THE FEATURES AND RELATIONSHIPS OF REASONING, PROVING

FOU-LAI LIN, KAI-LIN YANG and CHUANG-YIH CHEN

THE FEATURES AND RELATIONSHIPS OF REASONING, PROVING AND UNDERSTANDING PROOF IN NUMBER PATTERNS

ABSTRACT. Three issues about students’ reasoning, proving and understanding proof in number patterns are investigated in this paper. The first is to elaborate the features of junior high students’ reasoning on linear and quadratic number patterns. The second is to study the relationships between 9th graders’ justification of mathematical statements about number patterns and their understanding of proof and disproof. The third is to evaluate how reasoning on number patterns is related to constructing proofs. Students in this study were nationally sampled by means of two stages. Some new findings which have not been discovered in some past researches are reported here. These findings include (1) checking geometric number patterns appears to have different positions between the tasks of the linear and the quadratic expressions; (2) proof with the algebraic mode is easy to know but hard to do; (3) disproof with only one counterexample is hard to know but easy to do; (4) arguments with empirical mode or specific symbols were hard for students to validate but very convincing for them; and (5) reasoning on number patterns is supportive for proving in number patterns, and reasoning on number patterns and proof in algebra should be designed as complementary activities for developing algebraic thinking. KEY WORDS: number pattern, reasoning, proof and disproof

The Roles of Reasoning on Number Patterns in Junior High School The study and description of patterns and general rules has been included within the algebra curriculum in some countries for a long time. For example, the Curriculum and evaluation standards for school mathematics (NCTM, 1989) in the US, the National statement on mathematics for Australian schools (Curriculum Corporation for the Australian Educational Council, 1990), and the English National Curriculum (1991) recommended that algebra learning should be introduced by investigating geometric number patterns. These curricula believe that students’ algebraic thinking can be advanced through analyzing, generalizing and representing the functional relationships of number patterns. Junior high school students’ in western countries are familiar with a setting of geometric number patterns, a generalization perspective to introducing algebra. On the contrary, algebra in the Taiwanese curriculum mainly demonstrates the function of generalized arithmetic and provides a vehicle for solving word problems, but is seldom considered as a way of communiInternational Journal of Science and Mathematics Education (2004) 2: 227–256 © National Science Council, Taiwan 2004

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cating a proof of a conjecture or a representation for expressing structures and properties in beginning number theory. Students learn patterns from number series and the rule of judging whether a number is a given multiple, and learn algebra from solving equations or word problems. Therefore, exploration of geometric number patterns does not always stand on its own as a curricular topic or activity in Taiwan. Although there is currently a significant mathematics curriculum innovation under way in Taiwan, A draft plan of nine-year joint mathematics curriculum guidelines (Taiwan Ministry of Education, 2003) also highlights the recognition of regulations, algorithms of number series and symbolic expressions of relationships between patterns. The processes of generalization and representation, which incorporate exploring and searching for geometric number patterns, and explaining patterns verbally or diagrammatically, still remain neglected. A number of studies have revealed useful information about students’ reasoning on geometric number patterns and found several students’ strategies of generalization or representation (Bishop, 2000; Kuchemann & Hoyles, 2001; Orton, 1999; Stacey, 1989). Students attempted to generalize or represent from a single or successive terms with counting or proportional strategies. Students had difficulty in expressing a pattern algebraically even if they had the ability to perceive it. Students who worked backwards or solved equations to answer the questions of determining the number of the shape with a given perimeter had apparently bridged the cognitive gap by connecting the numbers and symbols of an equation to the situation (Bishop, 2000). However, few studies focused on the relationships between students’ generalization, representation and checking in geometric number patterns. One of the study purposes is to explore the relationships between students’ generalization, representation and checking in geometric number patterns. Learning to/about Proof and Disproof When validating and constructing mathematical arguments are set as an essential part of mathematics education, how students construct and understand proof or disproof is of concern. Some literature reports that most of their students construct inductive rather than deductive arguments to prove a conjecture (Balacheff, 1991; Healy & Hoyles, 2000). These western students had the chance to develop their informal methods under teachers’ strong encouragement. On the contrary, students in Taiwan lack such encouragement and are expected to extract the meaning of symbols from a lot of tasks of symbolic manipulations. The difference between students who learn algebra under more structural instruction and under more empirical instruction when they construct proofs seems worth exploring.

REASONING, PROVING AND UNDERSTANDING PROOF

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For simplicity, “proof and disproof” will be abbreviated as “p&d” hereafter. Learning about p&d is to understand what p&d is and to appreciate the need for p&d. Research indicates that students at both high school and university level have difficulty, not only in producing p&d, but also in recognizing what p&d is (Chazan, 1993; Zaslavsky & Ron, 1998; Balacheff, 1991; Moore, 1994). Furthermore, Healy & Hoyles (2000) have characterized that high-attaining 14- and 15-year-old students could recognize algebraic arguments as receiving the best mark but most of them still produced empirical arguments. Nonetheless, these views of proof were also related to students’ own proof constructions. Is there the same pattern in disproof? Understanding and appreciating p&d, in addition to its theoretical interest, is essential in order to give suggestions for the learning and teaching of p&d. We need empirical evidence to verify the relationship between students’ views and their ways in proving and disproving.

One Trend of Approaches to Algebraic Thinking: Generalization Recent research has focused on a number of approaches for developing meaning for the objects and processes of algebra. More emphasis has shifted from computational skills to effective reasoning about quantitative and qualitative relationships in school mathematics curricula (Thompson & Thompson, 1995; NCTM, 2000). The change in emphasis has contributed to a renewed interest in the teaching and learning of algebra. The generalization approach suggests that seeing a generality through the particular and vice versa, and expressing the underlying structure of geometric number patterns, numerical sequences, or the rules governing functional relationships, establish a foundation for algebraic thinking and proving (Bednarz, Kieran & Lee, 1996; Orton, 1999). Bearing in mind that mathematical proof is one threshold of mathematics education, learning proof should be considered from a wider perspective including learning for proving or disproving, and learning about proof or disproof, in addition to learning to prove or disprove formally. Learning for p&d is set to fill a gap from students’ spontaneous reasoning to formal p&d. Any learning activity promoting students’ reasoning is in preparation for learning to prove. For example, the cognitive unity between the phases of conjecture production and proof construction is proposed as a smooth approach to mathematical theorems (Boero, Garuti & Mariotti, 1996), and transformational reasoning may play roles of envisioning the transformation of a mathematical situation and the results of that transformation in making connections among mathematical ideas and validating mathematical ideas (Simon, 1996). Accordingly, to what extent the pattern

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generalization approach is efficient for proving in algebra is an unresolved problem. With respect to the pattern generalization approach to algebraic thinking, much debate has taken place about the unsatisfactory nature of the study of pattern in mathematics. One argument is that students encounter difficulty in reasoning strategies or expressions of patterns with words or symbols, and another is that the way students are taught may promote the learning of a routine procedure without understanding (MacGregor & Stacey, 1995). We agree that the emergent students’ algebraic thinking includes the understanding of signs’ meanings to achieve specific expressions of generality (Radford, 2000). Furthermore, specific expressions of generality can connect reasoning on number patterns and proving statements about number patterns but plays different roles in reasoning and proving. Generality means to generalize from a few of examples, and the symbolic sign is used as pattern generalizers, unknowns or arguments (parameters) while reasoning on number patterns. On the other hand, generality means to generalize and abstract the essential invariant which is encapsulated as a whole, and the symbolic sign can be structurally viewed as arbitrary marks on paper while proving mathematical statements about number patterns. Thus, it is necessary to investigate how reasoning on number patterns is related to constructing proofs. In sum, this study investigated the following research questions: (1) How are 7th and 8th graders able to reason on geometric number patterns when they have not learned this kind of task in school? (2) How do 9th graders construct and evaluate a proof or disproof while explaining whether a statement is right or wrong under a setting in which generalizing and representing number patterns, explanation and argumentation of conjectures are less emphasized? (3) How is 8th graders’ reasoning on geometric number patterns related to their proof-constructions?

L ITERATURE R EVIEW Strategies of Reasoning on Number Patterns Research distinguishes between different kinds of patterns, e.g., number patterns, pictorial/geometric patterns, patterns in computational procedures, linear and quadratic patterns, repeating patterns, etc. (Zazkis & Liljedahl, 2002). In what follows we focus on linear and quadratic geometric number patterns, i.e. number patterns with a pictorial setting. Stacey


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(1989) showed that 9–13-year-old students’ strategies of generalizing linear patterns could be categorized as counting, difference, whole-object, and linear methods, and the methods used by pupils were not affected by whether there was a pictorial context. The counting method and the second or third methods corresponded to concrete and proportional strategies respectively, and the linear method is subdivided into recursive and functional strategies (Bishop, 2000). On the other hand, being able to create a verbal description of relationships appears to be a critical factor that determines whether a student can write an appropriate symbolic representation (MacGregor & Stacey, 1995). Bishop had proposed a developmental sequence from the concrete, the recursive, to the functional category; however, the status of proportional category and reasoning on non-linear geometric patterns still required further research (Bishop, 2000). The students’ strategies should be encouraged when they learn reasoning on number patterns. What teachers can do is to design reasoning tasks and arrange the sequence of these tasks. Therefore, we converted our focus into exploring relations among different reasoning tasks in geometric number patterns: understanding, generalizing, representing, and checking. Besides, students’ reasoning strategies may be influenced by different items (generalizing, representing or checking) and structures (linear or non-linear). Linear and quadratic geometric number patterns are adopted to be the different problem situations. Understanding Formal Proof and Disproof in Algebra In recent research on mathematical proof, there is a broad range of approaches to this topic focused on better understanding students’ ideas of argumentation and proof. Hoyles et al. made a significant contribution to the field with their recent systematic investigation into junior high school students’ reasoning on geometric number patterns, understanding of logical implication, proof constructions, and views of proof (Healy & Hoyles, 1998, 2000; Kuchemann & Hoyles, 2001). The results showed that even high-attainers had great difficulties in generating proofs and are more likely to rely on empirical proofs. However, the majority of students valued general and explanatory arguments. Although around one third of English high-attainers had no idea of the validity of these empirical arguments, more than half gave completely correct evaluations. In disproof, many students are not convinced by a counterexample and view it as an exception that does not contradict the statement in question (Galbraith, 1981; Harel & Sowder, 1998). Even if students seem to understand the special role and status of counterexamples, they are unable to generate a correct counterexample. In an attempt to generate a coun-

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terexample, they either give an example that does not satisfy the necessary conditions or an example that is impossible (Zaslavsky & Ron, 1998). This research showed that students tend to argue that an invalid statement is right, view a counterexample as a non-example or exception, and are impeded by the ability to generate counterexamples. M ETHODS Ongoing projects on the development of mathematical argumentation in England and in Taiwan are conducted bilaterally. In Taiwan, the instruments were adapted from England and modified based on Taiwanese students’ responses. In addition, some new tasks were evolved from our local interviews, which enable the characteristics of students’ pre-formal reasoning to come through in both the instruments and coding system. For our study, we used six booklets comprised of questions in two domains of mathematics – algebra and geometry with respect to grades 7, 8 and 9 (roughly corresponding to 12 to 14+ years old) in Taiwan. Three pilot questionnaires were administered to 765 seventh to ninth graders from two typical junior high schools in Taipei. Not only were modifications made on the basis of students’ responses but the coding systems and some conjectures as to the relations among reasoning on number patterns and mathematical statements about number patterns were also formulated from the pilot study. The initial analysis of students’ proof-construction united the perceptions of protocols of experts, school teachers and postgraduates for fear of missing or misinterpreting students’ responses. Next, these compatible or incompatible perspectives were discussed to develop a coding system so that various reasonings could be consistently coded and interpretably decoded. In England (Healy & Hoyles, 2000), the survey only focused on highattainers (the top 20–25% of the student population) to investigate proof conceptions, because high-attainers would have been introduced to most of the content specified in the eight levels of reasoning in the National Curriculum in England and Wales. However, 7th to 9th graders in Taiwan are not introduced to different content about reasoning, and we believe most students can learn mathematical proof if the instructional activities are recursively designed and revised according to learners’ framework of reasoning, hence our study did not focus on high-attainers but mixed-ability students who are nationally sampled by means of two-stage sampling. The first stage was to divide our nation into six regions, and to sample schools from each region where the relative ratio of students of sampled schools in the six regions was approximately equal to the relative ratio of students


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in the six regions. The second stage was to equally distribute these classes of sampled schools into thirteen groups. Two of the thirteen groups were used as samples in a project on the development of students’ mathematical argumentation. Half the sampled students in each class answered the questionnaire booklet in Algebra (A), and the others answered the booklet in Geometry (G) according to their grade. This survey was completed in 45 minutes, and 1,181 seventh, 1,105 eighth and 1,059 ninth graders, respectively from 61, 60 and 61 classes in 18 schools, answered the booklets in algebra. Students’ judgments and explanations on conjectured statements were first coded and scored according to the information quality and reasoning validity of their arguments. The students’ scores of validity and explanatory power originally from Healy and Hoyles (2000) were adopted to assess their evaluation of generality and explanatory power of each argument. Students are asked to judge if an argument has a mistake and if an argument verifies the generality of a statement as to the validity phase. Students are asked to evaluate to what degree each argument explains the reason and convinces classmates. Descriptive statistics based on frequency tables, simple correlation and tests of independence or significance would be processed. The number patterns are labeled linear or quadratic according to their nth terms expressed as an + b or an2 + bn + c (Stacey, 1989). The features of the tasks are then classified into linear and quadratic forms which were used to represent generalized number patterns or the relation of number patterns in beginning number theory. These tasks include multiplication and addition of linear or non-linear structures, which represented the major content in beginning number theory (Campbell, 2002). For example, a statement like the sum of a multiple of 3 and a multiple of 6 covers a combination of multiplication and addition of linear structures, 3x + 6y = 3(x + 2y). Table I shows our framework of reasoning on number patterns, and proving or disproving statements about number patterns. Because the 8th graders are considered in transition from the stage of reasoning on number patterns to the stage of algebraic proof, this study investigates the relationship between reasoning and proving/disproving in number patterns with respect to them. To assess students’ reasoning on geometric number patterns, four types of items are taken into account: understanding the task, generalizing the number pattern, representing this pattern with symbols and checking if a given number fits into this pattern. Items for the checking component can specifically assess students’ reverse thinking competence of reasoning on number patterns or students’ use of alternative patterns of a given picture, which could not be assessed by items for generalizing number patterns

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TABLE I Framework of reasoning on number patterns and proving/disproving (p&d) Reason Linear geometric

Quadratic geometric

Subject

∗U

G

R

C

U

G

R

C

Grade 7 Grade 8

– –

A1(a) A1(a)

– A1(b)

A1(c) A1(c)

A2(a) A2(ai)

A2(b) A2(aii)

– A2(c)

– A2(d)

p&d Linear conjecture Subject Grade 7 Grade 8 Grade 9

∗ Concept.

– – A1(p), A5(d)

Non-linear conjecture

Validation

Const.

Concept.

Validation

Const.

– – A1(p)

A6(p&d) A6(p&d) A4(p), A6(p&d)

– – A7(p)

– – A7(p)

– – A8(p)

∗ U, G, R and C denote understanding, generalizing, representing and checking respectively. Concept. and Const. denote p&d-conception and p&d-construction respectively.

and representing a pattern with symbols. After identifying the four types of items, exemplary items are presented as Figures 1 and 2. Referred to Figure 2, when presented with a sequence of dot configurations or a figural pattern, students were expected to calculate the number of dots or a sub-figural pattern for the fifth (understanding), twentieth picture (generalizing), to represent the number of dots for the nth picture with symbols (representing), and to determine if a given number can fit into some term in the sequence or sub-figure (checking). Questions A1 and A2 are concerned with reasoning on number patterns in a setting (tile patterns) familiar to English students (Kuchemann & Hoyles, 2001) but unfamiliar to Taiwan students. An approach to seeing a pattern is suggested only in quadratic geometric number patterns (e.g., A2(a)), and students are required to calculate the number of dots for the fourth (e.g., A2(ai)) or fifth picture in the understanding item. The approach provides a hint to focus on the relation between the number of terms and the number of dots within each pattern. Undoubtedly, understanding the meaning of the task is necessary before students can generalize, represent or check the sequence of patterns. In particular, we provide the checking items, A1(c) and A2(d), in addition to


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Figure 1. Question A1 in Grade 8.

Question (b) is similar to question (a) but provides an approach of moving dots. Figure 2. Question A2 in Grade 8.

the items in the English study. Students’ responses to the generalizing and checking items were respectively coded into six categories (Table II). Proofs produced by the students are scored from 0 to 3 according to both the relevance of information and the rigor of deduction. The coding system of items for generalizing and checking is shown in Table II. This coding system would allow us to find the students’ different performance among seeing, recording or using a pattern via the generalizing, representing or checking items respectively. Students who saw or used a proportional relation between the gray and the white tile numbers with examples were categorized in Code 1. Students who saw or used par-

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TABLE II Response code for the generalizing and checking items Code

Key character of response

Exemplary response (item number)

1

Seeing (using) an improper pattern

12 × 10 (A1(a)); 24 × 5 (A2(aii)); It can be divided by 3 (A1(c)).

2

Seeing (using) some useful but incomplete pattern or correct result with incorrect explanation

40 × 2 (A1(a)); 20 × 20 − 1 (A2(aii)); However you arrange, 195 is impossible (A1(c)).

3

Seeing (using) a complete pattern only with correct arithmetic or photo-picture(manipulation)

84 (A1(a)); 21 × 21 − 1 (A2(aii)); odd numbers are impossible (A1(c)).

4

Seeing (using) a complete pattern with correct result and verbal explanation

The dot number of length is 1 more than the corresponding term (A2(aii)); 191 (195-4) cannot be divided into two equal parts (A1(c)).

5

Seeing (using) a complete pattern towards correct algebraic strategies

(n + 1)2 − 1 (A2(aii)); (n + 1)2 − 1 = 9999 (A2(d)).

6

Not showing to see (use) any pattern

4 + 20 (A2(aii)); Misunderstand 9999 as the 9999th figure (A2(d)); or no response.

tial relation, e.g., focusing on the relation between the tiles in the middle part but ignoring the tiles in the two sides, mismatching the number of dots and terms, or giving pseudo-manipulation as reason were categorized in Code 2. The Codes 3–5 are correct responses with arithmetic, verbal explanation and algebraic strategy respectively. Referring to Healy & Hoyles (2000), three phases of characteristics of p&d were taken into account: proof/disproof-conception, proof-validation and proof/disproof-construction in this study. Two kinds of proof-conception are modes of arguments that would be nearest to the students’ own approaches and those they believed would receive the best mark from their teachers. The modes of arguments are classified into empirical including examples and a counterexample, narrative and symbolic modes which may be valid or invalid. Three invalid arguments, in Figure 3, with symbolic forms derived from the Taiwanese students’ responses in the pilot study are included in addition to all arguments in the English version. Proofvalidation is related to students’ assessment of these arguments in terms of their validity (proofs that prove) or explanatory power (proofs that ex-


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Figure 3. Three invalid arguments derived from Taiwan students.

Figure 4. The linear conjectures to be proved or disproved in Item A6.

plain). Proof/disproof-construction takes account of what and how students justified a conjectured statement about number patterns. Taking into account that proof in algebra is not included in the Taiwan curriculum, the mathematical statements of Item A6 about the linear combinations of integral multiples are employed to investigate how students prove in algebra because of students’ familiarity with integral multiples and the operational and structural dualism of representing integral multiples with mathematical symbols (Sfard, 1991). Item A6, shown as Figure 4, is an additional question in the Taiwan versions for Grades 8 and 9 not in the English version. A similar Item A6 in the Taiwan version for Grade 7 is to argue patterns about the sum of a multiple of 2 and a multiple of 4. Items A1, A4, A7, A8 in the Taiwan version for 9th graders correspond to Items A1, A4, A6, A7 in the English version (Healy & Hoyles, 2000). A true statement about the sum of any 2 even numbers and different modes of arguments are presented in Item A1, a false statement about the result of multiplying any 3 consecutive numbers and different modes of arguments are presented in Item A7. Students are asked to make a choice for own approach and best mark and then to evaluate the validity and explanatory power of each argument in Items A4 and A7. Students are

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asked to prove whether it is true or false: “The sum of odd numbers must be even” in Item A4, and to prove whether it is true or false: “p and q are any two odd numbers, (p + q)(p − q) must be a multiple of 4” in Item A8.

R ESULTS WITH D ISCUSSION Reasoning on Geometric Number Patterns as Meaningful Learning Activities We first compare 7th and 8th graders’ spontaneous reasoning on geometric number patterns with respect to the items of generalizing and checking the linear and quadratic geometric number patterns. Table III shows the distribution of their responses to the generalizing items. While generalizing the linear geometric number patterns, 35.4% of Grade 7 and 52.7% of Grade 8 could perform correctly, but 30.3% of Grade 7 and 14.3% of Grade 8 used TABLE III Distribution of students’ responses to the generalizing items Subject Grade 7 (N = 1181)

Grade 8 (N = 1105)

Structure of geometric pattern (percentage)


Generalize

Linear

Quadratic

Linear

Quadratic

C1. C2.

30.3 7.2

11.4 10.3

14.3 7.1

1.4 13.2

34.9

29.8

45.0

50.1

0.4

6.4

7.1

14.1

0.1

0.1

0.6

0.1

27.0

42.1

26.0

21.1

C3.

C4.

C5.

C6.

Seeing an improper pattern Seeing some useful but incomplete pattern or only with correct result Seeing a complete pattern only with correct arithmetic or photo-picture Seeing a complete pattern with correct result and verbal explanation Seeing a complete pattern towards correct algebraic strategies Not showing to see any pattern

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TABLE IV Distribution of students’ responses to the checking items Subject

Checking C1. C2.

C3.

C4.

C5.

C6.

Using an improper pattern Using some useful but incomplete pattern or correct result with incorrect explanation Using a complete pattern only with correct arithmetic or photo-picture Using a complete pattern with correct result and verbal explanation Using a complete pattern towards correct algebraic strategies Not showing to use any pattern

Grade 7 (N = 1181)

Grade 8 (N = 1105)



Linear

Linear

Quadratic

38.4 5.4

6.9 6.5

1.4 19.2

22.5

35.7

23.0

3.2

1.8

2.1

1.4

11.3

11.2

29.2

37.9

43.0

the incorrect proportional reasoning strategy which was prevalently used by the English high-attainers (Kuchemann & Hoyles, 2001). While generalizing the quadratic geometric number patterns, 36.3% of Grade 7 and 64.3% of Grade 8 could answer correctly. In particular, 8.9% of Grade 7 generated improper patterns in which either only one dimension, e.g., the number of rows, columns or diagonal dots, or misused partial information (the number of terms and dots) was focused on. More 7th or 8th graders gave the correct answer while generalizing in quadratic than in linear geometric number patterns. This may result from the fact that we provided an approach in quadratic geometric number pattern, or that more students were attracted to the proportional relation between gray and white tiles in the linear geometric number patterns. Table IV shows the distribution of 7th and 8th graders’ responses to the checking items. While checking the linear geometric number patterns, 32.5% of Grade 7 and 55.3% of Grade 8 could at least use some useful but incomplete pattern to check. But 24.6% of Grade 7 and 5.6% of Grade 8

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used the proportional reasoning strategy (e.g., it can be divided by 3) to check this pattern. While checking the quadratic geometric number patterns, 36.3% of Grade 8 could answer correctly. Although 8th graders performed better in the generalizing and checking items than 7th graders in general, we draw attention to the fact that the percentage of responses not showing to see or use any pattern in the linear geometric number patterns did not decrease with grade. Therefore, it will be necessary to study why more 8th graders did not show to see or use any pattern than 7th graders in the checking but not in the generalizing item of the linear geometric number pattern. Under the classroom organization with small groups 4 to 5 students in each group, if there are two or more students in each group who have relational understanding of certain tasks, the others in each group will have much more opportunity to carry on meaningful learning through peer interactions. Therefore, reasoning on geometric number patterns is suggested to be the initial activity for learning algebraic thinking in Grade 7 based on the fact that about or above 30% of 7th or 8th graders almost correctly understood, generalized, represented and checked the linear and quadratic geometric number patterns unfamiliar to them, and that representing with symbols can be shifted to representing with students’ alternative notations in instruction for 7th graders. Relations of Students’ Performance among the Four Types of Reasoning Items In the following, we further investigated the hierarchical relations among the four types of items for reasoning on geometric number patterns with respect to the linear and quadratic structures provided in this study. It is noted that the linear and quadratic geometric structures mean that nth terms can be expressed as an + b or an2 + bn + c but not imply the different performance resulted from only this feature. In the quadratic geometric number patterns, Table V shows that most 7th and 8th graders who correctly generalized this pattern also correctly understood it. Table VI shows that most 8th graders who correctly symbolized this pattern also correctly understood (or generalized) it. In the linear geometric number patterns, Table VII shows that most 8th graders who correctly symbolized this pattern also correctly generalized it. The MacNemar’s test result (χ 2 = 14.6, N = 1105, p < 0.001) suggests that the frequencies of different responses between generalizing and representing linear geometric number patterns (Table VII) are significantly different. The results of Tables V–VII seem to suggest that a hierarchy proceeds from understanding, generalizing to representing linear or geometric number


TABLE V Percentage of students’ responses to understanding and generalizing the quadratic geometric number patterns Understanding Grade 7

Grade 8

Generalizing

Correct (%)

Incorrect (%)

Correct (%)

Incorrect (%)

Correct Incorrect

33.9 35.4

2.5 28.3

57.9 12.1

0.7 29.3

TABLE VI Percentage of students’ responses to representing, understanding and generalizing the quadratic geometric number patterns Grade 8 Understanding

Generalizing

Representing

Correct (%)

Incorrect (%)

Correct (%)

Incorrect (%)

Correct Incorrect

36.2 33.8

1.3 28.7

35.3 23.3

2.2 39.2

TABLE VII Percentage of students’ responses to representing and generalizing the linear geometric number pattern Generalizing

Representing

Correct (%)

Incorrect (%)

Correct Incorrect

46.5 6.5

2.9 44.1

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TABLE VIII Percentage of students’ responses to checking and generalizing the linear geometric number pattern Generalizing Grade 7

Grade 8

Checking

Correct (%)

Incorrect (%)

Correct (%)

Incorrect (%)

Correct Incorrect

18.7 16.5

8.4 56.4

38.5 14.2

10.3 37.0

patterns. This sequence is not surprising and not beyond common sense. On the other hand, the position of checking in this sequence is not the same in the tasks of the linear and quadratic number patterns. In the linear geometric number patterns, Table VIII shows that the percentage of students who correctly checked and incorrectly generalized the same pattern was more than 8%. After further analyzing the responses, most of them used a useful but incomplete pattern to achieve the correct answer, but this strategy is insufficient to generalize correctly. After combining the above results and the result of Table VIII, we diagram a hierarchy as follows and conjecture that the activity of using a pattern to check may be helpful in inducing a strategy for generalizing this pattern if the functional relation within a geometric number pattern is simple.

In the quadratic geometric number patterns, Table IX shows that above 40% (16.0/37.4) of 8th graders who correctly represented the pattern with symbols were unable to correctly check it. However, Table X shows that about 85% (30.7/36.3) of 8th graders who correctly checked the pattern also correctly generalized it. After combining the above results and the results of Tables IX and X, we diagram the hierarchical relations among the four components as follows and conjecture that the activity of using a


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TABLE IX Representing and checking the quadratic geometric number pattern Checking

Representing

Correct (%)

Incorrect (%)

Correct Incorrect

21.4 14.8

16.0 47.8

TABLE X Checkingand generalizing the quadratic geometric number pattern Generalizing

Checking

Correct (%)

Incorrect (%)

Correct Incorrect

30.7 28.0

5.6 35.8

functional relation to check a pattern after generalizing it may initiate the development of representing it with symbols.

Easy to Know and Hard to Do in Proof with Algebraic Mode The distributions of students’ choices of arguments for own approach and best mark in the linear (the sum of 2 even numbers is an even number) and non-linear (the result of multiplying 3 consecutive numbers is a multiple of 6) conjectures are presented in Table XI. The different forms of arguments provided in this study can be referred to the paper of Healy and Hoyles (2000). The proving modes students chose for their own approach were different, but the proving modes students chose for the best mark were similar between the linear and the non-linear conjectures. When choosing for their own approach, the algebraic proofs (45.4%) were most

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TABLE XI Distribution of students’ choices of arguments for own approach and best mark in linear conjecture (A1) and non-linear conjecture (A7) N = 1059

Form of proof Linear conjecture (A1)

Own approach (%)

Best mark (%)

Empirical Narrative (factor) Narrative (exhaustive) In(incorrect)-algebraic (xyz) S(specific)-algebraic (a, a + 2) S-algebraic (a − 1, a + 1) S-algebraic (2a, 2a) Algebraic (2a, 2b) Other

25.6 3.8 17.8 4.4 8.3 12.0 5.4 15.3 7.4

7.5 7.0 10.5 12.2 13.0 17.0 7.9 16.3 8.6

Non-linear conjecture (A7)

Own approach (%)

Best mark (%)

Empirical Algebraic and empirical Narrative Algebraic and narrative Other

28.4 17.2 28.8 17.8 7.8

6.1 17.2 14.7 53.4 8.6

25.6 21.6 4.4

25.7 15.3 7.4

7.5 17.5 12.2

37.9 16.3 8.6

popular in the linear conjecture, and empirical (28.4%) and narrative (28.8%) proofs were most popular in the non-linear conjecture. When choosing for the best mark, algebraic proofs were chosen by most students both in the linear (66.4%) and non-linear (53.4%) conjectures. In the linear conjecture, students were significantly more likely to select empirical arguments for their own approach than for the best mark, and to select algebraic proofs for the best mark than for their own approach. In the non-linear conjecture, students were significantly more likely to select empirical or narrative arguments for their own approach than for the best mark, and to select algebraic proofs for the best mark than for their own approach. The proving modes selected for their own approach and for the best mark were significantly different but not independent. This shows that views of proofs from “own approach” and “best mark” perspectives are correlated with each other. The percentage of the Taiwanese students who chose algebraic arguments for their own approach were above 35%, which was even more than the percentage of the English high-attainers (Healy & Hoyles, 2000).


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TABLE XII Distribution of students’ proving modes Grade 9 (N = 1059)

Linear (A4)

Quadratic (A8)

Modes of argument

No.

%

No.

%

Empirical Narrative Symbolic None Other

451 83 334 183 8

42.6 7.8 31.5 17.3 0.8

523 137 121 268 10

49.4 12.9 11.4 25.3 0.9

Those Taiwanese students have interiorized symbolic proofs as what they want to produce in proving. It is worthy of further investigation to compare the difference between students who have different “own approach” and “best mark” perspectives when they learn proof in algebra. On account of a statistically significant association between students’ “own approach” and “best mark,” we shall interpret the result of no significant variation in students’ choices for own approach according to the teacher variables reported by Healy and Hoyles (2000). Intuitively students’ perspectives on “best mark” are mediated by teachers, and the variation in students’ choices for own approach according to the teacher variables may have been included in students’ responses to the best mark. This provides another explanation for the fact that no significant associations were found when students’ choices for best mark had been added to the multinomial models of student choices for own approach in the English study. The distributions of proving modes and qualities of students’ responses to proving in the linear and the quadratic conjectures are presented in Tables XII and XIII. Producing empirical examples was the most popular mode of argument used by Grade 9. The 9th graders were more likely to use symbolic expressions than to give narrative explanations in the statement of the linear structure. However, symbolic expressions were not more likely than narrative explanations to be produced in the quadratic conjecture. On the contrary, English students (high-attainers) were more likely to give arguments expressed informally in a narrative style than to use algebra formally in both the linear and the quadratic conjectures (Healy & Hoyles, 2000). This different performance between English and Taiwanese students might result from the different patterns of instruction: more structural in Taiwan and more empirical in English classroom. One-third and one-fifth of the 9th graders could construct at least partial proof in the linear and the quadratic conjectures respectively as shown in

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TABLE XIII Distribution for students’ proving quality Grade 9 (N = 1059) Quadratic (A8) 0

No basis for the construction of a correct proof 1 No deduction but relevant information presented 2 Partial proof, including all information but some reasoning omitted 3 Complete proof Total

Linear (A4) 0

1

16.6

14.3

5.6

0.9

37.4

26.0

11.0

2.5

43.0

1.7*

10.7

0.8

13.7

0.7* 42.6

3.9* 31.2

1.4 5.7

5.9 100.0

3.5* 0.5* 0.0 20.6

2

3

Total

∗ Students scored higher in the quadratic conjecture than in the linear conjecture.

Table XIII. It is no surprise that students’ performance in the linear conjecture is significantly better than in the quadratic conjecture. Also, 10.3% of students scored higher in the quadratic conjecture than in the linear conjecture, and 62.4% of these students proved the quadratic conjecture with the narrative mode. It may imply that some students who can correctly argue the linear conjecture with the narrative mode attempted to argue with the symbolic mode. In Taiwan, the means of scores among students constructing narrative and algebraic arguments for the linear conjecture were respectively 2.41 and 1.95. However, the means of scores among students constructing narrative and algebraic arguments for the quadratic conjecture were 1.94 and 2.06 respectively. Regarding those data, there was no consistent evidence shown as in the English study that the narrative proofs were more likely than the algebraic proofs to be correct. Students’ choices of symbolic arguments for best mark are adopted to represent if they know symbolic arguments are regarded as formal proof. We contrast students’ proof-conception for best mark and proof-construction with respect to valid or invalid symbolic (algebraic) mode. In Table XIV, over 40% of the 9th graders knew that an algebraic proof for the linear or quadratic conjecture was required, but still had difficulty constructing a proof with symbols. Therefore, this pattern is depicted as easyin Chinese) in formal proof, and the to-know and hard-to-do ( teaching of proof in algebra should place more emphasis on the meaning of symbols.

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TABLE XIV The contrast between proof-conception and proof-construction Formal conception A1 (L) Construction

Yes

Able

A4 (Linear) A8 (Quadratic)

25.6

A4 (Linear) A8 (Quadratic)

41.0

Unable

No

A7 (Q) Yes

No

5.9 7.6

3.8

45.9

42.7

27.5

Hard to Know and Easy to Do in Disproof with Counterexamples Next, we focus on the distributions of 9th graders’ choices of arguments for own approach and best mark when faced with a false conjecture (when you add any 3 consecutive numbers, your answer is always even) is presented in Table XV. Whether choosing for own approach (40.5%) or for the best mark (64.3%), the algebraic and narrative disproof was most popular. Only 25.5% of the 9th graders chose the argument with a counter-example for own approach, and less than 10% of the 9th graders chose it for the best mark. This result was similar to the English high-attainers’ choices, which indicated that students were unlikely to choose a simple counterexample TABLE XV Distribution of students’ choices of arguments for own approach and best mark in false conjecture False conjecture (N = 1059) Form of proof

Own approach (%)

Best mark (%)

Supportive examples (empirical) Algebraic and empirical (incorrect formal) Narrative Algebraic and narrative (correct formal) Counterexample Other

14.6 9.4 5.1 40.5 25.5 4.9

8.3 8.5 5.0 64.3 7.5 6.4

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TABLE XVI Distribution of students’ disproving modes 6

9

Form of disproof

No.

%

No.

%

Non-example Supportive example Incomplete narrative or algebraic Complete narrative or algebraic Counterexample + Other Counterexample Other Total

7 59 168 50 307 177 291 1059

0.7 5.6 15.9 4.7 29.0 16.7 27.5 100.1

11 136 190 38 225 136 323 1059

1.0 12.8 17.9 3.6 21.2 12.8 30.5 99.8

TABLE XVII Distribution of students’ disproving quality N = 1059 (%) A multiple of 6 0

No basis for the construction of a correct disproof 1 No deduction but relevant information presented 2 Partial disproof, including all information but some reasoning omitted 3 Complete disproof Total

A multiple of 9 0

1

2

3

Total

26.1

0.8

0.5

0.8

28.2

0.4

3.6

0.2

1.4

5.6

1.1

0.1

12.4

2.3

15.9

4.0 31.6

8.3 12.8

4.9 18.0

33.2 37.7

50.4 100.1

for the best mark, although they were significantly more likely to do so for their own approach (Healy & Hoyles, 1998). The distributions of disproof modes and qualities of students’ responses to disproof in the linear conjecture are presented in Tables XVI and XVII. Producing counterexamples was the most popular mode of argument used by Grade 9. Two-thirds and half of the 9th graders could construct at least partial disproof in the two invalid conjectures respectively shown in Table XVII. It is noted that students will also get a score of 3 if they generate counterexamples and do not choose that the conjecture is right.

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TABLE XVIII The contrast between disproof-conception and disproof-construction Counterexample conception Disproof construction Multiple of 6 or 9

Able Unable

Yes

No

4.1 3.5

46.4 45.9

While students generated one or more counterexamples, about 1% of them thought that the conjecture is still right, and about 50% of them thought that the conjecture is uncertain. This implies that many students did not view the counterexamples sufficiently to refute the conjecture, although generating counterexamples about the two linear conjectures in algebra is not as hard as previous research had found. Students’ choices of a counterexample for best mark are adopted to represent if they know a counterexample is regarded as formal disproof. We contrast students’ disproof-conception and disproof-construction with respect to the counterexample mode of arguments. In Table XVIII, 46.4% of the 9th graders could give a counterexample to refute a conjecture but did not choose a counterexample for the best mark. It implied that students did not know the only thing they needed to do was give a counterexample even if they could produce a counterexample. Therefore, this pattern is in Chinese) in disdepicted as hard-to-know and easy-to-do ( proof with a counterexample. The teaching of disproof in algebra should focus more on students’ perception of the mathematical norm where one counterexample is sufficient to refute a mathematical statement. The Inconsistency between Validity Scores and Explanatory Power Table XIX shows the distributions of students’ validity scores and their evaluation of explanatory power for each argument. While comparing the distributions of the scores and the power, we can observe the opposite result between the score 2 and power 2 columns. In the linear conjecture, the 9th graders were much more likely to evaluate arguments with specific symbols (A1-6,7, and 8) as very powerful explanation than to correctly rate the validity of those arguments. The feature of those arguments is to use one symbol, x − 1 and x + 1 or 2x and 2x, instead of two symbols to represent two even numbers. Students misinterpreted any two even numbers either with two consecutive even numbers or with the same expressions. In addition to specific symbolic arguments, empirical arguments, to the

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TABLE XIX Validity scores and explanatory power of empirical, narrative, and (incorrect or specific) symbolic arguments, in percentage Grade 9 (N = 1059)

Linear conjecture (A4) Validity scores

Explanatory power

Form of proof

0

1

2

0

1

2

Empirical (A1(2)) Narrative with factors (A1(3)) Narrative with 0,2,4,6,8 (A1(4)) Incorrect symbolic (A1(5)) Specific symbolic (A1(6)) Specific symbolic (A1(7)) Specific symbolic (A1(8)) Symbolic (A1(1))

80.2 64.2 51.6 64.4 82.9 82.4 82.2 50.2

10.4 15.0 15.5 18.4 11.7 11.7 10.4 18.3

9.4 20.8 33.0 17.2 5.9 5.9 7.5 31.4

22.5 48.3 29.1 62.4 35.3 32.8 37.6 30.2

28.0 28.7 23.1 23.4 25.0 25.6 23.3 33.2

49.6 22.9 47.8 14.1 39.7 41.6 39.1 36.5

Non-linear conjecture (A7) Validity scores

Empirical (A7(2)) Symbolic and empirical (A7(3)) Narrative (A7(1)) Symbolic and narrative (A7(4))

Explanatory power

0

1

2

0

1

2

75.6 76.0 50.3 44.3

14.8 16.8 17.6 21.1

9.5 7.2 32.1 34.7

27.2 40.5 31.0 36.1

27.6 29.0 29.7 39.8

45.2 30.5 39.4 24.1

Note. N = 1059 for each argument. There are three evaluations of the validity of each argument (mistake, always true, sometimes true): 2 = correct evaluation, 1 = partially correct evaluation, 0 = incorrect. There are two evaluations of explanatory power (show you why, explains to someone in your class): 2 = explains private and public, 1 = explains private or public, 0 = does not explain.

linear conjecture (A1-1) and to the non-linear conjecture (A7-1), were regarded as very powerful explanation but incorrectly validated by most students. Moreover, the argument of most explanatory power was empirical examples in both linear and non-linear conjectures. Most students did not evaluate the arguments with empirical modes as specialization, and evaluate arguments with the specific symbolic form as a valid proof. In general, arguments with empirical mode or specific symbols were hard to validate for most students but highly explanatory for them.


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Why are most students not aware of the specificity of empirical arguments or specific symbolic arguments? It may result from the algebra curriculum in Taiwan which mainly focuses on algorithms and formulas in problem solving. Students may over-generalize from the experience of solving problems with situations about the two same or consecutive numbers and infer that x means any number so the other x is also any. They also lack the experience of using variables to represent the structure of whole numbers. On the other hand, students thought that these arguments were very powerful to convince oneself and peers on the possible basis of the usual contact with symbols. If mathematics is a special subject for learning logical reasoning, then the important issue of validity should be emphasized even more. Another problem of coordinating inconsistency arises when students encounter a proof they thought as explanatory is not valid. Hanna & Jahnke (1993) suggested that the role of proof in mathematics classroom should be presented as an indispensable tool rather than as the very core of that science. Nonetheless, how to bridge the gap between the acceptance by a learner and rigor in mathematics is still a pending problem. Generalization, Representation and Checking for Learning Proof and Disproof Students’ argumentation in Item A6(a), the sum of a multiple of 3 and a multiple of 6 must be a multiple of 3, was classified into 6 modes. Modes of 0, 1, 2, 3, 4 and 5 indicate no basis for proof, empirical proof, incomplete narration, inappropriate symbols, acceptable narration and acceptable symbols respectively. Modes of 0, 1, 2, 3, 4 and 5 were scored as 0, 1, 2, 2, 3 and 3 respectively. The proof-construction score of each student was employed to contrast the proving performance between the two groups of students who passed and failed in each kind of the reasoning tasks. Table XX showed that the mean of the proof-construction score among students who passed in each kind of the reasoning tasks was significantly higher (p < 0.001) than the mean among students who failed. This may suggest that students’ competence of proof construction in algebra can be enhanced to some extent as their competence of reasoning on number patterns is promoted. Therefore, reasoning on number patterns is supportive for proof in algebra. We further suggest that generalizing number patterns may be helpful for perceiving the structure of integers, and representing the generality of number patterns may be helpful for operating the representation as a whole while students prove. We singled out the percentages of the 8th graders who correctly generalized, represented with symbols, or checked, relative to the number of students producing some mode of proof in Item A6(a), because these

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TABLE XX Means of proof-construction scores among students who passed or failed in each kind of reasoning tasks Structure of expression in geometric number pattern

Reasoning task

Mean of proofconstruction scores among students passing in each kind of reasoning tasks

Mean of proofconstruction scores among students passing in each kind of reasoning tasks

Linear

Generalization Representation Checking

1.35 1.39 1.35

0.87 0.86 0.84

Quadratic

Generalization Representation Checking

1.36 1.46 1.47

0.78 0.92 0.93

TABLE XXI Percentages of students being able to reason on number patterns relative to the number of students producing some mode of proof in A6(a) Mode of proof 0 1 2 3 4 5 (n = 278) (n = 451) (n = 267) (n = 70) (n = 24) (n = 15)

Grade 8

(N = 1105)

Linear pattern

Generalization 30.2 Representation 24.8 Checking 28.8

53.4 49.4 60.3

68.9 66.7 70.0

64.3 68.6 67.1

70.8 66.7 83.3

86.7 80.0 86.7

Quadratic Generalization 27.7 pattern Representation 12.2 Checking 12.2

64.5 41.2 37.7

72.7 45.7 47.6

75.7 60.0 62.9

79.2 79.2 58.3

93.3 73.3 73.3

percentage patterns can be used to derive implications for the relationships between reasoning on number patterns and argumentation with acceptable symbolic mode. Students were grouped into six categories according to the mode of their proof construction. Table XXI showed the percentages of students who succeeded in generalizing, representing with symbols or checking geometric number patterns among each group of students.


253

Based on Table XXI, we may discuss whether students can reason on number patterns when they are able to construct acceptable proof with narration or symbols. Students classified as Mode 5, acceptable symbols, need to produce symbolic signs and expressions, i.e. 3a + 6b, then to operate the expression into 3(a + 2b), and to interpret a + 2b as a whole. However, about one fifth and one quarter of these students classified as Mode 5 failed in representing the linear and quadratic tasks with 2n + 4 and (n + 1)2 − 1, respectively. Students classified as Mode 4, acceptable narration, need to perceive the invariant or common relation between a multiple of 3 and a multiple of 6 and describe it with narration. However, about one third and one fifth of these students classified as Mode 4 failed in generalizing the functional relation of number patterns in the linear and the quadratic tasks respectively. It may imply that learning to construct acceptable proof is not completely sufficient for being able to reasoning on geometric number patterns. Therefore, reasoning on number patterns and proof in algebra should be designed as complementary activities for developing algebraic thinking. S UMMARY AND I MPLICATION In sum, the new findings in this study include (1) checking geometric number patterns appears to have different positions between the tasks of the linear and the quadratic expressions; (2) proof with the algebraic mode is easy to know but hard to do; (3) disproof with only one counterexample is hard to know but easy to do; (4) arguments with empirical mode or specific symbols were hard for students to validate but very convincing for them; and (5) reasoning on number patterns is supportive for proving in number patterns, and reasoning on number patterns and proof in algebra should be designed as complementary activities for developing algebraic thinking. In regards to designing instruction activities, where learning goals are set as being able to pose, to judge, and to argue conjectures about linear or non-linear number patterns, we argue that reasoning on number patterns can appropriately be used as initial activities for learning mathematical argumentation in Grade 7. We believe so because more than one third of Grade 7 and half of Grade 8 who did not learn reasoning on number patterns in school could correctly generalize linear geometric number patterns. However, about a quarter of the 8th graders who got the top proofconstruction score failed to represent or check geometric number patterns. We also need to design instruction activities for promoting students’ ability at representing and checking geometric number patterns.

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On the other hand, the relationships between checking and the other reasoning tasks appear to be different between the linear and quadratic structures of number patterns. We propose that there are two levels of checking in reasoning on geometric number patterns. One is to find a strategy for generalizing number patterns, and the other is to initiate the development of symbolization (Gravemeijer & Stephan, 2002) after it is integrated with the generalizing activity. However, the checking item is provided by our problems in this study and students may automatically use the checking strategy for understanding or generalizing number patterns, which can be regarded as another level of checking. The relations between the three levels of checking, understanding, generalizing and representing need to be further investigated. If the competence of generalizing, representing and checking functional relation of number patterns is emphasized, learning proof in algebra can not be substituted for the activities of reasoning on number patterns. We suggest that the teaching of proof in algebra may proceed from generalizing number patterns and representing the structures of number patterns with symbols in order to highlight the meaning of symbols, and the teaching of disproof in algebra should focus more on students’ perception of the mathematical norm where one counterexample is sufficient to refute a mathematical statement. How can we coordinate the inconsistency between explanatory and valid proof? Learning about the nature of proof needs to pass through several stages: the transitional stage of convincing oneself, then convincing friends, and finally convincing enemies (Mason, 1985). Convincing oneself needs explanatory proof coming from the heuristic idea, but convincing enemies needs valid proof conforming to the procedural idea. The key idea of each proof, providing the link between the heuristic idea and the procedural idea (Raman, 2003) may imply an approach for the transition. Going beyond informal reasoning, four issues coming out of this study are proposed for a journey from reasoning on number patterns to formal proof and disproof. First, which roles can the checking activity play in order to understand, generalize and represent number patterns? Second, to what extent does questioning a mathematical conjecture motivate students to justify or to refute and further to modify the conjecture? Third, how can students understand the dual significance of empirical arguments, sufficient when disproving “for all” and insufficient when proving “for all”? Last, how can the teaching of proof in algebra coordinate students’ inconsistency between the validation and explanation of proofs?


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ACKNOWLEDGEMENTS This work was funded by the National Science Council of Taiwan (NSC 91-2522-S-003-002). We are grateful to the members of the research team – Tam, H.P., Yu Wu, J.Y., Chen, E.R., Lin, C.Z., Hao, S.C, Liang, H.J., and Chang, C.H. – who contributed by analyzing protocols and commenting on drafts of this paper. The views and opinions expressed in this paper are those of the authors and not necessarily those of the NSC.

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