Five positive student outcomes from the textbook-free algebra class

We’ve got just 4-5 weeks left in the semester and until the textbook-free Modern Algebra course will draw to a close. It’s been a very interesting semester doing the course this way, with no textbook and a primarily student-driven class structure. In many ways it’s been your basic Moore Method math course, but with some minor alterations and usage of technology that Prof. Moore probably never envisioned.

As I mentioned in this lengthy post on the design of the course, students are doing a lot of the work in our class meetings. We have course notes, and students work to complete “course note tasks” outside of class and then present them in class for dissection and discussion. The tasks are either answering questions posed in the notes (2 points), working out exercises which can be either short proofs or illustrative computations (4 points), or proving theorems (8 points). We have a system for choosing who presents what at the board — I won’t get into the details here, but I can do so if somebody asks for it in the comments.

So the class meetings consist almost entirely of students presenting work at the board, where their responsibility is to make their work clear, correct, complete, and coherent — and ruggedized against the questions that I inevitably throw at them.

I was thinking yesterday that this method of doing class has really done a lot of good for the students in the class, in several key ways.

  • Students ultimately rely upon the soundness of their own work. The students can work with others or with print or electronic resources — although with no textbook, they have to learn how to find those resources and tell the good ones from the bad ones, which is a great skill by itself. But it boils down to presenting that work, on your own and with nobody there to bail you out, in front of your professor and peers. I think this is a good antidote to the occasional over-reliance on cooperative learning that we (in education as a whole, and in my department) have. Group work is all well and good, but to be a complete learner you have to be able to rely on your wits and your skills and not just prop yourself up on the strength of peers.
  • Students prepare for class in advance, several days in advance, every night. To do reasonably well on course note tasks, students need to plan on successfully completing 15-20 course note tasks throughout the semester, which comes out to about 1-2 per week. Combine that with the fact there are 8 students in the class all trying to do this, and it’s easy to see that working ahead is really essential. You want to get so far out in front of the class that you have no competition for a particular range of problems. Very often in college, there is no sense that you have to get ready for class the next day — unless there’s an assignment due — and we profs reinforce this by running classes that do not penalize the lack of preparation. (It’s not enough to reward the presence of preparation.) The course design here, though, rewards the students who have read and practiced ahead and learned on their own.
  • Students become skeptical and tough-minded about their own work. It’s quite common in traditional math courses for students completing an assignment to simply barf up something on a piece of paper, hand it in, and see how many points it gets. When you are presenting work before a class, that route leads only to embarrassment. When most of the class time is spent doing these presentations, students learn something I didn’t learn until graduate school — that if you are going to hand something in or present something with your name attached to it, you had better make very sure that it works. I’ve noticed the students anticipating not only the fact that I will be asking them penetrating questions about what they are presenting, but also what those questions are. At that point they are learning to think like mathematicians.
  • Students pay (more) attention to detail, especially terminology and the sensibility of a proof. It’s easy to write a proof or a solution to a problem that has no coherence or sense to it at all — but that incoherence and senselessness vanishes the moment you do something as simple as reading the solution aloud. Which is what these folks are doing every day. Example: A colleague told me a story of a student who was asked whether or not two groups G and G’ were isomorphic. The student answered, “G is isomorphic, but G’ isn’t.”
  • Students base their confidence on the math itself, not on an external authority. Students aren’t allowed to ask me “Is this right?” or “Am I on the right track?” To clarify, they can ask me those questions, but I will only greet them with more questions — mainly, “What justifies this step?” or “How do you know this?” It’s not about me or what I like or what makes me happy with regards to their work — it’s about whether each step of the proof follows logically from the one before it, and whether that logical connection is clearly validated. Students know pretty well now when they have got something right and when they don’t, and if they don’t have it right they have a better sense of what’s missing or incorrect and what they need to do to fix it.

A lot of these effects I’m describing are just embodiments of what it takes to be successful in math after calculus in the first place.


Filed under Abstract algebra, Education, Higher ed, Math, Problem Solving, Teaching, Textbook-free

5 responses to “Five positive student outcomes from the textbook-free algebra class

  1. Those are pretty impressive outcomes. Do you think the results are only a result of a textbook free course? I think there is probably more to it than that though. I’m assuming you stressed sharing the mathematical authority, among other things.

    Were there any negative outcomes? Anything you didn’t like as a result of choosing to structure the course in this manner?

    Would you do this again? What would you keep the same? What would you change?

    Again, impressive outcomes – keep up the good work!

  2. Jennifer

    I agree, those outcomes are quite impressive — congratulations to you and to your students!

    I stumbled on your blog last week and have been reading about this modern algebra course with great interest. It sounds like a great experience for the students especially, but also a lot of fun for you.

    I’m just starting to toy with the idea of teaching a course next fall using some of these kinds of ideas, so I am curious about the detail of how you (or the students?) choose who presents what.

  3. Jackie: I think these outcomes are more the result of basing the course on *problems to solve* rather than *material to cover*, than they are the result of not having a textbook. But clearly having a textbook tends to place the emphasis on covering material, at least if the textbook is traditional and has the usual 5:1 ratio of text to problems.

    When the main gist of the course is to make it through a laundry list of topics, the students have a lot less invested in the course because the material is all cut, dried, and ready for them. There’s no incentive to prepare ahead, manage the details, or any of that stuff because the material will get covered whether the student does it or not. And a textbook tends to enable the “covering material” approach just because there is a table of contents there which serves as a master list of topics to cover.

    I’m not saying that a prof or teacher can’t make a class problem-solving oriented with a textbook present. But I do think it’s a lot of work to do so, much more work than if you just ditch the book and rely on library sources for your information and make the class focused on problems.

    Negative outcomes? Well, we didn’t cover material as quickly as I would have liked, and so some of the seniors who are taking MFAT exams had to hit the books pretty hard to learn some algebra that normally they would have gotten by now. And I’m not going to be able to cover some of my favorite stuff, like finite fields. Also, I think this system tends to favor students who live on campus (proximity to library) and don’t have a busy non-academic schedule.

    I’d do this particular class this way again in a heartbeat. Other senior-level courses in the major I’d consider as well. I briefly looked in to doing my sophomore-level differential equations class like this, but I don’t think this way of doing things is well-suited for a more computational course like that. I’d very much like to try calculus this way sometime, though…

    As to what I’d do differently, etc., I think I’ll hit that with a post later rather than prolong this already lengthy comment!

  4. Jennifer: Briefly, the selection process worked like this.

    (1) Every day I made up a new randomized list of students in the class.
    (2) I would announce the next task from the notes that was up for grabs and ask who had completed that task.
    (3) If no hands go up, we table that course task until the next class. If the course task had already been tabled the previous time, it is “retired”. So there was a two-day expiration date on all course tasks, just to prevent us from getting bogged down in one section forever. Sometimes I would present solutions of retired problems myself, or post them on the course wiki.
    (4) If one or more student raises a hand to present work on the course task, I’d go to the first person on the randomized list whose hand was up. If that person had not presented yet today, they get the task. If they had presented already, I go to the next person on the list, continuing until I either assign the task to a person who hadn’t gone yet today, or until I loop around to the beginning of the list.
    (5) If I make it to the beginning of the list again, the problem goes to the first person on the list whose hand was up.

    Special case of (5) — if only one person raises her/his hand for a course task, they get it. One day there was a student who racked up 20 points on four different tasks, presenting one after the other, because she was the only one who had worked that far ahead in the notes.

    I might try illustrating this in another post, too.

  5. elementaryteacher

    I think this is a wonderful approach. I can see that it would be difficult to cover all the” material. I think every teacher has to make that choice–to go more in-depth on less material, or rush through all the material in order to “cover” it.

    Dedicated Elementary Teacher Overseas