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.

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.

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.

"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?