Elementary Math Education 300-level

ANALYZING MATHEMATICAL CONCEPTS AND PROBLEM-SOLVING PROCESSES

My idea in teaching and learning has a lot to do with reflection. As a teacher I think back on the day and ask "why does that work?," "why didn't it work?," "how could I have done that differently?" It's a matter of looking back on what's happened. Part of the larger process I want my students to be involved in is looking back on what they've done and making some kind of analysis, seeing what kinds of questions they have.-- Professor Joseph Zilliox

The journals helped me learn more about myself. Math was fun. I learned that children see math differently from me -- I need to think about relating to children. My expository writing has improved. I can express myself more easily on paper than before. --Student

COURSE GOALS

The focus of the course is to encourage students to question the past and present pedagogy in mathematics education. Students reflect on their own processes for problem-solving and relate their experiences to the learning processes of their future students. Through a workshop approach, students discuss mathematical concepts and operations appropriate for the elementary school; students collaborate on class activities to develop teaching strategies and problem-solving methods.

WRITING ACTIVITIES

1. PERSONAL RESPONSE IN THE LEARNING LOG

Students' write their reflections on course content and methods and respond to ideas emerging from class interactions and assigned reading. Students may also include responses to particular issues raised by the instructor. Although one log is required each week, students are encouraged to write more often. Logs are collected occasionally, responded to by the instructor, then returned to students. Logs are not graded but failure to keep logs will negatively affect a student's grade. Students write their logs on loose leaf paper or on the computer and collect them in a binder or folder.

In the following example, a student asks a real-world question in the learning log: How do we [teachers] do it all?

Logs helped because the writing forced me to reflect on what I was doing in the class and helped me get an idea of the progression of my thoughts about the teaching of math.--Student

. . . . There is a lot of preparation that I need to do before I even think of doing the heavy-duty geometry. I have a big question for you. How do we do it all? I mean, how do we add in geometry while teaching addition, subtraction, division, and multiplication? In my third grade OP class, most of the kids barely get to multiplication by the end of the year. I really want to put more geometry in the classroom, but as far as "sequencing" is concerned, I find it hard. Any suggestions?

The instructor responds directly on the learning log entry with these suggestions:

Throw out some of the computation. How much practice do we need? Connect the geometry to the operations. Ex: Build rectangles with 36 small squares. What is the length and width of these rectangles? 2x18...........3x12..........4x9...........6x6

As much as I don't want to write them at times, logs have proved to be a great resource. Logs catalog events along the way; teachers can look back at daily log entries and know where to modify class teaching and techniques. Children can dialogue privately with the instructor and take notes for future reference.--Student
PURPOSE: The learning logs provide students with opportunities to explore ideas, clarify thinking, pose questions, express concerns and interests, and assess the course, the instructor, and themselves. The logs allow the instructor access to information about the student unavailable in class discussions or in group activities. The instructor writes comments or poses questions in the logs. Students write frequently about how they are thinking about mathematics and about the teaching of mathematics. Logs may also be used as notes for the mid-term and final exams.
2. PROBLEM-SOLVING TASKS
Every two weeks the instructor distributes a problem-solving task dealing with mathematical ideas related to the content of secondary and elementary school mathematics (e.g. solving a magic square, finding and applying the rules for divisibility). Students may work individually, in pairs, or in groups of three. The problem-solving report, required to be written on a word processor, must include the solution(s) and details of the solving process; personal reflections on what the student(s) experienced in attempting to solve the problem; descriptions of areas where the student(s) got stuck or felt frustrated; descriptions of particular strategies used; a sentence or two where the problem-solving task fits into the elementary mathematics curriculum; suggestions for making the problem easier; suggestions for making the problem more challenging (click here to go to an example of a problem-solving write-up). Students are encouraged to submit illustrations or scratch work of solutions to the problem. The instructor evaluates the reports based on the completeness and attention to details suggested in the description of the assignment rather than on the correct solution to the problem. The problem-solving activities were refreshing; I had forgotten how frustrating it can be to solve a seemingly simply math problem. It also made me more aware of the steps a student goes through as he/she solves a problem.--Student.
PURPOSE: Students identify and develop a set of strategies for solving problems, develop skills for posing and editing problems, and reflect on their individual approach and style in problem solving. Developing metacognitive skills rather than finding the correct answer is the emphasis of problem-solving.
3. READING COMMENTARIES
Students are required to submit three reading commentaries on journal articles focusing on elementary mathematics content or methods. Each student selects two articles, and the instructor assigns the third article. Commentaries are comprised of two sections: the first part is a one-page summary of the article and the second is a one-page reaction which should include comments, criticism, and questions of the article. The instructor suggests sources of articles, such as the Arithmetic Teacher, the Journal for Research in Mathematics Education, For the Learning of Mathematics, or students may find other appropriate journals. Students are invited to the instructor's office where they may select copies of journals from a large selection or borrow journals from the library.

The third commentary is assigned at the end of the second quarter. Since course participants include professional teachers as well as preservice teachers, the instructor provides textbooks relevant to the student's interest in teaching level. The first part of the commentary consists of an outline of the content: What is covered in the grade level? How much is new information? How much is repeated? The second part of the commentary is a reaction to the textbook in terms of topics explored in the curriculum course: What should be covered in the grade level? How are materials used? What is the role and function of small group work if it is encouraged in the text? What is the importance of context, concept development, soft algorithms, and Do-Say-Write (the practice of acting with materials, talking out loud about the actions, and recording the action)? Through class discussions of their findings, students review important mathematics content and methods topics.

All aspects --- reading, logs, problem-solving, and group work --- complemented and reinforced important concepts of mathematics as well as of human interaction -- two facets that teachers must combine in order to keep learning meaningful. -—Student
PURPOSE: The first two commentaries provide students with a context for analysis and discussion of the teaching of mathematics or theoretical concepts. In the third commentary students interact with the text, examining the curricular content, methods, and its underlying principles.
RELATED ACTIVITIES
1. COLLABORATIVE LEARNING GROUPS
In addition to problem-solving and reading commentary assignments which may be written collaboratively, most of the class activities are planned for groups of three to four students. The instructor provides mathematical tasks that can be approached with the use of manipulatives -- unifix cubes, straws, geometric figures -- which help students deal with abstractions. Every member is responsible for contributing some insight, question, or solution.

PURPOSE: The class is designed as a workshop so that students will experience many of the same activities they will be providing in their own classrooms. Assuming the role of teacher as student also helps them understand the thinking processes necessary for developing mathematical concepts. Students are expected to experience what can be learned through cooperation. They can discover that collaborative learning promotes active learning and that the individual contributions of group members help to teach one another in unique ways other than the teacher-directed model.

I feel that the group work with different people contributed to my overall learning in mathematics. If I can learn more through this approach rather worksheets, then so should my  students.--Student
2. COMPUTER EXPERIENCE

The instructor introduces students to the computer lab located in Wist Hall where class will meet occasionally for large group instruction. Students learn how to use "Superpaint" to enhance class presentations or to create student activity sheets which include one page of illustrations and one page of text. These activity sheets may be collaborated on with another student; the work must be titled, complete, organized, and neat. Other programs involving the use of a spread sheet are used for investigating mathematics content. The instructor also requires students to submit all written assignments (except weekly logs) using any word processing program.

PURPOSE: Students are expected to develop some competency and comfort using a word processing program as well as other computer applications. The computer is another writing tool students need to use more frequently to aid their writing and thinking.

3. BIWEEKLY QUIZZES

Students write a brief response with diagrams to a hypothetical situation encountered in a classroom. Quizzes may be open-book and/or open-notes.

4. MID-TERM and FINAL EXAMS

The mid-term is an open-book, open-notes examination during the regular class period. The final is a take-home examination; students are encouraged to work collaboratively in groups of two or three on the exam.

PURPOSE: The goal of both examinations is to foster careful thinking about the process of mathematical learning, concepts, and operations. Collaborative learning, which has been the instructional mode throughout the semester, is reinforced once more in the final exam format.

I am now more aware of the importance of providing mathematical experiences for kids and am no longer in favor of only computation drills. . . I also believe that the emphasis should be on the thinking instead of an isolated answer because kids need to feel confident that they have a chance to explain their reasoning and thought processes without being downgraded simply because their answers don't match a teacher's answer key . . . --Student

The group work was valuable because it gave us a chance to learn from one another. It was interesting to see how each person had a completely different approach to math.--Student

Professor Zilliox comments on his class (excerpts from an interview):

I see the role of writing in my class in many ways. Students must express their ideas, their opinions, and the writing gives me access to things that I don't have access to otherwise in class. Also I see a connection between the writing and reflection in class, and it took me along time to buy into that. I used to hate to write and still hate to write formally. In terms of writing for myself I find it easier to write my thinking down on the computer. Very often when I start to see my thinking in print, I see that's not what I meant or sometimes I think so fast as I'm getting it down on paper, that when I'm re-reading it I can keep sections or trash the file. It's one way of organizing, and I see writing as having that potential. I haven't decided what comes first with writing, and I'm having to work that through, what's comfortable, and what comes naturally. I do have sympathy for students who struggle with writing because they don't know what to write and because I was in that position.

My idea in teaching and learning has a lot to do with reflection. . . Because of the WI status we have certain expectations to get more writing into the course. I wasn't sure of the value of these things, but I had tried some things and I liked the responses. . . I'm not sure what it has meant for the students, although at the end of the semester I find so much that I keep as an evaluation for myself and the course. Some students respond very positively to the logs; others say it's a waste of time. Right now I have four classes using logs and one of the criticisms was "I really like your feedback." Students felt writing the logs was hard work, but there was a that I had through the logs that I don't have in class. . . They can say things to me directly through the log that they can't say face to face. . . I'm disappointed that I don't have enough time to keep up with the logs, but of course, it's my own management problem. This semester it's been working! Some use the logs as notes for the mid-term. Most of the students who do use the logs as notes are also the students who say they don't know what to write or say. Making a summary is easier to do than thinking about what it is they've done in class. I'd like to try doing a log together. . . I'd like to try log writing at the end of class; it might get them to think about the text.

One of the other things I like to use writing for is the problem-solving paper. The emphasis in the books of mathematics, the university, and education courses is problem-solving, not just practice. Many of these students have never been problem-solvers themselves, and I try to teach that or give them activities. . . The directions for the problem-solving papers are in the syllabus. . . One of the reasons for having students write in paragraph form is to have them reflect on what they're doing and why this is something even worth doing. We were doing something with magic squares; there was a lot arithmetic computation, and some only realized what they were doing only after they started thinking about it. One of the things I want students to do in mathematics is to have them do this kind practice -- a different sense of practice for students. This raises issues we talk about in class -- why are we in mathematics? so what if the answer isn't correct? These issues come up especially in the problem-solving paper. Sometimes when students start writing about what they're doing, they begin analyzing their process and discover where the problem is. Writing can be a way to get students to say "It wasn't until I was writing this up that I could explain it to you" and decide "Maybe I could have solved the problem this way . . ." Writing actually helps them solve the problems.

Students don't like doing these problem-solving papers because they're dealing with abstractions, and mathematics seems to aim for the one right answer. The goal is to get the right answer that the teacher already knows, and I don't know the answers to all the problems. There are bunches of mathematics problems still out there which haven't been solved for thousands of years. Some of them we don't know are solvable or we think they are, but no one really knows. But that's part of mathematics; it's not just one answer. Students believe there's some rule and if you know the rule and can apply the rule you'll get the right answer, and that's the end of it! However, in problem-solving there is no single rule or simple process. We can talk about what works or what doesn't work; we come to reasonable conclusions. This is one of the things I want to get across to my students, and I see it happening in the journal. I'm trying to get into my students' minds and getting them to question their own ways of teaching in general, teaching mathematics, and what mathematics is.