Student Feedback

Since I tried a few silly things with my class this semester, I wrote a couple of extra questions on the end-of-the-semester evaluations to get student feedback on them. In particular, I set up homework so that students were to choose which problems to do on an individual basis, and I orchestrated a class project to write our own textbook. Overall, student feedback was not as negative as I had anticipated, which was nice. Of course, only 19 of 39 students completed the evaluation...

Homework feedback was the least negative of the two. Of the 19 responses, 11 said they liked it. Others seemed to think it was ok, but didn't provide enough structure (not unexpectedly). A handful thought it would have been better if they were given some direction of things to focus on (I still don't see, for our class, how there was that much to choose from). One student apparently initially thought they would like the setup, but ended up not. A few flat-out didn't like it. Alas. I still like the basic idea.

The project was a source of frustration for all involved, which is too bad.

I'll begin with some of their feedback. A few students noted that they liked the idea of the project (and the project itself), working together to make something other students might use. And a few said they learned some things (mostly whichever mini project they were on), even if it wasn't related to the course content as listed on the syllabus. Some students found it tedious, and busy work, and would have rathered just do more homework problems (this confuses me, because assigning problems seems like busy work to me). A couple think in-class time to work with groups would have been good, which I can definitely see. And one student's comment, about wanting to see all the work at the end, makes me think another part could be added to the assignment, in which students would present their mini project work at the end of the semester - I think this would be valuable (even if the work is all accessible on the wiki anyway).

And now, my perspective. First off, there was no point in letting this project exist for half of the semester, as almost all of the work seemed to get done on due dates, right up to the minute. Apparently about two weeks probably could have produced the same result.

One of the parts I thought would be best was the "write questions" and then "write answers to classmates questions" parts. This seemed straight-forward enough to me, and the mostly clearly useful for our course content. Questions were due in two blocks, on a couple of Friday evenings. I'd then spend the evening (my life is that exciting!) formatting things so it all looked somewhat uniform, and randomly assigning problems to students. Due dates were such that after a block of questions were written, the answers were due two weeks later. I wanted to have a quick turn-around time from problems being written to problems being assigned, so students could have more time working on well-written answers, as well as working out issues with the questions (poor wording, ambiguity, problems in the wrong sections, etc). Of course, by my previous comment, I really could have spared more time before assigning answers. And I should have, using the time to edit the questions for clarity and content, since apparently students writing answers mostly didn't feel the need to do so (despite it being written as part of the assignment). I'm also concerned that I should have tried harder to see if questions (and, later, answers) were copied from the textbook, early on in the process. And for some reason I expected all of my students to follow the guidelines about how many of which questions (difficulty/content) to write (and write them on time), so that I when I redistributed them, everybody would end up writing the same number of answers. Silly me.

Clearly that part of the project would need to be tweaked (completely dismantled and reassembled) for future use.

The other useful part of the project was the MiniProjects, which students were put into groups for, based on initial project proposals they submitted. This part of the project was intended to involve group meetings about 2-3 weeks before the semester ended so students could show me rough drafts (ideally just little issues remaining). It became clear, the week of these meetings, that only very few of the groups had done anything at all (those that had done things had done well), which was hugely frustrating. Several students came to the meetings hoping I'd tell them what do to do get started, with no ideas of their own. This was weeks after their project proposals and group assignments. Perhaps a few more meetings would have been a good idea.

Anyway, eventually a whole-book pdf was compiled by one of the groups, of all of the sections and pages of questions and answers, and lots of appendices containing the further research found as part of many of the MiniProjects. It comes out to 126 pages, of which I wrote about 35-40. I'd share the book with you, but students at UVA own the intellectual property to their work, and I clearly can't force them to give me their work so I can just post it online for free for anybody, nor would I try. However, I did tell them I'd release all of my base effort under a CC license, and told students that if they wanted to contribute their work similarly, I'd be happy to compile it. I've got signatures from 18 students that I can gather their work up into a bundle, which I fully intend to do in the coming weeks. I'm sure you'll hear more about it, if you're still reading this far into this post.

Textbook Poll Results

Here are results from the first 77 responses to the textbook poll I mentioned recently. Charts are via Google's api, and numbers for the pie charts are percentages. I don't really know who ended up taking this, unfortunately, but my guess is that it is predominantly undergraduates at the University of Virginia.

1. How likely would you be to purchase an optional textbook for a math course, if it were a reasonable price?
   
   
   One of the fill-ins was: "Only if I perceived to be beneficial to my grade."
   
2. Would you be comfortable using a student compiled textbook for a math course if your professor used it as the main textbook for the course?
   
   
3. If you were taking a mathematics course and your professor provided you access to a free, online textbook in addition to your regular textbook, how likely would you be to reference the additional book for extra help and problems?
   
   
   
   With a write-in "Depends on how well I understand the respective material."
   
4. If a free digital copy of a textbook were legally available online, how likely would you be to buy a paper version?
   
   
5. To what extent do you use your textbook in a math course? (multiple answers allowed, counts are number of respondents). The available answers were: "I real all relevant sections", "I skim example problems", "To do extra unassigned problems", "I focus on highlighted formulas", and "Only to do assigned problems".
   
   
   Also a few fill-ins:
   
   • Whenever I find I don't understand a topic I find the book explains it in a very simple understandable way.
   • To try and find lost-related Equations
   • Reference tables and equations in the back of book
   • never
   • Paperweight
   • I do not own a textbook
6. What changes/improvements would you like to see in how textbooks are structured?
   
   Here's a sampling of answers:
   
   • A couple comments about digital versions:
     
     • "Putting textbooks online is hard for a lot of people (including myself!) to read from"
     • "would prefer more kindle/e-reader-friendly books".
   • Some about textbook usage in relation to class:
     
     • "Perhaps if the textbook order followed the order are curriculum is taught."
     • "Force students to use the textbook by alternating between online homework and maybe do quizzes based out of the book."
     • "More similarity between the material taught in class and the material in the textbook and more sample problems."
   • Some miscellaneous comments:
     
   • "Less space on corny math jokes. More space on actual math."
   • "More colorful; Black and white only is hard to look at and discourages you from using the book"
   • Most of the responses were about examples and solutions:
   • More examples
   • More and more-detailed examples of problems; detailed answers in the back of the book
   • More concrete examples and better solutions to the problems
   • To have more examples problems where the problem is worked out step by step and explained
   • clearer example problem presentation, clearer explanation of concepts
   • In the example section of a chapter, I would like to see questions that are actually challenging, something that will actually be on the test. The textbook companies always provide the most basic examples, which most of the time, are not helpful for actual application later.
   • When working out sample problems in each section split the problem into two sides. The left side would have the actual mathematical processes with each individual step shown. The right side would have the processes explained in words, not just symbols, which would make each step more understandable.
   • more steps to the answers
   • i would like to see example problems with full explanations of EACH step as well as a break down of definitions into simpler terms.
   • Solutions that show the work
   • More examples of how to solve hard problems, instead of just the basics.  If the point is to learn the material, why just assign really difficult applications of the principles as problems when you could just as easily show us how to solve them in the body of the text?
   • Many of the problems in the textbook seem rather easy compared to the problems on WebWork or questions asked on the test. It would be nice to have more problems in the textbook that would be comparable in difficulty to the UVA level of calculus.
   • Answers to every problem not just odds.
   • I think they are good but textbooks ought to have answers for all the problems - our textbook has answers for just the odd problems.
   • More examples would be nice, because for many people that is the best way to learn - by examining problems and understanding why they are done a particular way.  Also, better explanations of each step in given examples.
   • Make more sense with it. Show answers for EVERY PROBLEM and show how the answer is SOLVED. Don't just use one example and then hope that the people can figure out how to do the rest of the assigned problems. Show all the steps involved to do each problem. Make an ""answer"" textbook instead of putting the answers in the back, so that way people can have the assigned problems AND the answer textbook open so they can understand the steps required to solve the problems. It may sound like people are just going to copy the work, and maybe they will, but the answers will show a thorough way of how to solve the problems, which will help people do much better on quizzes and tests overall.
7. Do you have any additional comments or feedback?
   
   A sampling:
   
   • "The text book we have no has not helped me at all - my teacher explains everything we need to know, so the only use it has is practice problems. In the future I would suggest saving students thousands of dollars and not making a textbook mandatory"
   • "Try not to use all the example problems from the text book, it takes away a student's resource if the lectures are not helping."
   • "I haven't opened my textbook at all this year."
   • There were also two comments about how textbooks are too expensive.
   
   There was also a fairly lengthy response expounding on the virtues of e-books and the ePub format in particular. It mentioned how videos right in the textbook "could be a cool new way of learning mathematics." And apparently this student loves his or her iPad:
   

   iPad and tablets will run rampant in education/academia in the next 5 years. My iPad has transformed the way I take notes, do my reading for class, organize my life, and access my material. I have never been so prepared for class in my life, I don't necessarily do anything beyond what is assigned, I just now have the time to do it on-the-go, interactively, and efficiently. It's one of those cases where I am working smarter, not harder. And I owe it all to the iPad streamlining my academic life. Textbooks are going paperless and nearly half of mine are (all on my iPad).

Perhaps the thing that struck me the most about these results were all of the comments about examples. Many students seem to want answers to all of the problems in the book, not just the odds, and also want more detailed solutions. I can certainly see the appeal of this to students, and how it might seem helpful. However, I'm not sure that it actually would be helpful. Unfortunate as it may be, struggling through problems yourself seems to me to be a better way to really learn the content, instead of relying on textbook solutions. It's too easy to get stuck, look in the back, see what was done, and then think you understand what is going on. It's the same problem one of my students mentioned this semester: students may think they understand what's going on when they are listening to me talk about examples, but then feel stumped when they get to doing problems by themselves. Certainly I'm happy to hear that things make sense when I talk about them. But if students think they understand what's going because they can follow lectures and examples in the book, they may be fooling themselves. The only test is to struggle through problems without a guide (until you really need it).

A Homework Experiment

At the beginning of the semester I decided to try something new with homework. Basically, students were supposed to do as many problems as they thought necessary to get the understanding of the material that they desired. If a student notices that there are essentially one or two types of problems in a section (as is frequently the case with our textbook), they probably don't need to do many problems. Other students might require more practice to feel comfortable with material. I gave them the responsibility of deciding for themselves how much to do.

Each week, students turned in a sheet saying which problems they did, how they would grade their own understanding, what major and minor issues they had, any questions they had, and a "well-written" solution to one problem. Grading has been pretty easy, and I've basically just graded on completion. It's been nice to get questions on homework (and respond with individual answers), and to see students evaluating their own mistakes. Quality of "well-written" solutions varied a bit, but I did see many good ones.

Our first exam really didn't go so great, even though I hadn't seen many students rate their own understanding on homeworks much below a B. I think this was partially due to this relaxed structure of homework. After the exam I asked the students to tell me (anonymous feedback was fine) if they wanted to see anything change to make the class better for them, and I never heard anything. Our second exam, last week, went a bit better, and I like to think perhaps study habits had improved.

Anyway, part of what I wanted to see, doing homework this way, was if there was a relationship between number of problems done for homework and exam scores. I've been keeping track of how many problems each student did each week (probably miscounting slightly occasionally), and so after this last exam I broke out gnumeric. In the name of privacy (thanks to those in my twitter/facebook network for their thoughts here), I varied each point by some random small perturbation, and have removed axes labels and scales, with the following result:

The x-axis is number of problems done, and the y-axis is the sum of the two exam grades divided by the sum of the two best possible exam grades. It's nice to see that the linear fit has a positive slope, at least. The correlation coefficient for the actual scores was about 0.23, so not so great. What I find slightly interesting is that it just about looks like (and does in the actual scores too) there are 3 clusters... a large collection on the left, another 8 there in the middle, and 3 more at the end who did lots of problems.

I know it wasn't particularly scientific, or rigorous, but that's what I've got. I never told the students I was collecting this data (I'll show this to them in class tomorrow), in hopes that they wouldn't be just making up how much they did, but it's still a possibility. I almost wish I'd kept track of the scores students gave themselves, but it's too late now.

Speaking of class tomorrow, I better go sort out what we're doing...

Class Project

We're making a textbook. Or at least we're going to try.

On Friday I told my class I wanted to do a class project, where we all work together making a textbook. I told them I thought it would be fun and interesting and worthwhile. I told them that in addition to writing problems and solutions, like a textbook should have, they would also get to do something fun, of their own choosing, to make it a good project. I told them they had the weekend to think about it, and that we'd talk about it in class today. Attendance was unexpectedly low today, but the impression I got was that the majority of those in attendance were for the project. So we're going to try it.

The course I'm teaching is "Financial Math", and the textbook that has been used for this class in previous semesters, as decided by somebody else in the department some time in the past, is The Mathematics of Interest Rates and Finance, by Guthrie and Lemon (covering, approximately, chapters 1-6). I'm not a huge fan of this book, and I don't really hide that from my students. Rumor was that the department was considering finding a different book for this semester. I don't know how hard they looked, but we're using this book again. I'm pretty sure it's not because this is the best book we could be using. But what do I know. Anyway, the point is, this book is the basis for the course, and so will be the inspiration, if you will, for the book my class puts together.

I've been typing up my notes as I go, so I've got an outline and first draft of content for the first few chapters, so far. I told my students I'd write the first draft of the remaining content as well. There will be 3 parts of the project, from the perspective of it being a graded assignment. First, students will be put into groups and required to edit a few sections. I'm not much of a writer (in case you hadn't noticed), and I've done basically no editing of what I've written, so that'll be an important task. Next up, they will have to write some problems, and then I'll have them also write solutions for somebody else's problems. Finally, the fun part. They're to come up with a "mini project" that they want to do to make the book better. I've suggested things like:

• making diagrams, charts, graphics to accompany the text
• writing sections on using calculators, or spreadsheets
• writing historical or real-world content
• convert everything to other digital formats

But I'm hoping they come up with more. Things they actually want to do. I'm curious to see what they come up with. I told them to dream big, and don't worry if it was too big (we can trim it to something manageable for the purposes of grading). It'll be good to have ideas bouncing around.

The overall organization of this is still coming together. There's a bit of work that seems to need to happen on my end to get things off the ground. Next week we've got a midterm, and then we've got a little fall break, so hopefully by the time we return, in just over 2 weeks, things will be all organized for the class.

I've been LaTeXing my notes, but it seems like using the wiki feature of our course management system thanger will make it the most easy to access and keep together for everybody. So I dropped all of my current sections into the wiki. I cleaned up the first one a little, so it's now wiki-formatted, instead of LaTeX, and was working on the second section when I realized that I could just make this part of the editing they're supposed to do anyway :) Give them some experience using wikis. I may do a quick tutorial in class at some point, if we've got time. I'll aim to get first drafts of the remaining sections written quickly, so that students can start editing and writing problems and things when we get back from break.

I decided that since there are 5-6 chapters, depending on how I organize things, for the problem/solution writing portion of the project, they can just write one problem/solution per chapter. I'll have to decide how to organize who does solutions for which problems, but I figure I've got a little time in that regard.

For the mini projects, I decided to make them write a (brief, informal) proposal for what they want to do. I made it due next week, which gives me a week to look at them during break (while I'm grading exams, writing the remaining sections, oh, and that thesis thing...). I'm guessing there will be some overlap among projects, so they might end up getting grouped. And it will hopefully give me a chance to make sure everybody is doing "enough", or at least comparable amounts. I told them if they had "big" ideas, and people to work with, their mini project could be in groups. There might be some students who don't want to come up with anything - I told them they'd be assigned a project (possibly just writing more problems/solutions).

When I get carried away, I dream that this project could all come together to something beautiful and useful by the end of the semester (I think some students may be thinking like this too!). That maybe the department would start using it. That maybe folks outside of the department could get something out of it. In the discussions in class, I've told them how I think it would be great to release it as a fairly open project, like under a permissive creative commons license. In class today, I told them to assume they're working on a project that will be released fairly openly, and that if they were opposed to this they should email me or send anonymous feedback. They can still work on things and get their grade, I just wouldn't include their work in a release version (kinda tricky for "editing"... I'm hoping this doesn't come up). I also told them that I wasn't sure what the University would think about such a project, or about such a project being released however we want. Maybe they'll claim it, or restrict how we can distribute it. I'm probably going to go for a "beg forgiveness" approach, instead of "ask permission". I took a quick look online, but didn't find anything about university policies that seemed applicable. Presumably I'm being irresponsible... it's not the first time.

So, anyway. Could be fun. Could be disappointing. Will almost certainly be more work than I've considered. Hopefully we make something useful...

If you've got comments, feedback, or suggestions, I'd love to hear it. I'm sorta making this all up as I go. If you'd like to pre-order your copy today... :) I jest. Mostly. (drop a comment, you might inspire my class)

New Homework

I'm thinking about trying something new with homework this semester. The goal is to have students evaluate their own understanding, and have each individual decide how much they need to do in order to attain the understanding they desire. So each week I will tell them to work on particular sections. What they will turn in to me will be something like the following

• I attempted problems: (list of problems)
• Overall (or perhaps per section), I would rate my understanding at: (letter)
• I had the following minor issues: (list of problem with issue, things like... algebra/computational mistake, counting mistake,...)
• I had larger issues with: (list of problems, with work, and a description of the issue - couldn't start, got so far but didn't know what to do next, finished but got the wrong answer and don't understand why)
• Here's a well-written (sentences, little algebra) solution to problem X: (well-written solution).

As far as grading goes, I will probably do something like check for completion - is the student showing me evidence that they are thinking about the problems outside of class? Perhaps I'll use a 0 (didn't turn anything in), 1 (didn't spend more than 5 minutes on what they turned in), 2 (gave it an honest attempt).

There's certainly lots of potential for students to not take this seriously. To some extent, I'm willing to let them, basically. If they choose to not think about the course material, they can see what happens when an exam comes. If they do fine on the exam, so be it. I'm not here to do much for the students who don't need me for help (ok, whatever, I should try to challenge everybody?). I could also use quizzes to see if students claimed understanding is accurate.

So we'll see how that goes, I guess.

Finishing Up

Still have one student that needs to take a make-up final, but I'm basically done for the semester. I all honesty, I probably checked out a few weeks ago. I've got something like a vaguely un-scheduled week ahead of me (besides a Flying Karamazov Brothers show!), until a math conference in Georgia in the second half of next week. People tell me it'll be good to go. That remains to be seen. And basically as soon as I get back, I start a summer programming internship at Rosetta Stone. Could be something. So with my loose week, I just gotta bang out a thesis so I don't have to come back in the fall...

Err, right. So this post was supposed to be about thoughts about my class this semester. Things that went well (it ended). Things that went poorly (the other parts). Things to do differently next time (next time‽). Something I can look back on in the fall for guidance.

So... reading assignments with homework due before we talked about it in class didn't go hugely well. I stand by the method. But I've also been reading about how homework itself, as a graded assignment, sucks (maybe start at joebower.org if you want to follow along at home), so I'm a little torn. Halfway through the semester I switched to the following setup: read the book for Tuesday, and submit a "discussion seed" based on the reading, as your homework assignment for Tuesday. I then used the those discussion seeds as what I brought up in class Tuesday, and then had written problems due Thursday. Seemed to go fairly ok, as a setup. Basing class time on questions students had actually submitted seemed like sort of a fun idea. Having the questions as a Google Docs presentation seemed to work ok.

My students really seemed intrigued when I showed actual housing cost data, so I should work to bring more actual data into my classroom. I was also thinking it might be cool to have guest speakers, maybe from banking or realty. Perhaps I'll look into that more for next semester. Probably the best thing about this class is that students can actually mostly see it as something useful. Putting money in an account, saving up... sorta easy to relate to, I think. There's some amount of intrinsic motivation there, if I can just capture it.

One important thing I didn't know going in to the semester was what sorts of students took the course. It's a 100-level course; I was expecting mostly freshman and sophomores. No dice. Probably half of my class was graduating seniors, many of whom already probably knew basically all of the material. I sorta have a hard time believing these students should actually be allowed to take this course... this is part of the stupid game I hate about education when I think about it these days. I had friends in undergrads who, as graduating actuarial science seniors, took 100-level "finite math" sorts of classes. This seems like what sorta happened with the class I taught this semester. The impression I get is that many of them were then upset with the class for being boring. What the hell do you want from me? Other students, in the last few days, have mentioned that they thought I did well trying to balance for these students along with those who actually didn't know much of the material coming in. I think it helps that mostly the students who didn't need to be there stopped showing up.

Another thing I didn't know coming in was the content of the course. I was reading the book along with the students. Turns out, there's very little material. In a few-minute wrapup I did on the last day, I told my students how essentially we did one thing all semester: move money around on a timeline. Compound interest is just iterated simple interest. Bank discount interest is just simple interest from a different perspective. Annuities formulas are just conveniences for dealing with many things moving around at compound interest. And that's all we did. I'm a little curious to see if I could teach all of the formulas in a week or two. Knowing this, I intend to go faster next semester and cover more material. I think this should help with some of the boredom issue.

I'm toying with the idea of using next semester, when I'll likely be teaching the course again, as a chance to have the students write their own textbook. There were enough errors and oddities in the textbook we did use, I have a hard time believing we couldn't do better. We'll do a big collaborative project, as the entire class, writing a book. I think a project like this could be good for having students involved and interested. I also think it should be a good way for students to learn the material - you learn best by teaching. And, finally, I think we could make a pretty decent book.

Each semester we math grad students are given a teaching request form, where we get to request what to teach the following semester. Basically it's one of the four calculus sections (two levels of calc 1 and calc 2), and then occasionally this financial math course, or a calc 3. On my form, I requested to teach this financial math class again. A friend thought I was joking when I told her. My advisor asked if I was a masochist. I think, having gone through it once, the next time can't help but go better, which will be nice. I know the material now, and might not be quite as caught off guard by my class being mostly seniors. Also, the class has no emphasis on algebra, my current frustration with calculus courses. I do need to try to design better assignments that have students explain their work more, but hopefully I can sort something out. Writing a book should be good for that.

Anyway, I'm off. Take 'er easy.

Teaching

I've started thinking that, perhaps, I don't actually enjoy teaching. Which is to say, there seems to be quite a bit about being a teacher that I really don't care for. Or I'm doing it wrong. Perhaps this is just how it all goes... nothing's perfect, right?

I love learning (my own, I'm not sure I can sense others'). I love reading. I think I enjoy meeting with individuals, or with small groups, who are truly motivated to learn something that I can help with. Not motivated in the sense of finishing an assignment, but in understanding material for understanding-sake. Perhaps it's content I already know, or something I can join them in learning. That might just be where my joy in teaching ends. Sadly, this doesn't seem like much of what school is (or, like I said, I'm doing it wrong).

I want to help individuals learn something that they are interesting in learning. I have no interest in talking to a classroom full of students about things few of them care about. I have no interest in trying to motivate them to care. I have a hard enough time with my own motivation to encourage others in that regard. And, looking back, I know that when I was a student, I didn't care about most of my classes, and don't expect I was interested in being told I should care. I didn't try to learn the material for its intrinsic value. This is especially true outside of math and computer science, but I expect even in some of those classes I was just getting through assignments (at least sometimes). Now that I'm about done with taking school, I'm finally ready to learn just about anything.

Probably I should just suck it up. There are always at least a few students who are paying attention, who are learning something, right? I should focus on them, right? I should figure out how to make things so exciting that people just can't help themselves but want to learn? I should continue giving homework and exams and playing the grading game, without it helping my students much at all. I should humor the notion, hinted at by award nominations, that I'm a "good teacher", despite knowing that I'm hardly doing anybody much good.

I should continue playing this game.

But I think I've realized about myself that I don't like other people's games. I like my own games. I think I might go find some of those...

Textbooks

(begin rant)

Two days ago, an email went around the math department here indicating that the higher ups were thinking about changing calculus textbooks, and that there would be a meeting today to talk about it. The meeting was essentially a presentation from a salesman from the textbook company about how great the new textbook is. A few minutes of discussion before-hand, and a comment that those making the change (I'm pretty sure it's a foregone conclusion) would hear our thoughts on the matter. I sorta felt, leaving the meeting, that we'd get a somewhat formal (at least an email) request for comments. Now, a few hours later, I'm guessing we wont. In fact, they probably just signed a contract while the guy was here today.

Sorry, I'm a bit cynical.

Before I go on, I'm probably supposed to make some disclosures. I have at times made (and am currently making) money doing things associated with the Webwork online homework system (admin, library tending). I grew up on Stewart's textbook (used it as a student, and it's what UVA has been using since before I got here). The textbook we are thinking about switching to (like, apparently, everybody else considering switching textbooks) is Briggs and Cochran. At the meeting today, I got a free copy. They also gave us an access code to their online software, MyMathLab.

In all fairness, the new book looks fine. It looks like basically every other recently-released calculus textbook I've looked at (don't take that to mean I've looked at a lot, of for large amounts of time). There are lots of pretty pictures, and apparently the author(s) are big on geometric intuition, which is great. We were also told to be impressed that in the worked examples, each step was given a little explanation. And, gosh, isn't this something amazing, the authors organized the problems at the end of each section carefully. We were told several times that this was not a lower level book, clearly something the publishers are worried about people thinking. It is, as @MitchKeller opined about the projects it has, "a bit hand-holdy".

Oh, and there'll be a new edition along every 3 years.

Of course, the textbook by itself isn't, I expect, what people are all giddy about. The textbook is paired up with MyMathLab. You can make online homework assignments and quizzes. There are little tools for the students to use to get help, like looking at worked examples, looking at a digital copy of the textbook, and probably some others I forgot. Students can also see a little "Study Guide", a sort of summary of what sorts of problems they missed, so they would know what to go back and look at. Oh, and some large collection of those pretty pictures in the text are pretty animations (calculus being about change, we were reminded by the salesman) in the MyMathLab system.

Students have the option to not buy the physical book (at least, in theory, who knows what our goofy school will require), and instead spend somewhat less and just use the e-book in MyMathLab. I seem to recall the e-book being in the $70 range, the first year textbook in the $90s, and the 3 semester book in the $130s, +$6 for the MyMathLab software if you buy the textbook. These numbers might all be before bookstore markup (we were also told to be pleased that the UVA bookstore markup of ~25% is really low... go team!), and we were told that these prices are cheaper than Stewart. The presenter today did this cute thing where he pretended not to already know the price of the books, and looked them up online.

Digital rights management didn't really come up at the meeting today, and I'm ashamed that I didn't bring it up.

Quite honestly, it seems like a not entirely bad thing. Switching textbooks isn't the worst thing we could do. Like I said on twitter after the meeting, though, it feels like shuffling deck chairs on the Titanic.

If I were given freedom to set up the course however I wanted, I wouldn't require that my students purchase this, or any other, textbook. There are simply too many free resources online for me to justify requiring students to spend that money. If students want to buy the book, or the software (more accurately: access (likely temporary) to it), good for them. Go for it. But if a student would rather use their brother's old textbook, say, or the free ones they find online, that's fine too. If instructors are going to lecture (I generally do, more than I'd like), many students can likely get by on class notes (their own, or a friend's), maybe in combination with some office hours.

Sure, the pictures in the book were nice. And the animations. But there are plenty of calculus animations online (I'm not gonna google that for you). In fact, I could roll my own (and have, happily), and would be delighted to teach students to do so as well. It'd be awesome to talk to students about pictures during class, draw them on the board, and then have them make a digital version of the picture, or an animation (for the ambitious) (and release them all under a CC license, wouldn't that be a fun thing to talk about in class). Heck, why not have the calc students write a book, complete with pictures? Write about the things they got hung up on, and what got them over it. Make a wiki. The instructor likely has access to several textbooks. Use them to put together an outline for the semester. Lecture if you want, or tell them to go see what they can learn online or at the library (then use class time to synthesize what was found). Looking up resources online, you also get to talk about evaluating resources, and how to compare different sources. You can build up a big huge list, and have students rate each source (and share your result with... everybody!).

And as for step-by-step explanations, WolframAlpha jumps to mind, but there are certainly others (I'm not gonna google that for you either). Don't like W|A's solution to a problem (they can be a bit... pedantic)? Have students talk about improvements. This also points out that those problems are pointless anyway. A computer does them more quickly and more accurately, so let it. Let's work on understanding concepts, instead of continuing to test pointless algebra tricks. Use the time that we're not doing mechanical manipulations to come to the geometric understanding slowly, guiding students to discover it for themselves.

Webwork has, for me, done a perfectly adequate job as an online homework system. If UVA doesn't think the problem library is nice enough, of the interface good enough, perhaps they should invest some money in improving it (I know they (at least, Dr. Jeff Holt, through a grant) have to some extent, and also that there is money floating around). Then even more people benefit (webwork is open source), and UVA gets some nice credibility. Webwork can do most of the things the guy today tried to sell us on for online assignments, though it's interface might not be as gosh durn pretty. It will not show students which section from the book they missed problems on (it's not tied to any one book). But I don't really see the problem here. I think it would be a great exercise to teach students how to evaluate for themselves which sections they need to look at.

I know I'm not going on much but youthful optimism and naivety. It's not the first time, nor likely the last. I have no expectation at all that the powers that be will take my thoughts seriously. At least here I can pretend that my thoughts are worthwhile (thanks Blogger). I'm leaving after next year anyway, and have seen no evidence (I suppose I haven't asked) that the younger grad students think like I do. I should probably just let it go (until I'm at another school). [As I was writing this post, I decided to email the math grads, to see if they'd like more flexibility in their teaching.]

All I want is for the department to allow instructors to try something different. Let us not require our students to purchase the (any) textbook. Get out of the publisher's pointless "new edition" cycle

Think about dropping the heavy emphasis on algebra.

(end rant (for now))

Project Wrapup

I want to share my thoughts on how my recent in-class project went. Basically I still think it's a reasonable project. It certainly needs some tweaking though. I'm kinda excited to try it again.

Recap

Class began and I started as close to on time as possible. I had already posted the assignment, and some students had seen it. I did a quick run-through of the setup, explaining the two parts (write your own problems, with solutions, and solve a different groups problems). As students divided themselves into groups I distributed printed copies of the assignment. Students got in and settled and started working pretty quickly, which was nice. Throughout, it seemed like students were doing a reasonable job staying focus on the task. I think the time pressure helps with that.

I let students work without any interruption for a few minutes. One student came to me and asked if the problem he had written was ok. It looked quite similar to textbook problems, which I told him, and he went away a bit disappointed. After a few minutes I started wandering around to each group, asking how things were going. Many of the groups were perfectly happy, and didn't have any questions for me. A few groups were having a hard time coming up with things. I tried to indicate how one might make up a new problem. For example, make up crazy rules a bank might use for some sort of account. Or: look at a textbook problem. They solve an equation for one of the 4 variables in our formula. Change the problem around so that a different variable is the one to be found. Sometimes this might not make a particularly new problem, but I think sometimes it could. The students I gave these suggestions to seemed to think they could make progress, and got back to work.

A few other students showed me problems that looked like textbook problems. One told me that it was different, and I challenged him to explain how. He noted that his problem tested the interpretation of the answers from some textbook questions. I can't really argue. He had 3 problems, 2 of which looked like textbook problems, and the third asked for the interpretation. Fair enough I guess.

At the half-hour mark, when students were supposed to be done writing problems, nobody was. I talked to them about it, and it was pointed out that probably solving problems that are already written would be quicker than writing your own. So the second half should go quicker. Makes sense. I told them all to take 15 more minutes on writing their own questions and solutions.

Just after the first group finished, it occurred to me that I should emphasize that they are all getting the same grade. So they should make sure to double check each other's work. I think many groups were divvying up work, so that each person wrote a solution to one problem. This is fine, but I think it is important that all of the group members double-check this work too. I made an announcement about this.

We were running out of time. With 25 minutes, groups were mostly starting to finish, and I re-distributed problems to groups that were done the first part. Since the time wasn't working out as I had anticipated, I told them to solve 2 of the 4 problems they were given.

With about 15 minutes to go, all but 2 groups had begun the second part of the assignment, solving another groups problems. One of the two remaining groups finished (group A), and the other was done 3 of 4 problems (group B). I took the 3 problems over to the group A to start the second part, but let group B continue working on their own problems. Group A eventually finished, about 5 minutes over time, and group B didn't have any time to work on group A's problems when they had finished writing their own. I'm still trying to decide what to do about that. [I assigned them some of the more challenging problems I had written]

While students were working on problems other groups had written, I decided it might be interesting to have them rate the creativity of the problems they were given. I told them to rate each problem they were given with either a 0 (this is a textbook problem), 1 (kinda new), or 2 (terribly interesting). I'm not sure how seriously they took this task, I have not yet (as I write this), looked at the work that was turned in. [Mostly I think this turned out ok, though some 2's were pretty questionable]

As students were finishing, I told them (a) I wanted feedback on the project, what they liked/disliked, how it could have been better, and (b) that I'd probably like to try this again, so to keep track of questions they think of outside of class.

After Class

I've been trying to write daily (we only meet twice a week) blog posts for my class, on whatever we talked about that day. I don't think any of the students are reading them, but I could be wrong. Today I posted questions I had dreamt up, to give students some sort of idea what I had in mind. Some of the questions I had though up before class, others were inspired by discussion in class.

As I was writing up my questions, it occurred to me that several of them weren't quite as original as I had originally (bam!) thought. I could see how to translate my problems into textbook problems. I do still feel like there would be a translation step though, and I guess that's part of the game. Or perhaps I'm fooling myself.

I emailed the class, specifically encouraging them to read my post, and also to provide feedback. Hopefully the reminder generates some feedback.

Initial Reaction

I mis-judged timing. I had originally given more time for doing other groups problems than writing one's own. Definitely backwards. Probably we could do 45 minutes writing problems, and 30 answering others'. Also, I think I need to be firm on the deadline if I try this again. At 45 minutes, you will have to give me your paper, and will lose points for not having written enough problems (well, you just don't earn the points you would have). Of course, that means the group that gets your paper isn't being graded out of as many problems, which must be accounted for. Hmmm.

I think trying this again, students may have a better sense about the project. Writing problems might come easier. Hopefully, then, timing would work out a little better. Perhaps showing students a list of questions before-hand would have been a good idea.

I didn't plan enough about organizing the work that got turned in. I'm sort of dreading looking at the pile. Here's how I think I'd organize it next time: Each group is assigned a number. There are then 4 things they turn in to me by the end of class:

1. The list of problems they wrote. This should have "Written by group N" on it. It should also have "Solved by group M", the group that gets these problems.
2. The list of solutions to their own problems. This should have "Written by group N" on it.
3. The list of solutions to the problems they were given. This should have "Solutions written by group N, for group M's problems" on it. These solutions should not be on the same paper with the questions, since it mucks up re-distributing papers (more on this below).
   
4. A paper with the group number and names of all of the group members. I suppose this information could just go on, say, the list of solutions a group writes for their own problems

I need to bring a stapler to class.

I sorta like the idea of getting students to rate the problems they are given for creativity. I can't quite decide what to do about giving points based on those ratings, as they'd be pretty easily gamed. One option I came up with: Have students rate the problems they were given on the 0-2 scale above. As I'm grading, type up all the problems that earned a 2. Distribute this list, without any identifying marks, back to the students, and have them pick their favorite n, say (outlawing voting for your own (more bookkeeping for me, but doable as long as I have them write their name on the paper)). Any problem that gets more than m "favorites" earns the authoring group p points (1, likely).

Grading

As I organized papers, and got to a point I could start grading, I realized that I could extend this project to have the students do the grading. Group A gives their answers to Group B, and when Group B is done, Group A grades the solutions. This could be a valuable exercise for the students, seeing how their questions were interpreted, perhaps seeing other ways to solve questions they had designed. Of course, it leads to problems about what grade gets written down in the grade book... suggestions?

As I was grading, I realized that students can game the grading system by writing easy questions. This guarantees 3 points for solutions to each problem the group writes, even if they lose a point or two for writing uninteresting problems (I was pretty relaxed about taking points off for this). Perhaps this can be corrected for by having other groups rate "originality" first, (maybe "difficulty" too) and then base the score solutions are worth out of that grade?

Maybe take interesting problems in to class, have everybody do them, and talk about answers. Also problems with difficult wording.

On re-distributing papers: Group A did Group B's problems. Giving A's solutions back to A, they probably won't be able to see B's questions, to look back at.

Student Feedback

At this point, I still haven't gotten much. One said they enjoyed the project, even if making up problems was difficult. Another suggested just writing up questions individually and exchanging papers with partners; that working in larger groups was hard. Also the timing issues were pointed out.

Current Thoughts for Next Time

Be more organized about what names I need to see on which papers. I found that the paper-shuffling aspect of grading was easier for groups where group A did group B's problems and vice-versa. I'm not sure that it matters too much though.

Perhaps break this project into two class period. In the first, groups will meet to try to create interesting problems that they could solve (though I won't ask for solutions just yet). The problems will be re-distributed around the room, and groups will rate the originality and difficulty of the problems they were given. I will then gather up all of the problems, and we'll end class. Before the next class, I'll go through all the problems, find all those that earned good originality/difficulty ratings, and compile them into a list. In class the next time, all of the students will work those problems, maybe just working with a neighbor. Then we can spend time in class discussing solutions to the problems, talking about different interpretations of questions, what makes questions well-written, etc. There's issues here about turning in solutions - if students just wait until we talk about solutions in class, they can just copy those.

Thoughts?

I'd really like to know what you think about any or all of this. I know it's rather a lot to read through, sorry.

Challenge Your Neighbor

I think I'm going to try a new exercise in my financial math class tomorrow. We've just finished Chapter 1, which is on simple interest (yes, a whole chapter on simple interest). If you are visualizing how the following project might go, here are some parameters: 45 students (we'll see how many actually show up), 75 minutes. If you've done a project similar to what follows, or even if you haven't, and you have some feedback for me (things to watch out for), I'd love to hear about it (before noon Thursday :)). Below is the assignment I'll be giving my students:

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The goal of this project is for you to write your own questions to challenge your classmates, and help review chapter 1 material.

Outline: In groups of 5, you will be given half an hour to create a list of 4 ``interesting'' questions that cover chapter 1 material. The questions must not use formulas or concepts that are not covered in chapter 1. Your questions must not be re-writes of textbook problems, simply obtained by changing numbers or dates. You must be able to solve the problems you create (with the aid of a computer if the algebra is difficult). The 4 questions may all be based around a central scenario.

You will then exchange your list of problems with another group, and, in turn, will be given the problems created by a different group. You will have the remainder of the class period to produce solutions to the problems you have been given.

Specifics: In the first half hour, you must produce 2 papers, both containing the names of all group members. One paper should contain your list of questions. This paper will be given to another group in the second part of this assignment, so should only contain the questions you have written. Questions must be written well enough to understand without further explanation. Your second paper should contain well-written solutions to the problems you have written. To save time, you do not need to copy the text of the questions to this second page.

In the second portion, you must write solutions to the challenge questions you have been given. You may use the paper you have been given. You must write the names of all your group's members on this paper. As it will have two sets of names on it (the problem authors, and the solution authors), insure that the two sets are clearly identified.

Grading: Unless it is clear that a group member is not contributing, all group members will receive the same grade for this assignment. You may earn 2 points for each problem you write, up to 8 points total. Points may be deducted if the problem is not relevant or is poorly written. The write-ups for the problems you author, and the problems you are given, will all be graded out of 3 points.

You will not earn extra points for writing extra problems. You may have points deducted if the problems you author are not distinct from textbook problems.

Letter to My Students

Today in Financial Math we were talking about approximate versus exact time, and ordinary versus exact interest. I was telling the class about my observations on the topic, and our textbook's handling on the matter. I don't think our textbook does a particularly good job. After this became clear in class, one student commented how it made him excited to try learning independently from it (this is my goal for the course, I've rambled about it recently). We all had a little chuckle. I think it was at this point that one student opened up a discussion on thoughts about the course structure. Mostly I don't remember exactly too much what was said. Students could not believe that I was expecting them to read the text, do problems on the reading before I had talked about the content, and then still come to class. At least one student made a comment about the price of the education I was apparently not giving the class.

I was a little bit entertained, to be perfectly honest. It seemed a pretty strong reaction to have after a single assignment. But ok, the course is different, it'll take getting used to. It helps (me, if nobody else) that I really do believe it's a good way to structure class.

On one hand, I seem to have students who were shocked that I am asking them try to figure things out for themselves. On the other, I apparently have students who feel that they can do this well enough to make class discussion worthless. My guess is that many of the students alternately believe both of these things. I just can't win :)

So anyway, after several minutes in class of many of the students sharing their thoughts, things calmed back down a bit, and we got back to the topic at hand. I heard some mumblings about how the course was "ridiculous". It might have been "absolutely ridiculous", I don't recall.

I want my students to embrace this course. I really think it can work. I decided that I should share more of my thoughts on the matter with them. I probably should have done more of this earlier. I did tell them how the course was set up on the first day. And I remember asking if I needed to try to convince anybody that learning independently was a valuable thing. At the time, nobody claimed to need convincing.

But anyway, I wrote a little response for my students, and have posted it on our course discussion board. I hope to continue the discussion with the students there. For whatever reason I also thought I'd share my letter here. So without further ado:

Dear Students,

I would like to take some time to tell you about my thoughts on why I have set up our course the way I have, as well as to give you some better insight into how I anticipate this course working and what I expect from you.

Here is how I envision this class working. Every Tuesday you will have an assignment due at the beginning of class. The assignment will say to read some sections of the textbook, and will require you to write solutions to exercises. The content will be things I have not talked about in class. I expect that in addition to reading the contents of each section, you will look at all of the exercises, not just those that are assigned. You should look at all of the odd problems, and all of the preliminary short answer questions, and try enough of them to test your understanding. When you can answer odd problems correctly without looking at the back of the book, you should feel like you are in good shape.

Your write-ups should be clear and well-written. Explain, in your write-up, where your formulas, and the numbers in them, come from. You might model your write-ups on the solutions in the back of the text, or the solution to example problems worked in the section. I anticipate choosing even-numbered problems from the text that are noticeably similar to nearby odd-numbered problems.

I have designated several hours on Monday for office hours. If you are stuck on homework problems, please feel free to come to any of the scheduled office hours. If you are working on things over the weekend and have a question, feel free to email me. Also, if you would like me to look at your work before you turn it in, Monday's office hours are a good time for that. As indicated on the syllabus, I am also happy to meet with you at other times throughout the week, by appointment.

There are some difficult problems in our text. I do not anticipate assigning particularly difficult problems on written homework. I would rather talk about these problems during class time, after giving you a chance to ask questions.

I expect you to come to class Tuesday with questions from the reading - things you didn't understand, or examples that didn't work out as you expected. If you work a problem in a manner different from that in the text, you should share it with us. As you are reading each section, keep notes on your thoughts and questions, they will make good discussion material. I will do the same.

Thursday's class will likely begin with a spill-over of any discussion from Tuesday. Also, I expect Thursday will be a good time to work on the trickier problems that were not assigned. Finally, as I intend to have homework graded for Thursday's class, we can spend time talking about any mistakes you (or I) made in our solutions.

I hope the above begins to address some of the concerns mentioned in class today about how coming to class would not be beneficial. I have a very hard time believing that you cannot benefit from discussion with 45 other individuals all thinking about the same content that you are. Had you really thought about all of the issues mentioned today about exact versus approximate time?

There remains the issue that I have designed this course to make you learn the material by reading the book. I expect that this is not the way many of your other courses are, or have ever been, structured. It's a good thing I got to you before you graduated.

I absolutely believe that reading a textbook is a good way to acquire an understanding of any new material. I believe that being able to learn independently will make you more attractive to employers. I believe, moreover, that this independence will help you in your personal life. It should inspire you to dream bigger, knowing that you can figure out the steps necessary to attain larger goals. As corny as that sounds, I stand by it.

I also believe that this sets up class time to be more beneficial than a traditional lecture. Why would you bother sitting through a class where an instructor will be delivering a lecture, the content of which is sitting in a book you could read when it best fits your schedule? How many of the lecture notes you have taken have been things that were written in your textbook? How is that an efficient use of in-class time? Doesn't it make more sense to have thought about material on your own, developed your own understanding and questions, and then use class time to address that?

Yes, this course is structured differently. Yes, it does require you to do work. I believe you can do it. And I will be there to help you when you need it. All I ask is that you try it by yourself first.

Step Stool

I have a memory of, as a child, having my own little step stool. If memory serves it was yellow, and made of fairly sturdy plastic. My sister had one too. The stools probably had our names on them. Mostly I seem to remember using mine to reach the sink, when brushing my teeth or washing my hands. I'm sure I wasn't conscious of it at the time, but that step stool had plenty to offer.

A step stool is sturdy and dependable. If my step stool had ever slipped out from under me, or broken while I was using it, I would have been nervous to try using it again (if, indeed, it was still usable). I would not have used my step stool if I had not been confident in it.

A step stool is easy to use. I probably didn't need much instruction in how to use my step stool. There should be nothing complicated about using a step stool. If my step stool had some sort of policies I needed to be aware of (e.g., ``Do Not Use the Top Step''), they were ideally clearly marked, easy to adhere to, and did not artificially limit the use of my step stool.

A step stool is easy to take with you. My childhood step stool was probably relegated to the bathroom, but had I needed it, I could have easily picked it up and used it anywhere I had in mind. Knowing that I had a step stool, one I was confident in using, I could explore further.

A step stool is easy to move aside. When I was ready to use the sink without it, my step stool wasn't in the way. If I wasn't as ready as I thought, my step stool would be there to give me the boost I needed.

A step stool is an appropriate height. A step stool that is already as high as I want to go only necessitates another step stool. A step stool that barely reaches above the floor doesn't offer much.

A step stool doesn't reach the whole way. Making me go the last step will increase my self confidence. Making me do work along the way will make attaining the goal something I can be proud of.

A step stool gives one a higher viewpoint. With it, I might be able to see my way from where I am to where I want to be. Indeed, with another viewpoint, I might begin to make new goals, more ambitious than those before, or in another direction. My step stool would let me look in any direction.

A step stool never stops being useful. My childhood step stool could still help me occasionally. Any time I wasn't using my step stool, it would work just as well for my sister.

A good teacher is a step stool. One that adapts and grows. That challenges itself as it offers and inspires challenges for those who use it. That delights in the success of those it helps, and offers encouragement and patience when things don't go well.

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How's that for a teaching statement? Better than last year?

First Day

Well, my Financial Math class started today. I am not convinced it went particularly well. I'm trying to remind myself that the real test will come Tuesday (our next class meeting), after they've done some reading and their first assignment. Today I just wanted to talk about my goals for the course, and how it was structured.

I guess that's not all I wanted. I wanted to start the class off with lots of discussion, since that's what I want the class to look like for the rest of the semester. Not just me talking. I remember this happening last semester, and thinking it was a great thing. Perhaps it helps that I started with content on the first day last semester.

Today, besides people saying "Here" or so as I read their names from the roster, I think approximately 3 people said something during class. That might include me, I can't quite remember. Not quite all I'd hoped.

When I wrote my last post about this class, just two days ago, I was, for whatever reason, working under the assumption that a majority of the class would be first year students or so. It is, after all, a 100 level course. It turns out that nearly my entire class consists of seniors. I'm trying to remind myself not to expect them to be lazy before giving them the opportunity to prove otherwise. I'm having a hard time of it, to be honest.

I wanted students to to tell me why they were taking this class, and what they'd heard about it (since I knew little about it). One student said it filled a general math/science credit. Another said that he had heard it was a good intro to finance and things. I should have pressed the issue... I'm starting to think this course is looked at as any easy way to fill one last graduation requirement. And I sort of hate that.

On my run after class, trying to clear my head a bit, I realized that perhaps I was being too idealistic about this (I thought that's what ivory towers were for!). Looking back, I know that as an undergrad I cared very little about anything besides my math and computer science classes. I took whatever else I had to, but have no memory of enjoying learning any of it.

Since starting grad school, I seem to have really grown into loving learning. I basically cannot stop reading things online. And they aren't all math and computers and teaching. Well, mostly they are. But I also feel like I would actually be excited to maybe learn some history (which I always hated), more science, more literature... anything. My friend got my a gift card for a book store, and I wondered at some point if there was a convenient way to pick a "nice" random collection of books from scattered topics to read (feel free to leave a comment!). Broaden my horizons and all that.

Even learning financial math has been not bad. I've always shied away from applications. Today I went to the library and got a few popular-level books on finance. I felt a little dirty, honestly. But I was also excited to learn about things I know nothing about.

Surely my students will share this enthusiasm, right? I'm reminding myself who I was years ago, and realizing that probably many wont. But I'm also trying to remind myself that if I go to class thinking that, things will turn bad (worse?) quickly.

I did, in class today, get a nice chuckle out of the class when I told them I know nothing about finance. I expect many were a bit shocked when I told them that while I'd read the first chapter a few times, and skimmed the next two, I had no idea about the other three we'll be covering. I'm hoping they appreciate my honesty, and don't lose respect for the course. I'm wondering if a pop quiz is in order for Tuesday, to make sure they know I'm taking this class seriously.

I'll keep you posted.

A Confused Teacher

This semester I'll be teaching a course in financial math (a first-year-level course offered by the math department). This is a bit of a change from what I've been teaching (calculus, for 5 years). Moreover, I basically know nothing about the subject.

Ok, sure... you put money in a savings account, it earns interest at some rate, compounding sometimes, and you end up with more money than you started with. Conversely, you use a credit card to buy stuff, and it costs more, due to interest. I've got some grasp of those concepts (though not much... I mean... why would a bank give me money for having money?), and the equations involved. However, some of the chapters in the book that I'm supposed to cover are titled: "Discount Interest", "Ordinary Annuities", "Other Annuities Certain", "Debt Retirement Methods", and possibly also a chapter on "Investing in Stocks and Bonds". Well, I think I can claim to have heard those words before (not necessarily together), but that's as much as I can honestly own up to.

So I'm supposed to go teach a class (45 students) on a topic I basically know nothing about. This has put me in a bit of a philosophical mood. Or, since I also won't claim to know what philosophy is, let's just say: I've been thinking about how I should set up a course like this, and why I think that.

[Just a heads up. This post has parenthetical comments within parenthetical comments. And it isn't about LISP. If that gives you some indication about how rambly this post is, and you don't have time for it... perhaps I'll do better next time. See you then.]

My thoughts are: I'll be learning this stuff essentially as my students do. I will be no more an expert on the subject than they would be after reading each section. It believe it's quite possible that some students will come in knowing more about the subject than I do. So why should I lecture on it (nevermind if I should be lecturing when I do have an understanding (calculus))? Shouldn't they be able to teach themselves, just like I'll be doing? And then we can all get together and see if we came to the same understanding?

I asked basically these questions on twitter, where @michiexile was kind enough to offer up some thoughts. He stated that I've got some advantage, having spent more time learning how to learn, and also that I'm there to be "a bad conscience", "a voice of authority", and to "give context".

I can certainly see the bad conscience part. That's the point of making assignments with due dates. I'm not sure how much context I can give, though perhaps I'll do more reading in places besides the text, so I'll have more to share (but, again, they can do this). I'm not a particularly authoritative person, but perhaps that's something I should work on in my role as an instructor (or perhaps not?).

Clearly his point about having more experience learning is valid. If you kept count that high, I think I'd be in 22nd grade (only taking 21 of them though :)). With my students who I'm expecting to mostly be in 13th grade, that's approaching twice as many years in school (ouch). Surely I've picked up something in all that time. How can I best impart this knowledge (which I don't have a ready grasp on) to my students as efficiently as possible? I have no idea. My consolation is a guess that I'm not alone in not knowing.

It seems fairly clear to me that knowing a bunch of things is useful. Surely (though I have no experience to base this off of) employers like to hire people who already know many things, and know how to do many things. However, there is absolutely no way an employer can expect a future employee to know everything about pertinent subjects. There's quite simply too much out there. And this holds true especially for students fresh out of college, who almost certainly have significantly less experience than individuals who have been working for some time.

So it seems to me that what employers should be hoping to find is individuals who have a strong ability to learn. I'm guessing this isn't much of a revelation to anybody. I think I'd like to claim something stronger though. I believe that an individual who has a strong ability to learn independently makes for a stronger job candidate.

(Please don't think that my whole outlook is helping kids land jobs. I don't even want a job :), though of course money is nice. I have little idea how to get one anyway - that's a portion of the reason I'm still in school. Also, I don't have much in the way of an idea how an employer would determine if one individual is a stronger independent learner than another. I do believe the internet can help, by giving people a place to show what they do. But I'm getting off track (seems to be what I do).)

I've been trying, at least a little bit, to think about why I believe that an ability to learn independently is an admirable quality. Possibly I'm behind, and people have decided that it's simply true, sort of intrinsically or something. Perhaps not. Part of my concern is that I like to think that I have at least some ability to learn fairly independently (I'm quite possibly fooling myself), and so perhaps I just want to believe that this is an admirable quality. Or, more drastically, perhaps I've got some psychological issues about relying on other people. I like to think this isn't the case, but what do I know?

Having not much in the way of "other opinions" on the subject, I'm pretty much going on the assumption that I'll be doing well if I can get my students to be better independent learners. I'm not saying total independence. I'm not going to cancel all the class meetings and just have a final exam, and expect everybody to be fine at the end of the semester (man, think of all the free time though...). I know that I am not that independent myself (and I'm sure my thesis advisor would be quick to agree), and I don't think it's even a reasonable goal.

So here's what I think I'm going to do with my class. We have 75 minutes class periods, one on Tuesday and one on Thursday. Each Tuesday there will be an assignment due, presumably consisting of exercises from the text (I'm still working on this part), on sections that I have not talked about in class at all. This will require the students to read the appropriate sections, carefully, so that they actually gain an understanding of the material. I'm going to have office hours on Monday, to help students who got stuck on the assignment.

This is the way my undergraduate math courses were structured, by the way. I'm excited about diving in on this policy. I've made attempts in the past, with my calc classes, but went too easy with it, I feel.

So, great, my course is set up. Except, there's one small issue. I've got 150 minutes in front my students each week. If I'm not talking about new material, and they just turned in an assignment showing that they understand old material... what do I do with this time? This part has me seriously nervous (so I'm writing about it out here, good use of time :)).

My guess is that students (including me!) won't understand all of the material from the section. I'm planning on spending as much class time as possible and necessary having a discussion with my students (hopefully with them doing most of the talking) about any questions we have from the reading. Hopefully some will be confused about examples in the text. Hopefully some will still have questions about how to work problems. Hopefully some will have questions about how to interpret answers to questions.

I'm trying to think about what to do if they don't have anything to say. Or if they pretend they don't. I'm guessing if I say "So, since none of you have any questions, maybe I should just give you a quiz on this for you to earn lots of points on, and then we can all go home", some hands will go up (and some people will think going home is a great idea).

One thing I think I'll do is to try to keep track of my own thoughts and observations and questions as I read, so that I can share that with the class if they don't have their own things to talk about. I may also come with some extra examples, to give them time in class to work on them (possibly in groups, or as part of a game or something) and get better at problems simply by doing more of them. Possibly I'll be inspired to look at other materials online or in other books, and can come in with something to share from that reading.

I think talking about the material the students (and I) have just read, after we have worked some problems on our own, will help with our understanding of the material. Surely that's a good goal for the course.

But I was rambling on just a few paragraphs ago about getting my students to be more independent learners. I do think that setting up the course as described above will give them experience trying to develop this skill. I also think, however, that this goal can be addressed slightly more explicitly during class time. I hope that by keeping track of my own "how I read this section" process, I will be able to share that with the class. I touch on this a little above. But I could take it further, I think. I could perhaps scan a section of the text, and keep my notes on the paper as I go. Then during class I could walk through how I read the section with my class. Perhaps they'll pick up on things. Things like... what questions do I ask while reading? How can I tell if I'm actually understanding this material? Where can I go to answer questions I have? Perhaps the exercise will make me more aware of the process myself, and I'll be able to point something out. Perhaps students will point out their own methods of gaining understanding.

At this point I've probably rambled on far too much. Clearly this semester will be (or, at least, could be) an interesting one. I believe that I will learn quite a bit. I hope that my students also do.

Changing Calculus

Apparently toward the end of every semester I decide to write a post about changing calculus. At least, this is my second go at it. But today's attempt is more a follow-up to my most recent post, about the role of higher education.

My initial motivation in asking the questions in my last post was to provide some structure for what a calculus class should look like. Is the role of calculus really to make sure that all freshman know how to compute integrals by hand, and can use this to find the volume and surface area of a solid of revolution? That rising second years can go through mindless algebraic manipulations to arrive at an answer to a question that was asked without any context or applicability? Moreover, that they need to be able to do so by hand? Quick, find the antiderivative of $\sqrt{3-2x-x^2}$, no computers. Who gives a crap? How many times will that function, or one like it, actually show up in the lifetime of my student? I get paid to do math, and the only time I've ever cared about the answers to these algebraic questions was when I was taking or teaching a calculus course.

Looking back, I'm a little ashamed that the problem above was actually one I assigned as homework for my students this semester. What good does it do to have them work this integral by hand? Surely the answer depends on the students, right? My students going away from the sciences have gotten nothing out of this assignment. My students staying in the sciences have spent another... 5 minutes? 30?... practicing their algebra skills. Hurray‽

Look, I'm not saying algebra isn't useful. But I really do think it is over-emphasized in calculus courses (the ones I've encountered). My calc II course covered: techniques of integration, improper integrals, arc length and surface area of revolution, parametric curves, polar curves, multiple integrals, and infinite sequences and series. Lots of these are wonderfully fascinating and fun topics. Parametric and polar curves force students to start thinking about curves differently, to start re-interpreting functions and points in the plane. Infinite sequences and series are so unlike anything that's ever come before that they can't help but be interesting. Add up infinitely many things that get infinitely small, and produce an infinite value? Or a finite one? How the hell does that work? What does it even mean?

If not a single one of my students can do the integral mentioned above, I'm not sure that I'd feel bad about my semester. If not a significant portion of my students could tell me what was new about a parametric or polar curve, I think I'd be fairly upset (admittedly, now I'm a little afraid to ask).

My problem with my course is that algebraic manipulations are essentially given precedence over conceptual understanding. This is done by making exams that test algebraic nonsense, forcing instructors to make sure their students can do the algebraic nonsense, taking up valuable time that could be better used in other ways. With less time spent on algebra, more time could be spend on concepts, and more time could be spent on improving student's writing. The time spent on concepts makes it a more mathematically interesting and, I'd argue, worthwhile course. Whatever negative impact on my students is brought on by slightly less algebra practice will be more than compensated for by having them spend time thinking about ideas. More time spent teaching students how to write mathematics will force them to think more precisely about what they are saying and how to say it. How could this not be advantageous to students, whether they will be taking more math or not? Unless, of course, thinking about these ideas and how to write them is not aligned with the goals of my university. If the role of higher education isn't something that thinks this would be better, then I'm getting off this boat before it sinks.

So why not go for it? Spend drastically less time on techniques of integration and series convergence tests. Force students to turn in well-written assignments. Spend class time talking about what well-written math is. Spend time letting students play with computers to get a feel for parametric and polar curves. Computers will be faster and more accurate in plotting curves than any student ever will be. Give students time to plot lots of curves, changing parameters to see the effect on the graph. Make students look up resources online to learn how to find arc lengths or surface areas, instead of just telling them the formula (or even deriving it) and having them memorize it. Spend class time talking about how to read the resources they find, and how to evaluate them for quality.

My advisor pointed out, during our conversation mention in my last post, that there are some issues with trying new things. At the lowest level: who gets to teach such a course, and where does their funding come from? But I think better concerns that he mentioned are two that bring me back to some of my earlier questions.

1. It is hard to test conceptual understanding. It is much easier to test if a student can do some algebra. With several hundred students going through calculus every semester, this is a practical concern. How does a university handle scale?
2. Institutional inertia would have to be overcome. Change is hard. We do this because this is how we've done this for a while now, and it is clearly the best way to do things and it's going ok and nobody has complained.
   

So let me ask again, what is the role of the university, and how will the university adapt to the changing world? Is the role to grant degrees, and so is the role of my lowly calculus class to funnel students on to further classes until we can get them out of here with their paper in hand? What market are universities in? If we know, then we can see if our institutional inertia is good or bad. If we don't know, we should probably get on that.

These questions brought me back, full circle, to my initial desire for some formal guidelines for courses. I think it would be hasty to change all of our calculus courses around right away. I think a better solution involves some experimenting. Make a new course, make grad students apply to teach it, arguing for what they want to do differently, and require the instructor to report back on how it went. I think having formal guidelines for courses would aid the experimentation, because it gives an objective measure for comparing a newly designed course with a more traditional one.

I know that much of what I'm saying is oriented toward the way things are set up in my department. How is your department set up differently, and what does that mean about how calculus courses are structured?

Thoughts? Objections? (non-spammy) Links?

Higher Ed

I've been wrestling recently with identifying the role of higher education. I'm not much of a wrestler, physically or intellectually, so I thought I'd see what I could put into words and have you folks comment on. Thoughts of yours, or links to thoughts of others are much appreciated.

What brought on my recent concerns, or possibly made the pot boil over, as it were, was some frustration in making a final exam for my calculus course. The course I teach is one section among several, and we all have common exams. When we went to make the exam this semester, there was some disagreement about what should be tested. The lack of a prescribed vision for the course (at least, one I was aware of) started to upset me. And then I decided that such a vision should be accompanied by a broader vision of the math courses offered by the department. It should include formal indication of how the courses fit together, what is assumed of students coming in to each course, and what is assumed of students passing a course (I'm thinking something more formal and precise than the short list of pre-reqs for a course). And then I wondered where this vision came from, if it was internal to the department or how much say other departments had in what they wanted math classes to require. I'm not expecting the English department to care at all what is covered in Calc I or II. But perhaps Economics has some input for us? Or other departments? And then I wondered what the point of any of it was. What is the role of higher education, and how does my Calc I or II class fit into that?

Clearly things were getting out of hand.

Luckily, my thesis advisor turns out to have the title of Director of Undergraduate Affairs. Furthermore, he was happy to talk to me about these things, and brought a welcome level-headedness up against my relative insanity. Of course, I'm sure he won't mind much when I get back to research... :)

He pointed out that probably having formal guidelines for all courses is a bit much - that professors teaching upper level courses should be granted plenty of flexibility. I can't really argue with that. I have vast respect for all of my professors. But this flexibility doesn't really help in getting a vision for Calc I or II together. And if the courses are coordinated among several sections, there should be some vision backing it, right?

Part of my initial inquiry was if there was a curriculum review that happened periodically, and how long ago the most recent was. My guess was that if there was such a thing, it happened before nearly all of my students had personal laptops (or possibly even personal computers in their dorm room). I know I'm a bit of a nerd, but I really think it is important to adapt to the changes brought about by each student having their own computer (that they maybe can carry in their pocket!) connected to the internet. The explosion of infinite goods and real-time global communications, for "everybody" (yes, I know I live in a privileged region, or whatever is the PC way to say it. I'm sorry?), has been changing, is changing, and will continue to change... well... nearly everything. When the fundamentals of economics and communication change, what isn't impacted? [Side note: I know very little about economics, I'm probably saying things incorrectly]

Bringing things back down closer (somewhat) to the level of a graduate student who knows basically nothing about anything useful... how does the internet change higher ed? What happens when lectures become an infinite good, available to be watched freely online? What happens when the content that makes up our textbooks is available in a multitude of places online? What happens if students have direct access to experts in any field at campuses on the other side of the globe? Gah, I'm getting out of hand again.

Two recent posts online have been on my mind a bit as I think about this question:

1. A post on ars technica about the difference between the movie industry and the music industry. For some reason this article made me realize that I probably wasn't thinking much about what was distinct about higher ed, as opposed to the entertainment or news industries, and what that meant about the impact of technology.
2. A video on techdirt explaining the innovator's dilemma. This video makes me think that in order to adapt, universities should be asking what market they are in. I've been phrasing the question in terms of what role they play, but I think maybe they're pretty similar questions. Are we in the horse-and-buggy market, following the example in the video, or the transportation market? Are we in the market of churning out degrees? Are we in the certification market? Is that our primary goal and purpose?
   

I truly hope that I'm not saying anything surprising to people involved in higher education (or other fields!), or asking questions you've not heard before. I really do hope that I'm late to this conversation - that smart, dedicated people in the right places are asking these questions. Moreover, that they are coming up with some answers. I hope that I am just not seeing this conversation at my university, or at others, because I am not involved enough somewhere.

Anybody have some thoughts or links for me?

This post is somewhat of a part 1 in a series. The second focuses more on teaching calculus.

Grade Eachother?

I started doing daily homework assignments for my calculus class when we got to series convergence tests. The idea is that they will read the section in the book and work some problems to turn in, before we talk about it in class. It isn't going well.

Mostly students are "getting the right answer," but the write-ups are fairly bad (mostly making me feel like understanding is pretty low). I spent about 20-30 minutes in class Monday telling them about the things they were doing that are driving me crazy.

We've got three more of these daily assignments due over the next week. I'm thinking about doing the following: During class, have the students gather themselves up in pairs and have each student write their name on their partner's paper. The students will spend time looking for mistakes on their partner's paper. And I'll tell them that whatever points I take off of a paper will also be taken off of the partner's paper.

Surely somebody out there has done something like this before. How does it go? Does it help? What do I need to watch out for? I'm a little worried that "correcting" won't happen as much as "copy down what is hopefully a better answer, without understanding it".