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### Question #1

##### DISCUSSION GROUP #1

Often teaching is seen as presenting material to students. But of course teaching includes many more small and large tasks—figuring out what students know, composing good questions, assessing and revising textbook lessons, and so on. What are some of these recurrent tasks of teaching that require the use of mathematics?

Leader: Mercedes McGowan; Members: Dan Burch, Michael Hynes, Shirley Smith, Jane Swafford, and Alan Tucker

##### ASSUMPTIONS MADE IN FRAMING THE DISCUSSION

We made the assumption that recurrent tasks of teaching implied examining and thinking about classroom practices that apply to the teaching of mathematics generally and that we were not to focus our discussion on unpacking our thinking about the tasks associated with the teaching of a particular topic or grade level.

##### SUMMARY OF THE MAIN POINTS OF DISCUSSION

Recurrent tasks of teaching that require the use of mathematics identified by our group included

• uncovering students' current base of knowledge and the common base of knowledge shared by the class;

• assessing the “generality of knowledge,” i.e., knowing where a mathematical concept fits into a sizeable, interrelated body of knowledge;

• selecting worthwhile tasks designed to provide experiences with fundamental concepts and techniques, active student participation, and abundant opportunities for students to make discoveries.

As we began unpacking our own knowledge of recurrent tasks of teaching, we addressed the question, “What is a worthwhile task?” There was a common understanding that worthwhile mathematical tasks enable students to build particular organizations and classification schemas that can be utilized to explain subsequent, more abstract ideas. Characteristics of worthwhile mathematical tasks were identified and included tasks that

• are open-ended—meaning that the solution should not be readily available;

• contain significant mathematics and have multiple pathways to the solution;

• develop understanding of

1. the meaning of operations,

2. the algebraic properties of numbers,

3. relationships among quantities that change,

4. the ambiguity of mathematical notation,

5. the degrees/levels of complexity in a given context domain;

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OCR for page 131

OCR for page 131
Knowing and Learning Mathematics for Teaching model and guide the construction of acceptable mathematical arguments and justifications; are accessible and challenging; promote flexible thinking; include time to reflect. During the two days, we continued to struggle to unpack our thinking about recurrent tasks of teaching that require the use of mathematics while avoiding discussion of a particular topic or content area. As we came back together after attending various breakout sessions, we synthesized the ideas and discussions of those various breakout sessions into our small-group discussions. We began to discuss “ the mathematics of teaching”—how our knowledge of mathematics influences the ways in which we assess our students, evaluate programs, assign grades, use a rubric, choose textbooks, envision a course, design a lesson, and select mathematical tasks for investigation. Our discussions also unpacked some of our personal underlying assumptions and beliefs about the nature of mathematics, how students learn, the role of the teacher, the role of technology, and the means of achieving skill competencies. Given the diverse backgrounds of the group members, it was not surprising that there was no consensus on these issues. ISSUES The issue of what mathematical knowledge a preservice teacher needs to know was a recurrent topic of discussion throughout the conference. One general consensus was there is no way to provide preservice teachers with all the mathematics content knowledge we would like them to know. Rather, we need to think more deeply about how to provide preservice teachers with “sufficient” mathematical knowledge and desire for life-long learning so they continue to grow in their understanding of mathematics and of teaching on the job. We identified the need to develop a coherent vision of the course(s) as an essential component of a teacher's planning for instruction if one is to break away from the “cake-layer ” mentality of disconnected courses and Skill 1 today, Skill 2 tomorrow, etc. We left the conference with the following questions unanswered: How do teachers' beliefs and attitudes constrain their ability to envision the course as a coherent entity? What might help teachers who lack a coherent vision about the courses as a whole avoid being caught up in the bits and pieces of curriculum? What is “mathematical instinct” and how is it nurtured? Where do we learn to ask questions that build on students' prior knowledge?

OCR for page 131