It’s been a while since I last said anything about the textbook-free Modern Algebra class experiment. This is mainly because the class itself is now underway, five weeks into the semester, and it’s only now that I’ve got enough perspective to give a reasonable first look at how it’s going. So, let me give an update. (Click to get the whole, somewhat lengthy article.)

After an initial idea that students would be doing a lot of library research to find and present basic course information, I decided to get away from any kind of lecture at all, whether it was given by students or by me. Instead, I ended up changing the whole structure of the course to be a sort of modified Moore method. I created (and am still in the process of writing) a large set of course notes that consist mainly of two kinds of things: information that I give the students, and stuff for the students to do. The information I give consists mainly of definitions, comments, and a few worked-out examples. The stuff for the students to do, which goes under the official name of “course tasks”, consists of three different kinds of things as well:

**Questions**for students to answer;**Exercises**to work; and**Theorems**(and lemmas and corollaries, etc.).

To give a flavor of what this looks like, here’s a shot of a sample page from Chapter 3 of the notes, which is on basic properties of groups.

The questions are usually simple but designed to get students to develop the habit of asking penetrating questions when they encounter a theorem or a definition. The exercises are a little more computational in nature, having students do some things and then perhaps make an observation. Some exercises are also simple results that require a proof, or perhaps a proof of a special case of a theorem. Theorems are, well, theorems — basic and foundational results that are sometimes easy to prove and sometimes devilishly hard.

They are collectively called “course tasks” because it’s intended that the students will be supplying all the answers to the questions, all the solutions to the exercises, and all the proofs to the theorems. The notes just list them; the students work them out at the board, and that makes up most of the meetings. That’s what makes this Moore-method-like.

What makes the course a “modified” version of Moore is that students were not intended to be doing all this work from the outset. Instead, I created a graduated schedule for when students would be responsible for various tasks. During the first week and a half, I did everything for them; by doing so, they could see what sorts of methods were involved in doing tasks in abstract algebra and what kinds of thinking are involved. Then, in week 2, I declared them to be responsible for answering all the Questions in the notes, while I still held responsibility for the exercises and theorems. Then, after another week or so, I added all the exercises to their list of responsibilities while continuing to prove all the theorems. Finally, about a week ago, I handed over the responsibility for proving the theorems, so that now they do everything.

I like this way of doing Moore method, because if I were to do it again with a different class, I could accelerate or decelerate the schedule of handing over task responsibilities based on the strength of the students in the class. If I have a class with a particular weakness in proof-writing, I can wait longer to hand over the responsibility for the theorems. Conversely if I have a stronger class, I can give them that responsibility earlier. But eventually they will all be responsible for every task.

And my responsibility is now more of a traffic cop, coach, and irritating question-asker. Usually when somebody gives a response to a course task, there is some kind of discussion, and it’s my job to see that everybody gets their questions answered. And if there are no questions where there ought to be, my job is to initiate that discussion. But I do not lecture; the only time I work things out for the students is if there is an exercise or theorem that we must do right away before moving on, and nobody has it; or if they have a question that needs further work to get a sensible answer.

I am breaking one of the fundamental rules of the Moore method with the theorems in that I am allowing students, if they choose, to go find the proofs of the theorems in books or online and simply explicate what they find. This may seem like institutionalized plagiarism, but really it isn’t; I’ll allow students not to come up with original proofs, but their grade on the theorems is mainly based on the clarity of their exposition and how well they field follow-up questions from me and the rest of the audience. In other words, you can copy a proof out of a book and give it if you want; but you had better really understand it. This helps students get the really tricky proofs out there, such as the proof of the Division Algorithm; and it helps justify the one-size-fits-all point value of 8 points for every theorem, regardless of difficulty. And it encourages research, which was the whole point behind not having a textbook to begin with.

I am also incorporating technology into the course in the way I have students write things up. When a student completes a result, that student is responsible for writing it up and putting in our course wiki. Over the summer, Wikispaces started offering native LaTeX support so that you can type in the LaTeX code and it will compile when you save the page. We’ve used this to post results that were done in class with actual math formatting. Probably one of the future assessments will be “open-wiki”; I do not give points for posting results nor do I subtract or withhold points for not doing so, but it will be that much less information for students to use on an open-wiki assessment if they don’t do it.

As far as assessing all this goes, answers to Questions are worth 2 points; solutions to exercises are worth 4; proofs to theorems are worth 8. I will calculate a course task grade at the end of the semester by taking either 100 points or the highest point total earned in course tasks by a student, whichever is greater, and using that as a divisor to compute a percentage. So if a student earns 88 points in course tasks by the end of the semester but another student earned 120, the first students course task grade is a 73%. The 100-point minimum is, to make an analogy with eBay, the “reserve” that must be met. (Right now, in week 5, the high course task score is 38 points. That will climb a lot higher since they were only given the responsibility for proving the theorems, 8 points apiece, last week.)

Students will also do between 5-6 problem sets (the problems are sprinkled throughout the notes at appropriate places), take two in-class tests, and take both a midterm and a final exam. That gives five distinct types of assessed work. I am letting students assign their own weights to these five types. They can weight each type at 20% of their grade; or 40% on the course tasks, 30% on the problem sets, and 10% each for tests, midterm, and final; or whatever. The only parameters are (1) the weights have to add up to 100% and (2) no single item may be weighted less than 10% or more than 50%. This allows the students some flexibility to target the course at their strengths; all aspects of the course are pretty rigorous and so there’s no obvious easy way out here. However, everybody has weighted their course tasks about 35%. Fine by me — the higher you weight the course tasks, the greater incentive you have to be the student with the highest point total and (presumably) the harder you will work at those tasks.

So far, I would say that this course has been one of the most fun, satisfying, and academically rigorous courses I have ever taught. Students are working hard and, even when they make mistakes, the mistakes become lessons by which students learn and improve. And there are some surprises, such as the Religious Studies major with a math minor who is currently outperforming many of the experienced math majors, or the student who works 40+ hours a week and has not done so well in past math courses who has brought in some beautiful work on theorems and exercises and has really become a lead expert in the subject matter among the students. I’ve been contacted by one of the students in the class who said that not only was this the most interesting and engaging class he’s had in the math program, but could I please consider adopting this course structure for some more courses I teach in the future. (I’m thinking about it; I have Differential Equations coming up next semester and that seems like a good place to try this out.)

I credit the positive experiences in the class to the level of activity that is required of the students in each class meeting. Students cannot consistently be passive recipients of this material without failing the course. They have to get in and get their hands dirty to hammer out the concepts and tasks on their own, or they won’t make it. Students are working ahead every day in the notes because they realize that doing so puts them in position to do well.

And I credit the activity in the class not so much to the absence of a textbook — because my course notes are really “the book” — but rather to the sparsity of the notes. The notes give a bare minimum framework of information, rather than working out every conceivable angle for students like textbooks usually try to do. The bulk of the notes are questions and things to do; they can be used as a reference (hence the numbering of all theorems and other tasks) but primarily it’s a workbook, not a textbook.

To give credit where credit is due: I do have an underlying philosophy of my own as to what appears where in the notes, and maybe I will go into that in another post. But I am pulling a lot of my information and the structuring of that information from Joe Gallian’s textbook with help from Lindsay Childs’ book and Hungerford’s classic text which I used in graduate school. In the future, I may go back and decouple my notes from all these other sources to make them truly my own, and then put them out on the web for public consumption.

Coming up for us in the class is our midterm exam, which will be the most “expensive” assessment yet, and a formative assessment to boot — so we will see how well students are learning the material when they are put to a test like this. But I think that they will be OK.

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What I really liked about your post was the question students are asked about “WHY are we proving this?” That’s what was always left out of my math courses as a kid. Even though I only teach Grade 3 math, I always try to show kids WHY we are studying something, and how it might be useful to their lives NOW. For example, in introducing number lines this week, I pulled down the world map and showed them how elevations are marked by color, above and below sea level, in various colors, in a sort of number line. I pointed out how a thermometer is a sort of vertical number line. I pointed out how in history, A.D. and B.C. are sorts of number lines. We spent nearly a period discussing USES of number lines before we went into working on mastering them.

Eileen

Dedicated Elementary Teacher Overseas

elementaryteacher.wordpress.com

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The “why” question is certainly important on the broad scale. However, here I was being a little narrower than maybe you are thinking. The point of Question 2 was to have students think about what the group axiom about inverses says, and more importantly what it does NOT say. The group axiom does not mention uniqueness at all. But to the casual reader — which I do NOT want my students to be — the Theorem they are proving is exactly the same thing as the group axiom. The real problem in proving the theorem is realizing what the difference is; once you realize it’s the assertion that the inverse of a is unique, you’ve got some idea of what you need to do.

We had a short but interesting discussion after that question in class, about why we didn’t just stick “unique” in the axiom to begin with and have done with it, and why we want the things we assume to be true without proof to be as minimal as possible.

Yesterday I ran through 5 statements with my geometry students, then told them that we would be assuming one was true, and using that to prove the others (as an aside, I pointed out that it didn’t really matter which we chose, that we would choose the same one as our text, and that Euclid himself had assumed another…

All smiles and nods.

And so we assumed one, and I asked them to prove another. We’ve proved little things before, but this was our first theorem or corollary. And they struggled with what they needed to prove, and why they didn’t already know it.

Not the same as your problem, but there is something similar in the difficulty understanding what is to be proven, how, and why.

(Our text substitutes “If 2 || lines are cut by a transversal, corresponding angles are =” for the parallel postulate. Lot’s of opportunity for proving little theorems)

I like this approach

I love the idea of Moore Method for teaching Maths. If students learn the power of independent application of exploration, they will become mathematicians rather than formula crunchers. But potentially even more important, they will become skilled modellers of the world they inhabit. The excellence of their mathematics then becomes a useful byproduct of their education:-)

My current interest is in teaching my 4 year old independent thought – ideally also to be a mathematician. I am interested in extending this type of approach to early school so my child never has to unlearn didactic approaches and hopefully becomes an explorer of the world in a way my school education never prepared me to do.