diff --git a/ch2.tex b/ch2.tex index bea9266..8c77c59 100644 --- a/ch2.tex +++ b/ch2.tex @@ -1,13 +1,15 @@ \chapter{Research Perspective and Epistemological Framework} This work is a phenomenographic study, adopting the epistemological framework of social constructivism. +Phenomenography, and its extension variation theory, are appropriate because the subject is the understandings, including those obtained after reflection, of proof present in the student population. +Social constructivism is used because of the nature of proof as a communication. Interaction between people composing and understanding a proof helps people learn how to compose and how to understand proofs. -A qualitative study ought, for transparency, to contain a description of the author's viewpoint, or at least, the viewpoint that was exercised in the study. Ernest\cite[p. x]{ernest1994constructing}, crediting von Glasersfeld\cite[p. 41]{von1987learning}, notes: -\begin{quote} -To introduce epistemological considerations into a discussion of education has always been dynamite. Socrates did it, and he was promptly given hemlock. Giambattista Vico did it in the 18th century, and the philosophical establishment could not bury him fast enough. -\end{quote} +%A qualitative study ought, for transparency, to contain a description of the author's viewpoint, or at least, the viewpoint that was exercised in the study. Ernest\cite[p. x]{ernest1994constructing}, crediting von Glasersfeld\cite[p. 41]{von1987learning}, notes: +%\begin{quote} +%To introduce epistemological considerations into a discussion of education has always been dynamite. Socrates did it, and he was promptly given hemlock. Giambattista Vico did it in the 18th century, and the philosophical establishment could not bury him fast enough. +%\end{quote} -The work of others has contributed to both the research perspective and the epistemological framework\footnote{In the psychological sense, rather than that of Brouwer}. +%The work of others has contributed to both the research perspective and the epistemological framework\footnote{In the psychological sense, rather than that of Brouwer}. Marton developed the phenomenographic research perspective, which is broadly applicable to education. @@ -126,15 +128,15 @@ Socrates is a man.\\ Socrates is mortal. -\subsection{Variation Theory and Conjunctions} +%\subsection{Variation Theory and Conjunctions} -Marton and Booth\cite{marton1997learning} have observed that increased differentiation, i.e., specialization, and also integration in the ways in which we experience the world are the results of learning. +%Marton and Booth\cite{marton1997learning} have observed that increased differentiation, i.e., specialization, and also integration in the ways in which we experience the world are the results of learning. -The mind quickly learns certain specializations. -This was predicted by Valiant\cite{valiant2000circuits}. -This was verified experimentally by Fried, using single neuron experiments\cite{fried2014single}. -Cognitive neuroscience predicts that specializations that are conjunctions of positive literals of existing concepts are easy to learn and that conjunctions containing literals that are not existing concepts, but are negations of existing concepts may not be\cite{valiant2000circuits}. -By examining the conceptualizations present in the population of learners, we can hope to find clusters from which we can learn features whose values differentiate the clusters. It is these features, called in variation theory "critical factors", which instructors should emphasize, showing in their positive and negative form. Showing this variable in its positive and negative literals, and the effect of this variation on the conjunction being studied, is expected to be very helpful to the students\cite{marton1997learning}. +%The mind quickly learns certain specializations. +%This was predicted by Valiant\cite{valiant2000circuits}. +%This was verified experimentally by Fried, using single neuron experiments\cite{fried2014single}. +%Cognitive neuroscience predicts that specializations that are conjunctions of positive literals of existing concepts are easy to learn and that conjunctions containing literals that are not existing concepts, but are negations of existing concepts may not be\cite{valiant2000circuits}. +By examining the conceptualizations present in the population of learners, we can hope to find clusters from which we can learn features whose values differentiate the clusters. It is these features, called in variation theory "critical factors", which instructors should emphasize, showing in their positive and negative form. %Showing this variable in its positive and negative literals, and the effect of this variation on the conjunction being studied, is expected to be very helpful to the students\cite{marton1997learning}. @@ -169,36 +171,36 @@ Thompson \cite{thompson1994students} states ``\ldots an instructor who fails to It appears that within constructivism there are degrees to which the teachers regard their role as facilitative. -Wegner\cite[p. 279--280]{wenger1999communities} notes "Constructivist theories focus on the processes by which learners build their own mental structures when interacting with an environment. Their pedagogical focus is task-oriented. They favor hands-on, self-directed activities oriented toward design and discovery. They are useful for structuring learning environments, such as simulated worlds, so as to afford the construction of certain conceptual structures through engagement in self-directed tasks." +%Wegner\cite[p. 279--280]{wenger1999communities} notes "Constructivist theories focus on the processes by which learners build their own mental structures when interacting with an environment. Their pedagogical focus is task-oriented. They favor hands-on, self-directed activities oriented toward design and discovery. They are useful for structuring learning environments, such as simulated worlds, so as to afford the construction of certain conceptual structures through engagement in self-directed tasks." -Brooks and Brooks\cite{brooks1999search} enumerate, in their description of constructivism applied in the classroom, five overarching principles: -Teachers seek and value their students' points of view.\\ -Classroom activities challenge students' suppositions.\\ -Teachers pose problems of emerging relevance.\\ -Teachers build lessons around primary concepts and "big" ideas.\\ -Teachers assess student learning in the context of daily teaching.\\ +%Brooks and Brooks\cite{brooks1999search} enumerate, in their description of constructivism applied in the classroom, five overarching principles: +%Teachers seek and value their students' points of view.\\ +%Classroom activities challenge students' suppositions.\\ +%Teachers pose problems of emerging relevance.\\ +%Teachers build lessons around primary concepts and "big" ideas.\\ +%Teachers assess student learning in the context of daily teaching.\\ -They\cite[p. x]{brooks1999search} go on to say "Engagement in meaningful work, initiated and mediated by skillful teachers, is the only high road to real thinking and learning." They illustrate the student's viewpoint of such interactions: "teachers \ldots made difficult concepts accessible by seeking to understand what [the student] knew at the time \ldots these remarkable teachers mattered so much because they were less concerned about covering material than they were about helping students connect their current ideas with new ones." +%They\cite[p. x]{brooks1999search} go on to say "Engagement in meaningful work, initiated and mediated by skillful teachers, is the only high road to real thinking and learning." They illustrate the student's viewpoint of such interactions: "teachers \ldots made difficult concepts accessible by seeking to understand what [the student] knew at the time \ldots these remarkable teachers mattered so much because they were less concerned about covering material than they were about helping students connect their current ideas with new ones." -Ben-Ari\cite{ben1998constructivism} articulated a slightly different version of constructivism: +%Ben-Ari\cite{ben1998constructivism} articulated a slightly different version of constructivism: -"Passive learning will likely fail, because each student -brings a different knowledge framework to the -classroom, and will construct new knowledge in a -different manner. Learning must be active: the student -‘must construct knowledge assisted by guidance -from’ the teacher and feedback from other students." +%"Passive learning will likely fail, because each student +%brings a different knowledge framework to the +%classroom, and will construct new knowledge in a +%different manner. Learning must be active: the student +%‘must construct knowledge assisted by guidance +%from’ the teacher and feedback from other students." The value of social interaction is shown clearly by an experiment by Bausell et al.\cite{bausell1972factorial}, carefully comparing tutoring with classroom instruction, in which tutoring produced significantly greater achievement. -Marton contrasts what he calls individual constructivism with his idea of social constructivism.~\cite{marton1997learning} +%Marton contrasts what he calls individual constructivism with his idea of social constructivism.~\cite{marton1997learning} \subsection{Social Constructivism} %Need to define social constructivism There are multiple perspectives on social constructivism. -Wenger\cite[p. 4]{wenger1999communities} states: "My assumptions as to what matters about learning ans as to the nature of knowledge, knowing, and knowers can be succinctly summarized as \ldots +Wenger\cite[p. 4]{wenger1999communities} states: "My assumptions as to what matters about learning and as to the nature of knowledge, knowing, and knowers can be succinctly summarized as \ldots \begin{enumerate} \item We are social beings. Far from being trivially true, this fact is a central aspect of learning. \item Knowledge is a matter of competence with respect to valued enterprises -- such as singing in tune, discovering scientific facts, fixing machines, writing poetry, being convivial, growing up as a boy or a girl, and so forth @@ -214,13 +216,13 @@ He\cite[p. 66]{ernest1994constructing} says, "In simplified terms, the key disti -Lev Vygotsky founded the idea of social constructivism, which can be summarized as learning is facilitated by interactions in a group. +%Lev Vygotsky founded the idea of social constructivism, which can be summarized as learning is facilitated by interactions in a group. -According to Cole and Scribner\cite[p. 1]{vygotsky1978mind}, Vygotsky "and his colleagues sought to develop a Marxist theory of human intellectual functioning". They say[p. 5--6] that "What Vygotsky sought was a comprehensive approach that would make possible description \textit{and} explanation of higher psychological functions in terms acceptable to natural science. To Vygotsky, explanation meant a great deal. It included identification of the brain mechanisms underlying a particular function; it included a detailed explication of their developmental history to establish the relation between simple and complex forms of what appeared to be the same behavior; and, importantly, it included specification of the societal context in which the behavior developed." +%According to Cole and Scribner\cite[p. 1]{vygotsky1978mind}, Vygotsky "and his colleagues sought to develop a Marxist theory of human intellectual functioning". They say[p. 5--6] that "What Vygotsky sought was a comprehensive approach that would make possible description \textit{and} explanation of higher psychological functions in terms acceptable to natural science. To Vygotsky, explanation meant a great deal. It included identification of the brain mechanisms underlying a particular function; it included a detailed explication of their developmental history to establish the relation between simple and complex forms of what appeared to be the same behavior; and, importantly, it included specification of the societal context in which the behavior developed." -According to Cole and Scribner\cite[p. 6]{vygotsky1978mind}, "In stressing the social origins of language and thinking, Vygotsky was following the lead of influential French sociologists, but to our knowledge he was the first modern psychologist to suggest the mechanisms by which culture becomes a part of each person's nature. Insisting that psychological functions are a product of the brain's activity, he became an early advocate of combining experimental cognitive psychology with neurology and physiology. Finally, by claiming that all of these should be understood in terms of a Marxist theory of the history of human society, he laid the foundation for a unified behavioral science." +%According to Cole and Scribner\cite[p. 6]{vygotsky1978mind}, "In stressing the social origins of language and thinking, Vygotsky was following the lead of influential French sociologists, but to our knowledge he was the first modern psychologist to suggest the mechanisms by which culture becomes a part of each person's nature. Insisting that psychological functions are a product of the brain's activity, he became an early advocate of combining experimental cognitive psychology with neurology and physiology. Finally, by claiming that all of these should be understood in terms of a Marxist theory of the history of human society, he laid the foundation for a unified behavioral science." -According to Cole and Scribner\cite[p. 7]{vygotsky1978mind}, " Vygotsky believed that the internalization of culturally produced sign systems brings about behavioral transformations and forms the bridge between early and later forms of individual development. Thus for Vygotsky, in the tradition of Marx and Engels, the mechanism of individual developmental change is rooted in society and culture." +%According to Cole and Scribner\cite[p. 7]{vygotsky1978mind}, " Vygotsky believed that the internalization of culturally produced sign systems brings about behavioral transformations and forms the bridge between early and later forms of individual development. Thus for Vygotsky, in the tradition of Marx and Engels, the mechanism of individual developmental change is rooted in society and culture." Vygotsky defined the "zone of proximal development. The \textit{zone of proximal development} [extends from] the level as determined by independent problem solving [to] the level of potential development as determined through problem solving under \ldots guidance or in collaboration with more capable peers.~\cite[p. 85--86]{vygotsky1978mind}. @@ -233,18 +235,22 @@ The zone of proximal development characterizes mental development prospectively. % % %From somewhere get support for people have to talk to each other. % % %Because it is the adopted theoretical perspective/epistemological framework -From this range of inquiry about and understanding of social constructivism, we extract a domain over which we can imagine the style of thinking our students are using. -Some students may wish to think individually, developing an opinion in their own way, exploring relevant materials and activities on their own, before engaging socially in discussion on a topic. -Other students might find exploration of possibilities in a social context helpful, as they are in the process of deciding how they are experiencing and integrating new concepts with what they already know. +%Marton and Booth\cite[p. 11]{marton1997learning} "prefer to use 'social constructivism' as an umbrella term for a rather diverse set of research orientations that have in common an emphasis on what surrounds the individual, focusing on relations between individuals, groups, communities, situations, practices, language, culture and society." They give further examples\cite[p. 201]{marton1997learning} "an emphasis on cultural, linguistic, social, historical situations". Of social constructivism, they\cite[p. 12]{marton1997learning} say "Individual constructivism is a form of cognitivism in the sense that it regards the outer (act, behavior) as being in need of explanation and the inner (mental acts) as explanatory, whereas, as we have pointed out, the reverse is true of \textit{social constructivism}." This understanding must be taken in context, because Marton and Booth go on to say\cite[p. 12]{marton1997learning} that "in this book the dividing line between 'the outer' and 'the inner' disappears. \ldots The world \ldots is \textit{constituted} as an internal relation between them." + +%It can also be observed that a person's inner life can be influenced by the external world, and that the inner life can in turn motivate an individual's behavior, and that the social surround of a person may react to that behavior, with the reaction impacting the individual. Thus, a cyclic feedback situation may result. The idea of a person embedded in a feedback situation seems to encompass more possibilities than either of the ideas Marton opposes, "individual constructivism" or "social constructivism". Moreover, feedback loops, which seem a feasible model for human interaction with society, enjoy more complex dynamics than open loop systems, which may endow them with greater explanatory power. + +%From this range of inquiry about and understanding of social constructivism, we extract a domain over which we can imagine the style of thinking our students are using. +%Some students may wish to think individually, developing an opinion in their own way, exploring relevant materials and activities on their own, before engaging socially in discussion on a topic. +%Other students might find exploration of possibilities in a social context helpful, as they are in the process of deciding how they are experiencing and integrating new concepts with what they already know. -In this study we employ a social constructivism into our theoretical and epistemological framework: We are aware the students can vary as to their ways of acquiring, consolidating, relabilizing and reconsolidating meaning. +%In this study we employ a social constructivism into our theoretical and epistemological framework: We are aware the students can vary as to their ways of acquiring, consolidating, relabilizing and reconsolidating meaning. -\section{Mathematics Education} -Part of the research perspective is formed by the goals for what students learning proof should know: according to Ball et al.\cite[p, 32 -- 34]{loewenberg2003mathematical} "These activities -- mathematical representation, attentive use of mathematical language and definitions, articulated and reasoned claims, rationally negotiated disagreement, generalizing ideas, and recognizing patterns -- are examples of what we mean by \textit{mathematical practices}. \dots These practices and others are essential for anyone learning and doing mathematics proficiently. \ldots investing in understanding these 'process' dimensions of mathematics could have a high payoff for improving the ability of the nations' schools to help all students develop mathematical proficiency". -Ball goes on to say\cite[p. 37]{loewenberg2003mathematical} "Another critical practice -- the fluent use of symbolic notations -- is included in the domain of representational practice. Mathematics employs a unique and highly developed symbolic language upon which many forms of mathematical work and thinking depend. Symbolic notation allows for precision in expression. It is also efficient -- it compresses complex ideas into a form that makes them easier to comprehend and manipulate. Mathematics learning and use is critically dependent upon one's being able to fluently and flexibly encode ideas and relationships. Equally important is the ability to accurately decode what others have written." +%\section{Mathematics Education} +%Part of the research perspective is formed by the goals for what students learning proof should know: according to Ball et al.\cite[p, 32 -- 34]{loewenberg2003mathematical} "These activities -- mathematical representation, attentive use of mathematical language and definitions, articulated and reasoned claims, rationally negotiated disagreement, generalizing ideas, and recognizing patterns -- are examples of what we mean by \textit{mathematical practices}. \dots These practices and others are essential for anyone learning and doing mathematics proficiently. \ldots investing in understanding these 'process' dimensions of mathematics could have a high payoff for improving the ability of the nations' schools to help all students develop mathematical proficiency". +%Ball goes on to say\cite[p. 37]{loewenberg2003mathematical} "Another critical practice -- the fluent use of symbolic notations -- is included in the domain of representational practice. Mathematics employs a unique and highly developed symbolic language upon which many forms of mathematical work and thinking depend. Symbolic notation allows for precision in expression. It is also efficient -- it compresses complex ideas into a form that makes them easier to comprehend and manipulate. Mathematics learning and use is critically dependent upon one's being able to fluently and flexibly encode ideas and relationships. Equally important is the ability to accurately decode what others have written." -Even more tightly focused on proofs, Ball continues \cite[p. 37--38]{loewenberg2003mathematical} "A second core mathematical practice for which we recommend research and development is the work of justifying claims, solutions, and methods. \textit{Justification} centers on how mathematical knowledge is certified and established as 'knowledge'. Understanding a mathematical idea means both knowing it and also knowing why it is true. For example, knowing that rolling a 7 with two dice is more likely than rolling a 12 is different from being able to explain why this is so. Although 'understanding' is part of contemporary education reform rhetoric, the reasoning of justification, upon which understanding critically depends, is largely missing in much mathematics teaching and learning. Many students, even those at university level, lack not only the capacity to construct proofs -- the mathematician's form of justification -- but even lack an appreciation of what a mathematical proof is." +%Even more tightly focused on proofs, Ball continues \cite[p. 37--38]{loewenberg2003mathematical} "A second core mathematical practice for which we recommend research and development is the work of justifying claims, solutions, and methods. \textit{Justification} centers on how mathematical knowledge is certified and established as 'knowledge'. Understanding a mathematical idea means both knowing it and also knowing why it is true. For example, knowing that rolling a 7 with two dice is more likely than rolling a 12 is different from being able to explain why this is so. Although 'understanding' is part of contemporary education reform rhetoric, the reasoning of justification, upon which understanding critically depends, is largely missing in much mathematics teaching and learning. Many students, even those at university level, lack not only the capacity to construct proofs -- the mathematician's form of justification -- but even lack an appreciation of what a mathematical proof is." %\subsection{Radical Constructivism Applied to Mathematical Education} %Lerman\cite{lerman2012articulating} tells us @@ -259,9 +265,9 @@ Rota\cite{rota1997phenomenology} points out that proofs that are perceived by ma Marton and Booth\cite{marton1997learning} have compared phenomenology and phenomenography. They say: "phenomenography has human experience as its object, as distinct from human behavior, or mental states, or the nervous system." They give a reason why phenomenography cannot be subsumed under phenomenology "The only demand is that this established science would have the object of research as its only defining attribute, and not methods and theories. \ldots with [phenomenology] are inextricably linked a set of methods of going about the study of experience and theories about its nature, which make its subsumption of phenomenography problematic".\cite[p. 116]{marton1997learning} This method is the study of oneself, which is not helpful for phenomenography, which studies others. Moreover, phenomenology studies prereflective experience and excludes conceptual thought; phenomenography includes both. For phenomenlogy, the goal is to find a singular essence within the prereflective experience. By contrast, for phenomenography, the goal is to find the variation of conceptions. -\subsection{Phenomenology Applied to Problem Solving} -Phenomenology has also been invoked by mathematician Alan Schoenfeld in modeling teaching behavior. -His lesson segments are chosen for phenomenological integrity.\cite[p. 91]{kaput1998research}. -He states \cite[p. 91]{kaput1998research} ``develop knowledge and skills, pursue connection, extensions, generalizations to know how to make good conjectures and know how to prove them, have a sense of what it means to understand mathematics and good judgment about when they do. Have the tools that will enable them to do so. That means having a rich knowledge base, a wide range of problem solving strategies and good meta-cognitive behavior''. -He had, earlier on the same page, described meta-cognitive behavior as reflecting and acting on what you know. +%\subsection{Phenomenology Applied to Problem Solving} +%Phenomenology has also been invoked by mathematician Alan Schoenfeld in modeling teaching behavior. +%His lesson segments are chosen for phenomenological integrity.\cite[p. 91]{kaput1998research}. +%He states \cite[p. 91]{kaput1998research} ``develop knowledge and skills, pursue connection, extensions, generalizations to know how to make good conjectures and know how to prove them, have a sense of what it means to understand mathematics and good judgment about when they do. Have the tools that will enable them to do so. That means having a rich knowledge base, a wide range of problem solving strategies and good meta-cognitive behavior''. +%He had, earlier on the same page, described meta-cognitive behavior as reflecting and acting on what you know.