Technology can be used to supplement a modern algebra course to increase exploration of the relationship of abstract algebraic structures to real life problems. We will look at the effectiveness of this in a Spring 2000 course (see http://www.mathsci.appstate.edu/~sjg/class/3110 ). We will explore Maple demos on the applications of modular arithmetic to check digits for money orders, and the application of the direct product of cyclic groups to HBO cable security. We will then talk about followup WebCT quizzes and activities, and discuss the students' experiences with the Maple demos and WebCT activities.
I also wanted students to learn some applications of algebraic structures to real life. This was done mainly via Maple demos, which are accessible from my web page. We also learned about the history of algebra through algebraists and their mathematical accomplishments.
During this talk, we'll examine my use of technology in the course, specifically Maple and WebCT, and the effect it had on the students.
Our first Maple demo took place during class in the computer lab about 5 weeks into the course. It was on modular arithmetic and was based on a section from Joe Gallian's Contemporary Abstract Algebra. Students were to work with one other person and each group was to type in their answers to questions on the computer. We first defined a mod n as the remainder of a divided by n, and we had some questions on this, which they did well on.
The demo explained that the US post office uses mod 9 for the check digit on money orders. The US post office check digit is a 10-digit number mod 9. So, you hand me any 10-digit number, for example, the number in this demo. You look at the remainder of it divided by nine and the answer, in this case 2, is the check digit.
The post office then uses the 11-digit number formed by a 10-digit code with the check digit appended onto the right side. The post office puts this 11-digit number on its money order.
So, if we wanted to check for a forgery or if a money order number was incorrectly entered into a computer, the program would remove the last digit and check to see that it equals the remaining 10 digits mod 9.
During the rest of the demo, students were asked whether certain typing errors could be detected via the check digit. As I came around the computers and looked at their answers, I noticed that they did very well. We went over the demo the next class day.
Subsequently they had a generalized version of check digits as a problem on their problem set. They had their first test a couple of weeks later that did not include this material. This first test, on proof writing, sets functions and rationality was based on class work and standard problem sets. Even though they had done a couple of revisions on each problem set, the material still had not sunk in - they did not do very well.
So, I decided to try something new with them to help them jell the material - WebCT.
This is what the students would see once they logged in using their password. I mainly used the bulleting board and online quiz features of WebCT to supplement the standard problem sets and Maple demos, which I kept doing.
If I student clicked on online quizzes from this page, they would see the following:
Notice that the first quiz was not offered until the end of March. That is because I decided to use WebCT for drilling material because of the results of the first test. I learned about WebCT details over spring break in a workshop my university offered. We are going to look at the short answer part of test 2, given a week after I started using WebCT in the course.
The great advantage of WebCT quizzes is that when the students are finished with the quiz, they click "Finish" and then WebCT instantly grades their work and offers instant feedback.
The students did better on this test than on test 1, especially on proof wiring questions. They said that the ability to keep redoing quiz 1 and the instant feedback had helped them a lot.
But, I decided to rethink my Maple demo strategy of having them answer questions and also going over the demo in the next class, since they did terribly on this problem on the check digit.
We'll carefully examine one of the questions that I asked them and their posted answers and discuss how their posts helped me identify problems tha they were having.
A binary string is a string of 0s and 1s. We can think of this as an element of the direct sum of Z_2, the cyclic group of order 2. Then, we can define an operation, *, on bineary strings via adding corresponding digits mod 2.
. We then went over some examples of this and Maple's syntax.
I wanted to prove this for them, since this is the bases of a data security system used by HBO to protect its television signals.
I just did one direction of the proof and asked the students if it was correct. They had been asked similar questions in the past, with my proofs sometimes correct but sometimes incorrect (ie they were not necessarily expecting to find an error).
So, we want to prove that if 2 binary strings sum mod 2 to the 0 string, then each corresponding binary string digit is equal.
So fix i. Now a_i, b_i in {0,1} means that we have 4 cases for a_i+b_i mod 2.
I asked the students if my proof was correct. Notice my typo of 2 instead of 4. I announced this typo in class when a student asked if this was the mistake they were supposed to find.
The rest of the Maple demo is about the relation of this to HBO scrambling. You can access this demo from my web page.
When I went around the room and talked to them about this question they told me that they thought the proof was correct except for the typo.
I created bulletin boards of different topics and the students were asked to post within the Maple Demos topic.
While Anna wrote: "The proof is more than likely correct since you have shown that 1 and 4 work, but what about 2 and 3? Would that not show a contradiction?"
In Monday's class we went over the fact that the given info told us that we had to be in either case 1 or 4.
A number of students had been confused about this, but hadn't brought it up in their group or to me. I was pleased they they brought it up in the newsgroup posts instead of just agreeing with other posts. For some reason, for certain, often quite, students, posting felt like a safe place for them to post their questions.
WebCT really helped the students in this course. They loved being drilled on webCT quizzes, and felt in evaluations that they learned a lot from this. It revitalized the class, which had been tired from working hard to learn proof-writing skills via standard problems sets and classroom presentations and group work. This newfound energy helped the class continue to work hard and succeed at proofs.
I think that it was a good supplement to the problem sets and Maple demos and did help to jell the material for them.
The Maple demos enabled me to communicate interesting topics to the students that would have taken a long time to do by hand. Maple was used interactively as much more than just a calculator (see below) While the bulletin board was used lightly, it also helped to reach more students by forcing them to communication their ideas in a different forum that especially normally quiet students found less intimidating.
I plan to use it again in this class.
html - Maple Demo on
the Insolvability of Quintics by Radicals
mws Maple file of this demo
see also web pages
on the Solution of the Quadratic, Cubic, Quartic and Quintic by Radicals
html-Maple Demo on 1-1 and Onto Functions
mws Maple file of this demo
html - Maple Demo on Sophie Germain's
Modular Arithmetic Work on Fermat's Last Theorem
mws Maple file of this demo
html - Maple Demo on Applications of Direct
Product of Cyclic Groups to Data Security
mws Maple file of this demo
html - Maple Demo on Quaternions
mws Maple file of this demo
In class, we discussed the relationship of quaternions to the space shuttle.
Quiz 1 Solution by radicals, Fermat's Last Theorem,
definitions of function, one-to-one, onto, and the negation of the
definition of continuity.
Quiz 2 Groups
Test 2 short answerInduction, real numbers,
modular arithmetic applications and Fermat's Last Theorem biographies.
Test 2 long answer Induction.
Note that WebCT can't
grade this to give instant feedback -
the instructor must go into webct to grade long answer problems
.
Quiz 3 Groups and Rings
Quiz 4 Algebraic Structures
Quiz 5 Algebraic Structures
Graded WebCT section of the final exam
What the students see after they submit. (This is set to reveal the
answers although usually I turn this feature off.) The remainder of the final
exam consisted of standard proof-writing problems.
E-mail questions or comments about these pages to Dr. Sarah Greenwald, greenwaldsj@appstate.edu