The preservice and continuing education of teachers of mathematics should develop teachers' knowledge of and ability to use and evaluate-

instructional materials and resources, including technology;

ways to represent mathematics concepts and procedures;

instructional strategies and classroom organizational models;

ways to promote discourse and foster a sense of mathematical community;

means for assessing student understanding of mathematics.

Elaboration

In mathematics the reflective process, wherein a construct becomes the object of scrutiny itself, is essential. This is not because, as so many people claim, mathematics is removed from everyday experience. It is because mathematics is not built from sensory data but from human activity (mathematics is a language of human action): counting, folding, ordering, corn-, paring, etc. As a result, to create such a language we must reflect on that activity, learning to carry it out in our imaginations and to name and represent it in symbols and images. (Confrey 1990, p. 107)

Mathematics pedagogy focuses on the ways in which teachers help their students come to understand and be able to do and use mathematics. This standard identifies several components of pedagogy that are essential to quality teaching. These components act as a series of lenses through which teachers filter their knowledge of mathematics and of students in order to enrich and enhance their teaching of mathematics.

Teachers are responsible for posing worthwhile mathematical tasks.They may choose already developed tasks or may develop their own tasks to focus students' mathematical learning. To do so, they often rely on a variety of instructional materials and resources, including problem booklets, concrete materials, textbooks, computer software, calculators, and so on. Teachers need a well-developed framework for identifying and assessing instructional materials and technological tools, and for learning to use these resources effectively in their instruction.

Such a framework is built from teachers' own understanding of mathematics and what constitutes worthwhile mathematical tasks as well as their knowledge of ways to represent mathematical ideas. Modeling mathematical ideas through the use of representations (concrete, visual, graphical, symbolic) is central to the teaching of mathematics. Teachers need a rich, deep knowledge of the variety of ways mathematical concepts and procedures may be modeled, understanding both the mathematical and developmental advantages and disadvantages in making selections among the various models. In addition, teachers need to be able translate within and between modes of representations in order to make mathematical ideas meaningful for students (Heid 1988).

Representations serve as vehicles for examining mathematical ideas. Not only do teachers need to be familiar with a variety of representations, they must be comfortable with helping students construct their own representations. Designing instruction involves a variety of decisions about the role and use of representations: Should structured representational materials be used? If so, which representational materials provide the most appropriate model for helping develop the concept at this point in instruction? If not, how can students refine the existing models they have been using or develop new models for themselves? Of the various options, what representations are most familiar to students and, therefore, will make sense to them? Numerous other questions surface before instructional choices are complete. Choosing, modifying, or constructing representations are central pedagogical considerations that must be addressed continually.

Other capabilities suggest more meaningful learning. Can the child ... illustrate the rule with physical objects? ... give one or more reasons why the rule is true? ... set up a pattern (using objects or using numbers) through which the rule can be discovered? [These] capabilities go beyond computation; they involve connections between numerical symbols and nonnumerical domains, and they make explicit reference to reasoning processes as well as products. (Goldin 1990, p. 46)

Mathematical instruction often is approached in terms of stating and exemplifying rules-the "tell, show, and do" model. Based on the assumption that information can be presented by telling and that understanding will result from being told, such an approach does not work because it frequently overlooks two crucial developmental components: the process of assimilation and the issue of readiness. Essentially, in this approach, students are "ready" intellectually when the teacher is ready for them to receive the information. Leaming through such an approach often fails to promote a lack of transfer of mathematical information to new situations.

Teachers need to employ alternative forms of instruction that permit students to build their repertoire of mathematical knowledge and their abilities for posing, constructing, exploring, solving, and justifying mathematical problems and concepts. Promising models for such instruction are all highly interactive. In such models, teachers both model and elicit mathematical discourse by asking questions, following leads, and conjecturing rather than presenting faultless products (Ball 1990; Noddings 1990).

Teachers need to focus on creating learning environments that encourage students' questions and deliberations-environments in which the students and teacher are engaged with one another's thinking and function as members of a mathematical community. In such a community, the teacher-student and student-student interaction provides teachers with opportunities for diagnosis and guidance and for modeling mathematical thinking, while, at the same time, it provides students with opportunities to challenge and defend their constructions.

Teachers need to employ strategies that will help them develop the participation essential to engaging students in mathematics. Increasing the amount of time students spend working together supports the development of discourse and community. Working in groups, students gradually internalize the discourse that occurs, challenging themselves by asking for reasons and, in general, accounting for their own mental work. Another practice that supports students' participation involves shifting responsibility from teacher to student for control of learning by expecting students to make commitments to their answers. Further, students' reflective processes can be developed by focusing their efforts on interpreting problems, describing strategies for solutions, and justifying and defending the results.

Teachers' willingness to be flexible and curious about mathematics with their students is central to their ability to promote mathematical discourse. Engaging in personal discourse with other colleagues about and mathematics instruction and establishing a classroom hat encourages engagement in discourse helps teachers deepen, extend, and enhance their knowledge of mathematics and of their students' knowledge of mathematics. Teachers need to experience and reflect on discourse and their own efforts to promote discourse in order to identify what works or does not work, enriching and extending heir capabilities to involve their students in mathematical ideas.

As we need standards for curricula, so we need standards for assessment. We must ensure that tests measure what is of value, not just what is easy to test. If we want students to investigate, explore, and discover, assessment must not measure just mimicry mathematics. By confusing means and ends, by making testing more important than learning, present practice holds today's students hostage to yesterday's mistakes. (National Research Council 1989, p. 70)

Assessment should be an integral part of mathematics teaching. Through assessment, teachers learn how students think about mathematics and what they are able to accomplish. Moreover, students obtain feedback in order to make adjustments and deepen their understandings of mathematics (Stenmark 1989).

Assessment should focus on addressing students' development of mathematical power: their understanding of mathematical concepts and procedures and the relationships between them, and their abilities to reason mathematically and apply their knowledge to a variety of problem situations. Teachers need to align assessment with instructional goals and to consider their purposes in assessment as they select or develop the means of assessment. In addition, teachers need to understand the issues surrounding assessment in general, the arguments related to these issues, the distinctions between classroom assessment and accountability testing, and proposed alternatives for unifying instruction and assessment.

Every teacher is continually offered a wealth of assessment information during the process of instruction. Many act on this information but few document it. It is through our documented assessment that we communicate most clearly to students which behaviors and learning outcomes we value. (Clarke 1988, p. 19)

The results of assessment-formal and informal-need to be used and communicated. The communication may involve only the teacher using information collected to provide direction for working with an individual student, group of students, or class of students in continuing instruction. Teachers also need to communicate with one another about learning and teaching. Results from assessment often provide the catalyst needed for jointly diagnosing students' understandings and misunderstandings, designing curriculum, planning instruction, or initiating further assessment efforts. Finally, teachers need to evaluate students and communicate this information to students, parents, and others in a school district to provide feedback of a more formal kind that indicates students' understanding of mathematics.

Assessment has a central role in effective teaching of mathematics. Too often, teachers' experiences with methods of assessment are limited to the more traditional "testing and measurement" strategies provided through a preservice course. Given the growing awareness and efforts for change, there is a strong need to integrate the understanding and use of alternative methods of assessment as an ongoing topic throughout teachers' educational life.

The aspects identified in this standard as "mathematics pedagogy" are integral to the effective teaching of mathematics. Teachers' knowledge and their ability to use and evaluate these components develop over time. Decisions about instructional materials are intimately associated with decisions about ways to represent mathematics concepts and procedures. Choices for instructional strategies and classroom organizational models both evolve from and influence such decisions. Finally, the discourse of the classroom and the need for ongoing assessment also are part of this process of dynamic interaction that results in knowing mathematical pedagogy.

Vignettes

4.1 As part of the state's efforts to align school assessment with NCTM's Curriculum and Evaluation Standards for School Mathematics, a two-day conference is being held. Many schools have sent teacher representatives who will, in turn, sit on district committees to design K-1 2 assessment programs. The conference comprises a number of minisessions, each demonstrating an alternative form of assessing what children know and understand about mathematics.

We first visit the session on interviewing.

Presenter: Assessment is often thought of as being synonymous with paper-and-pencil testing. We would like to broaden this conception to include a multitude of ways of determining what a child actually understands.

The presenter first shows the audience a subtraction paper completed by Myra, a first-grade student. There are twenty-five problems dealing with subtracting a single digit number from a number in the range 5-19.

If paper-and-pencil tests are the only strategy for gathering information, teachers may know very little about their students' understandings of mathematics.

A more open-ended task allows the student flexibility in interpreting the task and demonstrating her understanding, and the teacher can quickly identify information for instructional planning.

Note: Myra counted to 28. Then she jumped to 97, followed by 96.

Myra has four problems circled that have incorrect answers. The audience is asked to hypothesize about what might be Myra's difficulties. After some discussion, it seems clear that the errors are inconsistent, and most agree that probably Myra just got bored.

Next, they view a videotape of a three-minute interview with Myra in her classroom. In the tape, Myra is presented with a large number of cubes (in this case 36) and asked how many there are. We see Myra counting as she points to individual cubes:

Myra: 1, 2, 3, 4, ..., 26, 27, 28, 97, 96, 95, 98, 99. Yes, there are 99 cubes.

Interviewer: Myra, you remember how we grouped cubes by making sets of ten. (Myra nods enthusiastically.) Do you think you could do that for me right now?

Myra begins to put cubes together. The tape zooms in as Myra is counting the remaining single cubes, having made three sets of ten cubes.

Myra: 1 ... 2 ... 3 ... 4 ... 5 ... 6. Six. (Myra looks up and smiles triumphantly).

The interviewer deliberately maintains a supportive and encouraging attitude. The goal is to find out what the child understands and not to intervene.

Quick interviews can provide a degree of knowledge about students' understanding that cannot be provided through paper and-pencil tests.

Interviewer: That's very nice. Now, can you tell me how many you have all together?

Myra wrinkles her forehead in concentration.

Myra: (Smiling). There are nine all together.

Some of the teachers in the audience react to the tape.

Lauren: Look! She counted the six single cubes and the three sets of ten cubes, which results in her answer of nine. I don't think she has a sense of the structure of base-ten place value.

Ed: Yes, the fact that she counted in this way, even after she had actually
put them together herself, makes me think she doesn't understand grouping.

Performance-based assessment involves students, working individually or in groups, solving a problem that may take 15-30 minutes up to a few days.

Such problems are open-ended, permitting students to explore a variety of options. Without a prescribed "rule' or limited expectations concerning results or solution strategy, teachers can find out quite a bit about how their students think.

Next we visit a session on performance-based assessment:

With the help of several high school students in grades 10-12, the presenter engages the teachers in a performance-based assessment activity. In small groups the teachers gather around two or three students and watch as they work on the following problem:

You are given a square that is 5 units on a side. Your task is to draw another square inside this square that is half its area. Write explanation of how you know that your new square is half the area of original square.

Afterwards the teachers are free to ask the students questions about their strategies and solutions. This is followed by a discussion during which the teachers share and compare what they feel they learned from watching the students work.

Marc: This really is a good problem for a variety of students. You could
do it with little "formal" mathematical knowledge.

Allison: It also was interesting to see the students' false starts and the difficulties they encountered. Listening to the students' explanations was particularly telling about what they did and didn't understand. It's so easy to accept an answer that looks good without investigating further.

Lu: I really need to think about how I would use this to assess students' knowledge. I could consider what we know about students' mathematical and problem-solving abilities based on their solutions. Clearly, using just one problem is not the answer. I do need to find more time to think about performance-based assessment in greater detail.

4.2 Fred is a first-year teacher. His preservice preparation has successfully convinced him that he needs to create a problem-solving focus in his teaching. Yet he is not prepared for the ways his students react to his efforts to involve them (Brown, Cooney, and Jones 1985).

In a series of lessons, he has students experiment with dice in order to help them appreciate probability and, eventually, insurance and mortality tables. Both the topic and the methods are not successful, and we hear him discussing his difficulties with a mentor mathematics teacher, Jim, who has been assigned as part of a special program in the school district directed at helping new teachers.

Fred has begun without assessing his students' understanding and conceptions of mathematics.

Fred feels a need to meet his students "on their level." However, the students don't feel as though Fred is addressing their need to learn "real mathematics." The environment in the class is not working well.

"My students don't view probability as real mathematics. They consider the activities with dice as playing games and think we are wasting time. It is really ironic, because if I was doing the types of things they wanted to do, they would be turning around in their seats and talking. So it's a no-win situation."

Jim, a teacher who is known for his ability to employ a variety of strategies to engage his students in mathematics, is sympathetic. He was a first-year teacher himself. And he certainly does not want to discourage Fred's desires to go beyond the textbook to engage his students' interest.

Fred continues, "My professors in college modeled and expected us to think about and use a variety of problem-solving techniques to engage students. When they were around-and my supervising teacher was around-I felt a little rough around the edges, but I didn't expect this to happen. Somehow, we never talked about this kind of student reaction. How can I tackle something that they consider "real mathematics" and do it in a problem-solving way?"

Fred is experiencing the first-year syndrome of "flying solo" without the ready availability and commitment of several professionals to assist him. He also views traditional textbook mathematics as not applicable to problem solving.

The mentor teacher identifies the two extremes of Fred's understanding of what constitutes successful ways to promote students' involvement.

Jim offers an observation. "It sounds like you may be thinking about problem solving and what your students call 'real mathematics' as two distinctly different ways to think about mathematics instruction. It is possible to rephrase questions to encourage more open-ended problem investigations even with the more traditional topics in mathematics."

Jim opens the textbook to a page that discusses finding the perimeters of a number of different figures. In each case, the figure is shown and each side of the figure is labeled with its respective measure.

Jim notes, "This is pretty straight-forward. But what if you posed some different questions? How about this one: Draw a rectangle with an area of twenty square units. What's the smallest perimeter you can have with a rectangle having an area of twenty square units? What's the largest?

The mentor teacher helps Fred consider ways to engage students in problem solving while he includes more traditional content. Question posing is an essential element in building classroom discourse. Jim's instructional strategies provide Fred with a window to make some changes in what and how he teaches.

The mentor teacher intends to help Fred use his instruction as a vehicle for assessing
students understanding so that he can find ways to modify and adapt his teaching more appropriately.

Alternatively, you can have your students construct rectangles with perimeters of twenty units. Now, what's the largest area possible? The smallest area possible? Going further, what happens if we consider other values for the areas or perimeters? What theories can the students build?"

Fred is silent for a short while. "Well, that is a good strategy. But I really could use some help looking at the textbook and thinking about integrating experiences with problem solving."

Let's plan to meet during our joint planning session tomorrow," Jim responds. "We'll begin to map out what content you want to consider during the next few weeks. Together we'll explore different instructional strategies that may help you engage your students in more problem solving even while they are doing what they call 'real mathematics.' As part of our discussion, let's also talk about how you can do some informal assessing of your students' learning during this process."

As a culminating activity the university students have been assigned the task of choosing a rich problem and using it in an interview with a student or pair of students during one of their pre-practicum visits to a secondary school. They are to prepare a written report on the qualities the students' understanding and problem solving.

Three of the university students have chosen to interview tenth-grade students using the following problem, which they found in one of the sources provided by Dr. Glendon.

This problem has the potential to shed light an secondary school students' understanding of several different mathematical ideas, their problem solving processes, and their ability to make connections.

In this activity, the students consider questioning techniques. These techniques, being developed in the context of evaluation, can also be applied in teaching.

A textbook is opened at random. The product of the numbers of the facing pages is 31 92, To what pages is the book opened ?

The three students, Michelle Tremblay, Peter Marshall, and Ruth Wong, have decided to make a joint report on their findings and have chosen to organize their presentation using Polya's description of the problem solving process, beginning with understanding the problem.

Michelle reports that the student she interviewed at first seemed confused by the problem and reluctant or unable to get started with it, So she asked, "Can you explain in your own words what the problem is telling you?" and later, "What operation was done with the two page numbers to get the answer 3192?" Michelle comments, "The questions I asked weren't really hints; they were just the encouragement the stunts needed to get into it."

Michelle reports that about half the students interviewed used an algebraic approach; some did so on their own initiative, while others did so following a general hint. "They didn't have any trouble deciding to let x be one page number and, (x + 1) be the other, and they all got x(x + 1) = 3192," she says. "But from there, different students had different problems. Beth wrote

but didn't know how to go on from there.

Bruce wrote

The interview also provides information about the students' persistence when they encounter problems that are structurally similar but numerically more difficult than typical ones.

but got stuck when he tried to factor the quadratic. I think he just gave up because 3192 was such a large number. The coefficients in the quadratics he had seen in class were always much smaller numbers."

The prospective teachers have identified another approach to this problem. This approach represents a mathematical connection that they as teachers may want their students to explore.

Selected follow-up questions help evaluate what Polya has called "looking back.' Another way of thinking about these questions is to see them as helping students to develop mathematical connections.

Peter considers the idea of developing a plan. "Michelle, Ruth, and I thought that students might use a factorization approach in solving this problem, but none of them did. We expected that the students might try to break down 3192 into its prime factors and then reassemble the pieces to make two factors that are consecutive numbers."

Ruth notes that most of the tenth-grade students interviewed by the three university students used calculators as they tried to solve the problem by a guess-and-test approach, which her student, Marc, referred to as "the long way." However, none of them thought of the square root operation. "I was a little disappointed that no one realized that would be a good estimate for the page numbers," she says. "As a follow-up question, after Marc had found the correct page numbers by trial-and-error, I asked, 'If you had known the value of the square root of 3192, would that have helped you to solve the problem?' But he didn't seem to see the connection I was hinting at."

Cross-Referencing Standard 4 with Vignettes in Sections 1 and 2

The vignettes presented in the first section, "Standards for Teaching Mathematics," reflect teaching episodes documenting the four components of tasks, discourse, environment, and analysis. As might be expected, these vignettes demonstrate teachers' applications of the components detailed in this standard as well. The chart below provides a cross-reference from the subtopics of this standard to the vignettes presented in the first section. For example, vignette 1.3 provides an example of using the calculator; as part of vignette 2.3, a teacher evaluates a mathematics textbook, making decisions about ways to integrate its use into her instructional plan. Both these vignettes demonstrate teachers using and evaluating instructional materials and resources.

As part of the preservice and continuing education of teachers of mathematics, the vignettes charted below may be helpful as part of teachers' efforts to develop the "knowledge of, and ability to use and evaluate" instructional materials and resources, ways to represent mathematics concepts and procedures, and so on.