The education of teachers of mathematics should develop their knowledge of the content and discourse of mathematics; including-

mathematical concepts and procedures and the connections among them;

multiple representations of mathematical concepts and procedures;

ways to reason mathematically, solve problems, and communicate mathematics effectively at different levels of formality;

and, in addition, develop their perspectives on-

the nature of mathematics, the contributions of different cultures toward the development of mathematics, and the role of mathematics in culture and society;

the changes in the nature of mathematics and the way we teach, learn, and do mathematics resulting from the availability of technology,

school mathematics within the discipline of mathematics;

the changing nature of school mathematics, its relationships to other school subjects, and its applications in society.

Elaboration

Knowledge of both the content and discourse of mathematics is an essential component of teachers' preparation for the profession. Teachers' comfort with, and confidence in, their own knowledge of mathematics affects both what they teach and how they teach it. Their conceptions of mathematics shape their choice of worthwhile mathematical tasks, the kinds of learning environments they create, and the discourse in their classrooms.

Knowing mathematics includes understanding specific concepts and procedures as well as the process of doing mathematics. Mathematics involves the study of concepts and properties of numbers, geometric objects, functions, and their uses-identifying, counting, measuring, comparing, locating, describing, constructing, transforming, and modeling. The relationships and recurring patterns among these objects and the operations on these objects lead to the building of such mathematical structures as number systems, groups, or vector spaces and the study of the similarities and differences among these structures. Mathematical concepts and structural properties are used to create powerful algorithms or procedures for solving whole classes of problems. At any level of mathematical study, there are important and appropriate concepts and procedures to be studied.

Such knowledge ought not to be developed in isolation. The ability to identify, define, and discuss concepts and procedures, to develop an understanding of the connections among them, and to appreciate the relationship of mathematics to other fields is critically important. Mathematics both arises out of, and influences continued development of, other fields. Advances in mathematical thought spur advances in physics. Advances in computer science raise new mathematical problems to be solved.

Somebody once quoted a student saying, "The reason I didn't go into mathematics is because my instructors were never interested in hearing what I thought. It was always what they thought." I think that interactions with students are absolutely critical. Interaction, engagement, listening to students, being co-learners with students ... these are important aspects. (A university mathematician]

Knowing mathematics also involves the larger context of mathematical discourse in which specific concepts and procedures are embedded. Discourse in mathematics centers on examining patterns, abstracting, generalizing, and making convincing mathematical arguments. It involves the role of definitions, examples, and counterexamples and the use of assumptions, evidence, and proof. Framing mathematical questions and conjectures, constructing and evaluating arguments, making connections, and communicating mathematical ideas all are important aspects of mathematical discourse. Engaging in mathematical discourse is central to how teachers come to know mathematics; to develop confidence in their own abilities to do mathematics; and to become aware of, and have an appreciation for, the place of discourse in the discipline of mathematics.

I do think a rigorous proof can be worked out by a group of students reasoning together. One student may pick out a nuance of a problem that triggers the key to the next step for another. Students can also learn there are different methods of approaching the same problem by working together. (Gilligan, Lyons, and Hanmer 1990, p. 295)

As part of the environment of discourse, the development of abilities in mathematical reasoning and problem solving are essential. Mathematical reasoning involves an interplay between intuitive, informal exploration and formal, systematic proof. All too often, the formal written record of mathematics is what teachers study. The struggles, the false starts, and he informal investigations that lead to an elegant proof frequently are missing. Teachers need opportunities to construct mathematics for themselves and not just experience the record of others' constructions. In addition, teachers need to interact with others to pose and solve problems in order to develop a repertoire of problem-solving strategies.

All my life I have known that I learned math by doing homework with my friends, comparing answers on the telephone, in the dorms, or on the way to school. This included male and female friends in high school and classmates in an all-female college. It was never something we would admit. If anyone ever found out we hadn't "done our own work," we felt wrong and accused of having cheated. Yet all real intellectual pursuits and learning take place with exchanges of information and ideas. We do not learn in a vacuum. There is as much learning that takes place in the small groups of two or three as there is that takes place at the individual desk. (Gilligan, Lyons, and Hanmer 1990, p. 294)

As an ongoing product of human activity, mathematics is a dynamic and pending system of connected principles and ideas constructed through exploration and investigation. Developing such a perspective includes an appreciation for the historical and cultural contributions made to the development of mathematics. It provides a provocative backdrop that may be useful in motivating students as they approach new subject matter and in encouraging the full participation and continued study of mathematics by all students.

Mathematics is a dynamic discipline that continues to grow and expand in its uses in our culture. Teachers will be called on to adapt to curriculum changes that this growth will entail. The study of some of the contributions made to the development of mathematics by different cultures should provide teachers with resources to use in motivating students as they approach new subject matter.

Technology is a vital force in learning, teaching, and doing mathematics, providing new approaches for solving problems and influencing the kinds of questions that are investigated. It should play a significant role in the teaching and learning of mathematics. There are a variety of ways technology may be used to enhance and extend mathematics learning and teaching. By far the most promising are in the areas of problem posing and problem solving in activities that permit students to design their own explorations and create their own mathematics.

Technology changes the nature and emphasis of the content of mathematics as well as the pedagogical strategies used to teach mathematics. Indeed, one central issue revolves around the fact that some of the computational procedures that have formed the basis for mathematics courses at all levels are no longer essential. Performing computational and representational procedures by hand is time-consuming, and students often lose sight of mathematical insights or discoveries as they become mired in the mechanics of producing the results. With the introduction of technology, it is possible to de-emphasize algorithmic skills; the resulting void may be filled by an increased emphasis on the development of mathematical concepts. Technology-computers and calculators-saves time and, more important, gives students access to powerful new ways to explore concepts at a depth that has not been possible in the past.

Central to the preparation for teaching mathematics is the development of a deep understanding of the mathematics of the school curriculum and how it fits within the discipline of mathematics. Too often, it is taken for granted that teachers' knowledge of the content of school mathematics is in place by the time they complete their own K-12 learning experiences. Teachers need opportunities to revisit school mathematics topics in ways that will allow them to develop deeper understandings of the subtle ideas and relationships that are involved between and among concepts.

Such opportunities should include developing broad understandings of significant mathematics concepts and how they are related to other parts of the curriculum. This includes opportunities to develop a substantial overview of the mathematics curriculum. At all levels, teachers need to see the "big" picture of mathematics across the elementary, middle, and high school years. To use a geographic analogy, teachers need to have a mental roadmap that shows the major cities (curriculum topics) and the roads (mathematical connections) among them. Such a mathematical map should also highlight the importance of connections between mathematics and other school subjects and between mathematics and situations in nonschool settings out of which mathematics arises or in which it is applied.

Common Experience in the Mathematical Education of Teachers

There are common experiences that should be ingredients in the ways teachers of mathematics build and extend their knowledge of mathematics. Regardless of the context, the following themes, as suggested in the Curriculum and Evaluation Standards for School Mathematics, should be prominent in these experiences:

• Problem solving in mathematics
• Communication in mathematics
• Reasoning in mathematics
• Mathematical connections (both within the discipline and to its uses in the world around us)

In addition, mathematical experiences for all teachers should foster-

• the disposition to do mathematics;
• the confidence to learn mathematics independently;
• the development and application of mathematical language and symbolism;
• a view of mathematics as a study of patterns and relationships;
• perspectives on the nature of mathematics through a historical and cultural approach.

These experiences may occur in mathematics courses, workshops, conferences, or other professional development activities. In the process of constructing and developing these experiences, appropriate attention to, and use of, mathematical modeling and technology should be included to enhance the teaching and learning of the mathematical ideas. To this end, teachers should become familiar with instructional technologies that provide powerful numerical, symbolic, and graphical tools for the exploration, investigation, and application of mathematics. These technologies should be incorporated in instruction and used for assignments whenever such inclusion is feasible and can add to student insight and understanding.

The discussion that follows identifies the mathematics content needed by all teachers in grades K-12, the additional mathematics needed by teachers in grades 5-8 and 9-12, and finally the additional mathematics needed by those who plan to teach mathematics in grades 9-12. This ensures that teachers at all grade levels have not only a thorough understanding of the mathematics they are teaching but also a vision of where that mathematics is leading.

Mathematics-- for All Teachers in Grades K-4, 5-8, and 9-12

Foundational knowledge in mathematics is essential both for those teaching mathematics at grades K-4 and for those teaching mathematics at grades 5-8 and 9-12. With regard to specific content preparation, the mathematical education of all K-1 2 teachers should include the following:

Number systems and number sense. Teachers of mathematics should have a well-developed number sense (including mental mathematics, estimation, and reasonableness of results] and an understanding of the use of number concepts, operations, and properties (including basic number theory), of the role of algorithms, and of place value. In setting the view of these ideas in the curriculum, teachers should be able to extend the number systems from the whole numbers to fractions and integers, then rationals and real numbers, including a discussion of the extension of the operations, properties, and ordering. Notions of fractions, decimals, percents, ratio, and proportion should be developed through problems with an applied flavor.

Geometry. Young students have an informal and intuitive idea of size and shape. Teachers need to build on this informal background in the area of geometry. Teachers of mathematics should understand how geometry is used to describe the world in which we live and how geometry can be used to solve real-world problems. Analysis of two- and three-dimensional figures should include the study of tessellations, symmetry, polygons, polyhedra, and curved shapes. Synthetic, coordinate, and transformational geometry should be used to provide opportunities for teachers to solve problems and to hone their skills in building justifications and coherent arguments for the plausibility of conjectures. Throughout the experience, spatial visualization should be emphasized.

Measurement. The concept of measurement needs to be understood from the perspective of its historical development. The attributes of what we measure include length, area, volume, capacity, time, temperature, angles, weight, and mass. Teachers should understand that the units to record measure are different from the process of measurement itself. These ideas should be reinforced through varied experiences, using both standard and nonstandard units where students learn to estimate lengths, areas, and so on. Of particular importance should be an understanding of the Systeme International d'Unites (the metric system). Derivations of the formulas for the perimeter, area, and volume of common figures should be approached through meaningful explorations. Indirect measurement and its many applications should be studied.

Statistics and probability. Teachers should have a variety of experiences in the collection, organization, representation, analysis, and interpretation of data. Key statistical concepts for all teachers include measures of central tendency, measures of variation (range, standard deviation, interquartile range, and outliers), and general distributions. Representations of data should include various types of graphs, including bar, line, circle, and pictographs as well as line plots, stem-and-leaf plots, box plots, histograms, and scatter plots. Probability of simple and compound events and its use in quantifying uncertainty should be built into these experiences. Students should have opportunities to explore empirical probability from simulations and from data they have collected and to analyze theoretical probability on the basis of a description of the underlying sample space. Probability trees and simulations using objects such as spinners, dice, slips of paper, and so on should be used to solve problems.

Functions and use of variables. Teachers need to experience the development of mathematical language and symbolism and how these have influenced the way we communicate mathematical ideas. Also, experience in representing and solving problems requiring the use of variables is important. To build bridges for their students to the mathematics that comes later in the school curriculum, teachers must have a basic understanding of the concepts of functions and their use in the growth of mathematical ideas. Understanding different representations of functions (tabular, graphical, symbolic, verbal), how to move among these representations, and the strengths and limitations of each is fundamental. The distinction between continuous and discrete approaches in the solution of mathematical problems should also be a part of the experiences provided for these teachers and should be introduced initially at an intuitive and informal level.

Additional Mathematics for Teachers in Grades 5-8 and 9-12

Teachers of mathematics at grades 5-8 and 9-12 must present mathematics that builds on the students' background established in the elementary grades. New mathematical knowledge should deepen the understandings of the topics already noted and introduce new and worthwhile mathematical ideas. With regard to specific content preparation, the mathematical education of teachers at the 5-12 level should include and extend the material described earlier by including the following:

Number systems and algebraic structures. The system of real numbers should be extended to the complex numbers. Investigations of selected algebraic structures should include concrete examples such as clock arithmetic, modular systems, and matrices. The properties of the operations in these structures and how they are reflected in the number systems of school mathematics should be investigated, especially the use of matrices and matrix operations to record information and to deal with solutions of systems of equations.

Geometry and measurement. Geometry should focus on intuitive, "common sense" investigations of geometric concepts in such a way that general properties emerge and are used as the basis for conjectures and deductions. Later, observations and deductions can be studied more formally as part of a mathematical system. Tessellations, symmetry, congruence, similarity, measurement, trigonometry, and other notions can be investigated through two- and three-dimensional physical models, drawings, and computer graphics, emphasizing visualization. Synthetic, coordinate, and transformational geometry should be revisited with an emphasis on solving problems. The need for assumptions, for more formal arguments, and for formulating, testing, and reformulating conjectures becomes more evident. Taxicab geometry and geometry on the sphere can be used to study alternatives to Euclidean plane geometry. Dimensional analysis can be used to solve more complex problems involving measurement and attendant conversions.

Statistics and probability. Teachers should learn to use key concepts of descriptive statistics, culminating in personal research projects that include experiences in collecting, organizing, analyzing, and interpreting data and in communicating the results of descriptive statistics to others. The concepts of dispersion and central tendency should be represented using techniques from exploratory data analysis. Relationships between two variables should be represented with scatter plots, and visual techniques for approximating a line of best fit through a scatter plot should be introduced as well. Potential misuses of statistics and common misconceptions of probability should be discussed. The power of simulation as a problem-solving technique for making decisions under uncertainty should be a prominent experience. Experiments involving dice, spinners, random numbers, and computer programs should be used to simulate probability and statistics problem situations. Other topics that should be introduced include fair games and expected value, odds, elementary counting techniques, conditional probability, and the use of an area model to represent probability geometrically.

Concepts of calculus. Teachers should acquire conceptual knowledge of the process of differentiation and integration, including examples of applications of these ideas in the sciences and in modeling and solving problems in mathematics. Functions, graphs, and the notion of limits should be explored, starting with concrete problems such as maximizing the volume of a box that can be folded from a rectangular sheet of grid paper. The rate of change of the volume of the box as a function of the height of the box can be investigated in a way that introduces the concepts of differentiation and integration in an appropriate manner for teachers of grade 5 and above. The concepts of limit and infinity should also be explored for their role in the history of the development of calculus and in the study of geometry.

Additional Mathematics for Teachers in Grades 9-12

Teachers of mathematics in grades 9-12 build on the knowledge students have obtained in grades K-8, provide students with broad experiences in the range of applications of mathematics, and help students extend and formalize their thinking and reasoning. With regard to specific content preparation, the mathematical education of teachers at the 9-1 2 level should include and extend the material described earlier by including the following:

Number systems, number theory, algebra, and linear algebra. Further study of the system of complex numbers should include both geometric (vector) and polar representations of complex numbers and the interpretation of complex solutions to equations. Also, investigations of selected algebraic structures should include groups, rings, integral domains, and fields (including order relations). Topics in number theory should be explored, including modern topics such as coding theory. Because of its wide application, linear algebra should receive extensive treatment. In addition, functions acting on these structures, such as isomorphisms of groups and linear (matrix) functions acting on vector spaces should be investigated.

Geometry. Geometry should be extended to include vector geometry and additional work in synthetic, coordinate, and transformational geometries. Alternatives to the parallel postulate provide opportunities to reveal non-Euclidean geometries. An introduction to the foundations of geometry can provide students with insight into the power of the axiomatic method. The study of geometric transformations, an important manifestation of the function concept, shows the interplay between algebra and geometry. Experiences with linear algebra should be applied to the study of matrix representations of transformations and can shed light on the geometric effects of transformations and the algebraic structure of a set of transformations.

Statistics and probability. For mathematics teachers in grades 9-1 2, the study of probability and statistics should include both descriptive and inferential statistics and probability from both experimental and theoretical viewpoints. The theoretical probability and 'statistics should include both discrete and continuous probability distributions and use such distributions to make inferences about populations. On the experimental side, teachers should have extensive experiences using and creating simulations of probability and statistics experiments, both with concrete objects such as dice and spinners and with computer programs. Misuses of statistics and common misconceptions of probability should be discussed. Descriptive statistics should include exploratory data analysis, including the median fit line for a scatter plot, as well as the traditional measures of dispersion and central tendency. Other statistics topics should include confidence intervals, hypothesis testing, correlation, and regression.

Calculus and analysis. Teachers of mathematics in grades 9-12 should have a firm conceptual grasp of the notions of limit, continuity, differentiation, and integration and a thorough background in the techniques and applications of calculus. The development and use of calculus to model and solve problems involving rates of change, optimization, and measurement need to be appreciated as fundamentally important intellectual achievements in the history of mathematics.

Discrete mathematics. The tools and modeling processes of discrete mathematics have gained increased prominence in applications to realworld problems, including those in computer science. Thus it is essential that the mathematical background of secondary school mathematics teachers include attention to symbolic logic, induction and recursion, relations, equivalence relations and functions, and sequences and series. A wide range of modeling applications of graphs and trees should be explored, along with properties of graphs and trees, matrix representations of graphs, and incidence paths in graphs. Other topics should include difference equations and an introduction to combinatorics.

Recommendations for Coursework in Content Mathematics

For teachers of grades K-4, a sufficient understanding of the mathematical topics described (see pages 135 -137) cannot be attained with less than nine semester hours of coursework in content mathematics. These mathematics courses assume as prerequisite three years of mathematics for college-intending students or an equivalent preparation.

For teachers of grades 5-8, a sufficient understanding of the mathematical topics described (see pages 135 -138) cannot be attained with less than fifteen semester hours of coursework in content mathematics. These mathematics courses assume as prerequisite four years of mathematics for college-intending students or an equivalent preparation.

It is expected that teachers of mathematics in grades 9-12 will have the equivalent of a major in mathematics to gain sufficient understanding of the recommended mathematics (see pages 135 -139). It is recommended that experiences showing the variety of applications of mathematics in other disciplines be integrated throughout their study of mathematics. In addition, an emphasis on problem solving and the history of mathematics is essential. The coursework for teachers at this level assumes as prerequisite four years of mathematics for college-intending students or an equivalent preparation.

Since the spirit and content of the coursework described above can be very different from traditional courses, every effort should be made to develop new courses that reflect these differences.

Given the nature of mathematics and the changes being recommended in the teaching of mathematics, teachers at all levels need substantive and comprehensive knowledge of the content and discourse of mathematics. In addition, teachers need to view mathematics through a variety of lenses, including the role and impact of culture, society, and technology and the place of school mathematics within the discipline of mathematics

Vignettes

2. 1 In a university mathematics class for preservice elementary teachers, students have been studying concepts in number theory. The instructor, Dr. Ong, has posed the following locker problem, and students, grouped in teams of four or five, are working on the problem.

One thousand students have lined up in a very long hall with 1,000 closed lockers. One by one the students run through the hall and perform the following ritual: The first student opens every locker. The second student goes to every second locker and closes it. The third student goes to every third locker and changes its state. If it is open, the student closes it; if it is closed, the student opens it. In a similar manner the fourth, fifth, sixth .... students change the state of every fourth, fifth, sixth .... locker. After all 1000 students have passed down the hall, which lockers are open?

Wanda, Craig, Dina, and Mario have been working together for several minutes, as Dr. Ong circulates among the groups.

Wanda: So the ninth person goes to locker nine and opens it.

Craig: What about the factors involved?

Dina: Seven is going to change the state of .... Seven stayed open until the seventh person got there. Five stayed open.

Wanda: Those are primes.

Mario: So all primes stay open until the person changes the state. So we know that all primes are closed.

Dr. Ong approaches the group.

Craig: One, four, and nine are open.

Dina: These are perfect squares.

Wanda: Let's try four squared.

Craig: Just do sixteen.

Wanda: You can't just do sixteen, because you might have multiples you have to close or open before sixteen.

They see making conjectures as a natural part of their work in this class.

The instructor helps students develop a disposition to question, investigate, and justify. He connects students' discoveries and generalizations to further questions.

Students frame their own mathematical questions and apply other mathematical knowledge as they investigate the problem.

Wanda (turning to Dr. Ong): We're going to conjecture that perfect squares are open. Primes are closed.

Dr. Ong: Why? (When the group offers no explanation, Dr. Ong continues.) You have an interesting conjecture, but why? What is so special about square numbers? What is it about the structure of primes that causes lockers with those numbers to be closed? (He moves on.)

Dina: Well, the primes get touched by only that person.

Wanda: But why are the square numbers open?

Dina: Let's look at composite numbers.

Mario: A composite gets hit for each factor.

Craig: Six is two and three, but four is two and two and nine is three and three. Then why aren't all composites open as well?

The group pursued this problem together for nearly thirty minutes before they had an explanation that satisfied them. Later, when they shared their results with their classmates, they recalled a particular breakthrough.

Wanda: It took us a long time to consider the importance of all factors of the number.

Mario: Yeah, four has one, two, and four as factors. But six has one, two, three, and six as factors.

Dina: Now it seems so simple. Squares have an odd number of factors and other composites have an even number of factors.

Craig: That repeated factor gave us fits.

Students recognize a need for providing convincing arguments to support conclusions reached through inductive reasoning.

Hearing and discussing other people's ideas and questions stimulates students to generate questions that extend the problem.

Yuko had been working in a group that also had found patterns among the squares, but she wasn't entirely convinced by the discussion so far: "I can see how it works for squares like 4, 9, and 25, but how do you know that a very large square number like 576 would be open?"

Other members of Yuko's group added their own questions:

What if there were more than 1000 lockers?

Even if there were only 1000 lockers, what would be the largest locker number that would be open?

What about the largest number of different people that touch the same locker?

2.2 A group of high school mathematics teachers has been meeting twice a month at their school for a seminar with mathematicians and mathematics educators from a nearby university. These teachers have been using computers in their geometry classes for the past year and a half, and the seminar provides them with opportunities to discuss what is happening in their classrooms as they think about new ways of teaching and learning.

The seminar has been a place where teachers can share their struggles with colleagues and university faculty and develop meaningful activities for their students. For many of the teachers, one of the most valuable aspects of the seminar has been the opportunity to extend their own understanding of geometry.

2.3 Dr. McPhee teaches a course on the teaching and learning of secondary school mathematics in a university's teacher education program. All his students hold bachelor's degrees and have taken a considerable amount of mathematics, at least the equivalent of a mathematics minor. Many of the students have a major in mathematics.

These students' background in mathematics is generally broad and deep, but they are only beginning their formal study of school mathematics as a subset of mathematics.

Statistics is an area of mathematics in which there have been substantial changes in the nature of the subject and its applications in society.

The prospective teachers' prior study of mathematics has prepared them to be flexible in understanding new developments and applications.

One focus of this course is the prescribed mathematics curriculum for grades 8-12. The students each have their own copy of the recently revised curriculum guide and are given assignments designed to familiarize them with its organization of topics into strands and scope and sequence.

As they begin their study of the curriculum strand called "Data Analysis,"some students are curious about terms such as stem-and-leaf diagram, box-and-whiskel-splot, 90% box plot, and median-median line. In class Dr. McPhee asks the students to indicate their familiarity with these terms. In almost all cases the terms are new to the students, but many think they can figure out the meanings from the context and on the basis of their previous study of statistics.

Dr. McPhee plans class activities that will introduce current conventions of terminology and practice in statistics. Although the focus of the activities is on the mathematics of the curriculum strand being studied, the tasks are taken from resource books written for teachers in which the problem settings are likely to be interesting to secondary school students.

The students see timely and relevant applications of mathematics in society.

The students consider the changes in the ways mathematics is taught resulting from the availability of technology.

Students observe the connections (similarities and differences) between the median-median line, which is new to them, and the least-squares line, with which many of them are familiar.

In one of the activities, students are provided with data about several communities' exposure to waterborne pollution from an atomic energy plant that produces plutonium and the number of deaths due to cancer in each community. This plant, which is located about 500 km from the university, has recently been in the local news as questions have been raised about airborne pollution from the plant.

As part of the class, students prepare a scatter plot of the data and carry out the steps of the paper-and-pencil procedure for drawing the "median-median line" to show the relationship between the index of exposure to pollution and the number of cancer deaths. The students also use computer software provided with the resource book to execute this procedure. One student in the class has a hand-held calculator with the regression function built in and uses it to perform a similar analysis based on "least squares."