Question #4
DISCUSSION GROUP #4
What are some promising ways to help teachers learn mathematics that also help them develop mathematically? What are the key features of what makes an approach promising?
Leader: Dwayne Channell; Members: Hy Bass, Michaele F. Chappell, Ed Esty, Don Gilmore, Margaret Jensen, Lena Khisty, Marie Sheckels
ASSUMPTIONS MADE IN FRAMING THE DISCUSSION
The group focused discussion on ways that tasks in mathematics content courses could be designed to improve the mathematical understandings of preservice teachers. While admitting that many suggestions could be implemented in either a methods or a content course, the group believes that all suggestions are appropriate for inclusion in content courses. It is also recognized that successful implementation of some suggestions will necessitate cooperative efforts among the faculty who teach mathematics content and those who teach mathematics methods.
SUMMARY OF THE MAIN POINTS OF DISCUSSION
Current shortcomings in the teaching of mathematics content. The discussion group identified two overriding problems that need to be addressed:

Preservice teachers (PSTs) typically possess a superficial understanding rather than deep understanding of elementary mathematics.

PSTs typically have not experienced making sense of mathematics and do not believe that sense making is a necessary component of understanding mathematics.
Participants suggested that fault for these shortcomings might be attributed to instruction that values speed and efficiency in arriving at “the” answer without regard for the need to reason and develop understanding.
Discussion focused on methods of situating mathematics instruction to require PSTs to examine their mathematical beliefs and understandings and encourage them to shed their reliance upon procedures without meaning.
The group believes that an important goal in the mathematics education of PSTs is to help them develop as teachers who value sense making and reasoning as necessary components of mathematical understanding and who instill those values in their own students. Consequently, instructional and assessment experiences need to be designed to permit PSTs to identify their own “concep
tual images” of mathematics and foster a change in those images so they are more conducive to meaningful learning. Mathematical experiences designed for PSTs should often reflect the ways in which they will use their mathematical knowledge in classroom teaching, for example, designing assessments, evaluating the written work of others, interpreting and assessing the validity of verbal arguments, and modifying mathematical tasks. Many of these “core teaching tasks” were identified in the conference plenary sessions and were echoed in breakout group discussions.
Promising approaches for teaching and situating content. In many, if not most, universities, the mathematics content preparation for PSTs is centered around the content in commercially published textbooks that, by the nature of the market, cover an immense amount of subject matter but do not always engage students in ways that promote a broad, thorough, and deep understanding of mathematics as discussed at this conference. It was suggested that students may better develop mathematically if faced with fewer topics investigated in much greater depth. In doing so, instruction could be based around several big ideas or underlying structures in mathematics. PSTs could be immersed in rich problem settings that require them to approach a task or problem situation from multiple perspectives, draw connections among those perspectives, and strengthen their flexibility and fluency as mathematical thinkers and communicators. As was frequently mentioned at this conference, such experiences would permit PSTs to develop “knowledge packages.” The group realizes that determining core essential content knowledge would be a very political process with more than one feasible solution. We also recognize the need to involve stakeholders from several factions in the process. However, the difficulties inherent in such a task do not lessen its importance.
Several alternatives for situating mathematical learning experiences for PSTs were suggested. Each suggestion offers the possibility of deepening the PSTs mathematical knowledge and perhaps strengthens their abilities to recognize the relationship of this knowledge to their future classrooms.
Some of the suggestions center around modifying traditional mathematical tasks from a focus that is strictly on the development of content knowledge to tasks that also emphasize the development of pedagogical content knowledge. Examples include:

Present PSTs with a mathematical task and require them to create new, related questions that remodel the task.

Have PSTs analyze the mathematics a student would need to know to investigate a task and delineate the mathematics they would learn by working through the investigation.

Provide PSTs with a situation and have them create a variety of questions that could be asked about the situation (continue with the previous suggestion by having them make an analysis of the needed mathematics and the learned mathematics from each question generated).

Provide PSTs with several correct approaches to solving a given task including, as appropriate, pictorial mental reasoning strategies, traditional procedural algorithms, and alternate sensemaking algorithms. Have them discuss and critique the advantages and disadvantages of each, including any implications for promoting student understanding.
Other suggestions focus on injecting explicit connections of mathematical knowledge to the teaching of K8 students. Examples include the following:

Present PSTs with examples of K8 student work that reflects invalid or incomplete reasoning or alternatives to traditional methods of solution. Engage PSTs in an analysis and critique of the work. Why do they think a student might reason in a particular incorrect way? Why do the students not understand or what are the origins of their confusion? Presentation of the student work could be made available in the form of individual written work or in videotapes of actual classroom episodes involving the interaction of a teacher with several students.

Require PSTs to design an assessment instrument for testing knowledge of content they have recently discussed. Develop a key (rubric) for assessing student performance on the instrument. Administer the instrument to other class members and evaluate their performance.

Present PSTs with some traditional mathematics for which they possess a memorized algorithm that could be used to solve the problem. Require the PSTs to abandon reliance upon the algorithm and use sensemaking strategies to explain a solution that could be understood by elementary/ middle school students who do not possess knowledge of the algorithm.
Key features of suggested approaches. Each of the alternatives suggested here situates mathematics in tasks that are motivating and relevant to future elementary/middle school mathematics teachers while maintaining an emphasis on critical thinking and mathematical understanding. While the advantages of such tasks over more traditional approaches may seem apparent to some, widespread acceptance of such suggestions is unlikely unless each is given a realistic evaluation in mathematics classes with preservice teachers.
Question #4
DISCUSSION GROUP #7
What are some promising ways to help teachers learn mathematics that also help them develop mathematically? What are the key features of what makes an approach promising?
Leader: Nancy Edwards; Members: Ava BelisleChatterjee, Robert Howard, Shinying Lee, Carol Midgett, Joyce Miller, Pat O'Connell Ross, Al Otto
ASSUMPTIONS MADE IN FRAMING THE DISCUSSION
Liping Ma's treatise on Knowing and Teaching Elementary Mathematics (1999) informed our discussion by enabling us to think systemically about a framework for content preparation of elementary teachers in mathematics.
Ma's insightful analysis of U.S. teachers' emphasis on procedure that results in a telling of rules and an accounting of stepbystep processes led the group to discuss ways to break from this approach. The group felt most content instructors and methods teachers have focused on an enormous number of concepts to teach prospective teachers, resulting in a “piece by piece” arrangement of the knowledge needed to teach students. The group found the thought structure of the Chinese teachers who emphasized the grouping of knowledge into packages as a revolutionary way of thinking that holds great potential for teacher content and methods preparation. These packages underlie what our group referred to as “core tasks” that must be present if prospective teachers are to understand the meaning behind that which they teach. The group also discussed the nature of mathematical problems that would give conceptual support to the core tasks, such as the subtraction problem described in Ma's work.
A discussion of Ma's research caused the group to see research as a key to change the approaches of content instructors and methods teachers. The group concluded that some of the most promising ways to help prospective teachers learn mathematics and develop mathematically are to evaluate research findings that indicate the core tasks and mathematical problems prospective teachers need to know and do to accomplish their work.
SUMMARY OF THE MAIN POINTS OF THE DISCUSSION
Ball's presentation on Teaching Mathematics for Understanding, Schifter 's Learning Mathematics for Teaching, Torres' Remodeling Mathematical Tasks, and Bastable's Analyzing Student Thinking, all formed the basis for the summary list of characteristics needed for maximum effectiveness in identifying and using core tasks and mathematical problems:

The best core tasks should aid prospective teachers to experience doing mathematics in the same way that mathematicians do it—i.e., in a way that makes sense to them. This involves investigating, conjecturing, and justifying the nature and uses of mathematics. The content should be taught in such a way that students struggle and engage in dialogue as mathematicians do. This leads to new mathematical understandings. The dialogue should foster responsibility among prospective teachers as a community of learners.

The best core tasks should build upon “real world” applications (contextualized in reality), with prospective teachers able to foster a seamless transition between concrete and abstract mathematical thought. Attitudinal change can come when prospective teachers see uses for mathematics in realworld models.

The best core tasks need to be taught the way we want teachers to teach, making lessons very explicit to the goals to be accomplished.

The best core tasks should lead to experiences where the parts are learned very well with depth, coherence, and reasonableness, as opposed to the study of too many topics superficially learned and quickly forgotten. Prospective teachers need experience in learning some aspect of mathematics very well and to reflect on the structure or schema that enabled them to reach that level of interest. They should then be encouraged to apply that aspect to solve problems in diverse mathematical contexts. This should lead to a sense of how to learn mathematics on one's own, with an awareness of the need for continued study after completing mathematics courses.
Instructors need to research what misconceptions elementary teachers have and design tasks that get at those problems. Promising ways include the following:

The use of problem sets and case studies.

The use of standard problems suggested by research mathematicians as well as mathematics educators. Some “classic” (i.e., standard) problems have started to appear in the literature as problems actually researched with varying populations to see if those problems really do lead to greater mathematical understanding among prospective teachers as well as students. It would seem advantageous to use proven problems rather than trying to have individual instructors invent new ones that may or may not help rectify the misconceptions that elementary teachers have.

The inclusion of more examples of children's mathematical thinking through videos and work samples studied in the content courses (not just in the methods courses). This sends a powerful message that children's mathematical thinking is worthy of indepth responses from the mathematical community.

The remodeling of tasks to get at the problem areas, challenging the prospective teachers to remodel tasks as assignments to see early on if they understand and use the structure of mathematics in this remodeling.

The use of multiple representations moving beyond using concrete models for the sake of using them. Such models can have the same fallacies as symbolic representations if not done for the right reasons.

Completing only 35 problems a semester. (The thinking was that a threecredit

course probably could handle no more than two indepth problems per week in a 16week semester with the possibility of an introduction problem and one presented as a midterm and final exam question.) Some problems will last over a period of days, fostering quality over quantity in learning.
The plenary session on the Promising Approaches for Helping Prospective Teachers Learn Mathematics for Teaching aided the group by identifying ways to affect change in the teacher preparation content, including the following:

The establishment of a network to share what misconceptions elementary teachers have and design tasks that get at those problems, combining best practice and the research.

The support of a Master TeacherinResidence Program at the college level—to team teach and keep content instructors realistically aware of the needs that will be faced in the elementary classroom. Such a program could foster mutual respect among the content mathematician, the mathematics educator, and the master classroom teacher. Prospective teachers in such programs witness collegial networks with potential for ongoing support in their own future classrooms. This sends a powerful message of the importance of mathematics on all levels of development—from the very young child to the adult.