CUNY Practicum: Mathematics

On "Constructivism Basics"

Among the 12 principles of constructivist teaching I chose:

6. Encouragement of students to engage in dialogue, both with the teacher and with one another.

and

7. Allowances for wait time after posing questions.

They are both practices I don't regularly do in my classroom. Also, part of the reason I chose them is because I can't really relate to most of the other points due to differences in discipline (There are no open-ended questions in math, I can't think of useful mathematical metaphors, etc). But I think the two principles I chose will do some good. I've noticed that due to time pressure and commitment to cover the entire syllabus I've become impatient. Though I ask questions I don't wait for the students to process them. It's like I'm defeating the purpose of asking the questions in the first place. Also, I'm beginning to think that student dialogue might actually do some good.

Group discussions in practice

My students seemed to like the group discussions. When I asked them for feedback they all agreed that the group discussion were helpful. This is what I did: I let them get into groups of three or four and then I wrote some problems on the board for them to do together. Then I visited each group and tried to get involved in what they were doing. I found that most of them were doing the problems individually. I think the first problems I assigned were too easy for them. The second time I tried the group activity I made the problems more challenging so that they may have to discuss. However, as soon as I got them into groups they began to challenge me and complain. That day was disastrous and I felt like a monkey in the zoo. As I was going around the groups around one third of the class began to make fun of me amongst themselves. It was a really horrible experience. But I thought I'd try the groups one more time. The third time they got into groups we were doing factoring problems which are challenging in nature. I gave them moderately difficult problems and except for a few students who just have no respect for the professor people seemed to do fine. They were solving and discussing questions. Some of them who understood quickly were explaining how to solve the problems to their other group members. I believe it will have positive effects and I hope to see them in the tests.

I found that the positive but scary thing of doing group activity is that it gives the students voice. They start feeling more confident about asking questions, which is always a good thing, and they also feel more confident about voicing their opinions, which is sometimes not so good. Ultimately I think for group activities to be successful professors need to be well-prepared and also command authority in the classroom.

Response to discussion and practice

One of the reasons I thought group activities would not work in math classes was because I think math is learned through reflecting by oneself. However, after Maria's presentation and after grading my students' first exams I decided I might give it group discussions a try. (You can guess what a disaster the first exams were).  I'm planning to get my students into groups of three and have them solve a number a questions. I might even give them types of problems I have not worked out for them so they can do their own little research. The goal is to have them more involved and give myself extra opportunities to engage with them. Also I hope they can get to know each other so that they may not be as shy to speak out in class.

ESL

By the way, I just thought about this one Korean student who does nothing in class. When I tell the students to solve a given problem, he doesn't do it. He doesn't even try. Also, when I ask him some question he only says "I don't know" like he doesn't know any other phrase. I'm trying to decide whether it is a good idea to pick on him, because he seems to be an ESL student. But I suppose he should have no problem doing the problems in class. I'm thinking of pulling him aside one of these days and have a chat with him. Maybe I'll wait and see how he does on the first exam.

Plans for next week

I've announced to the class that the first exam will be on Thursday. I'm thinking of giving a review on Wednesday. I thought it would be a good idea to write out a review sheet with problems similar to those they'll see in the exam, but I wanted my students to get it before we do the review. Unfortunately, I miscalculated. Since we don't meet on Monday I'll have to send the review sheet by e-mail. Hopefully they will check their e-mails and bother to try to do the review sheet. So on Wednesday I'm planning to go over that review sheet with the class.

First three weeks

We've completed three weeks of this semester and I find that I'm struggling on finding a proper balance in speed. The first few classes I was going very slow because I wanted to be, in some way, very constructivist, by making sure students understood all the material: Going over many different homework exercises, picking on students to think and solve problems, and always making sure everyone was happy before I moved on to the next section. However, I quickly realized that I was going too slow and I was getting behind by many hours. So the past few classes I feel like I've been rushing through the material. I've sort of come back to the traditional fully lectuing math class. Now that I've almost caught up with the schedule laid out in the syllabus I hope to stabilize.

Excuses?

After the class on Wednesday, one of my students came to me and told me that she wanted to let me know that she had a "severe learning disability." "I just wanted to let you know," she said. I told her that she must have done well in the Math Compass exam for her to be able to be in Math 120. She said that she failed it before, then she passed after. Then I said, ok. After the class I kept thinking that maybe I made a mistake in even agreeing. First of all, I don't know that anyone is allowed to take the Compass exam after doing poorly on it. Is it even possible to fail it? Also, by her telling me that she has a "severe learning disability" is she not just making for herself a cushion just in case she does badly in the class? She would then be expecting me to show some leniency to her and consequently she would not try as hard. I think I could have said something other than "ok".

Several negatives and a few positives

I'm finding that changing the structure of the class is not so easy. After each of the three classes this week I kept thinking about Bass' statement that every time he changes the structure of his class, the student evaluations drop. Today I went over homework problems as I had planned. I assigned five students to the board to solve simultaneously some of the homework problems. Two things happened: First, only one student got her problem correct. Second, I spent the whole hour going over homework and had no time for new material. Five classroom hours have passed by and if I follow the hour assignments on that big departamental syllabus, I have only covered two hours. This means that I have three hours to catch up to and this is only the first week! And why is it that, even though I spent so much time explaining a two hour session in roughly four hours, only one out of five students got her problem right? Maybe the problems I assigned were too difficult? Or maybe it is my lecturing that has a problem. One positive thing is, however, that I was able to see that my students are having trouble understanding before I get a shocking surprise when grading midterms. At least, now, I can do something about this. I realize how easily a professor can be deceived by the wide eyes and smart questions students may portray during lectures to think that they are really getting it.

Constructivism and my syllabus

The idea of implementing constructivism in my class made my thoughts wander. I thought about having small groups discuss homework assignments that might have challenged them. Then I remembered that I actually tried something similar in an Algebra course I taught at Hunter College. We made small groups of 4 at the beginning of each class and students were supposed to discuss homework assignments they had difficulty with. I think it went ok in the beginning, at least I saw students doing what they were supposed to do. But as the semester progressed, I observed that more and more students were discussing something other than math in their groups. I think now that this was bound to happen because many of the students don't do their homework and the students that do find that the groups are just a waste of time. So the small groups won't work. I think Belle's idea of having students present their work in the board will be good. I think this is as constructivist as I can get.

Also, in response to Bass' article, I thought it would be a good idea to have some sort of quiz in the first lecture. The idea is to get some sort of assessment of the students' analytical skills. Now I just have to work on how I actually construct the quiz so that it dosn't test their math knowledge as much as it does their math skills.

Class Observation

I observed Prof. Sandra Peskin's class on an introductory algebra course. When I came into the class my mindset was on observing the students rather than the professor. Partly, this is because I know what to expect in an abstract mathematics class: The professor goes over homework problems on the blackboard, once in a while students raise questions, professor corrects their misunderstandings, etc. There are not too many ways in which you can teach mathematics, especially an abstract class like algebra. I was also interested in observing the students because I wanted to know what to expect next semester.

I was surprised at the students' enthusiasm to learn. Unlike what I expected, most students were eager to participate and to ask questions. The reason could have been that they had an exam the next day. My guess is that Prof. Peskin made them feel comfortable to participate. It made me wonder if this is an attribute due to the personality of the teacher or to techniques of teaching. Or maybe it is not even the teacher, but the students who are eager to learn by intereacting.

As to Prof. Peskin's teaching styles or skills, I cannot say she does anything differently from other professors of introductory algebra or even from my own style of teaching algebra. I could point out that she was very patient and willing to reteach many times over as long as there seemed to be misunderstandings from the students. One thing I did learn from her skills is that she stresses identifying the type of problem that is dealt with. Some math problems look identical even though they may be entirely different types of problems. For example, I could write a long mathematical statement, and the students in general would not know whether they have to solve for a variable or simplify the formula. I was many times annoyed when grading exams by students who solved when they should have simplified or simplified when they should have solved. I suppose Prof. Peskin had similar experiences and she decided to emphasize reading the instructions carefully to identify the type of problem.