Groupprops:Introduction for teachers

Dear teachers!

A very warm welcome to Groupprops. Most of my knowledge of, and passion for, group theory comes from the great teachers I've had, some who've taught me personally, and some who have taught me through their books or other indirect support. I appreciate the crucial role you play in developing an understanding and passion for mathematics for your students. I hope you find the content and organization of this wiki interesting.

What the wiki model means for you
How does Groupprops affect your teaching activities? I've been working hard for the last few months to figure out how Groupprops can be made a better resource for teachers. A lot of resources and ideas are still in a development stage, but I'm convinced that the impact on your teaching activities can be a lot.

One of the problems I faced as a student, attending a lecture and trying to take notes, was that when I came back from the lecture, I sometimes couldn't make sense of my notes. What could I do? Read the textbook -- the textbook is long, and covers things in a somewhat different order. Or bother the teacher or teaching assistant? Not for every little thing.

Another problem was remembering the little examples and counterexamples, and looking up references. Teachers make painstaking efforts to look up references for students, but despite my best intentions, I've hardly ever looked up the references for the little examples and counterexamples. It's too much work to locate a book in the library and find the relevant page. Besides, I've to do that every time I need to look up the little example or counterexample.

Yet another problem is that of problem sets and exercises. A huge amount of preparation time for teachers goes into preparing and grading assignments. And students also don't have quick access to a nice common problem pool. Each book has its own problems, but the problems are localized to the books; hopping between books, assignments and other sources for problems is a pain.

I've tried to use Groupprops as a solution to these problems at two, somewhat different, levels. At one level, the structure and organization of the wiki itself means that after the lecture, the student can go back, look at the lecture notes, and check whatever was unclear on the wiki. Didn't understand a proof? No longer a need to open up a book and be told that the proof follows from Lemma 2.3, Proposition 4.1 and Corollary 4.3. Instead, just type the name of the theorem in the search bar, and read a fully detailed article on the proof, with links to all relevant references in books and perhaps even to papers written about the theorem!

To see the way things are organized here, check out definitions like characteristic subgroup or permutable subgroup or proofs of facts like characteristic implies normal. Or a counterexample page like normality is not transitive. Many of these pages are undergoing significant reworking, but I think the current state of these pages shows how easy it is to look up and cross-check facts.

The second level at which I've been trying to solve this problem is the creation of add-ons. Loosely speaking, I'm working on articles that are not part of the main wiki but that use content on the wiki to provide resources useful to students. Some of these developments are:


 * Problem sets: These are huge lists of problems in relevant topics in group theory. You'll see that many of the principles of instructional design have been merged smoothly with the organizational principles on the wiki. The solutions not only give a quick explanation but also link to relevant term and fact pages so that the student can take the opportunity to read more about things as he or she solves the problems. see, for instances, problems in elementary group theory (still in development).
 * Entertainment: Entertainment is important, and I've been working on some ideas of how entertaining articles can help cement a student's love and understanding of group theory. See, for instance, what on earth is a group? (still in development).
 * Dynamic guided tours: Here, students are taken through truncated versions of pages in a select sequence, with summaries given periodically. In between, entertainment pages are shown for short breaks, and problem sets are administered regularly. Learners arealso encouraged to take detours into related articles to get a better feel of what's going on.

We're different from Wikimedia projects
We differ significantly from projects like Wikipedia, Wikibooks, or Wikiversity. For one, we are very few right now -- this wiki has approximately 15 registered users and most of the edits are done by one person (that's me). Moreover, our policies differ significantly.

Too much nonstandard material and too much content per page
It could be extremely confusing for students to go through the Groupprops wiki without guidance, because we have a lot of nonstandard terminology (terminology local to this wiki). Moreover, even the pages on simple topics often contain details that the student might find distracting.

The first problem can be solved somewhat, through the dynamic guided tours, and by encouraging students to concentrate on these categories:


 * Category:Basic definitions in group theory
 * Category:Semi-basic definitions in group theory
 * Category:Basic facts in group theory
 * Category:Semi-basic facts in group theory
 * Category:Elementary non-basic facts in group theory
 * Category:Results with direct proofs

On balance, though, I think it's good to give students access to more information and ideas; it encourages them to explore freely. Those who aren't interested in more will anyway concentrate on the definitions. But those who are interested have a wonderful forest of ideas to explore.

Too many proofs
Even the simplest statements have proofs, as indicated in Category:Results with direct proofs. As the number of fact articles expands, you may find that many textbook problems can be solved just through a quick search on the wiki. Does this discourage original thinking? Hardly.

The presentation and layout of the proof pages has been designed to really encourage the student to think about why the proof works, to consider related proofs, to understand ideas behind the proofs, and to revisit the definitions and the lemmas and ideas used. The purpose isn't just to create a solution booklet: the purpose is to spur the student's creative thinking. As a resourceful teacher, you'll be able to come up with problems that cannot be solved through a direct search on the wiki. In fact, we do not put facts on the wiki that amount to trivia, whereas assignments could contain trivia problems that use deeper ideas (like the problem of proving that the quotient of the free group on all English letters by the equivalence relations induced by similar-sounding letter combinations, is the trivial group). A student confronted with such a problem must then identify resources on the wiki that are most helpful, and in the process, learns a lot of important results as well.

Dry style and too few survey articles
We're working on more survey articles! For instance, varying group, arithmetic and normal subgroup structure, varying normality, or characteristic versus normal. Your suggestions for what kind of survey articles are needed, is most valuable.

See Category:Survey articles for a complete listing.

Also, our development has been somewhat lopsided, and we are working to correct that.

Thanks, and hope you have a nice time around the wiki!