diff --git a/ch2.tex b/ch2.tex index e6318e3..31b291a 100644 --- a/ch2.tex +++ b/ch2.tex @@ -22,37 +22,56 @@ Significant work has been done on teaching computer science students about mathe %student and the material. %Then there was variation theory. -Svensson\cite{svensson1997theoretical} has written on the theoretical foundations of phenomenography. From Svensson we learn that +Svensson\cite{svensson1997theoretical} has written on the theoretical foundations of phenomenography. +Svensson tells us that the name phenomenography was coined in 1981 by Ference Marton. +From Svensson we learn that phenomenography is a research orientation. -This orientation has as its purpose, to describe conceptions. The context in which these conceptions are of interest is mainly, but not entirely, education. +This orientation has as its purpose, to describe conceptions, +or categories of conceptions. +Moreover, phenomenographic descriptions must be amenable to comparison. +The context in which these conceptions are of interest is mainly, but not entirely, education. This orientation is associated with an approach; this pair forms a research specialization. -Svensson also tells use that the name phenomenography was coined in 1981 by Ference Marton. The approach in phenomenography is about the way of arriving at the descriptions of conceptions. - It is a kind of contextual analysis. -Phenomenographic descriptions are amenable to comparison. These descriptions make use of categories. An open, explorative form of data collection is used. During analysis, the data are interpreted. The nature of the conceptualizations which are collectively the object of the description is the meaning that something has to the individual, i.e., the individual's understanding. Thus it is empirical and subjective. Moreover, the conceptualizations are regarded as occurring in a social context, and are qualitative in nature. Phenomenography does not include a position on the nature of reality.\cite[p. 165]{svensson1997theoretical} Nevertheless, phenomenography is concerned with the relation between dependence of conceptions / knowledge upon external reality. Because our conceptions are about mathematical proof, we may evade the distinctions between ideal and real external world, as we are dealing with mental constructions. However, the social nature of creating convincing arguments ensures that these mental constructions must be shared. +It is a kind of contextual analysis. +The approach in phenomenography begins with data collection. +An open, explorative form of data collection is used. +During analysis, the data are interpreted. +The nature of the conceptualizations, +which are collectively the object of the description, +is the meaning that something has to the individual, i.e., the individual's understanding. +Thus it is empirical and subjective. +Moreover, the conceptualizations are regarded as occurring in a social context, and are qualitative in nature. + +Phenomenography does not include a position on the nature of reality.\cite[p. 165]{svensson1997theoretical} +Nevertheless, phenomenography is concerned with the relation between dependence of conceptions, or knowledge. upon external reality. +Because the conceptions in this study are about mathematical proof, we may evade the distinctions between ideal and real external world, as we are dealing with mental constructions. +However, the social nature of creating convincing arguments ensures that these mental constructions must be shared. The approach will necessarily involve descriptions of meaning, and of similarities and differences in meaning. -By empirically determining similarities and differences of meaning, we might detect the absence of general agreement on meaning. -Categories of description are developed. These cluster more similar conceptualizations, so that distinctions among categories emerge. Because the category represents its members, abstraction occurs, reducing the number of individual ideas being considered, and summarizing the data. +By empirically determining similarities and differences of meaning, we might detect the absence of general agreement thereon. +Categories of description are developed. +These cluster together relatively similar conceptualizations, +so that distinctions among categories emerge. +Because the category represents its members, abstraction occurs, reducing the number of individual ideas being considered, and summarizing the data. Dahlin\cite[p. 328]{dahlin2007enriching} recounts that variation theory was a development upon phenomenography that brought in dynamic elements to the description of conceptualizations. "The concepts of discernment, variation and simultaneity are the core of variation theory. In order for learning to take place, the learner has to discern a critical aspect or dimension of variation in the phenomenon; she has to see how this aspect can vary; and she has to become simultaneously aware of the possible 'values' along this dimension of variation in order to compare them." -Variation is seen as occurring among conceptualizations, and as occurring during the learning. That is, the teacher may emphasize variation of an aspect of the material being taught, and may emphasize that values taken on along this dimension of variation are significant for the material being taught. This emphasis serves to help students discern not only the dimension of variation, but the factor that is changing; change of the factor calls attention to the factor. Were that factor constant, it might not be noticed. Variation among the categories of description extends the outcome space, such that more distinct conceptualizations are found.\cite[p. 124--125]{marton1997learning}. The outcome space is a complex of categories of description, capturing different ways of experiencing, comprising distinct groupings of aspects of the phenomenon and relationships among them. Often, but not always, in the form of set inclusion, these relationships can capture conceptualizations that are more includsive, or complex, or built including more layers of experience. +Variation is seen as occurring among conceptualizations, and as occurring during the learning. That is, the teacher may emphasize variation of an aspect of the material being taught, and may emphasize that values taken on along this dimension of variation are significant for the material being taught. This emphasis serves to help students discern not only the dimension of variation, but the factor that is changing; change of the factor calls attention to the factor. Were that factor constant, it might not be noticed. Variation among the categories of description extends the outcome space, such that more distinct conceptualizations are found.\cite[p. 124--125]{marton1997learning}. The outcome space is a complex of categories of description, capturing different ways of experiencing, comprising distinct groupings of aspects of the phenomenon and relationships among them. Often, but not always, in the form of set inclusion, these relationships can capture conceptualizations that are more inclusive, or complex, or built including more layers of experience. -Variation theory\cite{marton2013meanings} suggests that critical aspects, which are particular ideas, are necessary\cite{marton2006some} for meaning making (understanding) to progress from one level of conceptualization to a more advanced level. +Variation theory\cite{marton2013meanings} suggests that critical aspects, which are particular ideas, are necessary\cite{marton2006some} for meaning-making (understanding), to progress from one level of conceptualization to a more advanced level. The research approach associated with variation theory has the goal of identifying these specific ideas, which, on the basis of an identification of the conceptualizations present in a student population, empirically are seen to differentiate one level of conceptualization from another. Variation theory uses these so-called critical factors. They are emphasized in teaching, specifically by varying them, and considering the consequences. For example, we may highlight the significance of climate zones by considering the variation in annual rainfall from one zone to another. We may make salient the distinction between the ideas of language as contrasted with speech, the difference between speaking (in general), and speaking in a specific language (in particular) by acquainting children with the existence of a second language. (In the context of only one language, the distinction still exists, but might not be so readily described or learned.) -Marton and Pang\cite{marton2006some} "identify some necessary conditions of learning. To learn +Marton and Pang\cite{marton2006some} identify "some necessary conditions of learning. To learn something, the learner must discern what is to be learned (the object of learning). Discerning the object of learning amounts to discerning its critical aspects. To discern an aspect, the learner must experience potential alternatives, that is, variation in a dimension corresponding to that aspect, against the background of invariance in other aspects of the same object of learning. (One could not discern the color of things, for instance, -if there was only one color.) The study results illustrate that what students learn in a sequence +if there was[sic] only one color.) The study results illustrate that what students learn in a sequence of lessons is indeed a function of the pattern of variation and invariance constituted in that sequence. All teachers make use of variation and invariance in their teaching, but this study shows that teachers informed by a systematic framework do it more @@ -62,7 +81,7 @@ systematically, with striking effects on their students' learning." A commonly used example of a proof utilizing one application of modus ponens is: -All men are motal.\\ +All men are mortal.\\ Socrates is a man.\\ Socrates is mortal.\\ @@ -102,8 +121,9 @@ Socrates is mortal. 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. -We know this from single neuron experiments\cite{fried2014single}. -Consistent with this, from cognitive neuroscience we know 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}. +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}. @@ -120,7 +140,11 @@ aggregating new information onto their present conceptions. Whilst part of what we perceive comes through our senses from the object before us, another part (and it may be the larger part) always comes out of our own head\\--William James \end{quote}\cite{van2011slow} -\subsection{Constructivism of Piaget} +There are many specializations of constructivism.\cite{ernest1994constructivism} +These range over the radical constructivism of von Glasersfeld\cite{von2013radical} and social constructivism.\cite{vygotsky1978mind} +Some debate has gone on, in the mathematics education literature as to which type of constructivism is most suitable for mathematics education.\cite{lerman2012articulating,ben1998constructivism} + +\subsection{Piagetian Constructivism} %didactical obstacle see McGowan Tall 2010 Jour Math Behav McGowen and Tall\cite{mcgowen2010metaphor} suggest that "it is even more important to take into account the particular mental structures available to the individual that have been built from experience that the individual has 'met-before'." They say [p. 170] "New experiences that build on prior experiences are much better remembered and what does not fit into prior experience is either not learned or learned temporarily and easily forgotten." @@ -128,13 +152,24 @@ McGowen and Tall\cite{mcgowen2010metaphor} suggest that "it is even more importa %McGowen and Tall\cite{mcgowen2013flexible}, citing -Thompson \cite{thompson1994students} states ``\ldots an instructor who fails to understand how his/her students are thinking about a situation will probably speak past their difficulties. Any symbolic talk that assumes students have an image like that of the instructor will not communicate. Students need a different kind of remediation, a remediation that orients them to construct the situation in a mathematically more appropriate way % Thompson 1994 p. 32. +Thompson \cite{thompson1994students} states ``\ldots an instructor who fails to understand how his/her students are thinking about a situation will probably speak past their difficulties. Any symbolic talk that assumes students have an image like that of the instructor will not communicate. Students need a different kind of remediation, a remediation that orients them to construct the situation in a mathematically more appropriate way" % Thompson 1994 p. 32. %Thompson P N 1994 Students, functions and the undergraduate curricular in Dubinsky, Schoenfeld and Kaput Research in collegiate mathematics education I, CBMS issue in math education vol 4 pp 21-44. Marton contrasts what he calls individual constructivism with his idea of social constructivism.~\cite{marton1997learning} \subsection{Social Constructivism} %Need to define social constructivism + +BenAri\cite{ben1998constructivism} articulated the version of social constructivism we adopt for this study, in his description 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." + + 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." @@ -148,15 +183,23 @@ Vygotsky defined the "zone of proximal development. The \textit{zone of proxim The zone of proximal development characterizes mental development prospectively.~\cite[p. 87]{vygotsky1978mind}. +%Wishing to nurture the growth of students' understanding via guidance by or collaboration with more capable peers, +%and to facilitate the construction of new knowledge in the context of the knowledge framework brought to the classroom by the students, +% % %From somewhere get support for people have to talk to each other. +% % %Because it is the adopted theoretical perspective/epistemological framework \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 developer 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." +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." + +%\subsection{Radical Constructivism Applied to Mathematical Education} +%Lerman\cite{lerman2012articulating} tells us + \subsection{Phenomenology Applied to Mathematical Proofs} We wish to point out a distinction between @@ -164,21 +207,12 @@ phenomenography and phenomenology. Phenomenology might be more familiar: it has been used by mathematician Gian-Carlo Rota to describe the beauty in mathematics, particularly in proofs. Rota\cite{rota1997phenomenology} points out that proofs that are perceived by mathematicians as beautiful are easier to remember. + +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 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. -\section{Cognitive Science} -Cognitive science includes modeling plausible functioning of the brain. It helps us notice constraints that computational complexity illuminates, on how learning, including memory and retrieval of information, might be carried out. - -\section{Neuroscience / Neurophysiology} - -Though Marton and Booth\cite[p. 114]{marton1997learning} state that description of experience is autonomous from the student's nervous system, nevertheless we believe that supportive evidence from natural science can be of interest. - -In education with assessment, instructors transmit information towards the student and attempt to elicit signs of learning from the student. -Neuroscience enables us to investigate stages of progress in learning that may occur starting from the moments at or just after the transmission through the event of assessment. -For example, single neuron measurements show that conjuncts of variables having positive literals, as described by Valiant\cite{valiant2000circuits} in cognitive science, are learned in one showing, into medial temporal lobe tissue. -Functional magnetic resonance imaging (fMRI) combined with electro-encephalography (EEG) show that slow wave sleep is a process that supports memory formation in other locations in the brain. -Moreover, these technologies show us that exposing students to educational stimuli of multiple modalities rather than single modalities (e.g., sight and sound vs. only sight) changes the area within the brain in which the information is stored. These show us that transmitting with multiple modalities increases the ease of recall of information. These show us that if we wish information to be subject to recall when relevant, rather than what Whitehead\cite{whitehead1959aims} calls inert knowledge, we should concern ourselves about how the information is transmitted. These provide guidance about how we can plan for better assessment results by facilitating recall, in turn by shaping our transmission strategies. diff --git a/literature.bib b/literature.bib index 60c3372..aa61542 100644 --- a/literature.bib +++ b/literature.bib @@ -123,6 +123,31 @@ author={Hilburn, Thomas B and Ardis, Mark and Johnson, Glen and Kornecki, Andrew journal={repository.cmu.edu}, year=2013 } +@book{von2013radical, + title={Radical constructivism}, + author={Von Glasersfeld, Ernst}, + year={2013}, + publisher={Routledge} + +} +@inproceedings{ben1998constructivism, + title={Constructivism in computer science education}, + author={Ben-Ari, Mordechai}, + booktitle={Acm sigcse bulletin}, + volume={30}, + number={1}, + pages={257--261}, + year={1998}, + organization={ACM} +} +@article{lerman2012articulating, + title={Articulating theories of mathematics learning}, + author={Lerman, Stephen}, + journal={Paul Ernest, Studies in Mathematics Education Series}, + volume={4}, + pages={41--49}, + year={2012} +} @article{moreno2012balancing, title={Balancing software engineering education and industrial needs}, author={Moreno, Ana M and Sanchez-Segura, Maria-Isabel and Medina-Dominguez, Fuensanta and Carvajal, Laura}, @@ -917,6 +942,23 @@ publisher={Elsevier} % % % % % % % % % % % % % % % % % % % % % % % % % % 1995 % % % % % % % % % % % % % % % % % % % % % % % % +@article{ernest1995one, + title={The one and the many}, + author={Ernest, Paul}, + journal={Constructivism in education}, + pages={459--486}, + year={1995} +} +@article{ernest1994constructivism, + title={Constructivism: Which Form Provides the Most Adequate Theory of Mathematics Learning?}, + author={Ernest, Paul}, + journal={Journal f{\"u}r Mathematik-Didaktik}, + volume={15}, + number={3-4}, + pages={327--342}, + year={1994}, + publisher={Springer} +} @article{selden1995unpacking, title={Unpacking the logic of mathematical statements}, author={Selden, John and Selden, Annie}, diff --git a/thesis2.pdf b/thesis2.pdf index 1ea0131..f6e6a9a 100644 Binary files a/thesis2.pdf and b/thesis2.pdf differ