Knowledge Connectedness in Geometry Problem Solving

Lawson, M. J., & Chinnappan, M. (2000). Knowledge connectedness in geometry problem solving. Journal for Research in Mathematics Education, 31, 26-43.

Advisor: Ming-Puu ChenReporter: Lee Chun-Yi

Doctoral Student at Department of Information and Computer Education, National Taiwan Normal University.

Introduction

• Our concern in this study was to examine the relationship between problem-solving performance and the quality of the organization of students' knowledge. – We report findings on the extent to which content and

connectedness indicators differentiated between groups of high-achieving (HA) and low-achieving (LA) Year 10 students undertaking geometry tasks.

– The HA students' performance on the indicators of knowledge connectedness showed that, compared with the LA group, they could retrieve more knowledge spontaneously and could activate more links among given knowledge schemas and related information.

– Connectedness indicators were more influential than content indicators in differentiating the groups on the basis of their success in problem solving.

Literature Review

• Hiebert et al. (1996) have recently interpreted this aim (Mathematics Education) as meaning that students in mathematics classes "should be allowed and encouraged to problematize what they study, to define problems that elicit their curiosities and sense-making skills" (p. 12).

• In statements on mathematics teaching, teachers have been asked to help students to "develop multiple representations and connections, and construct meanings from new situations" (National Council of Teachers of Mathematics, 1989, p. 125).

• It is argued that the better the quality of the students' problematizing and of their knowledge connections, the more powerful will be the knowledge representations that can be called upon during a problem-solving episode.

Literature Review

• These studies (self-explanation) have shown that the more detailed and focused the constructive activity generated by the students' ongoing self-explanations, the more successful have been their later problem-solving activities (Bielaczyc, Pirolli, & Brown, 1995; Chi, Bassok, Lewis, Reimann, & Glaser, 1989; Chi, De Leeuw, Chiu, & LaVancher, 1994; Renkl, 1997).

• What this self-explanation research does not provide is information about the organizational state of the knowledge representation that can be drawn upon during problem solving.

• The focus of this article is an investigation of the relationship between success in problem solving and the quality of the knowledge connections that have been developed by students when they have tried to make sense of their mathematics.

Literature Review

• In Anderson's (1990) two-- concept theory of memory, components of this associative structure (connections in memory ) are conceptualized as varying in both states of activation and strength. – Activation refers to the momentary availability of a knowledge

component, whereas strength describes the durability of the knowledge component over the long term (Anderson, 1990).

– Anderson's discussion of strength and activation is useful here because it suggests that researchers attempting to gain information about the state of organization of the knowledge base need to probe the nature of the connections among knowledge components in a manner that provides opportunities for the student to access components that are low in activation level.

– If such probing does not occur, statements about the state of connectedness of knowledge may be based on incomplete information because knowledge that is available to the student may remain inert and is likely to be regarded as missing from the student's knowledge base.

Literature Review

• We (Lawson & Chinnappan, 1994) showed that a group of less successful problem solvers at high school level failed to use a substantial body of their available knowledge during attempts to solve geometry problems, yet they could access that knowledge when prompted to do so. – Important components of knowledge remained

inert in these students until they were given cues by the researcher.

Literature Review

• We contended (Chinnappan & Lawson, 1994) that failure to access available knowledge might arise from three aspects of students' processing activity: – the students' dispositional states, – the strategic nature of their memory-search activity, – and the quality of organization of the knowledge relevant

to the problems being considered.

• Access failure might result from one or more of the following problems: – lack of persistence with the solution attempt due to low

self-efficacy, – ineffective use of cues provided in the problem statement, – or lack of strongly connected knowledge relevant to the

problem.

Literature Review

• We report the results of a study that was designed to provide further information about the relationship between indicators of the knowledge states of students and their problem-solving performance.

• This relationship is examined through comparison of the performance of groups of high-achieving and low-achieving students.

• In the design of the study we have made a distinction between performance measures that indicate what problem-relevant knowledge is available to the student and measures that allow us to draw inferences about the state of connectedness of that knowledge.

Literature Review

• Content Indicators: the most commonly used indicators of the states of students' knowledge bases are what students do and what they say during a problem-solving episode.– Written actions – Verbal actions (think-aloud )

• The recall or recognition tasks provide evidence only about the student's knowledge in a discrete form and do not require the student to show relationships between a knowledge component and other related components.

Literature Review

• Connectedness Indicators– Response time: the longer the response time, the less

strongly related and the less accessible is the problem-specific knowledge (Anderson, 1990).

– Hinting Task: Campione, Brown, and Ferrara (1982) developed a graded hinting task that was designed to provide a measure of which knowledge schemas could be accessed and used on transfer tasks.

• We (Lawson & Chinnappan, 1994) adapted this task for use in examining the levels of connectedness of knowledge schemas used in solving simple geometry problems.

• Use of this measure provides information about the relatedness of components in a knowledge schema, information that is not available using presence/absence indicators (the internal connectedness of a schema).

Literature Review

• Connectedness Indicators– In our study of geometry problem solving

(Lawson & Chinnappan, 1994), we developed two tasks to examine relationships among knowledge schemas (external connectedness).

• Application tasks: this task required students to move beyond simple accessing of a schema to embed the complete schema in an appropriate problem framework.

• Elaboration tasks: with this task we investigated how different theorem schemas that were relevant to a particular problem could be related, one to the other.

Literature Review

• The recognition, hinting, application, and elaboration tasks provide information that more directly indexes the state of organization of knowledge than do the content indicators generated from observations of problem-solving and recall performance.

• We contend that effectively organized knowledge will be readily accessed and more richly connected, internally and externally.

Literature Review

• In this study we compared the performance of groups of high- and low-achieving students on sets of content and connectedness indicators.

• The comparison between groups differing in level of problem-solving performance was set up to facilitate the investigation of the influence of the two sets of indicators on problem-solving performance.

Method

• Participants

• Procedure

Participants

• The participants were 36 Year 10 male students from a private college in metropolitan Brisbane, Australia;

• The college curriculum required that all students complete a topic involving trigonometry and geometry during Years 8, 9, and 10.

• High-achieving students (HA: n = 18) came from the upper two Year 10 streams. The low-achieving students (LA: n = 18) came from the three classes of the lower streams.

Procedure

• All students participated individually in two 60-minute sessions. – During the first session, students were required

to complete four tasks: the Free Recall Task, the Problem Solving Task, the Geometry Components Task, and the Hinting Task.

– During the second session, students were required to complete three tasks: the Recognition Task, the Geometry Application Task (Application), and the Geometry Elaboration Task (Elaboration).

First Session

• The Free Recall Task (Recall) required students to identify known geometry theorems and formulas. – Students were asked to recall any geometry

theorems that they knew, and they were told that they could identify the theorems by verbal and written statements or through use of diagrams.

– Better recall performance could reflect the existence of either more extensive available knowledge or more effective recall of available knowledge.

First Session

• The Problem Solving Task consisted of four plane geometry problems that can be solved by the use of theorems and formulas that are taught in the first 3 years of the high school mathematics curriculum. – The task provided a sample of students'

problem-solving performance during which their accessing of problem-relevant knowledge would be cued by the problem statements and by their own problem-solving actions.

– This observation of performance was necessary to provide an estimate of students' knowledge activation when they worked unaided on typical problems.

Problem-Solving Task

First Session

• The Geometry Components Task was developed to examine students' knowledge of parts of geometric figures and of the theorems or rules that are represented by these figures. – Students were shown figures related to the

problem shown and were required to identify the parts of the figure (Forms) and to produce a rule or theorem that was illustrated by the figure (Rules).

– Students saw one figure at a time and were shown five figures during this task.

Geometry Component Task

First Session

• In the Hinting Task students were provided with a sequence of graded hints on the basis of a commonly adopted solution path for three of the problems. – When students failed to produce the complete

solution for one of these problems, they were required to attempt to solve that problem with the help of hints given by the investigator.

– Each hint within a sequence provided a student with an increased level of assistance.

Hinting Task

Second Session

• The Recognition Task developed for this study was computer-presented and computer-controlled and was based on HyperCard software that recorded the time taken by a student to correctly identify a particular geometric form or relationship. – Each display in this HyperCard program was

developed to represent a geometric schema commonly taught in the classroom, for example, right-angled triangle and its properties.

– The scores used for this indicator were the mean recognition times for correctly recognized components.

Recognition Task

Second Session

• In the Application Task students were required to report or illustrate in a diagram an example of use of each of five theorems or formulas. – In undertaking this task, the students

were required to display knowledge of links among components within schemas associated with the theorems or formulas.

– If students were unable to produce an example, they were provided with simple problems to solve.

Application Task

Second Session

• In the Elaboration Task, the investigator presented pairs of theorems or formulas to the students, one pair at a time. – Students were required to generate a

problem that involved use of both theorems.

– This task was designed to provide an estimate of the extent to which students could establish and exploit connections among related schemas.

Elaboration Task

Results

• The performances of the two groups on each of the sets of content and connectedness indicators were compared using separate one-way multivariate analyses of variance.

• Because the F values for both sets of indicators were significant beyond the .05 level, the initial analyses were followed up with univariate t tests of the differences between group means on each indicator within a set.

• In judging the significance of the individual univariate comparisons, we made a Bonferroni adjustment, so that the alpha levels set for significance were .017 and .012 for the content and connectedness indicators, respectively (Stevens, 1996, p. 160).

Results

• Content indicators: – Rules Task (Geometry Components Task), – Recall Task, – Recognition Task.

• Connectedness indicators:– Hinting Task– Elaboration Task– Geometry Application Task– Problem Solving Task (Response Time)

• Discriminant Analysis

Content Indicators

• The difference between the groups for the set of content indicators was statistically significant (Multivariate F(3, 32) = 3.72, p < .03), suggesting, in general terms, that the HA group was able to spontaneously access a wider range of problem-- relevant knowledge.

• The univariate comparisons suggest that it was performance on the Rules Task that contributed to the multivariate significant difference found between the groups on these indicators. – The pattern of performance on these tasks suggested that the

difference between the groups in terms of content was not simply in ability to recognize the simple geometrical forms that provide the basis of knowledge relevant to this area of problem solving.

– Instead, differences between the groups were more apparent in the more complex relationships represented in the Rules Task.

Content Indicators

Connectedness Indicators

• The multivariate test of difference between the groups on the connectedness indicators was also significant (Multivariate F(4, 31) = 4.52, p <.01). – We interpret this difference as pointing to a superiority in

organization of the knowledge of the HA group.

• When the individual univariate comparisons were considered, the t values for both the Hinting and Elaboration comparisons were significant at the adjusted alpha level. – The HA group required less assistance in the form of

graded hints to access relevant knowledge. – The HA students also showed greater evidence of

external connectedness among schemas in the Elaboration Task.

Discriminant Analysis

• In this case the purpose of the analysis was to gain information about which indicators were most important in predicting the membership of the two groups observed in this study.

• For this analysis all the content and connectedness indicators were used as predictors and were entered into a direct discriminant-analysis procedure using the SPSS Discriminant program.

• Apart from the Rules score, the connectedness indicators contribute more strongly to separation of the groups than do the remaining content indicators (look at structure coefficients and standardized weights).

Discriminant Analysis

Conclusion and Discussion

• In both free-recall and prompted situations, the HA students were able to access a wider body of knowledge of geometry facts and theorems than the LA group.

• The groups did not differ in their recognition of geometric forms, but they did differ significantly in their spontaneous accessing of geometric rules.

Conclusion and Discussion

• The results of the Hinting Task showed that the LA students required more assistance to access relevant knowledge that had not been accessed spontaneously.

• The LA group appeared to have knowledge available that was not accessed until they were provided with cues that facilitated memory search.

Conclusion and Discussion

• Use of Hinting task allowed us to make two important judgments about the students. – First, we could provide evidence that supported

the expectation that the more successful students would suffer less from the problem of inert knowledge.

– Second, and more important, by probing systematically for students' knowledge in this Hinting Task, we were able to make reasonable claims about the quantum of knowledge available to the students in this area of geometry.

Conclusion and Discussion

• The response-time data provided further evidence suggestive of the more effective organizational state of the knowledge bases of the HA students. – These students were able to correctly recognize

relevant knowledge components more quickly. – This finding also suggests that some feature of

the state of organization of their geometry knowledge, possibly strength, allowed more rapid access to this knowledge.

Conclusion and Discussion

• The results of the Application and Elaboration Tasks address the issue of knowledge organization more directly.

• We contend that students with high scores on these tasks show evidence of being able to activate wider networks of geometry knowledge.

Conclusion and Discussion

• Configuration model of geometry-theorem proving developed by Koedinger and Anderson (1990)– expert geometry-problem solvers

organize their geometry knowledge in clusters of facts "that are associated with a single prototypical geometric image" (geometric perceptual chunks).

– Our results suggest that the perceptual chunks of the HA group are of better quality than those of the LA students.

Conclusion and Discussion

• The HA students' performance on the indicators of knowledge organization used in this study showed that, compared with the LA group, they – (a) were able to retrieve more knowledge spontaneously

and – (b) could activate or establish more links among given

knowledge schemas and related information.

• The results of this study suggest that successful problem-solving performance is associated with a knowledge base that is better organized and more extended, supporting the views expressed by Prawat (1989) and Larkin (1979).

Implications

• First, our results showed that the organizational quality of geometry knowledge constructed by the high achievers confers on them an advantage over the low achievers in the solution of problems. – The challenge for mathematics educators and classroom

teachers is to devise strategies for helping all students to improve the state of connectedness of their knowledge bases, but particularly to assist the less effective problem solvers to exploit more of the knowledge they have acquired.

– The findings of the present study suggest that less effective problem solvers might need extra time and discussion to set up the types of connections and representations that lead to effective accessing of knowledge.

Implications

• A second major implication of this study concerns assessment of school mathematics, especially geometry. – Senk, Beckmann, and Thompson (1997) found

that high school teachers tended to assess students' understandings from a narrow base of standardized tests and argued for the need to use more open-ended tasks.

– The tasks that we have developed and used in this study, especially the Elaboration and Application Tasks, appear to provide a wider and possibly more productive environment in which students could display their geometrical knowledge.