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This paper focuses on identifying features of a mathematical argument for it to count as a proof in elementary school mathematics. The author inquires whether the argument qualifies as a proof by analyzing four features of an argument foundation, formulation, representation and social dimension. Furthermore he bases is analysis on the theoretical framework that consists of the intellectual honesty principle and the continuum principle. The intellectual honesty principle states that the conceptualizing the notion of proof in school mathematics should be acceptable to the norms of proof in mathematics and the current understanding or the theoretically accessible understanding of the learner with the help of the teacher or peers. The continuum principle states that there should be uniformity in the way the notion of proof is be conceptualized over the different grades.
The data of this analysis was derived from a wide range of resources collected from a database of the University of Michigan. The author investigates the episode of a third-grade class where a student (Betsy) uses the definition of odd and even numbers to come up with an argument for the conjecture ' odd + odd = even'. The author analyses Betsy's argument examining its four major elements against the intellectual honesty principle and the continuum principle. The foundation, formulation and representation of Betsy's argument when analyzed using the intellectual honesty principle and the continuum principle stands a strong chance of qualifying as a proof. Moreover Betsy's argument is made stronger by it being recognized as an example of the thought experiment which is the highest level in the cognitive development of proof according to Balachff's hierarchy of arguments. According to the social dimension point of view the argument should not only be accepted as a proof by the wider society but also by the community in which it was built. However Betsy's argument did not convince many students in the classroom as they had different rules of discourse about proving. Thus although Betsy's argument was compatible with the wider society it was not socially accepted by the classroom community. Therefore from the social dimension point of view the augment could not qualify as a proof as the rules of discourse of the classroom community was not in line with the socially accepted rules of the wider community. At this point the importance of the teacher's role is brought in to light. If the teacher had intervened in the development and sharing of the rules for disclosure in the classroom community Betsy's argument could have qualified as a proof since it already qualified as a proof with respect to the first three structural element of the argument. In conclusion the author states that the social dimension should not be a determinant of whether an argument should count as a proof but rather, it is the duty of the teacher to help the students to build and socially share the rules of disclosure that would help them to decide whether the argument is a proof or not.
Characteristics of the of the paper in terms of methodological and ideological paradigms
In this paper Stylianides uses data retrieved from the database of the 'mathematics teaching and learning to teach project' (Stylianides,2007, p. 4) carried out by the University of Michigan. He gathers information from a wide range of resources which consist of 'videotapes of lessons, field notes, student work and the teacher's journals with the lesson plans and teaching reflections' (ibid). The method of research used by Stylianide is qualitative research method as he does not use quantities nor uses 'statistical techniques' (Hammersley, 1994, p. 25) but with 'verbal descriptions ' (ibid) and focuses on a significant player ( Betsy) in the scenario (Cohen, Manion, & Morrison, 2007, p. 461).
According to my opinion the method of investigation used by Stylianides is the case study method. Romberg (1992, p. 57) states that in a case study the researcher conducts 'an in-depth story about a particular case'. Similarly Stylianides conducts an in-depth analysis of the protocol of a classroom episode of a third-grade class. Furthermore he states that he studies the classroom activity from outside trying to understand general issues of the notion of proof (p.5). Therefore Stylianides does not make any judgements about a program nor does he is he test any theoretical assumptions (Romberg, 1992).
In my opinion the ideological paradigm of this paper is Social Constructivism as the students try to construct knowledge through the activities and the social interaction facilitated by the teacher. Stylianides (2007, p. 5) notes that the students explore mathematical problems individually or in pairs, in groups and finally as a whole group. Moreover, the classroom episode shows that Betsy illustrates conjectures by using the drawing on the board. Then the other student expressed their opinion about Betsy's arguments. The discussion continues although most of the children disagree with Betsy's argument. This behaviour clearly shows the social interaction, particularly which learning occurred through the process of interaction, negotiation and collaboration (Palincsar, 1998). Furthermore Ball (1993) in Stylianides (2007, p. 5) states that the teacher modelled her 'classroom as a community of mathematics discourse' where the 'validity of ideas rested on reason and mathematical argument rather than on the authority of the teacher or the answer key' (ibid, p.5) which means that the students played the key role as well as the teacher. Furthermore the teacher does not try to cultivate her opinion, rather continuously inquires the opinion of the students ('what do other people think?') (Stylianides, 2007, p. 7) or questions the students about the reasons for the explanation ('why not Mei?') (ibid, p.7) which indirectly encourages them to express their ideas. This I think is a clear example of the teacher acting as facilitator encouraging the students to take an 'active role in their learning, to explain their ideas to one another, to discuss disagreements, and to cooperate in the solution of complex problems' (Palincsar, 1998, p. 355). Moreover, I feel that this paper is not social cultural as Stylianides (2007) does not emphasise on the tools used or the culture of the classroom.
Critique of the paper
Stylianides is one of the important researchers who has focused greatly on the teaching and learning of the fundamental mathematical concept of proof in both school and teacher education settings. This article 'The notion of proof in the context of elementary school mathematics' is one of the most interesting articles written on proof in elementary school. The class episode used and the way the episode was analysed have made the article more interesting to the readers. However, reading and then giving a critical review on this article was done with great enthusiasm and pleasure.
The quality of the evidence is an important aspect when 'building an argument about the finding of an investigation' (Romberg, 1992, p. 58) as it effects the accuracy and the applicability of the conclutions made at the end of the research. In particular, validity and the reliability of the collected data needs to be considered. Validity means to what extent does the research tool actually measuer what it is supose to measuere (Wellington, 2000). Reliability is the extent of which the research tool would give the same result if it is used by different researchers or in a different setting (Wellington, 2000). Although validity and reliability is discribed seperatly they do not have seperate identity as some of the aspects of validity is inherent in reliablity (Ridgway, 1988). According to Golafshani (2003), although many researchers have argued that reliability and validity are not applicable to qualitative research, they have at the same time realised the necessity of having some kind of qualifying check or measure for their research. Moreover, Wellington (2000) states that an instrument can never be valid with total certainity as the instrument can only measuer the result on a certain day at a particular time under given conditions. Thereby, we can only claim that the instrument is valid to a certain extent.
Stylianides (2007) uses various means to overcome the weaknesses and biasedness of the methods used thereby in my opinion the data used in this research paper is high in validity and reliablity. According to Cohen and Manion in Wellington (2000) 'triangular techniques' (p. 24) can be used to describe the 'richness and complexity of human behaviour' (ibid, p.24) by studying from diffrent viewpoints. Stylianides (2007a) does not rely on a single data collecting method to gather evidence. He uses 'videotapes of lessons, field notes, student work and the teacher's journals with the lesson plans and teaching reflections' (p.5) retrieved from the data base. This is clear evidence for 'methodological triangulation' (Wellington, 2000, p. 24) as he uses more than one method to gather data. Moreover he analyses the episode using the four elements of an argument side by side with theoretical framework that comprises of the intellectual honesty principle and the continuum principle which clearly indicate that Stylianides uses more than one theoretical scheme in the interpretation. This gives evidence of 'theory triangulation' (ibid, p.24) as alternative theories are used in the same situation. As explained above the use of triangulation increases the credibility and validity of the results as a weakness in one method could be avoided by using the other method.
Stylianides (2007a) does not engage in data collection but uses the data collected by the University of Michigan which makes the data he is dealing with secondary data as it is collected by another party. Relying on secondary data could weaken the validity and reliability of his data as he cannot manipulate or control the tools and as he does not know how reliable the other part is. As an example he does not have the freedom to control or train or instruct the teachers or to choose the questions discussed in class. However, although he uses secondary data he actually looks into the original sources or documents: 'videotapes of lessons, field notes, student work and the teacher's journals with the lesson plans and teaching reflections' (p.5) which makes the data which he is actually using for the analysis primary data. This enables him to get an insight as to what really happened in the classroom. Moreover, he is also able to crosscheck the information without relying on one source. This in my opinion has helped him to overcome the weakness of not collecting his own data.
Stylianides (2007) emphasises on the importance of incorporating proof into the students mathematical experiences throughout their schooling starting from the elementary grades. According to him elementary mathematics focuses mainly on 'arithmetic concepts, calculations and algorithms' (ibid, p.2) and the sudden introduction of writing proofs in the secondary school creates a 'didactical break'. This results in a difficulty to understand and write proofs by the secondary students. Thereby, it is important to introduce proofs in the early grades so that the students are familiar with the proof concept. When reading the Stylianides (2007) the reader forms the opinion that this is the only reason for the introduction of proofs from lower grades. However, when reviewing the literature it is evident that this is not the only reason for introducing proof from lower grades. In my opinion Stylianides (2007) should have presented the other reasons as well so that the audience would be able to get a correct perception. However Stylianides later on, in one of his own articles Stylianides & Stylianides (2008) states two other main reasons for the emphasis in teaching proofs in early grades. Firstly, proofs should be introduced in elemantary level as 'proof is the fundamental to doing mathematics' (ibid, p.104). Ball & Bass (2003) uses two class room episode based on Ball's third grade class records where he looks at two segments five months apart. This study elaborates that teaching proof in elementary grades improves mathematical understanding and develops their knowledge, establish, negotiate and communicate mathematical knowledge. Secondly, Fawcett in Stylianides & Stylianides (2008) that increase in student proficiency in proof would increase the proficiency in mathematics because proof is important and used in all circumstances when decisions or conclusions are to be made. In my opinion if these points were brought illustrated the reader would be able to have a broader Stylianides had given
Stylianides (2007) identifies four features of an argument needed to qualify as a proof by looking at how proofs and arguments are conceptualised in the 'discipline of mathematics' (p. 3). The four main features of the argument are foundation, formulation, representation and social dimension. However in my opinion the explanation lacks sufficient evidence as to both whose work has shaped his ideas and his basis of deriving the above features. Stylianides acknowledges the fact that there can be more than one definition as different researchers look at this issue from different angles which helps in developing research questions (A. Stylianides, 2007b). He further states that he believes that there are more than one way of classifying a 'tight' argument and that he has chosen the one that believes are more valid logically 'tight' to meet the standards of proof (A. Stylianides, 2007b). He also states that he does not devalue other mathematical activities and forms of reasoning which support the development of proofs. As there could be more than the above given method he should have explained his basis more clearly.
Stylianides in one of his own papers Stylianides (2007b) however explains in detail when he builds the definition of proof based on the notion of proof in elementary school mathematics. Foundation is the basis on which the argument is built on. These are statement which are already accepted by the classroom community as true or can be used without any further justification (A. Stylianides, 2007b). Formulation is how the argument is developed using different forms of reasoning which are 'valid and known' (ibid, p. 291) by the classroom community. Representation is the way an argument is expressed. However different 'forms of expressions' (ibid, p. 291) that are 'appropriate and know' (ibid, p. 291) or which are reachable by the classroom community. Social dimension is the acceptance of the argument as a proof by social context of the mathematical community.
Stylianides analyses the episode using the three features; foundation, formulation and reperesentatation and makes his argument stronger by recognizing it as an example of the thought experiment of Balacheff. He also analyses the episode using the theoretical framework which also recognises the Betsy's argument as a proof with consideration to the three features. However when Betsy's argument analysed from a social dimension point of view, did not qualify as a proof as it did was not accepted by the mathematical community in which it was built in. Stylianides emphasises a lot on social dimension throughout the latter part of the paper although it finally did not count as a feature of an argument to qualify as a proof. Finally it was evident that it is the teachers duty to help the children to build socially shared rules of discourse. However, even if an argument qualifies as a proof in the classroom community we cannot be certain that every individual learner in the community accepts or understands the proof (A. Stylianides, 2007b). This is due to the fact that the knowledge which is shared within the community would not reflect the individual understanding of each and every student in the classroom community. But what is important is that the whole community follow a continuum of understanding even though individuals may follow a different pace or speed.
Stylianides (2007) uses the two principles intellectually honesty principle and the continuum principle to have an insight into whether an argument would qualify as a proof. In my opinion I strongly agree with Stylianides in using these two principles. According to continuum principle there should be continuity in what is taught throughout the grades. In my own experience as a primary teacher and a lower secondary teacher I have faced instances where my own teaching in a lower grade has become a 'didactical obstacle' (Warfield, 2006, pp. 27-28) to the students when they come to a higher grade. Didactical obstacle is where a particular mathematical teaching later on becomes an obstacle for further learning. As an example in lower grades when teaching subtraction I frequently mentioned that "you cannot subtract a larger number from a smaller number" to help the students when doing standard written subtraction. However, later on in a slightly higher grade when teaching negative numbers there were many instance where children still try subtracting the smaller number from the larger number irrespective to what the question says. This was due to the fact the fact that the student had formed a strong idea that a larger number cannot be subtracted from a smaller number. Likewise even when teaching proofs in elementary levels teachers should be careful not to develop conceptions of proof which need to be undone when they reach higher classes. Thereby obstacle need to be taken in to account when situations are designed (Warfield, 2006).
According to the intellectual honesty principle the conceptualisation of proof in elementary school mathematics should be honest to mathematical norms of proofs and it should be known or should be accessible to the learner with the help of teacher or peers. The intellectual honesty principle is inspired by Brunner (Stylianides, 2007). Brunner(1960) highlights that depending on the stage of development the child view the word and understands it in different ways. Therefore when teaching the child the structure of the subject should be present in a way that it is compatible with the child's way of viewing things. Moreover when these ideas are 'represented honestly and usefully [ ] these first representations can later be made more powerful and precise the more easily' (Brunner, 1960, p. 33). As an example: According to Betsy's development stage she is in the concrete operational stage according to Piaget developmental stages. At this stage the child cannot form abstract ideas. Therefore, we cannot expect Betsy's argument to be like a mathematician's standard algebraic proof. Due to this reason even though Betsy's argument was presented using everyday language it had to be accepted as a valid argument. Moreover, the argument should also be correct mathematically. If the argument is not compatible with the mathematical norms the students will form false ideas which would form misconceptions in the minds of students. These, type of misconception in the minds of students would create confusion and difficulty in further learning. As an example: if the students believe that a well chosen arguments constitute a proof, they will find proof in higher grades difficult (Martin & Harel, 1989).
One other important aspect that Stylianides (2007) discusses in this paper is the role played by the teacher in teaching of proofs. As given in the article the main reason for Betsy's argument to not qualify as a proof is because it was not able to gain social recognition among the classroom community as it did not convince many students in the class. The students in the class shared 'different rules of disclose' in fact they had conflicting ideas about methods of obtaining desired statement. The teacher did not intervene in sharing the rules of discourse thereby the classroom episode ended without the children recognising whether Betsy's argument is a proof or not. The lesson ended without 'making the knowledge the students have constructed available and useful' (Warfield, 2006, p. 27) which means the episode has ended without a 'situation of institutionalisation' (Stylianides, 2007). In my opinion this is not a good way of ending a lesson.
In an elementary class the students have very limited experience and knowledge in mathematics as they have just started their education and as the elementary school curriculum gives very little attention to proof (Martin & Harel, 1989). The classroom teacher becomes the primary source of students experience with verification and proof (ibid). Thereby, the classroom teacher plays a crucial role as a mediator 'in influencing the mathematical aspects of the knowledge the students construct' (Yackel & Cobb, 1996, p. 474). The teacher should not be telling or showing but rather should interact with the student to constitute what 'counts as an acceptable mathematical explanation and justification' (ibid, p. 461). The teacher needs to play an active role rather than a passive role in the development of the students proving abilities (A. Stylianides, 2007b) . Hanna and Jahnke in A. Stylianides (2007b) emphasises if the teachers play a passive role the students would not have access to methods of proving and that we cannot expect the students to know 'sophisticated mathematical methods' or 'accepted methods of argumentation' (ibid, p.318) . Cobb, Yackel & Wood in (ibid) also confirms that as mathematics is a social practice where teachers need to intervene with the student construction of mathematical knowledge which is compatible with those of the wider society.
Stylianides (2007b) suggest that a teacher needs to judge whether the arguments that the student bring qualify as a proof, decide what arguments could qualify as a proof and decide on actions need to be taken to improve mathematical resources related to proofs when playing an active role. As an example the teacher should be knowledgeable enough to not to accept a few well chosen example to constitute a proof as an empirical argument would not count as a proof. (decide what arguments could qualify as a proof) Moreover in carrying out this active role the teachers knowledge about proofs is a crucial factor (A. Stylianides & Ball, 2008). Teachers knowledge about proofs would help them to decide whether a conjecture could be 'refuted' or 'verified' (A. Stylianides & Ball, 2008, pp. 327-328) and to help the students in the process of verification or refutation (ibid). As an example in the episode given by Stylianides(2007) the students were trying to refute or verify whether Betsy's conjecture was a proof. Where the students were trying to verify the conjecture, the teacher should have had the knowledge to from a mathematical point of view as to how to verify a statement that was addressing an infinite number of cases. The teacher should also have the knowledge of different mathematical issues involved in deciding what the sequence the students should have taken in the proving process (A. Stylianides & Ball, 2008). The refute or justification of the conjecture by thinking and reasoning results in a mental process where conflicting ideas are resolved (Wood, 1999). This results in a conceptual change and a progression of thought of the student (ibid).