Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop (2001)

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

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.

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;

• 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.

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:

1. How do teachers' beliefs and attitudes constrain their ability to envision the course as a coherent entity?

2. 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?

3. What is “mathematical instinct” and how is it nurtured?

4. Where do we learn to ask questions that build on students' prior knowledge?

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: Deann Huinker; Members: Carne Barnett, Helen Gerretson, Kay Sammons, Mark Saul, Betty Siano, and Gladys Whitehead

We agreed that teachers engage in numerous recurrent tasks as they plan for the teaching of mathematics and facilitate student learning of mathematics. The tasks involve both short-term and long-term planning and reflection, as well as on-the-spot decision making. Many of the on-the-spot decisions made by teachers are unconscious.

We brainstormed and listed many recurrent tasks and discussed which required the use of mathematics. From the lists, six categories of recurrent tasks that required the use of mathematics emerged. The categories included (1) managing class discussions, (2) establishing a classroom culture for mathematical reasoning, (3) designing and selecting tasks, (4) analyzing student thinking and work, (5) planning instruction, and (6) assessing student learning. Table 1 provides a list of tasks for each category. We realize that some tasks may fit into more than one category; however, we placed each task within the one category we felt made the best fit.

Several questions were raised concerning the relationship of recurrent tasks to teachers' mathematical knowledge. Do any of these recurrent tasks require the use of mathematics? Does the mathematical content knowledge of teachers impact their decision making as they engage in these tasks? What mathematical content knowledge is needed to make good decisions? What day-to-day tasks and decisions are difficult for teachers to make when they lack specific mathematical knowledge?

Three assertions emerged from our discussions regarding the recurrent tasks of teaching that require use of mathematics. These involved the level of teachers' mathematical knowledge, the impact of this content knowledge on recurrent tasks of teaching, and the preparation of teachers.

Teachers need to develop a deep, interconnected understanding of the mathematical content knowledge they are expected to teach. Teachers need to know more mathematics than their students. However, it is even more important that

teachers understand the interconnectedness of mathematical ideas and develop knowledge packages (Ma, 1999) for key mathematical topics. This will allow teachers to more clearly identify students' current understandings and the direction in which to further that understanding.

The power of mathematical knowledge for teachers becomes apparent in their decision making as they interact with students. We posit that a strong relationship exists between teachers' mathematical knowledge packages and their ability to make “good” decisions that push student learning. Teachers make numerous onthe-spot decisions as they interact with students. Deciding what questions to ask students, when to provide an example, what diagram or model to use, and when to let them struggle are all examples of decisions that can be impacted by the depth of teachers' mathematical knowledge. It is also likely that a teacher's mathematical content knowledge plays a major role in deciding how much time to spend on a topic or what to emphasize regarding that topic. For example, teachers that lack an understanding of geometry are probably more likely to skip the topic or teach it at a low level of reasoning with an emphasis on memorizing definitions and formulas.

The preparation of teachers needs to explicitly connect mathematical knowledge to the recurrent tasks of teaching. Taking more mathematics content courses is not sufficient preparation for teaching. Prospective teachers need to examine the recurrent tasks of teaching in relation to mathematical knowledge. For example, rather than examining assessment strategies in general, discussions could focus on how to design assessment strategies to target specific aspects of mathematical knowledge. Then after using the assessments, discussions could focus on what next steps could be taken to further students' mathematical understanding based on the results.

Methods courses and field experiences, as well as content courses, should be examined to determine whether there is a better way to help prospective teachers make a connection between their mathematical knowledge and the recurrent tasks of teaching such as we explored during the sessions. For example, mathematics methods courses are often organized by mathematical topics. Methods courses could be organized by the recurrent tasks of teaching. Even though the tasks of teaching are interconnected and support each other, the organization by specific tasks could provide a framework for teacher learning. If methods courses remain organized by mathematical topics, then greater attention to the tasks of teaching need to be addressed within each area. These courses could make use of written and video cases to analyze student work, student thinking, and teacher decision making. The specific cases should be selected to bring out a discussion of mathematical content, not just pedagogical issues. Discussions could then center on why teachers made decisions in relation to the mathematical knowledge students demonstrated.

In examining the recurrent tasks of teaching that require use of mathematics, we were forced to look at the work of teachers rather than just examining their mathematical content knowledge. We struggled but came to understand that mathematical knowledge is connected to the day-to-day work and decision-making of teachers. However, we also found it difficult to articulate the connections

between the recurrent tasks of teaching and mathematical knowledge. The challenge we leave for others to consider and explore is to reveal and make explicit the connections between mathematical knowledge and the tasks of teaching.

Ma, L. ( 1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum.