A follow-up assessment of a second-grade problem-centered mathematics project

P A U L C O B B , T E R R Y W O O D , E R N A Y A C K E L ,

A N D M A R C E L A P E R L W I T Z

A FOLLOW-UP ASSESSMENT OF A SECOND-GRADE

PROBLEM-CENTERED MATHEMATICS PROJECT

ABSTRACT. Five second-grade classes in two schools participated in a project that was generally compatible with a constructivist theory of knowing. At the end of the school year, the students in these classes and their peers in six non-project classes in the same schools were assigned to ten textbook-based third-grade classes on the basis of reading scores. The two groups of students were compared at the end of the third-grade year on a standardized achievement test and on instruments designed to assess their conceptual development in arithmetic, their personal goals in mathematics, and their beliefs about reasons for success in mathematics. The levels of computation performance on familiar textbook tasks were comparable, but former project students had attained more advanced levels of conceptual understanding. In addition, they held stronger beliefs about the importance of working hard and being interested in mathematics, and about understanding and collaborating. Further, they attributed less importance to conforming to the solution methods of others.

In this paper, we report a follow-up assessment of a second-grade mathematics project s whose development was guided by a constructivist theory of knowledge. In particular, we compare the arithmetical learning, beliefs, and motivations of project and non-project students at the end of third grade after both groups had received a year of traditional, textbook-based instruction in the same classrooms. During the discussion of the results, we will refer to a previously reported assessment conducted at the end of the second-grade school year (Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz, 1991). One central issue will be the extent to which project students' demonstrated superiority in terms of conceptual understanding and problem-solving in arithmetic which is upheld after a year of traditional instruction. In addition, an analysis of students' responses to the beliefs and personal goals will clarify how project students coped with the contingencies of traditional instruction.

T H E O R E T I C A L O R I E N T A T I O N

Two years prior to this assessment study, we conducted a year-long classroom teaching experiment in collaboration with one teacher, in the course of which we developed a set of instructional activities for all areas of second-grade mathematics. In doing so, we were guided by a constructivist view of learning (Piaget, 1971, 1980; von Glasersfeld, 1990a) and by detailed cognitive models of young children's mathematical learning (Cobb and Wheatley, 1988; Steffe and Cobb, 1988; Steffe, von Glasersfeld, Richards, and Cobb, 1983). These

Educational Studies in Mathematics 23: 483-504, 1992. �9 1992 Kluwer Academic Publishers. Printed in the Netherlands.

484 P. COBB, T. WOOD, E. YACKEL, AND M. P E R L W I T Z

models specify both the conceptual advances that children make and the processes by which they might make them. In addition, two types of instructional settings - - pair collaboration and class discussion - - were developed to facilitate the occurrence of learning opportunities during communicative interactions in the classroom. The rationale for this instructional approach was derived from analyses of the relationship between social interaction and conceptual development in general (Barnes, 1976; Blumer, 1969; Mead, 1934), and in mathematics in particular (Bartolini Bussi, 1991; Confrey, 1990a; Perret-Clermont, 1980; Smith and Confrey, 1991; Yackel, Cobb, and Wood, 1991).

In general, the instructional approach is compatible with the constructivist view that mathematics learning is a process in which students reorganize their mathematical activity to resolve situations that they find personally problematic (Confrey, 1986; Thompson, 1985; von Glasersfeld, 1984a). As a consequence, all instructional activities, including those involving arithmetical computation and numeration, were designed to be potentially problematic for children at a variety of different conceptual levels (Cobb, Wood, and Yackel, 1991a). In addition, the activities reflect the view that what are typically called conceptual and procedural developments should, ideally, go hand in hand (Cobb, Yackel, and Wood, 1988; Silver, 1986). For example, a situation in which a student's current computational procedures prove inadequate can give rise to a problem, the resolution of which involves a conceptual advance that makes possible the construction of more sophisticated algorithms.

In this approach, it is students' experiential realities rather than formal mathematics that constitute the starting point for developing instructional activities (Treffers, 1987). Where possible, research-based models of children's mathematical development were used to guide the development of instructional activities. These included the models of children's arithmetical learning developed by Steffe and his colleagues (Steffe and Cobb, 1988; Steffe, von Glasersfeld, Richards, and Cobb, 1983). However, in some non-arithmetical areas of second- grade mathematics, where research on children's learning was not as extensive or detailed, we relied primarily on intuitions derived from our own prior experiences with young children. For example, the only relevant discussion of the conceptual construction of units of time we identified was a general theoretical analysis conducted by von Glasersfeld (1984b). In general, the research-based models were used to anticipate what might be problematic for students at a variety of conceptual levels and what mathematical constructions they might make to resolve their problems (Cobb, Wood, and Yackel, 1991a). A description of the instructional activities, developed to facilitate students' construction of arithmetical thinking strategies and computational algorithms, can be found in Cobb, Wood, and Yackel (199 la).

P R O B L E M - C E N T E R E D M A T H E M A T I C S 485

Social Interaction and Mathematical Learning

Thus far, we have spoken as though problems arise for students only when they make theft independent, solo interpretations of instructional activities. In doing so, we have underemphasized the argument that mathematical learning is an interactive, as well as an individual, constructive activity (Bauersfeld, 1980, 1988; Confrey, 1990a; Sinclair, 1990; Steffe, 1987; Voigt, 1985; Yackel, Cobb, and Wood, 1991). An emphasis on social interaction brings with it the notion that mathematical learning is a process of enculturation in which students come to be able to participate increasingly in the mathematical practices institution- alized by the wider society (Bishop, 1988; Bruner, 1986; Cobb, Wood, and Yackel, in press-a; Minick, 1989; Rogoff, 1990; Saxe, 1991). Learning opportunities can then be seen to arise for students as they and the teacher interactively constitute taken-as-shared 2 mathematical interpretations and understandings. These taken-as-shared mathematical ways of knowing, themselves the products of prior interactive negotiations, both make mathematical communication possible in the classroom and serve to constrain individual students' mathematical activity (Mehan, 1979; Voigt, 1989). In this characterization, mathematics as an individual, experientially based constructive activity and as a process of enculturation are but complementary sides of the same coin. Students' individual cognitive constructions both contribute to and are constrained by the classroom community's negotiation and institutionalization of mathematical meanings and practices (Cobb, Wood, and Yackel, in press-b).

Three points follow from this discussion on the role of social interaction in mathematics learning. The first relates to the well-documented finding that mathematical activity in school is typically a matter of following arbitrary procedural instructions for many students. As a consequence, one of our goals when we conducted the classroom teaching experiment was to develop an alternative mathematics tradition in the classroom that had the characteristics of what Richards (1991) called "inquiry mathematics." As Richards noted, the dis- tinction between inquiry mathematics and traditional school mathematics is analogous to that between the so-called "logic of discovery" of research mathematics and the "reconstructed logic" of journal mathematics. In contrast to the co-construction of procedural instructions, an inquiry mathematics tradition is characterized by the teacher's and students' interactive constitution of truths about a taken-as-shared mathematical reality (Cobb, Yackel, Wood, and McNeal, 1992). The development of a mathematics tradition of this type in the classroom requires that students have frequent opportunities to discuss, critique, explain, and, when necessary, justify their interpretations and solutions. To facilitate the occurrence of such opportunities in project classrooms, we encour-

486 P. COBB, T. WOOD, E. YACKEL, AND M. PERLWITZ

aged students to work collaboratively in small groups and to participate in whole-class discussions of their problems, interpretations, and solutions.

The second point concerns the nature of the teacher's and students' roles as they do and talk about mathematics. In all instructional settings, there is an essential power imbalance between the teacher and students (Bishop, 1985). It is the manner in which the teacher expresses his or her power in action that is important. It is one thing for the teacher to cue students until they can act as though they have learned what the teacher had in mind all along (Bauersfeld, 1980; Brousseau, 1984; Voigt, 1985) and another for the teacher to express his or her authority in action by initiating and guiding the explicit negotiation of mathematical meanings. Ideally, classroom interactions about mathematics should be characterized by a genuine commitment to communicate in which the teacher assumes that students' mathematical actions are reasonable from their perspectives even if that sense is not immediately apparent to the teacher (von Glasersfeld, 1990b; Wood, Cobb, and Yackel, 1992). The teacher's role in initiating and guiding the negotiation of mathematical meanings is therefore highly complex and involves a variety of activities:

�9 framing conflicts between alternative interpretations or solutions as problems to be resolved (Lampert, 1990)

�9 helping students develop small-group relationships (Wood and Yackel, 1990)

�9 facilitating mathematical dialogue between students (Wood, Cobb, and Yackel, 1990)

�9 implicitly legitimizing selected aspects of students' contributions to a discussion in light of their potential fruitfulness for further mathematical learning (Wood, Cobb, and Yackel, 1991)

�9 redescribing students' explanations in more sophisticated terms that are inferred to be comprehensible to students (Newman, Griffin, and Cole, 1989)

�9 guiding the establishment of particular ways of representing and symbol- izing mathematical activity (Cobb, Yackel, and Wood, 1992; Confrey, 1990b; Kaput, 1991)

The third point concerns the development of classroom social norms that make an inquiry mathematics tradition possible (Cobb, Yackel, and Wood, 1989). For example, the establishment of social norms that enable children to engage in small-group work without constant monitoring by the teacher is essential to the success of the collaborative learning approach. Social norms for small-group work include persisting to solve personally challenging problems, explaining personal solutions to partners, listening to and trying to make sense


of a partner's explanations, and attempting to achieve consensus about an answer, and, ideally, a solution process in situations where a conflict between interpretations or solutions has become apparent. Social norms for whole-class discussions include explaining and justifying solutions, trying to make sense of explanations given by others, indicating agreement, disagreement, and lack of comprehension, and questioning alternatives in situations where a conflict between interpretations or solutions has become apparent.

As Mehan (1979) and Voigt (1985) note, social norms are not static prescrip- tions or rules to be followed but are, instead, continually reconstructed in the course of the classroom interactions. Thus, teachers would not be effective if they merely gave their students a list of rules or principles to follow to engage in inquiry mathematics; indeed no such list could ever be complete. Instead, our observations indicate that specific incidents are typically framed as paradigm cases in which teachers and students discuss what their obligations are to each other and what it means to do mathematics (Cobb, Yackel, and Wood, 1989). In the process, students come to view mathematics as an activity in which they are obliged to resolve problematic situations by constructing personally meaningful solutions that they can explain and justify.

PRIOR R E S U L T S

We inducted 18 second-grade teachers into the project during the year following the classroom teaching experiment. At the end of their first year of participation in the project, an assessment of students' arithmetical learning, beliefs, and motivations was conducted in the three schools in which there were both project and non-project classes.

Arithmetical Learning

Both the previously reported second-grade assessment and the third-grade assessment discussed in this paper focus on students' arithmetical computation abilities and their related numeration and whole number conceptions. This focus was chosen because arithmetical computation is of central importance to many constituencies and because computational proficiency seemed, a priori, to be one of the most difficult goals to achieve in an instructional approach compatible with constructivist theory.

Two tests of arithmetical competence were administered in the prior second- grade assessment study. The first was a state-mandated standardized achievement test (ISTEP) composed entirely of multiple choice items. The mathematics portion of this test was composed of two subtests, Computation, and Concepts


and Applications. The mean grade equivalent scores of the two groups of students were almost identical on the Computation subtest (3.50 and 3.51 respectively), but the performance of project students was significantly superior on the Concepts and Applications subtest (4.54 and 3.73 respectively).

The second arithmetic test, the Project Arithmetic Test, was developed by the project staff. It was composed of two scales, the Instrumental and Relational scales. The label "Instrumental" indicates that it is possible to perform well by using standard computational algorithms without conceptual understanding. In contrast, items on the Relational scale were designed to assess both students' conceptual understanding of place value and their computational abilities in non-textbook formats. The specific items on both scales are described by Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz (1991). The mean scores of the project and non-project students were similar for the Instrumental scale (1.28 and 1.31 respectively, scored out of two), whereas project students' mean score on the Relational scale was significantly superior to that of non- project students (1.00 and 0.62 respectively). Taken together, the results for ISTEP and the Project Arithmetic Test indicate that project students had con- structed more advanced arithmetical conceptions than had non-project students.

Personal Goals and Beliefs about Mathematics

In the process of attempting to initiate a tradition of doing inquiry mathematics in their classrooms, the second-grade project teachers guided the renegotiation of classroom social norms and thereby influenced their students' motivations and their beliefs about their role, the teacher's role, and the general nature of mathematical activity. The first aspect of students' views about learning mathematics in school, assessed in the prior study, concerned their motivational orien- tations or personal goals during mathematics insl~uction. Students responded to the items on the personal goals questionnaire on a five-point scale under each question: "Yes, yes, ?, no, No" (i.e., strongly agree, agree, undecided, disagree, strongly disagree).with the two extreme options being in bolder type. The questionnaire was composed of the four motivational scales shown in Table I.

TABLE I

Personal Goals Scales

(The stem for all items is "I feel really pleased in math when ...")

Effort M I solve a problem by working hard. The problems make me think hard. What the teacher says makes me think. I keep busy. I work hard all the time.


T A B L E I ( c o n t ' d ) Understand and Collaborate M Something I learn makes me want to find out more. Everyone unders tands the work. We help each other f igure things out. Other s tudents unders tand my ideas.

Ego M I know more than others. I finish before my friends. I get more answers right than my friends. I am the only one who can answer a question.

Work Avoidance M It is easy to get the answers right. I don ' t have to work hard. All the work is easy. The teacher doesn't ask hard quest ions.

The personal goals assessed by these scales are:

�9 working hard (Effort M) �9 making sense and collaborating (Understand and Collaborate M) �9 being superior to peers (Ego M) �9 not having to do any work (Work Avoidance M)

The results indicated that both project and non-project students were motivated to work hard and to understand and collaborate. The only significant difference on this group of scales showed that project students were less motivated to be superior to their peers (Ego M).

The second set of scales focused on students' beliefs about what one has to do to succeed during mathematics instruction. The five scales that composed the Beliefs questionnaire are shown in Table II. Both the belief and motivation scales were developed in collaboration with John NichoUs by adapting scales originally designed to assess the personal theories of older students.

T A B L E I I

Beliefs About the Reasons for Success Scales

(The s tem for all items is "Students will do well in math if . . .")

Effort B They work really hard. They always do their best. They l ike to think about math. They are interested in learning.

Understand and Colla borate B They try to explain their ideas to other students. They try to unders tand each o ther ' s ideas about math. They try to unders tand instead of jus t getting answers to problems. They try to f igure things out. They don ' t give up on really hard problems.

490 P. C O B B , T. W O O D , E. Y A C K E L , A N D M. P E R L W I T Z

T A B L E II ( c o n t ' d ) Conform B They solve the problems the way the teacher shows them and don ' t think up their own ways. They all solve the problems the same way and don ' t think up different ways. They try to find their own ways of doing problems, a They like to find different ways to solve problems, a

Competitiveness B They try to get more things right than the others. They try to do more work than their friends. They are smarter than the others.

Extrinsic B They are jus t lucky. Their papers are neat. They are quiet in class.

a These i tems were reversed when scored.

As before, students responded to each item on the Beliefs questionnaire using a five-point scale. These scales assessed students' views about the following reasons for success in mathematics:

�9 working hard and being interested in mathematics (Effort B) �9 persisting and collaborating to understand (Understand and Collaborate

B) �9 conforming to the solution methods of the teacher or peers rather than

developing one's own solution methods (Conform B) �9 being superior to peers (Competitiveness B) �9 being lucky, neat, or quiet (Extrinsic B)

The results showed that both groups of students believed that success stems from working hard. This is consistent with their similar motivational orienta- tions (Effort M). However, project students saw significantly less value in being superior to their peers (Ego M), a result that is consistent with the finding that they were less motivated to be better than others (Competitiveness B). In addition, project students rejected the conjecture that conformity to the teacher's or peers' solution methods leads to success (Conform B). In contrast, non-project students tended to agree with this conjecture. Project students also believed more strongly that the mathematics classroom is a place where success derives from attempts to understand and to explain one's thinking to others (Understand and Collaborate B).

Taken together, the responses to these last two scales suggest that what is meant by doing and understanding mathematics differed in project and non- project classrooms. This inference is consistent with project students' superior


performance on the Relational scale. Thus, although both groups of students were motivated to work hard (Effort M) and to understand (Understand and Collaborate M), they differed in the nature of the activity in which they engaged as they strove to understand and to do their best. This conclusion is consistent with an interactional analysis of mathematical activity in project and non- project classrooms (Cobb, Yackel, Wood, and McNeal, 1992). These findings also lead to the speculation that there might have been significant differences between the expectations of project students and their third-grade teachers at the beginning of the school year. The analysis of data collected at the end of third grade will allow us to infer the extent to which project students modified their beliefs and goals as they attempted to be effective while interacting with their teachers and non-project classmates.

METHOD

Subjects

The principals of each of the three schools that participated in the prior second- grade assessment study assigned students heterogeneously to third-grade classes on the basis of reading scores. The ratios of project to non-project second-grade classes in the three schools were 5:2, 3:2, and 2:4. The students in the latter two schools participated in the current study. Consequently, between one third and two thirds of the students in each of the ten third-grade classes in these two schools had been in a project second-grade class. Complete data sets from the third grade were obtained for 79 former project students and 111 former non- project students.

The third-grade teachers used the Addison-Wesley (1987) third-grade textbook as the basis for their approximately 45 minutes of mathematics instruction each day. They were aware that some of their second-grade colleagues participated in the project, and some had voluntarily attended a half-day workshop conducted by project staff. No systematic observations were made of the third-grade classrooms. There was, however, no indication that any of the ten teachers departed significantly from their teachers' guide when they taught arithmetical topics.

Instruments and Procedures

Arithmetic Tests. Both the state-mandated achievement test (ISTEP) and a Project Arithmetic Test were administered. As at the second-grade level, the mathematics portion of the third-grade ISTEP was composed of two subtests, Computation and Concepts and Applications. All items on both subtests were


taken from the California Achievement Test. This test was given to students in the first week of March by the classroom teachers, who followed a test administration manual. The students' responses were computer-scored by the testing agency.

The Instrumental scale of the third-grade Project Test consisted of three two- digit addition and three two-digit subtraction tasks, all but one of which involved regrouping. The items of the Relational scale were grouped into ten subscales as listed below.

1) Sequence: Two tasks that involved extending a number sequence (e.g., "52, 45, 38, " ) .

2) Everyday Language: Three addition and three subtraction tasks posed in everyday language (e.g., "How many do you have to add to 75 to make 1047").

3) Horizontal Sentences: Four missing addend, missing subtrahend, or missing minuend tasks posed symbolically as horizontal sentences (e.g., " + 28 = 54, 34 - = 1 6 " ) .

4) Multiplication Picture: Two tasks in which a multiplication word problem was accompanied by a picture (e.g., A picture of five boxes and three apples accompanied the task statement, "Each box has 4 apples in it and there are 3 more apples. How many apples are there altogether?").

5) Multiplication-Division: Four multiplication and four division word problems (e.g., "Six pizzas are cut into pieces. The same number of pieces are cut from each pizza. There are 48 pieces in all. How many pieces are there in each pizza?").

6) Money: One addition and one subtraction story problem that involve money (e.g., "Maria spends 87 cents. How much change does she get back from $1.00?").

7) Numeration: Three addition and two missing addend tasks adapted from interview tasks developed by Steffe and Cobb (1988) that involve visible and screened collections (e.g., "There are 25 cubes hidden in the box. How many cubes are there altogether?" The accompanying picture shows a large box, five stacks of ten cubes, and nine individual cubes).

8) Puzzle: One task in which students were asked to find the sum of the numbers on each side of a triangle (see Figure Ia). A second task in which they were asked to find a missing number so that the sums of the numbers on each side of a triangle would be the same (see Figure Ib).

9) Less challenging Word Problems: Five addition and subtraction word problems that exemplify the lowest level of Carpenter and Moser's (1984) classification scheme.


(a) (b) Fig. 1.

10) More Challenging Word Problems: Seven addition and subtraction word problems that exemplify the higher levels of Carpenter and Moser's (1984) classification scheme.

The Project Arithmetic Test was given to all ten third-grade classes by a member of the project staff in the first week in May. Each item was scored as either correct or incorrect.

Goals and Beliefs Questionnaires. The same questionnaires used by Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz (1991) were administered in the first week of May by a member of the project staff. The items and the scales they compose are shown in Tables I and II. The administration procedure is described by Nicholls, Cobb, Wood, Yackel, and Patashnick (1990).

DATA A N A L Y S I S AND R E S U L T S

Arithmetic Tests

The students' achievement on the ISTEP Computation and on the ISTEP Con- cepts and Applications subtests were compared by running ANOVAs with instruction as the main effect and grade equivalent scores as the dependent measure. Differences were not significant on either subtest. The means and standard deviations are given in Table III.

ANOVAs of the same format were run to compare performance on each scale of the Project Arithmetic Test with raw scores (scaled out of 10) as the dependent measure. Differences in favor of project students proved to be significant at the p < .05 level on six of the ten subscales of the Relational scale. In addition,


differences on the Horizontal Sentence scale were very close to achieving significance. The means and standard deviations of the raw scores for both groups of students are given in Table III.

T A B L E III

Means (and Standard Deviations) for ISTEP and the Project Arithmetic Test

Group F P

Project Non-Project (1,189)

ISTEP a

Computation 4.66(1.21) 4.51(1.27) 0.66 .42 Concepts and Applications 6.17(2.74) 5.67(2.55) 1.69 .19

Project Ari thmetic Test b

Instrumental Scale 8.58(2.02) 8.28(2.08) 0.93 .33 Rational Scale

Sequence 6.70(3.57) 5.95(3.88) 1.81 .18 Everyday Language 5.93(3.22) 4.75(2.88) 7.06 .01 * Horizontal Sentences 3.93(3.64) 2.98(3.11) 3.58* .06 Multiplication Picture 7.10(3.36) 6.80(3.79) 0.33 .57 Multiplication Division 5.13(3.00) 4.13(3.02) 3.65 .05* Money 7.25(3.19) 6.10(3.90) 4.72 .03* Numeration 6.64(2.80) 5.32(3.22) 8.55 .00 ' Puzzle 6.70(3.42) 5.55(3.48) 5.11 .03* Less Challenging Word

Problems 8.28(2.58) 8.28(3.12) 0.00 .99 More Challenging Word

Problems 6.59(2.98) 5.43(2.87) 4.81 .03*

Note: n = 79 for project students and 111 for non-project students. a Grade-equivalent scores, b Raw scores converted to a scale from 0 to 10. * p<.05.

Personal Goals and Beliefs about Mathematics

Students' responses to each of the five beliefs scales and four personal goals scales were compared by using ANOVAs with instruction as the main effect. The results for the treatment comparisons are shown in Table IV. Higher scores on the five-point scale indicate agreement and lower scores, disagreement. The analysis of the reasons-for-success scales indicates that project students valued working hard and being interested in mathematics (Effort B) and attempting to understand and collaborate (Understand and Collaborate B) more than non- project students, but saw less value in conforming to others' solution methods (Conform B). With regard to the motivational orientation scales, both groups of students were motivated to work hard (Effort M) and to understand and collaborate (Understand and Collaborate M).


T A B L E IV

Means (and Standard Deviations) on Beliefs and Personal Goals about Reasons for Success Scales

Group F P

Project Non-Project (1,189)

Reasons for Success Effort B 4.71(0.54) 4.42(0.75) 3.50 .06 Understand and Collaborate B 4.34(0.74) 4.05(0.72) 7.54 .01" Conform B 3.66(1.11) 4.52(1.05) 29.84 .00" Competitiveness B 3.12(1.04) 3.12(0.99) 0.00 .99 Extrinsic B 2.73(1.14) 2.64(1.12) 0.25 .62

Personal Goals Effort M 4.13(0.84) 3.94(0.97) 1.94 .17 Understand and Collaborate M 4.48(0.63) 4.35(0.79) 1.99 .16 EgoM 3.34(1.04) 3.31(1.18) 0.44 .51 Work Avoidance M 3.01(1.22) 3.24(1.18) 1.25 .27

Note: n = 79 for project students and 111 for non-project students * p<.05.

D I S C U S S I O N

Arithmetic Tests

The previously conducted comparison of students' performance at the end of second grade supported the conclusion that project students had generally attained more sophisticated levels of arithmetical reasoning than had non- project students (Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz, 1991). It is not possible to directly compare differences in the two groups' mean scores at the end of third grade with those at the end of second grade because different Project Arithmetic Tests were used, and because students who participated in the current study constitute only a subsample of those who participated in the previous study. The most that can be concluded from the students' performance on the Relational scale of the third-grade Project Arithmetic Test is that project students maintained their superiority over non-project students to some degree after a year of traditional instruction. Differences on all but one of the ten subscales, Less Challenging Word Problems, favored project students and seven of these differences either attained or approached statistical significance. It is also noteworthy that the three subscales on which there was no significant difference in performance were generally composed of less conceptually challenging items. For example, mean scores on the Less Challenging Word Prob- lem subscale were identical whereas project students significantly outperformed non-project students on the More Challenging Word Problems subscale. Simi- larly, the difference in group means was significant for the Multiplication-


Division subscale but not for the Multiplication Picture subscale where word problems were accompanied by a picture.

The relatively small magnitude of the difference in group means on the ISTEP Concepts and Applications subtest is consistent with the observation that project students tended to significantly outperform non-project students on the more conceptually challenging tasks. Many of the items on the ISTEP Concepts and Applications subtest could be solved by following the idiosyncratic conventions of traditional, textbook-based instruction. These conventions were, in all probability, directly taught to all students during the third-grade year.

The performance of the two groups of students on both the ISTEP Computa- tion subtest and the Instrumental scale of the Project Arithmetic Test indicates that their ability to complete routine computational tasks presented in familiar textbook formats were similar. The same conclusion was reached when computational proficiency was compared at the end of second grade. There, it was argued that the similarity in computational performance on tasks presented in the column format masked differences in the extent to which students' algorithms were conceptually based. The same conclusion seems warranted a year later. Project students significantly outperformed non-project students when additive tasks were presented in four other formats (Everyday I.amguage, I-Iori- zontal Sentences, Money, and More Challenging Word Problems). The non- project students' greater format-dependency in computational situations can be at least partially attributed to their less sophisticated understanding of place value as indicated by their performance on the Numeration subsc~e

Beliefs and Motivations

The five belief scales shown in Table II were developed to assess students' interpretations of their classroom realities (Nicholls, Cobb, Wood, Yackel, and Patashnik, 1990). The finding that there were significant differences on three of the scales as shown in Table IV might at first seem surprising given that project and non-project students sat side by side in the same third-grade classrooms. Such differences are, however, reasonable from a constructivist perspective which emphasizes that interpretation is an active process. Recall that during the second-grade school year, project teachers initiated and guided the renegotiation of classroom social norms that constituted the social reality of their classrooms during mathematics instruction. For their part, project students modified their beliefs about their own role, others' roles, and the general nature of mathematical activity as they participated in this renegotiation process (Cobb, Wood, and Yackel, 1991b). As a consequence of these modifications, which were documented in the prior assessment, there were general differences in the interpre-


tive schemes within which project and non-project students made sense of classroom events when they entered third grade. The current assessment study documents the ways in which the two groups of students modified their initially differing beliefs during the third-grade school year as they strove to give a meaning and coherence to classroom events (cf. von Glasersfeld, 1984a, 1990b). The differences in the beliefs of the two groups of students at the end of the third-grade year demonstrate that they generally dealt with the issue of how to be effective in different ways, and indeed had differing understandings of what it meant to be effective.

With regard to the specific findings, the greater value that project students placed on doing their best and being interested in mathematics (Effort B), attempting to understand and collaborate (Understand and Collaborate B), and developing personally meaningful solution methods (Conform B) indicates that they had maintained some of the beliefs that the second-grade project teachers had attempted to foster. These results are consistent with the general patterns of performance on the arithmetic tests. In particular, project students' superior conceptual understanding and problem solving is reflected in the beliefs that supported the development of those capabilities. The responses of the two groups of students to the two remaining belief scales were similar, with both groups being generally neutral about the importance of showing they were superior to others (Competitiveness B) and being lucky, neat, and quiet (Extrinsic B).

The students' responses to the Conform B scale are disturbing despite the relatively large difference in group means. The means on this scale at the end of the second-grade year were 2.01 and 3.64 for project and non-project students respectively. These results indicated that project students generally rejected the conjecture that success in school mathematics stems from attempts to conform to the teacher's or other students' solution methods. The third-grade means indicate that project students now tend to agree with the conjecture and non-project students very firmly agree with it. Again, it should be noted that the students who participated in the current study are a subsample of those who participated in the prior second-grade assessment. Nonetheless, the magnitude of the change between second and third grade strongly indicates that both groups came to value conformity more strongly as a consequence of their instructional experiences during the third-grade year.

A case study of one third-grade classroom conducted by McNeal (1991) during the first six weeks of the school year goes some way towards explaining the students' decrease in intellectual autonomy. In the classroom that McNeal stud- ied, the teacher legitimized thinking strategy or derived fact solutions at the beginning of the school year when she and her students reviewed the basic addi-


tion and subtraction facts. However, the teacher rejected efficient, non-standard algorithms when she reviewed two-digit addition and subtraction. From the teacher's point of view, it was essential that her students demonstrate that they were following the steps of the standard algorithms by recording the conven- tional regrouping procedures. More generally, the teacher's actions served to foster a view of mathematics as an activity that involves following procedural instructions. The establishment of this interpretive stance was not restricted to the algorithmic aspects of mathematics, but also occurred when supposedly more conceptual topics such as place value numeration were dealt with and when manipulatives were used. It should be stressed that the teacher was a sin- cere professional who genuinely wanted her students to learn with understanding and who would be judged as highly competent by all traditional means of assessment including those derived from the effective schools research. She was merely doing her job as she understood it, an understanding that is compatible with that of most parents, administrators, test constructors, and, apparently, textbook authors. In doing so, she attempted to induct her students into the interpretive stance of traditional school mathematics, and, in the process, her students came to see increasingly less value in their own non-standard solution methods.

The most surprising finding concerning the students' beliefs about reasons for success is that both groups agreed that attempting to understand and collaborate is important, albeit with differing degrees of firmness (Understand and Col- laborate B). At first glance, this interpretation of the social reality of the classroom would seem to be contradicted by their responses to the Conform B scale. If, however, we assume that the students' responses are sensible from their points of view, then the apparent conflict allows us to clarify students' notions of what it might mean to understand in mathematics. In particular, the results suggest that, for non-project students, to understand meant to enact and perhaps describe the steps of procedural instructions. Project students' less than whole-hearted belief in the importance of conforming to the solution methods of others suggests that mathematical understanding might, in their view, involve a more extensive conceptual aspect. In general, the students' understandings of mathematical understanding reflect the accommodations they made during the school year as' their teachers attempted to induct them into the tradition of school mathematics.

In conlrast to the differences in their beliefs about the reasons for success, the personal goals of the two groups of students were similar. Both groups generally achieved a sense of satisfaction when they worked hard (Effort M) and when they attempted to understand and collaborate (Understand and Collaborate M). Further, both groups were generally ambivalent about whether doing better than others (Ego M) and not having to do any work (Work Avoidance M) were

P R O B L E M - C E N T E R E D MATHEMATICS 499

pleasing experiences. It should also be noted that the students' responses to the Understand and Collaborate M scale call into question a speculation made by Cobb, Wood, Yackel, NichoUs, Wheafley, Trigatti, and Perlwitz (1991). There, it was found that the interpretations of the reality of the classroom made by non- project students were less closely aligned with the personal goal of understanding and collaboration than were those of project students. It was therefore suggested that the persistence of such a discrepancy might lead some non- project students to revise their commitment to understand and collaborate, thus giving rise to negative attitudes towards, or alienation from, mathematics. This does not appear to have occurred during the third-grade year despite the fact that non-project students continue to attribute less importance both to understanding and collaborating (Understanding and Collaborating B) and to developing non-standard solution methods (Conform B). Instead, it would seem that these students and, to a lesser extent, the project students coped with a tension between their personal goals and their beliefs about what they have to do to be effective in their classrooms by modifying their understanding of what it means to understand in mathematics. In other words, the students continued to hold understanding and collaboration as a personal goal, but they modified their conceptions of mathematical understanding as they interacted with teachers who unknowingly attempted to induct them into the school mathematics tradition (cf. Cobb, Yackel, Wood, and McNeal, 1992; Lampert, 1990).

CONCLUSIONS

The data presented in this paper give further credibility to the claim that a problem-centered instructional approach which emphasizes mathematical argu- mentation is feasible in public school classrooms. In particular, the finding that project students' conceptual understanding and problem solving in arithmetic was superior to that of non-project students after a year of traditional instruction adds credibility to the claim that the problem-centered approach facilitated students' construction of increasingly conceptually sophisticated operations. This conclusion is further substantiated by the finding that project students tended to outperform their non-project classmates on more conceptual challenging tasks. In this regard, Carpenter, Fennema, Peterson, Chiang, and Loef (1989) reported that the performance of students who had participated in their first-grade Cognitively Guided Instruction (CGI) project was superior to that of non-CGI students on more challenging addition and subtraction word problems. In contrast, CGI and non-CGI students' performance was similar on less challenging word problems that exemplify the lowest level of Carpenter and Moser's (1984) classification scheme. The results presented in this paper complement those of

5 0 0 P. C O B B , T. W O O D , E. Y A C K E L , A N D M. P E R L W I T Z

Carpenter, Fennema, Peterson, Chiang, and Loef (1989) by indicating that differences of this sort can be maintained to a significant degree when students are subsequently assigned to textbook-based classrooms.

The differences in project and non-project students' responses to the belief scales also corroborate conclusions drawn from the prior second-grade study. It seems clear that students who participated in the prior study did not respond to the beliefs questionnaire by reporting an official classroom doctrine that was unrelated to their mathematical activity. Instead, the findings of the prior and the current study overwhelmingly indicate that the second-grade project teachers were reasonably successful in influencing some of their students' fundamen- tal beliefs about mathematics and themselves as learners. More generally, the results indicate that project instruction encourages the development of intellectual autonomy whereas textbook-based instruction as it was realized in the third-grade classrooms fosters intellectual heteronomy (cf. Kamii and de Clark, 1985).

At the outset, we noted that the development of the project's instructional approach was informed by a constructivist theory of knowledge. This should not be interpreted to mean that the study reported in this paper constitutes a test of constructivism. As a self-referential theory of knowing, constructivism implies that the issue of whether people really do construct their own ways of knowing is beside the point - - it is not worth asking the question (cf. Rorty, 1983). The interesting question is instead whether the metaphor of active construction is a useful and ethically appropriate way to view people for particular purposes. Readers whose views about what is important in students' mathematics education are similar to our own might conclude that constructivism offers some promise as a theoretical orientation within which to frame questions, develop alternative forms of classroom practice in collaboration with teachers, and analyze learning and interaction in the classroom.

N O T E S

1 The research reported in this paper was supported by the National Science Foundation under grant No. MDR 885-0560 and by the Indiana Department of Education. The opinions expressed do not necessarily reflect the views of either NSF or IDE.

2 The term "taken-as-shared" refers to whatever notions or meanings an individual believes are equivalent to or shared with others. This term is preferred to the more usual "shared" because we do not have direct access to each others' experiences and, consequently, cannot directly compare meanings. The belief that meanings are shared can be thought of as a potentially revisable assumption that makes communication possible. The term "taken-to-be-shared" was used by Schiitz (1962) and was subsequently changed to "taken-as-shared" by Streeck (1979).


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Paul Cobb

Vanderbil t University

Nashvil le , Tennessee

U.S.A.


Terry Wood and Marcela Perlwitz Department of Curriculum and Instruction Purdue University West Lafayette, Indiana, 47907-1402 U.S_4.

Erna Yackel Department of Mathematics, Computer Science, and Statistics Purdue University-Calumet Hammond, Indiana, 46323-2094 U.S.A.

A follow-up assessment of a second-grade problem-centered mathematics project

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