Using Knowledge of Children's Mathematics Thinking in Classroom

Using Knowledge of Children's Mathematics Thinking in Classroom

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DOCUMENT RESUME ED 292 683

AUTHOR TITLE SPONS AGENCY PUB DATE GRANT NOTE PUB TYPE

EDRS PRICE DESCRIPTORS

IDENTIFIERS

SE 049 064

Carpenter, Thomas P.; And Others Using Knowledge of Children's Mathematics Thinking in Classroom Teaching: An Experimental Study. National Science Foundation, Washington, D.C. Apr 88 MDR-8550236 64p.; Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 5-9, 1988). Reports Research/Technical (143) -Speeches /Conference Papers (150) MF01/PC03 Plus Postage. Addition; Elementary Education; *Elementary School Mathematics; *Instructional Improvement; Mathematics Achievement; Mathematics Curriculum; Mathematics Education; *Mathematics Instruction; Number Concepts; *Problem Solving; Subtraction; *Teacher Behavior; *Teaching Methods Mathematics Education Research

ABSTRACT

This study used knowledge derived from classroom-based research or teaching and laboratory-based research on children's learning to improve teachers' classroom instruction and students' achievement. Twenty first-grade teachers, assigned randomly to an experimental treatment, participated in a month-long workshop in which they studied findings on children's development of problem-solving skills in addition and subtraction. Other first-grade teachers (N=20) were assigned randomly to a control group. Although instructional practices were not prescribed, experimental teachers taught problem solving significantly more and number facts significantly less than control teachers. Experimental teachers encouraged students to use a variety of problem solving strategies, and they listened to processes their students used significantly more than did control teachers. They believed that instruction should build upon students' existing knowledge more than did control teachers, and they knew more about individual students' problem-solving processes. Experimental students' exceeded control students in number fact knowledge, problem solving, reported understanding, and reported confidence in problem solving. (Author)

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Using Knowledge of Children's Mathematics Thinking In Classroom Teaching: An Experimental Study

Thomas P. Carpenter

Elizabeth Fennema

University of Wisconsin-Madison Penelope L. Peterson

Michigan State University

Chi-Pang Chiang

Megan Loef

University of Wisconsin-Madison

U S. DEPARTMENT OF EDUCATION Off e of Educational Research and Improvement ED CATIONAL RESOURCES INFORMATION CENTER (ERIC)

This document has been reproduced as received from the Person or organization

originating it. O Minor changes have been made to improve reproduction Quality

Pointsot view or opinions statedinthe document do not necessarily reoreaent official

OERI position or policy

"PERMISSION TO REPRODUCE THIS MATERIAL HAS BEEN GRANTED BY

-Mama R rp TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC)."

Running head: Using Knowledge of Children's Mathematics Thinking Assisting in all phases of the research were Deborah Carey, Janice Gratch and Cheryl Lubinski. Both the experimental treatment and the data collection were facilitated by Glenn Johnson and Peter Christiansen III of the Madison, WI Metropolitan School Dibtrict and Carolyn Stoner of the Watertown, WI Unified School District.

The research reported in this paper was supported in part by a grant from the National Science Foundation (Grant No. MDR-8550236). The opinions expressed in this paper do not necessarily reflect the position, policy, or endorsement of the National Science Foundation. Paper presented at American Educational Research Association Annual Meeting in New Orleans, LA, April, 1988.

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Using Knowledge of Children's Mathematics Thinking Abstract

This study used knowledge derived from classroom-based research on teaching and laboratory-based research on children's learning to improve teachers' classroom instruction

and students' achievement. 20 first-grade teachers, assigned randomly to an experimental treatment, participated in a month-long workshop in which they studied findings on children's development of problem solving skills in addition and subtraction. Other first-grade teachers (N = 20) were assigned randomly to a control group. Although instructional practices were not prescribed, experimental teachers taught problem solving significantly more and number facts significantly less than control teachers. Experimental teachers encouraged students

to use a variety of problem solving strategies, and they listened to processes their students

used significantly more than did control teachers. They believed that instruction should build upon students' existing knowledge more than did control teachers, and they knew more about individual students' problem-solving processes. Experimental students' exceeded control students in number fact knowledge, problem solving, reported understanding, and reported confidence in problem solving.

1

Using Knowledge of Children's Mathematics Thinking In Classroom Teaching: An Experimental Study

Thomas P. Carpenter Elizabeth Fennema University of Wisconsin-Madison Penelope L. Peterson

Michigan State University Chi-Pang Chiang

Megan Loef

University of Wisconsin-Madison

One of the critical problems facing educators and researchers is how to apply the rapidly expanding body of knowledge on children's learning and problem solving to classroom

instruction. Theory and research on cognition and instruction have now reached a point where they can be used profitably to develop principles for instruction to guide curriculum

development and the practice of teaching. Implications for instruction do not follow immediately from research on thinking and problem solving, however, and explicit programs

of research are needed to establish how the findings of descriptive research on children's thinking can be applied to problems of instruction (Romberg & Carpenter, 1986). Thus, researchers need to investigate how research-based knowledge of children's learning and

cognition can be used and applied by real teachers to instruction of real children in actual classrooms with all the complexity that is involved.

To undertake such a task requires building on knowledge derived from classroom-based research on teachers and teaching as well as knowledge elrived from laboratory-based

research on children's learning and cognition. Traditionally, research on children's learning and research on classroom teaching have been totally separate fields of inquiry governed

Using Knowledge of Children's Mathematics Thinking

by different assumptions, asking different questions, employing different research paradigms,

requiring different standards of evidence, and conducted by different groups of researchers (Romberg and Carpenter, 1986). The present investigation was unique because we, as researchers from these two distinct paradigms, came together and used knowledge derived

from both paradigms to attempt to improve the teatting of actual teachers. In the discussion that follows we describe how the present investigation built on knowledge derived from each of these research paradigms.

Research on Children's Thinking

Research on children's thinking has tended to focus on performance within a specified content area, and the analysis of the task domain represents an important component of

the research. This study draws on the extensive research on the development of addition and subtraction, concepts and skills in young children. Researchers have provided a highly structured analysis of the development of addition and subtraction concepts and skills as

reflected in children's solutions of different types of word problems. In spite of differences in details and emphasis, researchers in this area have reported remarkably consistent findings

in a number of studies, and researchers have drawn similar conclusions about how children solve different problems. This research provides a solid basis for studying how children's

thinking might be applied to instruction. (For reviews of this research see Carpenter, 1985; Carpenter and Moser, 1983; or Riley, Greeno, and Heller, 1983). Analyses of Addition and Subtraction Problems

Research on addition and subtraction word problems has focused on the processes

that children use to solve different problems. Recent research has been based on an analysis of verbal problem types that distinguishes between different classes of problems based on

their semantic characteristics. While there are minor differences in how problems are categorized and some researchers include additional categories, the central distinctions

that are included in almost all categorization schemes are illustrated by the problems in

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Using Knowledge of Children's Mathematics Thinking

Table 1. Although all of the problems in Table 1 can be solved by solving the mathematical

sentences 5 + 8 = ? or 13 - 5 = ?, each provides a distinct interpretation of addition and subtraction.

Insert Table 1 about here.

The join and separate problems in the first two rows of Table 1 involve two distinct types of action whereas the combine and compare problems in the third and fourth row describe static relationships. The combine problems involve part-whole relationships within

a set and the compare problems involve the comparison of two distinct sets. For each type of action or relation, distinct problems can be generated by varying which quantity is unknown, as is illustrated by the distinctions between problems within each row in Table

1. As can be seen from these examples, a number of semantically distinct problems can be generated by varying the structure of the problem, even though most of the same words appear in each problem. Children's Knowledge and Strategies

These distinctions between problems are reflected in children's solutions. Even before they encounter formal instruction, most young children invent informal modeling and counting strategies for solving addition and subtraction problems that have a clear relationship to

the structure of the problems. At the initial lever of solving addition and subtraction problems, children are limited to solutions involving direct representations of the problem.

They must use fingers or physical objects to represent each quantity in the problem, and

they can only represent the specific action or relationship described in the problem. For example, to solve the second problem in Table 1, they construct a set of 5 objects, add

more objects until there is a total of 13 objects, and count the number of objects added.

To solve the fourth problem, they make a set of 13 objects, remove 5, and count the re-

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Using Knowledge of Children's Mathematics Thinking

maining objects. The ninth problem might be solved by matching two sets and counting the unmatched elements. Children at this level cannot solve problems like the sixth problem in Table 1, because the initial quantity is the unknown and, therefore cannot be represented

directl' with objects. Children's problem-solving strategies become increasingly abstract as direct modeling

gives way to counting strategies like counting on and counting back. For example, to solve the second problem in Table 1, a child using a counting-on strategy would recognize

that it is unnecessary to construct the set of 5 objects, and instead would simply count

on from 5 to 13, keeping track of the number of counts. The same child may solve the fourth problem by counting back from 13. Virtually all children use counting strategies before they learn number facts at a recall level. Although children can solve addition and subtraction problems using modeling and

counting strategies without formal instruction, the learning of number facts remains a

goal of instruction (NCTM, 1987). Number facts are learned over an extended period of time during which some recall of number facts and counting are used concurrently. Children

learn certain number combinations earlier than others. Before all the addition facts are completely mastered, many children use a small set of memorized facts to derive solutions

for problems involving other number combinations. These solutions usually are based on doubles or numbers whose sum is 10. For example, to find 6 + 8 = ?, a child might recognize

thi.t 6 + 6 = 12 and 6 + 8 is just 2 more than 12. Derived facts are not used by a handful of bright students, and it appears that derived facts play an important role for many children in the learning of number facts. Applying Cognitive Research to Instruction

Until recently, researchers on children's thinking and problem solving have focused

on children's performance without considering instruction. They have provided a picture of how children solve problems at different stages in acquiring skill in a particular domain,

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Using Knowledge of Children's Mathematics Thinking

but they have not addressed the question of how children learn from instruction. Researchers are beginning to turn their attention to this important question (Carpenter & Peterson, in press), but much of the work represents an extension of the carefully controlled clinical approaches employed in cognitive science (c.f. Anderson, 1983; Collins & Brown, in press).

Although we do not deny the importance of research that investigates how instruction

may operate under optimal conditions, it is not clear that the findings from this research always can be exported directly to typical classrooms. Some researchers have found that

instructional programs that were effective in tutorial settings or with small groups of children were much less effective when they attempted to train teachers to apply them in traditional classroom environments (Fuson & Secada, 1986; Resnick & Omanson, 1987). Thus researchers

need to begin to explore directly how to apply research on children's thinking to instruction in real classrooms with all the complexity that is involved. Several approaches for applying cognitive research to classroom instruction are consistent

with the findings of research on children's thinking and problem solving. One possibility would be to design instruction that directly teaches the knowledge and strategies required

for competent performance. An alternative would be to specify instructional sequences that lead children through the primary stages in the development of competence (Case,

1983). A third alternative would be to use the detailed knowledge that research provides about the errors that children make and the knowledge deficiencies that cause them (Brown & Van Lehn, 1982) to develop specific diagnostic procedures for teachers to assess children's knowledge and misconceptions so that instructional programs could be matched to outcomes

of the assessment to build upon children's existing knowledge or explicitly redress their deficiencies.

Although we concur that instruction should be matched to children's existing knowledge

and that instruction should take into account what researchers know about the thinking involved in the acquisition of competent performance, instructional approaches that attempt

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Using Knowledge of Children's Mathematics Thinking

to specify explicit programs of instruction ignore two critical variables in classroom instruction: 1) the classroom teacher and 2) the classroom setting. To address these two critical variables, we need to draw on knowledge derived from classroom-based research on teachers and teaching. Classroom-Based Research on Teachers and Teaching

Over the years, researchers on teaching have documented that to understand teaching in an actual classroom setting, one needs to recognize that teaching is an interactive process. In early process-product studies of teaching effectiveness, researchers such as Flanders (1970) and others developed teacher-student interaction observation systems to describe

the teaching-learning processes in actual classrooms. These researchers then correlated

observed teacher-student interaction patterns and behaviors with students' scores on standardized achievement tests and defined effective teaching practices as those that correlated highest with achievement (Dunkin and Biddle, 1974). Present day researchers on teaching

have continued to focus on the interaction of the teacher and students in their studies of classrooms. As Shulman (1986) put it, Teaching is seen as an activity involving teachers and students working jointly. The work involves exercise of both thinking and acting on the parts of all participants. Moreover, teachers learn and learners teach (Shulman, 1986a, p. 7). Findings from classroom-based research on teaching have demonstrated the importance

of studying teaching and learning in actual classroom settings. As Good and Biddle (in press) noted, "Early teaching effectiveness research in the 1970's was motivated by a dissatisfaction with previous research which had been conducted in laboratory settings... and

a dissatisfaction with solutions that were not based on an understanding of existing classroom practices and constraints" (Good and Biddle, p. 26). Another assumption shared by most researchers who have conducted classroom-based research on teaching is that the teacher has a central role in classroom instruction (Shulman, 1986a). In teaching mathematics in

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Using Knowledge of Children's Mathematics Thinking

the elementary school the teacher, ultimately, has the responsibility for planning, developing, and carrying out instruction that facilitates students' meaningful learning of mathematics. A Cognitive View of the Teacher

In keeping with the emerging cognitive view of the learner described above, researchers on teaching have begun to take a cognitive perspective on the teacher (Clark and Peterson, 1986; Peterson, 1988). Like their students, teachers are thinking individuals who approach

the complex task of teaching in much the same way that problem solvers deal with other complex tasks. Researchers studying teachers' thinking and decision making have documented that teachers do not mindlessly follow lesson plans in teachers' manuals or prescriptions for effective teaching (Shave lson and Stern, 1981; Clark and Peterson, 1986). ",:eachers

interpret plans in terms o.: .heir own constructs and adapt Irescriptions to fit the situation as they perceive it. Teachers' knowledge and beliefs affect profoundly the way that teachers

teach. Moreover, previous efforts at curriculum reform may have failed because reformers attempted to prescribe programs of instruction without taking into account the knowledge, beliefs, and decision making of the teacher implementing the program (Romberg and Carpenter, 1986; Clark and Peterson, 1986). Thus, teachers' knowledge, beliefs, and decisions have become important foci of study. Reseaxch_on Teachers' Knowledge. Beliefs. and Decisions

Previous research on teachers' decision making suggests that teachers do not tend to base instructional decisions on their assessment of children's knowledge or misconceptions

(Clark and Peterson, 1986). Putnam (1987) and Putnam and Leinhardt (1986) proposed that

assessment of students' knowledge is not a primary goal of most teachers. They argued that keeping track of the knowledge of 25 students would create an overwhelming demand

on the cognitive resources of the teacher. Putnam and Leinhardt hypothesized that teachers follow curriculum scripts in which they make only minor adjustments based on student

feedback. The evidence is far from conclusive, however, to support the belief that teachers

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Using Knowledge of Children's Mathematics Thinking

do not or cannot monitor students' knowledge and use that information in instruction. Furthermore, Lampert (1987) has argued that a concern for monitoring students' knowledge

should be related to a teacher's goals for instruction. Although teachers may be able to achieve short-term computational goals without attending to students' knowledge, they

may need to understand students' thanking to attain higher level goals of facilitating students' meaningful understanding and problem solving. Researchers have begun to investigate how teachers' knowledge of and beliefs about

their students' thinking are related to student achievement. In an earlier study, based on the same group of teachers as the study reported here, we measured 40 first-grade teachers' knowledge of students' knowledge and cognitions through questionnaires and an

interview (Carpenter, Fennema, Peterson, and Carey, in press). We found that these firstgrade teachers were able to identify many of the critical distinctions between addition and subtraction word problems and the kinds of strategies that children use to solve such problems. However, teachers' knowledge was not organized into a coherent network that related distinctions between types of word problems to children's solution strategies for

solving the problems, nor to the difficulty of the problems. In the same study, we found that teachers' knowledge of their own students' abilities to solve different addition and subtraction problems was significantly positively correlated with student achievement on

both computation and problem solving tests. Similar results were reported by Fisher, Berliner, Filby, Mar liave, Cahn, and Dishaw (1980), who found that teachers' success in predicting students' success in solving specific problems on a standardized test was significantly car -

related with their students' performance on the test. In a related study (Peterson, Fennema, Carpenter, and Loef, in press), we found a significant positive correlation between students' problem solving achievement and teachers'

beliefs. Teachers whose students achieved well in addition and subtraction problem solving, tended to agree with a cognitively-based perspective that instruction should build upon

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Using Knowledge of Children's Mathematics Thinking children's existing knowledge and that teachers should help students to construct mathematical knowledge rather than to passively absorb it.

In none of the studies cited above did researchers address the critical question of, "How might knowledge of the very explicit, highly principled knowledge of children's cog-

nitions derived from current research influence teachers' instruction and subsequently affect students' achievement?" Research provides detailed knowledge about children's thinking

and problem solving that, if it were available to teachers, might affect profoundly teachers' knowledge of their own students and their planning of instruction. Shulman (1986b) called this type of knowledge pedagogical content knowledge, and Peterson (1988) referred to it as content-specific cognitional knowledge.

The purpose of this investigation was to study the effects of a program designed to provide teachers with detailed knowledge about children's thinking. We have termea this approach to applying cognitive research, Cognitively Guided Instruction. Cognitively Guided Instruction

Cognitively Guided Instruction (CGI) is based on the premise that the teaching-learning process in real cla,sr)oms is too complex to be scripted in advance, and as a consequence, teaching essentially is problem solving. Classroom instruction is mediated by teachers'

thinking and decisions. Thus, researchers and educators can bring about the most significant changes in claszroom practice by helping teachers to make informed decisions rather than

by attempting to train them to perform in a specified way. The guiding tenet of Cognitively Guided Instruction :5 that teachers' instructional decisions should be based on the goals of instruction which can be achieved through careful

analyses of their students' knowledge. The goals of instruction include dev'lopment of problem solving, understanding of concepts and the acquisition of skills. To accomplish these goals, teachers must have a thorough knowledge of the content domain, and they

must be able to assess effectively their students' knowledge in this domain. Relevant know-

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Using Knowledge of Children's Mathematics Thinking

ledge integrates content knowledge with pedagogical knowledge. In the domain of early addition and subtraction it includes an understanding of distinctions between problems

that are reflected in students' solutions at different levels of acquiring expertise in addition and subtraction. Teachers must have knowledge of problem difficulty as well as knowledge of distinctions between problems that result in different solution strategies. Teachers' ability to assess their own students' knowledge also requires that teachers have an under-

standing of the general levels that students pass through in acquiring the concepts and procedures in the content domain, the processes that students use to solve different problems at each stage, and the nature of students' knowledge that underlies these processes.

Two major assumptions underlie CGI. One is that instruction should develop understanding by stressing relationships between skills and problem solving with problem solving

serving as the organizing focus of instruction. The second assumption is that instruction should build upon students' existing knowledge.

Several broad principles of instruction may be derived from these assumptions. The

fill/ principle that is embedded in all the other principles is that instruction should be appropriate for each student. A second prix .iple is that problems, concepts, or skills being learned should have meaning for each student. The student should be able to relate the new idea to the knowledge that he or she already possesses. Third, instruction should be organized to facilitate students' active construction of their own knowledge with understanding. Because all instruction should be based on what each child knows, the necessity for continual assessment is the fourth, principle. Teachers need to assess not only whether

a learner can solve a particular problem but also how the learner solves the problem. Teachers need to analyze children's thinking by asking appropriate questions and listening

to children's responses. Research on children's thinking provides a framework for this

analysis and a model for questioning. Fifth, teachers need to use the knowledge that they

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Using Knowledge of Children's Mathematics Thinking

derive from assessment of their children's thinking in the planning and implementation of instruction.

Purpose of the Study

The purpose of the study reported here was to investigate whether giving teachers' access to knowledge derived from research on children's thinking about addition and sub-

traction would influence the teachers' instruction and their students' achievement. Our hypothesis was that knowledge about different problem types, children's strategies for solving different problems, and how children's knowledge and skills evolve, would affect

directly how and what teachers did in the classroom. Perhaps more importantly, we hypothesized that such knowledge would affect teachers' ability to assess their own students,

which, in turn, would be reflected in teachers' knowledge about their own students. Teachers' knowledge of their own students would affect instruction, allowing teachers to better tailor instruction to students' knowledge and problem solving abilities. Thus, students' meaningful learning and problem solving in mathematics would be facilitated.

Aingof Paradigms in the Design of the Study Because we were attempting to build on research-based knowledge derived from both research on children's learning and research on classroom teaching, we drew on both paradigms in designing the study. We measured student achievement with standardized achieve-

ment tests in the trauition of classroom-based process-product research on teaching, but we constructed additional tests and scales that were sensitive to the distinctions between different levels of problem solving identified by cognitive research on children's learning. To assess learning, we interviewed children using techniques derived from research on

children's problem solving in arithmetic to identify the processes that children used to solve different problems. We developed two classroom observation systems that followed procedures commonly employed in studies of classroom instruction. However, we derived

many of the observation categories from a cognitive analysis of the content of instruction,

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Using Knowledge of Children's Mathematics Thinking

the strategies that children use, and different assessment and grouping practices that we hypothesized that teachers might employ in applying their knowledge about children's thinking. Our observation categories required observers to go beyond students' overt behavior to infer the cognitive strategies that teachers expected or encouraged students to use and

that students were actually using to solve problems. Finally, in this study we employed a large sample of classrooms and quantifiable measures to provide the kinds of evidence commonly reported in traditional process-product studies of teaching.

In spite of

some similarities in design and methodology to traditional process-product experimental

studies, such as that of Good, Grouws and Ebmeier (1983), the present study differed in several important respects from those studies. First, we did not specify a program of

instruction for the teachers. Teachers designed their own programs of instruction. A goal of the study was to investigate whether and how teachers applied knowledge about children's thinking and problem solving in their own classrooms. Second, we hypothesized

that the most critical influence on teachers' instruction would be their knowledge and learning, including the knowledge about students that teachers gained during the school

year as they taught their own students. Thus, the year of classroom instruction following the initial teacher workshop was not conceptualized as a separate implementation phase;

it was part of the treatment. Whereas we did not continue intensive work with the teachers during the instructional year, we assumed that teachers' knowledge and beliefs would continue to change as teachers gathered more knowledge about their own students. We collected classroom observation evidence not to assess fidelity of treatment implementation, but

rather to obtain quantifiable data that would help us understand what the treatment actually was.

Research Questions

In this study we addressed the following questions about teachers and their students. 1.

Did teachers who had participated in a program designed to help them understand children's thinking:

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Using Knowledge of Children's Mathematics Thinking

2.

a)

Employ different instructional processes in their classrooms than did teachers who had not participated in the program?

b)

Have different beliefs about teaching mathematics, about how students learn, and about the role of the teacher in facilitating that learning than did teachers who did not participate in the program?

c)

Know more about their students' abilities than did teachers who did not participate in the program?

Did the students of teachers who participated in a program designed to help them understand children's thinking: a)

Have higher levels of achievement than did the students of teachers who did not participate in the program?

b)

Have higher levels of confidence in their ability in mathematics than did the students of teachers who did not participate in the program?

c)

Have different beliefs about themselves and mathematics than did students of teachers who did not participate in the program? Method

Overviqw

Forty first-grade teachers participated in the study. Half of the teachers (N = 20) were assigned randomly by school to the treatment group. These teachers participated in a four week summer workshop designed to familiarize them with the findings of research on the learning and development of addition and subtraction concepts in young children

and to provide teachers with an opportunity to think about and plan instruction based on this knowledge. The other tea..:hers (N = 20) served as a control group who participated

in two 2-hour workshops focused on non- routine problem solving. Throughout the following school year, all 40 teachers and their students were observed during mathematics instruction by trained observers using two coding systems developed especially for this study. Near

the end of the instructional year, teachers' knowledge of their students was measured by as1;ing each teacher to predict how individual students in