Not everything is a question of knowledge. From what we have done together, what are some mathematical instincts, sensibilities, dispositions that seem crucial to teaching mathematics? What mathematics beyond what is taught in class must a teacher know to do a good job teaching mathematics in that class?

Leader: *Carol LaCampagne;* Members: *Don Balka, Irene Bloom, Carey Bolster, Blanche Brownley, Judy Kasabian,* *Cathy Liebars, Deborah Schifter*

Group 2 had some trouble distinguishing between necessary teacher knowledge of mathematics (Question 1) and mathematical instincts, sensibilities, dispositions that seem crucial to teaching mathematics (our Question 2). We believe that mathematical knowledge and these instincts, sensibilities, and dispositions are often inseparable. Surely both are needed in order for a teacher to have a profound understanding of elementary mathematics and its teaching. Thus we often intertwined knowledge and “habits of mind” in our discussion.

Group 2 identified five major instincts, sensibilities, and dispositions that we felt were crucial for the good teacher of mathematics to possess: mathematical habits of mind; ability to create a classroom community of mathematical discourse; view of mathematics as fluid, dynamic, and connected; ability to solve mathematical problems; and the ability to make wise classroom decisions.

*The good teacher of mathematics possesses mathematical habits of* *mind.* These mathematical habits of mind include the ability to reason mathematically; to construct a valid mathematical argument, to judge the validity of the mathematical arguments of others, and to understand the nature of proof. People with these habits of mind have authority over their own mathematical learning. They do not view mathematics as either black or white. People with mathematical habits of mind become confident in their own mathematical knowledge and ability. They develop a positive attitude toward mathematics. Such people are curious about mathematics and learn mathematics beyond what is taught to them in school or college. Such people become lifelong mathematical learners. These habits of mind were exemplified in the video segment presented by Ball and Bass.

*The good teacher of mathematics creates a classroom community of* *mathematical discourse.* Such teachers understand and assess the mathematical thinking of their

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Knowing and Learning Mathematics for Teaching
Question #2
DISCUSSION GROUP #2
Not everything is a question of knowledge. From what we have done together, what are some mathematical instincts, sensibilities, dispositions that seem crucial to teaching mathematics? What mathematics beyond what is taught in class must a teacher know to do a good job teaching mathematics in that class?
Leader: Carol LaCampagne; Members: Don Balka, Irene Bloom, Carey Bolster, Blanche Brownley, Judy Kasabian, Cathy Liebars, Deborah Schifter
ASSUMPTIONS MADE IN FRAMING THE DISCUSSION
Group 2 had some trouble distinguishing between necessary teacher knowledge of mathematics (Question 1) and mathematical instincts, sensibilities, dispositions that seem crucial to teaching mathematics (our Question 2). We believe that mathematical knowledge and these instincts, sensibilities, and dispositions are often inseparable. Surely both are needed in order for a teacher to have a profound understanding of elementary mathematics and its teaching. Thus we often intertwined knowledge and “habits of mind” in our discussion.
SUMMARY OF THE MAIN POINTS OF DISCUSSION
Group 2 identified five major instincts, sensibilities, and dispositions that we felt were crucial for the good teacher of mathematics to possess: mathematical habits of mind; ability to create a classroom community of mathematical discourse; view of mathematics as fluid, dynamic, and connected; ability to solve mathematical problems; and the ability to make wise classroom decisions.
The good teacher of mathematics possesses mathematical habits of mind. These mathematical habits of mind include the ability to reason mathematically; to construct a valid mathematical argument, to judge the validity of the mathematical arguments of others, and to understand the nature of proof. People with these habits of mind have authority over their own mathematical learning. They do not view mathematics as either black or white. People with mathematical habits of mind become confident in their own mathematical knowledge and ability. They develop a positive attitude toward mathematics. Such people are curious about mathematics and learn mathematics beyond what is taught to them in school or college. Such people become lifelong mathematical learners. These habits of mind were exemplified in the video segment presented by Ball and Bass.
The good teacher of mathematics creates a classroom community of mathematical discourse. Such teachers understand and assess the mathematical thinking of their

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students. This teacher moves students from learned helplessness about mathematics to independent mathematical thinkers. The good teacher is flexible and may change the lesson plan because of student discourse but ultimately keeps an eye on the “big mathematical picture.” This teacher chooses the appropriate language of mathematics for the discourse and encourages students to use common ways to talk about mathematics and common ways to judge the mathematical truth of an argument. The good teacher listens to student ideas. The video clip of Dr. Ball's class exemplified the creation of a classroom community of mathematical discourse
The good teacher of mathematics views mathematics as fluid, dynamic, connected, as a powerful group of ideas to be explored, not as a collection of disjoint facts to be memorized. Such a teacher views mathematics as a growing body of knowledge, not as a static body of knowledge set forth hundreds or thousands of years ago, recognizing the connection between mathematics and the sciences (both physical and social) and between everyday lives and the lives of the students.
The good teacher of mathematics is a problem solver and possesses the tools to solve mathematical problems. This teacher has the ability to tolerate frustration in working through a problem to a satisfactory solution, enjoying mathematical problem solving, and seeing mathematical problems in everyday life experiences. Such a teacher is mathematically curious. The pre-Workshop task and the accompanying chapters from Liping Ma's book motivated us to think of the teacher as problem solver and of the ability to use problem solving strategies to encourage students to also become good problem solvers.
The good teacher of mathematics possesses the ability to make wise classroom mathematical decisions—both in the planning of lessons and in on-the-spot classroom situations. This teacher has the ability to identify the big ideas in the mathematical curriculum and emphasizes them. The teacher can choose appropriate mathematical problems with multiple entry points that elicit a variety of student responses. A good teacher possesses the internal flexibility to manage the resulting classroom discussion, to question, to probe, and to elicit mathematical knowledge from students. Such a teacher probes student explanations and looks for counter-examples. The good teacher reflects on her/his teaching and the mathematical learning of the students and uses students ' mathematical ideas as a way to advance knowledge. The good teacher takes a variety of student solutions to problems and makes sense of them. The teacher detects the mathematical errors of students, helping them to understand and correct these errors.
ISSUES
These attitudes and skills are those that Group 2 would like to see built into the undergraduate preparation of teachers of mathematics. We acknowledge several roadblocks to building them into the undergraduate program. First, we feel that the undergraduate mathematics faculty would need much preparation themselves to conduct their classes in a way to encourage the development of such attitudes and skills into their students. Second, we acknowledge the need for students to have considerable practice as they develop these skills. We feel that some experience in teaching in the context of their own undergraduate classrooms via short presentations as well

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as early entry into the middle school classroom to observe and to begin teaching under supervision is needed. A greater collaboration between college and schools is also needed to ensure that beginning teachers receive the support and nurturing from both school and college to develop their teaching abilities. Nowhere was the honing in on teaching skills more apparent than in the presentation by Liping Ma concerning Chinese teachers.

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Knowing and Learning Mathematics for Teaching
Question #2
DISCUSSION GROUP #9
Not everything is a question of knowledge. From what we have done together, what are some mathematical instincts, sensibilities, dispositions that seem crucial to teaching mathematics? What mathematics beyond what is taught in class must a teacher know to do a good job teaching mathematics in that class?
Leader: Judy Sowder; Members: Benjamin Ford, Jim Lewis, Olga Torres, Rosamond Welchman
ASSUMPTIONS MADE IN FRAMING THE DISCUSSION
The use of the adjective “mathematical” in this question led to discussion of how to distinguish mathematical instincts, sensibilities and dispositions from pedagogical ones, such as the disposition to listen carefully to and reflect on students ' responses. It was often difficult to make this distinction because good pedagogy is combined with mathematical instincts, sensibilities, and dispositions when good mathematics is being taught. We all agreed that the following points attend to the mathematical focus of the question.
SUMMARY OF THE MAIN POINTS OF DISCUSSION
Teachers must be inclined and able to take the work of their students and explain or illustrate this work mathematically in different ways. A teacher should be able to follow the logic of a student's argument, whether correct or incorrect, to narrow in on an important mathematical idea of the argument and expand on it through questions that lead students to more fully understand the idea, or by representing the idea in a different way that will prevent overgeneralizing or undergeneralizing the idea. For example, many students will say that 34.5 is smaller than 34.42 because there are only three digits in the first number but four digits in the second number. These students have a weak understanding of place value and have overgeneralized a property of whole numbers. A teacher who recognizes this overgeneralization might illustrate the meaning of 0.5 and 0.42 using base 10 blocks. As another example, a student may divide to solve the problem “A certain kind of cheese costs $1.89 per pound. How much will a package weighing 0.79 pounds cost?” because he knows that the answer must be smaller than 1.89, and “knows” that “division makes smaller.” The teacher who recognizes this misconception as an overgeneralization of whole number arithmetic will be able to provide illustrations to show this student that “multiplication makes bigger and division makes smaller” do not always hold when rational numbers are involved.
This mathematical instinct to approach

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problems from many points of view is clearly required to adequately analyze student work on open-ended tasks. The breakout session analyzed student work involved in an orange-juice mixture problem that was approached by students in many different ways. The teacher who is unable to switch freely among the approaches of the students will be unable to adequately respond to their thinking. In another session, Hyman Bass pointed out that when mathematical reasoning is part of the classroom culture, the mathematical reasoning of third-graders can be consonant with the mathematical reasoning of professional mathematicians. Thus, the reasoning of third-graders needs to be respected and understood by the teacher.
A teacher must be able to break down a mathematical idea into its fundamental components. This ability implies that the teacher has a deep understanding of an idea's underlying mathematics. Some teachers, for example, know the algorithm for dividing fractions and explain it using division and multiplication as inverse operations. But a teacher with deep understanding knows there are several components that must be in place before division of fractions can be understood. She will know there are two underlying notions for division; partitive (e.g., 12 cookies shared by 3 children, how many cookies each?) and quotitive (e.g., 12 cookies, 3 to each child, how many children?). This teacher also has a deep understanding of fractions, including the fact that fractions are used to represent part-whole situations, division, measurement, ratios, and probabilities. The teacher will be able to present situations involving multiplication and division of fractions and understand the role of referent units and the confusion they cause for children. For example, suppose Jan is making candy that calls for ½ cup of corn syrup. She has ¾ cup of corn syrup on hand. How many recipes can she make? How many ½ cups of corn syrup does she have? (Notice that this problem uses the quotitive model for division.) The problem would be represented as ¾ × ½ = 1 ½. What does the ½ stand for? That is, what is the referent unit for ½? It is one cup (of corn syrup). What does the ¾ stand for? The referent unit for ¾ is also one cup. And what is the referent unit for the quotient, 1½? 1½ what? The quotient refers to the number of ½ cups, not to the number of cups. The teacher must ascertain whether or not children can interpret the quotient correctly in such a problem. She can lead students to understand why this problem calls for division and then, through several problems, lead them to a deeper understanding of the division of fractions algorithm. For example, suppose the teacher speaks about the problem 3 ÷ ½ in the following way: for each whole, there are 2 halves, so when I divide 3 by ½ , I can say there are 2 halves in each of the 3 wholes. This reasoning can eventually be extended to show why ¾ ÷ ½ = ¾ × 2.
The inclination to break a problem into smaller pieces is required when analyzing student work. In addition, remodeling tasks on the fly, as was examined in another of Saturday's breakout sessions, requires the ability to quickly discern those aspects of a problem that are causing it to be unsuccessful in getting at the desired concept.
A teacher must be disposed towards focusing on the major ideas of mathematics and know how they are developed. There are several fundamental or “big” ideas in the mathematics taught in K-8 classrooms, ideas such as place value, proportion, symmetry, and rate of change. When a teacher has a good grasp of these ideas, she can guide the instruction

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toward them, particularly at times when questions are asked that might at first seem to be unconnected to the progress of the lesson. A teacher with this disposition and with understanding of the content can find order when chaos seems to exist. She will not be afraid of dealing with a variety of student responses.
The teacher with this disposition is also likely to pick up on a student's comments that contain a hint of a connection to other mathematics. The teacher would recognize this connection, weigh the value of pursuing it, then be able to ask questions that would draw out the connection. Knowledge of the mathematics that comes before and after the class being taught will help this teacher make decisions on whether ideas are worth pursuing at the time they occur.
Liping Ma's work (1999), discussed in the opening plenary session of the meeting, emphasizes the importance of “fundamental mathematics.” The teacher with the disposition we are describing here has a deep understanding of fundamental mathematics and can quickly identify the core ideas in any topic.
Teachers should expect conceptual understanding of mathematics for themselves and for their students. A person who believes that mathematics makes sense is far more likely to seek out that sense and to be successful in mathematics. Teachers who believe that they are capable of understanding the mathematics of elementary schools will seek this understanding. This process begins when the teacher recognizes that understanding is not present. Although this point might sound trivial, many teachers believe they understand elementary school mathematics when they know how to carry out algorithmic procedures and can teach children these algorithms. In such cases, there is not so much an unwillingness on their part to understand as much as there is a lack of knowing that their content knowledge is incomplete. When teachers have opportunities to explore the mathematics that underlies procedures, this opportunity opens doors for them. They now know what it means to understand mathematics, which is the first step toward seeking further understanding. They have a “profound understanding of fundamental mathematics,” as discussed by Ma. Teachers who understand mathematical ideas can then judge whether or not their students are ready for these ideas and might delay instruction on certain topics until they know that the students can understand the mathematics involved.
A teacher should be disposed to enjoy mathematics and to have an interest in learning new mathematical ideas. The teacher who enjoys mathematics has enthusiasm and curiosity that can be infectious in the classroom. This disposition develops out of doing hard thinking about mathematical ideas in ways that lead to success and confidence. Teacher preparation courses and inservice programs must provide opportunities for teachers to tackle the mathematical ideas they are expected to teach in an atmosphere that allows these teachers room and time to grow and to succeed.
Enjoyment should not be confused with “fun.” Teachers sometimes want mathematics to be fun for students and seek out activities they think students will enjoy. Too often, the students enjoy the activity or game and not the mathematics, which gets lost along the way. But enjoying mathematics implies a willingness to undertake the hard work that can lead to wonderful mathematical discoveries and a strong sense of satisfaction. Building a classroom culture, such as that described in Ball's session, is a first step toward

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establishing a community in which mathematics can be enjoyed.
ISSUES
What opportunities do teachers have to develop the “mathematical instincts, sensibilities, and dispositions” that seem crucial to teaching mathematics? In many places, teachers have support for a few days of inservice work, but this amount of time is not sufficient to undertake the development spoken of here. The work described by Schifter, Bastable, and Lester in their session on alternative approaches to helping teachers learn mathematics is one promising approach to teacher development, but there are few such opportunities available to teachers.
Where do teachers find the time to plan classes that will lead to the teaching of mathematics as described throughout this workshop? In China, according to Ma, teachers spend a good part of their day in preparation for teaching, whereas elementary teachers in this country are usually teaching all day, and prepare on their own time. There is not an easy solution to this problem, but it is one that must be solved if we want teachers to be successful in understanding the mathematics themselves and leading their students to understand, appreciate, and enjoy mathematics.