. This leads to students being confused because they think about the same picture in different ways. I didn't have a lot to say in this area, but opened the door for them to discuss it among themselves. I hope they are able to go somewhere with it.

Overall, I think it is a net positive given the amount of time spent with them. I intentionally stayed away from pedagogy specific ideas because I'm not a pedagogist. I'm just a pure mathematician, but I think this is a helpful type of "outsider" perspective for them. I'm done with that project for now, but I left the door open to come back and talk to them about a different topic if they choose to invite me back.

Instant Gratification

Henderson College's Graduation Rate Disappointing

I found this to a surprisingly negative and pointless article. I accept that the numbers are bad and need to be better, but the article's shading is clearly tilted against the institution and puts it in an unfairly negative light.

1) "If no more finish over summer, Nevada’s newest public college will report a six-year graduation rate of just less than 16 percent — one-third of what California’s public state colleges achieve." -- This is comparing completely different types of institutions. In CA, it's a number taken (presumably) as an average over many established institutions. Here, we have a small, growing, start-up institution in a state with already very low academic standards.

2) "Some NSC students discover they are interested in majors the college does not offer, Stewart said. Others leave Nevada or transfer to UNLV. “ ... But NSC was not meant to act as a community college, preparing students to pursue an education elsewhere. A 2001 report supporting establishment of the new school said Nevada needed a state college to produce more college graduates." -- If anyone thought they would have a fully-functioning college offering the full selection of degree programs as a 10,000 student campus within 6 years, they were poorly mistaken. We're happy to finally have our first new building up, with plans to move in shortly.

3) Though only 10 students from NSC’s first full-time freshman class had graduated from the college as of spring, the school has conferred 586 degrees since its inception, with many going to transfer students. -- Thanks for putting this down in the 4th to last paragraph. It really helps to lead with "just 10 had graduated from the institution" and close with "by the way, there are almost 600 degrees that have been conferred."

4) "The California State University system graduates more than 45 percent of its freshmen within six years, a higher percentage than UNLV. The system’s newest campus, at Channel Islands, began accepting freshmen in 2003 and graduated 25 percent of them within four years."

Being from CA, I would guess that 90% of those students are coming straight out of high school and into college, and they are almost all full time students, as this is how most Californians do college. This is likely an unfit comparison as Nevada students are returning from perhaps several years of work and are in need of remediation.

Spring 2008 reflections

I don't think I put thoughts in this blog nearly as often as I did in the first semester, but now that the semester is over, I have the time and energy to put a lot of thought into how things went this semester.

First of all, Math 124 went a lot better this time around. I still don't think I connect quite as well with that class as I do with the Math 097 students. It is true that I go into the classes with different mentalities. Since Math 124 is a college level class, I treat them more like how I think college math students should be treated. I do less hand-holding and I let the students work on their own more.

Math 097 went very well this time around. I think I found the right level at which to try to meet the students and the introduction of more worksheets seems to have had the effect of helping the students to understand the material.

The homework cover sheet was a pretty good addition. Some students didn't really take to them, but I'm going to keep using them anyway. I changed them slightly so that they interact with the course content more, so that should be a positive change. I also need to write a more defined homework policy and let the students know much more clearly what I am really expecting from them in the homework.

There were a number of interesting comments that I received on the unofficial evaluations I handed out to the students, and even though the probability of them actually seeing them here is very small, I believe it is a useful exercise to reflect on them, and then to take my thoughts from here and put them into the next syllabus and work them into the class content. (Side comment: I'm always amused by the level contradiction from different students about how things have gone in a class.)

"I do not think quizzes you be a part of attendence." The same student also wrote "I like participation credit for [the] test." This is the type of comment that I feel deserve little attention. I know that Jason gives credit for attendence (he also has 'detention' where he forces students to show up to the tutoring center for specified periods of time to make up for missed classes), but it seems like extra paperwork to me. This is a college level course in which the emphasis is on learning content. What part of that description implies that credit should be given for simply showing up? It's very much like a work environment. Do you expect to get paid simply because you show up for work, and not because you actually perform your job competently?

"Quizzes should be at the beginning of class." Over the course of the last year, I have started to agree more and more with this, and will probably implement it during the summer session. This will also be used to make sure class actually starts close to 'on time' and only punish the students who show up late to class.

"Too much homework." No student actually used that phrase this time around, but a couple students commented on it. Students who say this are usually the ones who aren't very good at math, and they often spend far too much time on their homework because they spend all their time being lost and making zero progress. These also tend to be the students who don't come to office hours or ask questions to get help. Even though students will probably make this comment as long as I'm teaching math, unless the response is overwhelmingly stating that the assignments are too long, I will tend not to give much weight to these comments. (There was another comment from another student: "The amount actually helped me to learn the material." That's the whole point of the homework!)

I had a spot on the evaluation where I ask the students to write themselves a short note to themselves at the beginning of the semester. I find this quite amusing and I will definitely have to talk about this on the first day of class for the upcoming semesters. I added my own comments in parentheses:

• "Do your homework." (This was the most frequent comment)
• "Stop slacking!" (Do I need to say more?)
• "Do not miss any classes, you will get behind."
  
• "Be prepared for the quizzes." (If you spend the 5 minutes before class glancing over the homework, the quizzes will be significantly easier because the ideas will all be fresh in your head.)
• "Don't take this class if you are taking other hard classes." (Straight-forward, reasonable advice. Math classes take up a lot of time if you're not as natural with it, and this is worth considering at the start of the semester.)

Math 373 was a fun class this semester. With such a small class, I handled it very differently. I left a significant chunk of time in class to doing numerical explorations as these are really key in number theory. A lot of the results came out of 'playing' with the numbers, so that's how I tried to approach it.

90% Failure

Here are a couple articles from the local paper:

Math Tests Carry Shock Factor

Preliminary Math Test Failure Rates

Basically, 90% of the high school students are failing middle school algebra. There seems to be some controversy about how those tests were administered, but 90% is simply too large to be merely a statistical outlier. As a department, we agree that this is pathetic and that we want to do something about it, but we have no plans yet (we also have no funds -- but there's some work being done on that side). Our plan is to address this issue at the middle school level because we think high school is too late. Elementary school level would be better for getting students interested in math in general, but we don't think it will have as much of a lasting effect. I think we're going to target some middle school teachers, but this has to be thought out more carefully.

NSC on TV

I hope I got the link right...

NSC 5 Year Anniversary

You're not doing math

I had an interesting conversation with a student after class today. The essence of the conversation was that he did not receive full credit on a problem on his exam even though he arrived at the right answer. My reason for not giving him full credit was that he did not demonstrate how he arrived at his answer and that the work that he showed did not make sense.

The problem gives some relationships between various angles of a triangle and asks the students to compute the angles. This student's first step was to divide 180 by 3 because "there are three sides on a triangle." Now, it happens to turn out that one of the angles of the triangle is 60 degrees, but the fact that 180/3 = 60 has no bearing on this. Nevertheless, the student continued to insist that this makes perfect sense.

Upon further attempts to argue with me, he reached the point where he declared that I'm not giving him credit because he "did not do it [my] way." To this, my response at the time was that he did not receive credit because he did not demonstrate how his calculations give him the correct answer.

Afterwards, I decided that I had the option of taking the much more confrontational route, which is the "what makes you an expert in math?" route. Am I not the teacher in the class, the one who is being paid to determine what is and what is not math? Did I not spend 9 years of my life studying math? And yet he believes what he is doing is, in fact, math. And because he thinks it's math, I should therefore defer to his understanding of math? How absurd. Does this happen in other fields? Do physics students try to convince their physics professors that they know physics better? That doesn't make sense (but I'm perfectly willing to believe it happens).

Whether this second option is any better doesn't seem like a good debate. Obviously, it's more edifying to me to simply pull rank to shut him down, but it does little for the student besides pulling further into the grave he has dug himself by being unteachable. My basic conclusion is that I really don't care what this student thinks. He's wrong and he's unwilling to admit it. So how can you work with such a student?

I offered him a challenge: I changed the numbers slightly and challenged him to show me how dividing 180 by 3 is a correct first step towards arriving at the answer. If he can do that he will get back his two points on the test. I changed the numbers so that the answer will end up with fractions, so he's not going to accidentally stumble across the right answer. I gave him until next class to come up with something, so it should be interesting to see what he says. Part of me thinks that he's simply going to walk away from the class, which is perfectly fine by me.

There's a principle in play here that is part of Gershon Harel's formulation of learning mathematics. It is the "necessity principle" (if I'm remembering right). A mathematical concept will be accepted as true by a student until he discovers that it is necessarily false. Necessarily false means that the two ideas are inherently contradictory and so one must eliminate one of them in order to maintain a logically coherent understanding. The purpose of the challenge is to force the student into a position where he is unable to solve a problem using his random techniques. Until he realizes that his toolbox is insufficient for the task at hand, he will never look to new tools. So now I just sit back and wait to see what he comes up with.

Money issues

This is a news clip about the impending budget issues. This is one of those things that I didn't ever really think I would have to think about, but here it comes anyway:

Link to the clip

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.

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.