Category Archives: Abstract algebra

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.

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

Fear, courage, and place in problem solving

Sorry for the slowdown in posting. It’s been tremendously busy here lately with hosting our annual high school math competition this past weekend and then digging out from midterms.

Today in Modern Algebra, we continued working on proving a theorem that says that if a is a group element and the order of a is n, then a^i = a^j if and only if i \equiv j \ \mathrm{mod} \ n. In fact, this was the third day we’d spent on this theorem. So far, we had written down the hypothesis and several equivalent forms of the conclusion and I had asked the students what they should do next. Silence. More silence. Finally, I told them to pair off, and please exit the room. Find a quiet spot somewhere else in the building and tell me where you’ll be. Work on the proof for ten minutes and then come back.

As I wandered around from pair to pair I was very surprised to find animated conversations taking place about the proof. It wasn’t because of the time constraint — they’d been at this for three days now. For whatever reason, they were suddenly into it. One pair was practically arguing with each other over the right approach to take. By the end of the 10 minutes, two of the groups had come up with novel and mathematically watertight arguments. Between the two, and with a little bit of patching and a lemma that needs to be proven still, they generated the proof.

One student made the remark that she had been thinking of these ideas all along, but she didn’t feel like it was OK to say anything. This is a very verbal, conversational class done in Moore method style, so I can only interpret that comment to mean that she didn’t feel free enough, or bold enough, to say what she was thinking. The right proof was just bottled up in her mind all this time.

There’s something about our physical surroundings which figures in significantly to our effectiveness as problem solvers. Getting out of the classroom, for this one student at least, was tantamount to giving her permission to have the correct thoughts she was already having and to express them in a proof. I think our problem solving skills are highly inhibited by fear — fear that we will be wrong. And it takes a tremendous amount of confidence and/or courage as a problem solver to overcome that fear.

When you feel that fear in a classroom, it becomes compounded by the dread of looking like an idiot. Changing the surroundings — making things a little less cozy, a little more unusual and uncertain — doesn’t seem to make the fear go away as much as it helps us feel like that fear is perfectly normal and manageable, if not less fearful.

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Filed under Abstract algebra, Critical thinking, Education, Higher ed, Math, Problem Solving, Teaching

Textbook-free Modern Algebra update

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.) Continue reading

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Two thoughts on what Modern Algebra is

I’ve been working most of today on the foundations of my Modern Algebra course for the fall, which you will remember is being done without a required textbook. The last time I posted about this course, I made the point that not having a textbook around forces the professor to think about certain deep issues that often get glossed over — for example, what the course is all about and how the topics connect together and reinforce the main questions of the subject. Good college professors always think about these things regardless of textbooks; the absence of a textbook, then, forces the professor to think like a good professor. And that can’t be bad.

Along those same lines today, it occurred to me that there are two particularly useful ways to consider the entire subject of Modern Algebra.

1. Modern Algebra can be thought of as the study of why algebra, as the students know it, works. For example, we all “know” that if 0x = 0 for all x. But how many college students have thought about why this is true? It’s easy enough to give a hand-wavy explanation that ends up sounding like a 60’s hippie anthem (e.g. “Nothing times x can’t give anything but nothing!”), but is there a more solid mathematical explanation? The answer is yes, and surprisingly the thing that makes it work is the distributive property[1]. So we find that the distributive property of multiplication over addition is a deeper property of the real numbers than any supposedly mystical properties of the number 0.

This view of Modern Algebra makes the subject immediately applicable to education majors, who make up the majority of the students I’ve had who take this course. Some smart kid in their class someday is going to ask why 0x = 0. Don’t insult that kid’s intelligence by offering half-baked mysticism — it’s a real question, so give them a real answer. Or at least convey to them the idea that there are real reasons why things in math actually are true, and we don’t have to rely on blind faith or the instructor’s (or textbook’s) authority.

2. Likewise, Modern Algebra can be thought of as the study of finding the largest possible collection of structures for which algebra, as the students know it, works. I mentioned 0x = 0 above. But I didn’t say initially what kind of thing x is, nor what I meant by “0”. There are lots of things we denote with “0” in math — the zero vector, the zero matrix, the zero polynomial… What kind of thing is it now? And more importantly: Does it matter, or is this a property that is true for any situation where “0” and multiplication make sense?

On the other hand, everybody “knows” that if ax = bx and x is not zero, then a = b. Right? Well, what we find out is that it depends. It depends on what the a, b, and x represent. If they are integers, real numbers, or polynomials with real coefficients, then a = b always in this situation. But if they are 2×2 matrices, then it’s quite possible for a ≠ b — just let a and b be any nonsingular matrices and x any singular matrix. So it does matter here exactly what kind of object a, b, and x represent. So we make an important discovery: There is a limit to how far we may apply the notion that ax = bx implies a = b. It doesn’t always work. Which leads to the next question — under what conditions will this implication hold? What algebraic conditions have to be in place to use this simple cancellation idea? This is the right kind of line of questioning that a junior/senior in mathematics, especially somebody in math education, ought to be pursuing.

Taking Modern Algebra and casting it in terms familiar to students like I’ve done here seems to be crucial to crafting a course that isn’t just going to be written off by students once the exam’s over.

[1]: 0x = (0 + 0)x = 0x + 0x. Subtraction gives 0x = 0. This proof works in any instance where multiplication and addition are defined, additive inverses exist for each object, and multiplication distributes over addition. The ring is the kind of structure we usually think of here.

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The Big Picture of abstract algebra?

 English Images FarmarialI’ve been working here and there on my Modern Algebra class for this fall. As regular readers know, I am doing this course this time around without the use of a required textbook. One of the difficult, and good, things this approach imposes on me as the professor is that I cannot rely on the book to provide structure and order to the course. I have to do this myself. Before I can do any realistic planning, I first have to decide what I am going to cover and the order in which I am going to (try to) do it. And before I can do that, I have to face some questions that professors are surprisingly able to sidestep when using a textbook, namely: What is this course about? What themes unify, and therefore motivate, the material? And what are the core issues and questions that this course attempts to address?

Far too often, students can take a course in college or high school and make good grades, and even are able to do some of the tasks that the course outcomes in the syllabus require. But they are not able to explain at the end of the semester or school year just exactly what it is that they just studied. They are deprived of the Big Picture, the aerial shot that shows where everything is in context, because nobody ever thought to bring it up. They and their teachers just brachiate from one topic or calculation to the next without thinking of where it all fits and what it’s all about. Whether this is out of the teacher’s sheer laziness, incompetence in the subject matter, or just not having enough time to do it, is irrelevant — students can’t be expected to demonstrate real mastery of a subject if they don’t know what it’s about and what it was invented for. If they have that mastery in the absence of the Big Picture, it’s a coincidence.

That strongly worded statement out of the way, here is my initial attempt at articulating the big issues and questions in Modern Algebra.

Students in this course have, by the first day of class, completed a gamut of advanced mathematics classes including two semesters of calculus, a course in advanced problem solving, and a semester of linear algebra. In those courses, and going back to high school, we’ve seen all kinds of things one can do with mathematics. All of the things we do in math tend to revolve around certain kinds of objects that we do things to. Perhaps the six most prominent types of objects we encounter in math are

  • The integers
  • The rational numbers
  • The real numbers
  • The complex numbers
  • Matrices
  • Polynomials

And as far as doing things to these objects, there is — somewhat surprisingly — a lot of overlap. In particular, we have a notion of addition for each of these and a notion of multiplication. Although the specifics differ — multiplication of 2×2 matrices is a lot different than multiplying two polynomials, for instance — these two operations give us many things to do with all six of these objects. And high school algebra is spent pushing these operations to their limits, doing things like solving equations and factoring and so on.

The purpose of Modern Algebra is to find out what we can do with these sets and the operations of multiplication and addition, and — crucially — any other set of objects with similar operations that behave like these sets. Throughout the course, one takes the Big Six sets of objects and any other set with operations that behaves in some sense (to be defined later) like them — for example, matrices with only integer entries, or fractions whose denominators are powers of 2, or polynomials with matrix coefficients — and asks the following questions. These questions are the essential motivating themes of the course:

  1. What sorts of arithmetic properties hold? For example, do we have the distributive property, the commutative property for one or both operations, etc.?
  2. What kinds of arithmetic properties do all of our sets with operations have in common? Which ones come close?
  3. Can we solve equations with these objects and their operations?
  4. Is there a sort of “basis” for each set with operations, in the sense that any element in the set can be built out of “basis” elements using the operations provided? [Example: Prime numbers form a “basis” for the integers if you use multiplication (and include 0 and -1).]
  5. If so, then is it possible to take an element of our set and factor it into a combination of “basis” elements?
  6. If so, then is that factorization unique?

In other words, the whole course comes down to four basic problems:

  • Defining arithmetic and deriving arithmetic properties (or failing to do so, and taking note)
  • Equation-solving
  • Finding a “basis” for a set with operations
  • Factoring, with a view toward unique factorization (or determining that factorizations are not unique and examining the consequences)

I think those six questions and the four basic problems provide a nice umbrella for the entire course — which will run through basic group theory and a bit of rings and fields — in a way that definitely “feels like algebra” for the students. I’m still working on how best to run the day-to-day classes and how best to assess students, but I believe that every day and in everything students do, we’ll come back explicitly to these two lists.

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