If You Want To Know What Students Understand, Ask Them!

To gain insight into how much students actually understand, and what they have learned by working the problems, have them discuss the evolution of their ideas as they work on homework sets.

Background and Purpose

The University of Wisconsin Oshkosh is both a major undergraduate and regional graduate campus in the statewide University of Wisconsin System with a student population of about 10,000. Many of the students I work with are pre-service elementary and middle school teachers. I teach an Abstract Algebra class for prospective middle school mathematics teachers. This course is specifically designed as part of a mathematics minor for those majoring in elementary education.

In my classes I assign non-routine, challenging problems that are to be handed in and evaluated. These problems usually go a step beyond the material discussed in class and are designed to set up situations where students can discover some mathematics on their own. Most students are successful in working exercises similar to those done in class and examples in the text, while few hand in "good" solutions to the challenging problems. I found myself giving low scores to what looked like poor attempts and meager results.

Anyone familiar with the history of mathematics can think of several examples of mathematicians who have worked on a given problem for many years. Perhaps the original problem was never solved, but the amount of mathematics learned and developed in working on the problem was astounding. I applied this situation to my students. What is more important: that they solve a given problem, or they learn some new mathematics while struggling with a problem? I was looking for papers which were organized, logical, thoughtful and demonstrated students had engaged themselves with the course material.

Method

In many of the problems I assign in my Abstract Algebra course, I expect students to compare properties of abstract mathematical structures with more familiar structures (integers and rationals). However, these students are not yet ready to respond to this kind of assignment with a clear, textbook-like essay. To help them understand what I do expect of them, I hand out "Guidelines for Written Homework," which explains

Your papers can be as expository as you want to make them. Tell how you arrived at your particular solution. You can discuss how the problem relates to material in class and possible applications of the problem. Do you understand the class material better now by working on this problem? Explain. If you could not arrive at a solution tell what you tried and why it didn't work, or describe what you think may work. Vent frustrations! Identify obstacles you encountered and what you think you need to overcome these. The most important thing is to demonstrate to me that some thought went into your work and that you learned something!

I add that an expository paragraph describing their experience with the problem must be included with the solution.

For example, the following homework assignment was given while we were discussing groups, before they were aware that a group satisfies the cancellation property.

HOMEWORK #3

When working with numbers, we "cancel" without even thinking. For example, if 3 + x = 3 + y, then x = y, and if 3x = 3y, then x = y. The purpose of this homework problem is to get you thinking about this "cancellation process."

a. Let a, b and c be elements of Z₆ (integers modulo 6). Then if a b = ac, is it necessarily true that b = c? Justify your answer.

b. Let a, b and c be elements of Z₆. Then if ab = ac, is it necessarily true that b = c? Justify your answer.

c. Let (G, *) be a mathematical structure and suppose that a, b and c are elements of G. What properties must (G, *) satisfy so that a * b = a * c implies that b = c?

Findings

The expository portions of the first two assignments were quite unrevealing. Comments like, "the problem was hard," "I really didn't understand what you wanted," "I worked 2 hours on this." "this one wasn't too bad," etc., were common. I used the "power of the grade" to obtain more thoughtful responses. If, in my opinion, the expository section was too shallow, I would deduct a few points. I would also include a question or two which they could consider for the next assignment. Those who did not include any expository section, lost several points. After a few assignments, I began to get some interesting comments.

Assignment #3 described above was the first one on which I received good comments. Comments I received included:

I first thought in part (a) you just subtract a from each side and in part (b) and (c) you just divide by a on each side. When I mentioned this to you, you asked me if subtraction and division are defined in these structures. This helped me in part (b), because we did a problem in class about division in Z_n. But I am stuck with part (c). How can I define division in (G, *)? I am sure that when a, b and c are in Z_n, ab = ac implies that b = c if a is not a zero divisor because I can then define division. You said I am on the right track, but I still don't see it.

I didn't really have any trouble understanding parts a and b. I simply made the tables and checked to see if each element of Z₆ appeared at most one time in each row of the table. If this happens the element can be canceled. For example in Z₆, 30 = 34, but 04. In the "3" row, the number 0 appears more than once. However, part c was a different story. I now see what you mean when you say that it is different to show something which is an example and something is true in general. I could just make a table for Z₆ but how in the world do you make one for (G, *)? When I came and talked to you, you told me to think of the algebraic properties involved. I didn't know what you meant by this. You then asked me what are the "mechanics involved in going from 3x = 3y to x = y." I then started to see the picture. Is this what abstract algebra is like? Trying to decide what properties will allow certain results?

Neither of the students who wrote the above paragraphs above solved part c. But because of what they wrote, I know that the first student has a better understanding of what it means for division to be defined and the second appreciates the difference between working examples and justifying statements in general. Both of these tell me the student has learned something.

I realized that this was the first assignment where the student was given an open-ended problem in which they needed to make a conjecture. I didn't require proof. They needed to convince themselves that their guess was accurate. I also got something that I didn't expect. Here is what one student wrote about part c:

I feel that the examples we did in class on Friday helped serve as a guide for this problem. It wasn't really too difficult. I didn't spend very much time on it.

By itself this seems like a pretty insignificant comment. However the solution the student handed in was completely incorrect! Some students submit incorrect solutions knowing they are incorrect. That, in itself, shows that the student knows something. It is quite different, however, when a student believes an incorrect solution is correct. I simply wrote a note to this student and we discussed this problem in my office.

Later in the semester I gave an assignment which was designed to have the student conjecture when a linear equation ax + b = 0 (with a and b elements of a ring R) has a unique solution. In the first part, students were asked to find examples of a linear equation which has more than one solution, a linear equation which has a unique solution, and a linear equation which has no solution in each of the rings Z, Q, Z₄, Z₅, and Z₆ (if possible). Then they were asked to give a general condition which will imply that a linear equation ax + b = 0 over a ring R has a unique solution.

Most students could completely handle the integer and rational cases. Additionally, only a few struggled with the linear equations over Z₄, Z₅, and Z₆. Discovering the algebraic properties involved was the heart of the problem and the difficult part. One interesting comment I received indicated that the student never thought about the number of solutions an equation has and that this is somehow connected to the elements and properties of the structure. She just solved them without much thought. She also wondered if the same problem could be solved for quadratic equations.

A second student thought that a linear equation always had at most one solution. This proved to be true in Z and Q. She described trying to use "regular" algebra in Z₄ by subtracting b from both sides and then dividing by a. This caused problems when she found that division is not always possible in Z₄. So she chose numbers and substituted them for a and b in the equation ax + b = 0. She solved the problem in Z₄ by using the operation tables. She then went on to explain that this is where she figured out that if an element doesn't have an inverse, there can be more than one solution. She concluded that this is why in Z and Q there are unique solutions.

I feel like the first student's interest was sparked by this problem. The second has some misconceptions. She says that there are multiple solutions when an element is not invertible and comments that this is why equations over the integers have unique solutions (when solutions exist). Yet only two integers are invertible.

Several students told me they learned a lot by completing this assignment. That is insufficient. I want to know what they have learned.

For the most part, the student reaction to writing about their experience with a problem was quite positive. At the end of the semester I asked for students to comment on this process. Students said that the expository section of the homework really helped them to think about what they did and why it did or did not work. In general they felt they had an alternate route to demonstrate learning was taking place. They also appreciated getting credit even if their solution was not entirely correct.

Use of Findings

I have been learning about how students approach and think about problems. Before, I could tell whether or not a student could solve a problem, but that is all. I couldn't even tell if they knew if they were right or wrong. Students see this technique as a nonthreatening way to express their understanding. I use their paragraphs to construct new problems, to prepare for class, and to instigate classroom discussions. For example I wrote a problem asking when the quadratic equation will work in Z₄, Z₅, and Z₆ based on the student comment above. After grading a problem with poor results, I go over it in class. These paragraphs helped me to focus on what the students did and didn't understand. I could use their ideas to change a condition in the problem and asked them how the results change. We found ourselves asking "What if …?" a lot.

In addition, I now reward process, not just results. I try to assess the solution together with the exposition, giving one score for the combination. I try to convince my students that the most important part of their paper is to demonstrate to me that substantial thought went into their work. The paragraphs which addressed a student's own learning were particularly helpful with my assessment. In addition, students were using these papers to ask questions about material that was not clear. The method created a nonthreatening situation for the student to express confusion.

Success Factors

It is important to realize that this expository writing will not work for all kinds of problems, e.g., routine symbol manipulating problems. I had my greatest success on open-ended problems. Initially, I didn't get any really good responses, and some students never really cooperated. In the future I may include specific questions I want answered with a given problem. Some example of questions on my list are:

• Do you understand more or less about this topic after this problem?
• Explain how you arrived at your solution. Discuss trials that didn't work.
• Discuss possible alternate solutions.
• Discuss possible applications of this problem.
• Discuss the connections between this problem and our work on ____ in class.
• Describe what you were thinking and feeling while working on this problem.
• Can you think of a better problem to assign on this topic? If so, what is it?

These questions may be helpful at the beginning of the semester. Students will be encouraged to add additional comments beyond responding to the specified question. Then, depending on the results, I can allow more freedom as the class moves on and the students' confidence increases.