Torquetum

 

The author ran a large (N = 473) quantitative study of students’ perceptions of proof and then a smaller qualitative sub-study of proof perceptions and practices. He lists several tensions: between novice and professional, between informality and formality, between state of becoming and state of being. “From an educational point of view the prototypical proof practices of beginning undergraduates, despite being imperfect, ought to be initially accommodated simply because the learner believes in their veracity—for the student it is a proof. A tension also arises in the process of convincing the student of the necessity of making the transition to formal proof” (p. 870). Hmm, reference to Ruthven and Coe (1994). I should dig out work by these two if I do a study of my own; it looks intriguing. This study has a nicely described methodology. The qualitative part included students judging the efficacy of proofs across a range of representations, such as visual or abstract. There were also interviews. There is lots of detail on the qualitative stuff, I won’t summarise it here. One point: he finds the 2nd year students have a deeper understanding of proof than the 1st years, which is an encouraging result. The qualitative results show that most students have poor proof perceptions and practices despite apparent agreement with the principles encountered in the questionnaire. The author speculates that at least part of the problem is the difference between how maths is taught and how original mathematics is constructed. The author closes with a suggestions related to development of concept images. The paper has a detailed Appendix which could be of use of one wanted to replicate the study.

 

 

Almeida, D. (2003). Engendering proof attitudes: can the genesis of mathematical knowledge teach us anything? International Journal of Mathematics Education in Science and Technology, 34(4), 479-488.

 

The author “reports on how proof attitudes could be inculcated in students by offering them a course design that is faithful, to some extent, to the historical genesis of modern mathematics.” (p. 479). As some of the papers already discussed, the author gives a brief rundown on the history of proof. “If the sequence for understanding mathematics [intuition, trial, error, speculation, conjecture, proof ] as espoused by MacLane also applies in mathematics learning as well as in research and if the theory of levels in proving is accepted then there are implications for proof in mathematics education” (p. 482). The author describes a study where the students manipulated descrined shapes in a computer package and had to come up with a conjecture based on their observations and then prove the conjecture. The results were interesting, with students coming up with conjectures and providing proofs across a range of formality. Interestingly, the students displayed positive proof perceptions and practices.

 

 

Interesting: I am starting to see some themes running through these papers. There was the bit on empirical validation I mentioned in the paper just before this. Here I am seeing, again, the emphasis on how students don’t know what a proof is or what it is supposed to do. This is really important. It’s rearranging the deckchairs on the Titanic for us to quibble over misuse of symbols, for instance, in an algebraic proof, when the students don’t even know what constitutes a proof or why we bother to place value on them. Anyway, back to the paper.

 

The students involved in this study are pre-service mathematics teachers. If they don’t, as university students, have a good conception of proof, then they’re not going to be able to teach proof well to the learners in their classrooms. (Interesting segue into grading concept maps – something I’ve never thought about.) There was a correlation between the complexity of the proof concept map and the student’s passing grade on the course, however the author argues that neither passing grade nor complexity of map relates to being a better teacher. The author makes a good point that many of the studies on student’s conceptions of proof are based on small studies – not that he’s disagreeing with them. The author closes with a call to break the “vicious circle” of training maths teachers without attention to proof who then fail to pass on understanding of proof to their classes.

 

 

From the abstract: “This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus... We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof... We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them, i.e., determine their correctness” (p. 123).

Students can be confused between proof and evidence, also, some accept proofs as valid, but still like to see empirical evidence. There are some nice references to different ways to teach proofs to students. The authors look at students’ abilities to clarify the logical structure of mathematical statements in the context of constructing and validating proofs themselves. They found that this ability was not successfully represented by most of the students in the study. When judging the correctness of a proof, there are questions which need to be asked and answered by the person reading the proof. Their study showed that students could neither ask those questions nor answer them. The paper goes on to define and discuss terms like “proof”, “unpack” and “formal and informal statements”. Also a proof framework is defined: “By a proof framework we mean a representation of the "top-level" logical structure of a proof, which does not depend on detailed knowledge of the relevant mathematical concepts, but which is rich enough to allow the reconstruction of the statement being proved or one equivalent to it. A written representation of a proof framework might be a sequence of statements, interspersed with blank spaces, with the potential for being expanded into a proof by additional arguments” (p. 129).

 

The authors argue that students “who cannot reliably unpack the logical structure of informally stated theorems, also cannot reliably validate their proofs” (p. 130) and support their argument with discussion of proof frameworks. There is a lovely discussion of “statement images” which is a collection and network of all knowledge items associated with a particular statement and which can be very rich and detailed. There is reference to a Radford 1990 paper which I’d like to dig out. The data is presented in detail and finally three suggestions for teachers are made (summarised from p. 142):

First, since many undergraduate students have difficulty unpacking informally written statements, it might be useful to present theorems and definitions both in a more informal way and in a more formal way.

Second, it might be useful to offer university students some explicit instruction or advice on validation, an area currently more or less neglected. There is no reason for such instruction to be restricted to proofs.

Finally, because university students who still have weak validation skills may not be able to distinguish proofs from supplementary or explanatory comments, it might be good to present material in a way that makes this distinction clear.

 

 

Rigorous proof in geometry has existed since the time of the ancient Greeks. Rigorous proof in analysis and algebra was slower to develop, only becoming ubiquitous in the 19th century. “Curiously enough, the traces of these two different traditions can be seen to this very day in school mathematics, where “proving” has remained for the most part peculiar to geometry” (p. 421). During the 1960s and 1970s there was a worldwide attempt, the authors say, to include formal proof at the school level, however there was poor teaching and much complaint and the movement did not succeed. There has been a “shift to a pragmatic view of proof” (p. 422) in which application is key - both “extra- and intramathematical” applications (p. 426). The authors say, “in the classroom proof has a status in algebra quite different from and inferior to its status in geometry” (p. 423) which definitely then has implications for university teaching, especially if one’s students have done very little geometry. The authors draw distinctions between the role of proof amongst mathematical scholars and in the classroom. For the former, the deductive nature of proof and syntactic correctness are of greatest importance. In the classroom, the relation of mathematics to reality is of greater importance and “the complex relationship between deductive reasoning or symbolic manipulation on the one hand and its application on the other must be developed afresh for each individual case” (p. 433), making teaching proof multi-dimensional and hard to do successfully.

 

“Our first thesis … is as follows: communication in scholarly mathematics serves mainly to cope with mathematical complexity, while communication at school serves more to cope with epistemological complexity” (p. 433)

“Our second thesis can … be stated as follows: in order to understand the meaning of a theorem and the value of its proof, students must have extensive and coherent experience in the appropriate application area. This pragmatic foundation can and should be taught in conscious separation from the formal derivation. Only then will students be able to see the real point of a proof” (p. 434).

This is a lovely informative paper which I enjoyed reading a lot. There is a lot of detail on the history of proof which I have omitted in my little summary here, but this paper is definitely worth a reread should I ever need to do something on proof again.

 

 

There are a lot of similarities between this chapter and the 2009 paper previously discussed. The main difference is the discussion here on classroom design experiments – the things that need to be central and the pitfalls you need to watch out for. I haven’t read much on design experiments before (although I probably should have) and I found that part very interesting.

 

An important feature, for instance, of a classroom design experiment is to “develop, test and refine theories, not merely to empirically tune ‘what works’ ” (p. 224). Also, “instructional design and research are interdependent” (p. 224). The design serves as a context for research and the analyses inform ongoing design. There are typically phases of a classroom design experiment: preparing, experimenting, conducting retrospective analyses.

 

In preparing, determining the focus of the experiment is of vital importance and can be a challenge. You need to know what aspects of students’ learning you want to focus on and what is background or of secondary importance. Often such experiments have as focus students’ development of domain-specific skills and reasoning and students’ identification with classroom activity is secondary. Cobb et al. support making students’ identification with classroom mathematical activity the focus. In the second phase, the objective is not to test whether the instructional design works (although that will necessarily be assessed), but to improve the design “by testing and revising conjectures inherent in the design about both the process of students’ learning and the specific means of supporting it” (p. 225). These ongoing analyses are highly selective and often involve “implicit, unarticulated assumptions” (p. 225). Cobb et al. argue that it is critical to “explicat[e] the key constructs used when making these interpretations so that underlying suppositions and assumptions are open to public scrutiny and critique” (p. 226). They put forward their empirically grounded (developed “while conducting retrospective analysis of a classroom design experiment” (p. 226)) framework of the key identity constructs as a way of addressing this concern of explication.

 

I won’t go into detail here on the key constructs (personal and normative identities) as it has been discussed elsewhere, but there is a lot of detail present in the chapter.

 

Interestingly, they discuss Sfard and Prusak’s (2005) narrative framing of identity at some length and make a particular point of discussing their “designated identity” which is related to trajectories into the future. To my mind, this ties in directly with Cobb and Hodge’s (2005) “core identity” yet in this paper they regard it as an “unfortunate oversight” that they (Cobb, Gresalfi and Hodge) did not include a construct related to future trajectory in their interpretive scheme. Surely they did? In 2005? With “core identity”? I’m confused. Anyway, they tie Sfard and Prusak’s designated identity to D’Amato’s extrinsic and intrinsic value (structural or situational significance), which is something I’d like to read more about.

 

I like the fact that they regard Sfard and Prusak’s framework as complementary to their own. Of all the identity theories which I have read, Sfard and Prusak’s narrative framework and Cobb et al.’s key constructs are the ones which make the most sense to me. I’d just like to read more on this lost “core identity” though …

 

Potential further reading:

D’Amato, J. (1992). Resistance and compliance in minority classrooms. In E. Jacob & C. Jordan (Eds.) Minority Education: Anthropological perspectives (pp. 181-207). Norwood, NJ: Ablex.

Horn, I.S. (2006). Turnaround students in high school mathematics: Constructing identities of competence through mathematical worlds. Paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA.

 

 

Identity is a term thrown around a lot in educational and other literature. Sfard and Prusak felt that a definition of identity that is operational is needed if it is to be useful. First and foremost, they define identity as stories about people. The notion of identity differs from other terms such as character, nature and personality in that it is not as “tainted” with undertones of biological determinism and as such is better suited to research located in a sociocultural context. 

 

“[I]n particular, identity features prominently whenever one addresses the question of how collective discourses shape personal worlds and how individual voices combine into the voice of a community” (p. 15) [so group to individual and individual to group] and “the question of the mechanisms through which the collective and the common enter individual activities also lies at the center of educational research on learning” (p. 15). It is known that culture shapes learning. There is a “complex dialectic” between learning and its sociocultural context and identity seems tailor-made to bridge the gap. However, identity can only be really useful if it has an operational definition. The authors says “it is the activity of identifying rather than its end product that is of interest to the researcher”, whose focus is not on identity so much as the “complex dialectic between identity-building and other human activities” (p. 17). 

 

But why not beliefs or attitudes as the bridge between learning and sociocultural context? Beliefs are tricksy things as they imply an existence which is discourse-independent yet how is an analyst to perceive and objectively identify them? Attitudes are similarly tricksy. 

 

Sfard and Prusak refer to Blumer’s test of admissibility, which is something I have not read the original of and probably should seek out. For a concept to be applicable in research it has to meet 3 conditions: we have to know what to look at in order to pinpoint the concept, we need to know how to identify things which are not the concept, and it needs to “enable accumulation of knowledge” (p. 15). Beliefs and attitudes don’t meet these criteria and, in fact, identity is at similar risk unless an operational definition can be found. The authors criticise others’ usage of the word identity without an explanation of what it is or how to use it, however they like Gee and other’s use of the term or idea of narrativisation, of people telling stories about who they are. The flaw they find in this is the postulation of there being some sort of internal thing about which you are telling stories. They call this an “essentialist vision” of identity and consider the notion of identity being something that is extra-discursive and independent of action to be both untenable (it gives us no idea of what to look for) and potentially harmful (descriptors can become self-fulfilling). 

 

Instead of beliefs or attitudes, Sfard and Prusak support the use of identity as a descriptive tool. They provide a collection of definitions which operationalise identity. 

Identity: “we suggest that identities may be defined as collections of stories about persons or, more specifically, as those narratives about individuals that are reifying, endorsable and significant” (p. 16) 

Reifying: the use of verbs such as be, have or can (rather than do) and with the adverbs always, never, usually, etc. that stress repetitiveness of the action. 

Endorsable: the “identity-builder” would agree that the story “faithfully reflects the state of affairs in the world” 

Significant: any change in the story is likely to affect the storyteller’s feelings towards the identified person. Often the most significant stories are those about inclusion in or exclusion from a community. 

 

Actual identity: stories about the actual state of affairs; present tense; formulated as fact 

Designated identity: stories about a state of affairs expected to be the case in the future, having the potential to become part of one’s identity; future tense, or expressed in words evoking obligation, commitment, necessity, wish; not necessarily desired, but always binding; sometimes the person is not aware that there are alternatives; the most important source is narratives authored by others; 

Critical stories: “those core elements that, if changed, would make one feel as if one’s whole identity had changed” (p. 18); without these elements the person would find it hard to tell which stories about him/her were endorsable or not. 

Significant narrators: influential people, “carriers of those cultural messages that will have the greatest impact on one’s actions” (p. 18). 

 

They then define learning as closing the gap between actual and designated identities. 

 

Any narrative can be seen as a triple BAC where A is the identified person, B the storyteller and C the recipient. Without going on any further it is clear that there are therefore multiple identities for any one person, which really has to be part of any definition of identity. Whether any of these three are the same person or not allows us to define three “levels” of identity narrative: 

AAC: the storyteller is the identified person: first-person identity 

BAA: the story is told to the identified person: second-person identity 

BAC: all three are different people: third-person identity 

AAA is a particularly special identity, a first-person self-told identity which is possibly what many people mean when they speak of identity – it has potentially the most impact on one’s actions. Second- and third-person identities have the potential to be incorporated into our first-person designated identities (p. 18). 

 

A merit of this narrative definition of identity is that “human agency and the dynamic nature of identity are brought to the fore” (p. 17); the focus of attention is on things being said, not on something existing behind what is said. As stories, identities are human-made, have authors and recipients, and can change according to need, they are accessible and investigable. A criticism of this narrative definition is that “text” cannot replace experience. Wenger in particular stresses that the words used to express or represent identity are not the identity itself. Sfard and Prusak argue that, while they agree with Wenger that identities originate in the “experience of engagement”, it is the stories one tells about experiences, or the visions of these experiences, that constitute identities, not the experiences themselves. 

 

Of all the theoretical texts on identity which I have covered so far, it is this narrative one I like the best. One reason is that I am wanting to use theory to look at some data I have and that data is largely in the form of interviews. Perhaps on a different data set I might prefer a different theoretical tool. Still, this definition of identity as story, with a split into actual and designated identities with learning drawing those two together really works for me. I shall try to apply this to my data and see what comes out! 

 

Blumer, H. (1969) Symbolic Interactionism: Perspectives and Methods. Englewood Cliffs: Prentice Hall.

 

 

Black et al use concepts from cultural historical activity theory (CHAT) to develop a theory of identity which is associated with shifting hierarchies of motives within the individual. A student’s evolving sense of self and relationship with mathematics affects their mathematical identity and hence their level of engagement with mathematical studies. Black et al explore two key aspects, namely “how [students] perceive doing mathematics fits with ‘other’ motives – particularly, their understanding of themselves in relation to future aspirations” and “how a connection between future aspirations and ‘mathematical identity’ shifts or changes over time as students progress [through their studies]” (p. 56).

 

In CHAT there is a notion of leading activity (credited to Leont’ev). CHAT “views identity as emerging from engagement in joint object-oriented and socio-culturally mediated ‘activity’” (p. 56). The theory (as I understand it) is that we, throughout our lives, encounter a wide variety and great number of activities. We are a unique product of these “constellations” of activities and experience a variety of motives associated with these activities. These motives and associated “subjectivities” form a hierarchy. Each activity and associated motive forms part of an identity, where we each draw upon a hierarchical collection of identities. Some activities are more significant in one’s development than others, specifically any activity which brings about a change in motive as to why one is carrying out the activity is called a leading activity. Black et al, here adding to existing CHAT concepts, suggest calling the identity invoked by the leading activity and associated motives, a leading identity. A leading activity shapes “psychic processes to the extent that development is essentially dependent on such activities” (p. 57).

 

By understanding (naming? locating? identifying?) a student’s leading identity it may inform the teacher of the student’s understanding of self and his/her trajectory. The authors cite Leont’ev in saying “activities become leading when new motives are generated so that the original motive of actions is surpassed by a new motive, and hence, a new activity” (p. 57). The notion of leading activity can form a useful methodological tool for understanding periods of transition.

 

This approach to identity differs from others in two major respects

1. While the authors acknowledge multiple simultaneous identities as do other scholars, the notion of leading identity indicates an identity which is more significant that others in that it suggests changes in a hierarchy of motives and associated developmental change.

2. Framing developmental change and notions of identity through CHAT allows access to theories of social psychology and socio-situated processes.

 

Motives are derived from aspirations. In order to bring about a change in motive towards studying mathematics and development of a leading identity related to enjoyment of mathematics, there needs to be an aspiration to study mathematics for more than the “exchange value” of good grades. There needs to be an understanding of the long-term “use value” (and labour power) of mathematics and it is the challenge of the instructor and course designer to come up with and design activities which will become leading activities.

 

“[I]t is important that teachers, colleges and policymakers recognise the varied motives students have for studying mathematics. By seeking to understand the particular motive or identity which may be leading for the student at a given point in time, there is much to be gained in terms of offering tailored support and appropriate ways of teaching and learning mathematics. Therefore, we suggest a move away from strategies which focus on maximising the exchange value of grades” (p. 71).

 

Boaler, J. (2002) Exploring the nature of mathematical activity; using theory, research and ‘working hypotheses’ to broaden conceptions of mathematics knowing, Educational Studies in Mathematics, 51, 1-2, 3 – 21.

Boaler, J., Greeno, J. (2000) Identity, agency and knowing in mathematical worlds, in J. Boaler (Ed.) Multiple Perspectives on Mathematics Teaching and Learning (pp. 45 – 82). Westport: Ablex.

Solomon, Y. (2007) Not belonging? What makes a functional learner identity in the undergraduate mathematics community of practice? Studies in Higher Education, 32, 1, 79 – 96.

Williams, J. (2008) Towards a theory of value in education. Paper presented at the Society for Research in Higher Education Conference, Liverpool, December.

William, J., Davis, P., Black, L. (2007) Sociocultural and Cultural Historical Activity Theory perspectives on subjectivities and learning in schools and other educational contexts. International Journal of Educational Research, 46, 1-2, 1 – 7.