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From page 313... ...
In this chapter we turn from considering what there is to learn and what is known about learning to an examination of teaching that promotes learning over time so that it yields mathematical proficiency. instruction as interaction Our examination of teaching focuses not just on what teachers do but also on the interactions among teachers andstudents around contents Rather than considering only the teacher and what the teacher does as a source of teaching and learning, we view the teaching and learning of mathematics as the product of interactions among the teacher, the students, and the mathematics in an instructional triangle (see Box 9l)

From page 314... ...
314 ADDING IT UP Box 91 The Instructional Triangle: Instruction as the Interaction Among Teachers, Students, anal Mathematics, in Contexts SOURCE: Adapted from Cohen and Ball, 1999, 2000, in press. cal task, ask different questions, and complete the work in different ways.

From page 315... ...
Rather, the quality of instruction is a function of teachers' knowledge and use of mathematical content, teachers' attention to and handling of students, and students' engagement in and use of mathematical tasks. Moreover, effective teaching teaching that fosters the development of mathematical proficiency over time can take a variety of forms.

From page 316... ...
The first example (Box 92) is typical of much teaching that many American adults remember from their own experience in mathematics classes.3 Note how the teacher, Mr.

From page 317... ...
9 TEACH/NO FOR MATHEMAT/CAL PROF/C/ENCY 317 Writing "45,000," Mr. Angelo says, "Good, you are all seeing the trick.

From page 318... ...
He goes over the first exercise to make sure his students remember what to do. While the students work, Mr.

From page 319... ...
By giving the students a rule, he simplifies their learning, heading off frustration and making getting the right answer the point and likely to be attained. Concerned about the spring testing, he attempts to ensure that his students develop a solid grasp of the procedure and can use it reliably.

From page 320... ...
He focuses instead on ensuring that they can use it correctly. Other aspects of mathematical proficiency are also not on his agenda.

From page 321... ...
Jim raises his hand and offers 18 and 13 as equivalent fractions with a common denominator.

From page 322... ...
Satisfied that the students seem to understand and are able to carry out the procedure, she assigns a page from their textbook for practice. The assignment contains a mixture of problems in adding fractions, including some fractions that already have like denominators and many that do not, and in adding whole numbers as well as several word problems.

From page 323... ...
Lawrence's goals: The students spend time practicing mental computation, developing a general rule for adding fractions, explaining and making sense of others' explanations, and working with a partner to practice on more complex examples of what they were learning. The lesson proceeds at a steady pace, but one that affords time for developing the ideas.

From page 324... ...
When the students arrive at a numerical answer, he asks questions such as "Can you explain what that number refers to?

From page 325... ...
If students merely see different representations without explicit attention to their correspondences, the lesson he is teaching will not produce the learning that he is striving for. The discussion of multiple solution strategies at the overhead projector provides an opportunity for Mr.

From page 326... ...
Ms. Kaye asks him what he is going to do next.

From page 327... ...
Again, she works with individual students. Over the course of the class period, she is able to work individually with almost half the class; the next day, while working on the next set of problems, she will try to get to the rest.

From page 328... ...
Comparing the Lessons The four classroom vignettes provide snapshots of different ways in which students, teachers, and content interact to produce different opportunities for student learning, teaching practice, and curriculum content to be mani

From page 329... ...
She asks questions designed to take the lesson where she wants it to go; the students are expected to participate in that venture, answering questions and following the development of the ideas. What she makes mathematically central a procedure for adding fractions together with its justification melds conceptual understanding, procedural fluency, and adaptive reasoning.

From page 330... ...
Ms. Kaye's lesson also illustrates that how the development of the mathematical content in instruction can rest on the teacher engaging students in solving mathematical problems.

From page 331... ...
Lawrence's classes, the teacher is the source of the lesson substance, and the students engage less with one another as a source and medium of mathematical work. These vignettes help to show that the mathematical content and how it Is framed and formulated into instructional tasks make a difference for the learning opportunities provided in a lesson.

From page 332... ...
These snapshots of four classrooms are no more than glimpses into a complex set of interactions happening over time. They are segments from single lessons and, as such, provide a nearsighted view of school mathematics instruction.

From page 333... ...
Using the instructional triangle depiction of instruction in Box 91, we ask what is known about the impact on student learning of how teachers select and use content (the teachercontent side of the triangle) , how teacher and students interact (the teacherstudent side)

From page 334... ...
In one study of secondgrade classes, the average time allocated to mathematics ranged dramatically from a low of 24 to a high of 61 minutes a day for different teachers.~° In another study some "teachers spent as much as 40 percent of their time teaching mathematics; several others never taught mathematics in the twenty randomly chosen hours when our observers visited each classroom." That sort of variation is not unusual across classrooms and even within an individual teacher's practice. Teachers also vary in how they manage the time they have, sometimes focusing on one strand of proficiency and ignoring others.

From page 335... ...
Still, whatever task a teacher poses, its cognitive demand is shaped by the way students use it. In fact, tasks that are set up to engage students in cognitively demanding activities often degenerate into less demanding activities as teachers and students work together to help the student "understand." 14 Several factors have been identified as influencing the decline in cognitive demand from task setup to task enactment.

From page 336... ...
The discussion of multiple solution strategies at the overhead projector provides an opportunity for Mr. Hernandez as well as several students to model a high level of performance another factor that helps maintain engagement in cognitively demanding tasks.

From page 337... ...
Teachers' plans seldom elaborate the content that the students are to learn through their engagement with the proposed activities.~7 Other research suggests that teachers who make detailed plans can sometimes be relatively inflexible when students encounter difficulties or raise thoughtful questions. These teachers are committed to their plans and have difficulty making midcourse adjustments.

From page 338... ...
Low expectations can lead a teacher to interact with certain students in ways that fail to support their development of mathematical proficiency. For example, in comparison with their treatment of high achievers, some teachers consistently wait less time for low achievers to answer a question before calling on someone else.

From page 339... ...
Therefore, teachers can motivate students to strive for mathematical proficiency both by supporting their expectations for achieving success through a reasonable investment of effort and by helping them appreciate the value of what they are learning. Maintaining an expectation of success.

From page 340... ...
Other strategies include helping students to commit themselves to goals that are near at hand, specific, and challenging and then following up by helping them assess their performance in terms of their progress toward those goals rather than by comparing their performance to that of their classmates. In modeling their own mathematical thinking, in communicating expectations to students, and in socializing students' attitudes and beliefs, teachers should continually emphasize that mathematical proficiency is built up through experiences in learning and applying what has been learned (and are not innately given and limited)

From page 341... ...
Rather than motivate students through interest or intrinsic aspects of the intellectual work, he inspires confidence because the goal seems attainable. Teaching Students with Special Needs Although existing research does not provide clear guidelines for teaching mathematics to children with severe learning difficulties, existing evidence and experience suggest that the same teaching and learning principles apply to all children, including specialneeds children.

From page 342... ...
Consider the case of Ann, a Down syndrome child, who is placed in a regular eighthgrade mathematics class along with

From page 343... ...
. In brief, Ann's integration into the class is in name only and does almost nothing to foster her mathematical proficiency or even rote learning of mathematics.

From page 344... ...
For example, lowachieving minority students can do as well as other students when placed in more demanding programs.34 Also, in a study of teachers in schools serving children of poverty, higher achievement results were obtained when teachers placed more emphasis on meaning in their mathematics classrooms.35 Because the quality of the interaction of teacher and student around the content is so critical to the success of instruction, the most successful teachers are not merely sensitive to the cultural diversity of their students but use that diversity to enrich the learning experiences they provide to the class as a whole.36 Communities of Learners Creating classrooms that function as communities of learners has been the focus of much recent research and scholarship in mathematics education.37 In the research on teaching and learning mathematics with understanding, four features of the social culture of the classroom have been identified.38 The first is that ideas and methods are valued. Ideas expressed by any student warrant respect and response and have the potential to contribute to everyone's learning.

From page 345... ...
Managing discourse An important part of classroom instruction is to manage the discourse around the mathematical tasks in which teachers and students engage. Teachers must make judgments about when to tell, when to question, and when to correct.

From page 346... ...
Grouping by achievement level is more common in elementary and middle schools. At those grades, homogeneously grouped classes are usually taught essentially the same content, but the higher the level, the greater the depth and breadth of mathematical ideas and the more rapid the pace.

From page 347... ...
Any gains that might accrue to the high achievers are more than offset by losses to the low achievers and by the resultant perpetuation of social class, racial, and ethnic inequities in schooling.39 This controversy highlights a second point about grouping: Many studies on grouping have been conducted over the years (including studies on grouping for mathematics instruction) , but the results concerning effects on achievement have been both weak and mixed.40 The findings indicate that overall mathematical achievement is likely to be similar whether students are grouped homogeneously or heterogeneously, especially if the same curriculum is provided to all groups.

From page 348... ...
A third is to help students develop their social and collaborative skills and not just support their learning of content. Like most such techniques and tools, whether cooperative groups contribute to the development of mathematical proficiency depends primarily on how they are used.

From page 349... ...
Skills for working cooperatively have to be taught directly, and students need to be prepared for both the social and the cognitive demands of such work. Further, there is evidence that children's collaborative interactions vary across social and cultural groups.45 For teachers to use cooperative groups effectively, they also need to select, organize, and present tasks that are well suited both to collaborative work and to the curriculum.

From page 350... ...
Teachers' ability to interpret and make judicious strategic use of assessment information from many sources is a critical factor in their instructional effectiveness. Students and Content Students and Tasks How well a mathematical task works to support students' learning is a function both of its quality that is, of its potential for stimulating mathematics learning and of the ways students interpret and use it.

From page 351... ...
Textbook and worksheet exercises offer the most common kinds of practice used in U.S. mathematics classrooms.

From page 352... ...
Several useful purposes that homework can serve have been identified, including providing practice, preparing students for the next class, fostering traits such as responsibility and independence, and communicating with the home. Assigning homework for punishment, however, is always inappropriate.53 As a site for practice, homework can be used to increase procedural fluency and to maintain skill.

From page 353... ...
Beginning in the 1960s, manipulatives gained popularity in U.S. elementary school mathematics with the introduction of a variety of concrete materials, including base10 blocks, Cuisenaire rods, chips for trading, logic blocks, fraction pieces, and Unifix cubes, to name a few.

From page 354... ...
Although mathematics educators have advocated the appropriate use of calculators since the 1970s, persistent concerns have been expressed that an extensive use of calculators in mathematics instruction interferes with students' mastery of basic skills and the understanding they need for more advanced mathematics. A large number of empirical studies of calculator use, including longterm studies,62 have generally shown that the use of calculators does not threaten the development of basic skills and that it can enhance conceptual understanding, strategic competence, and disposition toward mathematics.

From page 355... ...
On the other hand, use of calculators enhanced basic skills acquisition by averageability students at all other grade levels, so the negative effect at fourth grade might have been an artifact of conditions specific to those studies that included fourth graders. For all ability groups at all grades, problem solving was improved by the use of calculators.

From page 356... ...
i Issues in Improving Instruction Research on teaching mathematics offers useful direction for developing nstructional practices that lead to mathematical proficiency. The studies we have cited, as well as others too numerous to include, offer a set of recurrent findings worthy of attention.

From page 357... ...
Comparative research that affords opportunities to learn about key elements of teaching and learning, as well as examining both practice and the environments that shape it, would be enormously helpful in developing a greater knowledge of teaching and learning for mathematical proficiency. Researchers need to address not just what the curriculum is but how it is used and what teachers and students do with

From page 358... ...
People seem to assume implicitly that instruction acts on students and that opportunities to learn are actually moments of learning. Research that examined both what students have to know and do in mathematics instruction and what teachers can do to enable all students to make use of that instruction would add significantly to the knowledge base on teaching and learning mathematics.

From page 359... ...
Instruction that develops mathematical proficiency is neither simple, common, nor well understood. It comes in many forms and can follow a variety of paths.

From page 360... ...
360 ADDING IT UP 7. Berliner and Biddle, 1995.

From page 361... ...
9 TEACH/NO FOR MATHEMAT/CAL PROF/C/ENCY 36' 39. Oakes, 1985: Oakes, Gamoran, and Page, 1992.

From page 362... ...
~ 1996~. Empowering children and teachers in the elementary mathematics classrooms of urban schools.

From page 363... ...
Paper presented at the meeting of the American Educational Research Association, San Diego, CA.

From page 364... ...
Paper presented at the meeting of the American Educational Research Association, New Orleans. (ERIC Document Reproduction Service No.

From page 365... ...
(1997~. Mathematical tasks and student cognition: Classroombased factors that support and inhibit highlevel mathematical thinking and reasoning.

From page 366... ...
American Educational Research Journal, 30, 328360. Mason, D., Schroeter, D., Combs, R., &Washington, K

From page 367... ...
~ 1996~. The QUASAR Project: The "revolution of the possible" in mathematics instructional reform in urban middle schools.

From page 368... ...
(1988~. A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers.

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