Working Group Session # 1 - Judith Sowder
The Role of Understanding Children's Thinking in Convincing Teachers that They Must Develop a Profound Understanding of the Mathematics They Teach

In this presentation I make the argument that prospective elementary teachers of mathematics should have access, during the time they are taking their first content course in mathematics, to ways that children think about mathematics. This access leads them to realize that they must come to a deep understanding of the mathematics they are learning. Their attitudes towards the mathematics content courses changes quite dramatically when they work with children or even view videotapes of children being interviewed about mathematics. The presentation is divided into two parts. In the first, I describe a large project in which this conjecture is being tested. Then the videos that were shown are described and the group discussed questions are listed.

Background for today’s session

The Integrating Mathematics and Pedagogy (IMAP) project at San Diego State University is funded by the Interagency Education Research Initiative (IERI, which includes NSF, the DoE, and NIH). Randy Philipp (Professor of Education) and Judith Sowder (Professor Emerita of Mathematical Sciences) are the principal investigators. I describe the project, then describe video clips shown during the presentation and the discussion questions for each.

The goals of the project are:

bulletTo undertake a series of experimental and qualitative studies that will lead us to better understand the role of carefully designed early field experiences, coupled with mathematics content courses, on the beliefs and the mathematical growth of prospective elementary teachers of mathematics.
bulletTo develop instruments that will (a) measure prospective teachers’ depth of understanding of the school mathematics they will be expected to teach, and (b) provide accurate measures of beliefs prospective teacher hold and how these beliefs change over time.

The expected research results include

bulletinformation based on experimental studies that can be used to inform policy and practice in the design of teacher preparation programs;
bulletresearch findings about changing beliefs and content knowledge of prospective elementary teachers of mathematics that will move the field forward and can assist other researchers.

Other materials expected for use in designing courses or in undertaking further research include

bulleta template and materials (including videoclips, guides for discussions, guides for prospective teachers [PSTs] to use in interviews of K-8 students) for an early field-based course that can be used in many types of elementary mathematics teacher preparation programs;
bulleta preservice elementary program effective in inducting teachers early into the culture of teaching within a discipline, that can be used as a model to undertake similar changes in secondary mathematics and in other discipline areas;
bulleta videoclip library of elementary mathematics teaching and learning situations that can be used by other researchers and mathematics teacher educators; and
bulletresearch-based aplets focused on conceptual understanding of place-value and fraction knowledge.

Thus far, we have

bulletpiloted different forms of the early field experiences,
bulletdeveloped a content test and rubrics for measuring understanding of place value and of rational numbers
bulletdeveloped a computer-based instrument and accompanying rubrics to measure student beliefs about what mathematics is and what it means to learn and understand mathematics

Progress of project

This semester, we have about 200 students, now taking the first of four mathematics content courses designed for elementary teachers, involved in a large-scale study. They are involved in one of the following treatments (with random assignment when possible):

bulleta highly structured early mathematical field experience in which preservice teachers meet for 3 hours per week at an elementary school. They interview elementary students and plan instruction to help children overcome problems noted in interviews, and discuss as a class their interactions with the elementary school students.
bulletA highly structured early mathematical field experience based on viewing and discussing videotapes of teaching and of student learning in elementary school mathematics
bulletWeekly visits to the mathematics classes of highly skilled teachers
bulletWeekly visits to the mathematics classes of a random sample of teachers
bulletControl students attend mathematics classes only

Viewing of Videoclips and Discussion

The remainder of this session was devoted to viewing and discussing videoclips illustrating student thinking, teacher comments, and prospective teacher comments on their experiences with children. The following is a summary of the video clips we saw, and the discussion points for each.

Clip 1: A 5th grade teacher comments on students’ need to create their own understanding.

Clips 2 and 3: Two students of this teacher, Everett and Rachel, are asked to 3 3/8 to an improper fraction. Everett does so effortlessly, and explains why his method works. Rachel tries to apply an algorithm, but multiplies 3 by 3 then adds 8. She immediately then goes on to explain her work by drawing 3 circles, separated into eighths, then draws a part of a circle and marks 3/8. This resulting answer is different from her original answer, and she ponders this, then corrects the algorithm and says her drawing is correct. She tells the interviewer that if she had learned the second way first, she would remember it and would not have gotten the wrong answer in the beginning.

Discussion Questions for this set of clips

bulletDo you agree with this teacher’s comments about learning? Why or why not?
bulletWhat do you think Everett and Rachel understood about fractions?
bulletWhy did Rachel have difficulty with the standard method for changing mixed numbers to improper fractions? Why did she hesitate when she found a conflict between her two answers?
bulletRachel had interesting comments about how to sequence instruction. Do you agree? Why or why not?
bulletWhat would you expect Rachel’s teacher’s reaction to be if she viewed this videoclip, based on the teacher’s earlier

Clip 4: Ally (5th grade) compares fractions then shows how she converts between mixed numbers and improper fractions. She consistently compares fractions using reasoning such as the following: Comparing 3/10 and 1/2, she says that 1/2 is larger “because I could just change the bottom number one more digit and it (the entire fraction) would be one (and whole numbers are always larger than fractions.) She is asked to change improper fractions to mixed numbers and vice versa. She has no idea how to do this.

Discussion Questions for Clip 4

bulletWhat does Ally “know” about fractions?
bulletThere is a consistent “rule” Ally uses to compare fractions. What is it? Where do you think it came from?
bulletAlly is quite poised. Do you think she feels she knows what she is doing?
bulletWhere would a teacher begin to remediate Ally’s problems?
bulletIf Ally were your daughter, what could you do to help her?

Clip 5: Ally’s teacher provides an analysis of Ally’s work in mathematics, and class instruction on fractions. She appears not to realize how weak Ally’s understanding of fractions is, and it also appears that she teaches by presenting rules.

Discussion Questions for Clip 5

bulletDoes this clip make you feel any differently about the first clip?
bulletWhat kinds of experiences do you think this teacher had, and that the previous teacher had, that led them to be quite different teachers, and what does this have to say about teacher preparation? What would you do to assure (?) that your students would do a better job teaching fractions to students like Ally?

Clip 6: Felisha (end of second grade) solves 1/2 + 3/4 using drawings. She was in a one-week summer session, focusing on fractions, given by a 1st grade teacher who worked with us on this project. She provides excellent explanations of her work.

Discussion Questions for Clip 6

bulletCompare Felisha’s understanding of fractions to that of the other students you have viewed thus far.
bulletHow do you suppose Felisha’s understanding of addition of fractions will relate to later instruction on the standard algorithm for adding fractions?
bulletCan you guess at what type of knowledge the teacher who gave the one-week course (a few hours per day) had of mathematics and of pedagogy? And what are the implications for what you might do when you prepare teachers?

Clip 7: Two prospective teachers attempt to help Talicia (3rd grade) think about place value while adding 638 and 476. Talicia says that 600 +400 is “ten hundred” and writes 110. She of course gets the wrong sum, and realizes it is wrong. But no matter what the prospective teachers do, they are unable to help her see her error.

Discussion Questions for Clip 7

bulletWhy do you think Talicia insisted on writing “ten-hundred” as 110?
bulletWere you surprised by the robustness of her thinking that 110 was the correct way of writing “ten-hundred”?
bulletWhat would you have done, that you think might have been more successful?
bulletWhat effect might this clip have if shown to preservice teachers as they study place value in their content class?
bulletWhat effect do you think this experience had on the prospective teachers shown on the video?

Clip 8: Two experienced teachers provide advice for prospective elementary teachers. They tell the prospective teachers to try to understand the mathematics they are learning; to learn one way to do something, then another way, and then another way. And if they do not understand, then find a way to take more mathematics courses until they do understand.

Discussion Questions for Clip 8

bulletWhat do you suppose the impact of what these teachers say would have on prospective teachers?
bulletWhat impact did it have on you?
bulletIs it worth the time, in a mathematics content class, to listen to teachers such as these?

Clip 9: Several prospective teachers talk about why it was important to them to work with students while taking their first mathematics content course. They say that the content course make a lot more sense to them, because they actually see students doing the things the instructor and text materials say they do. These prospective teachers are taking a mathematics class with students who are not involved in this early field experience, and they say that the other students don’t appreciate the fact that what they are learning is relevant and needs to be understood.

Discussion Questions for Clip 9

bulletDo you think that these prospective teachers understand better than most prospective teachers taking a math content course the necessity of developing a “profound understanding” of the mathematics they will teach?
bulletWhat do these remarks imply about having a context for the mathematics taught in elementary mathematics content courses?
bulletWhat do you think might be the effect of this experience on how prospective teachers view future mathematics content courses they take?
bulletWhy do you think this experience with children had such a profound effect on these prospective teachers?
bulletIs undertaking some sort of early field experience, in conjunction with a content course appropriate, or should work with elementary school students be postponed to the methods course?

All participants received copies of the video tapes that included these clips and more.

Some of the clips will be found on the IMAP website,