Designing and Implementing Interactive Textbook for High school Mathematics
Michal Yerushalmy University of Haifa, Israel
[email protected]
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“We simply cannot limit our explorations of possible future illiteracies to extrapolations of what we think we understand about literacy today.” diSessa 2000 Changing Minds About what settings or infra structures we should think regarding learning cooperatively with external materials (artifacts)?
• Education (and specifically science and math education) will change because the material basis for scientific thinking is changing. • A computational medium must be expressive – it must extend minds with new ways of thinking and knowing. Expressiveness cannot remain just being able to lay thoughts out concretely.
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The structure of the presentation • Electronic books – setting expectations • Hyper text and hyper diagrams – definitions and an example • The instructional affordances of pre-constructed examples genres of composition diagram & text • Contribution for further research and development 3
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)Barker
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Math texts and computers • Computers fueled two lines of development for traditional physical textbooks. One is the use of software to pose questions and generate tasks. The text, usually paper, is still needed for these tasks although its status as the leading narrative might change. • More recently, a new trend in mathematics textbooks provides digital versions of the text and illustrations. Not many electronic publications actually take full advantage of the media to support interactivity and dynamic options. The domain of such interactive environments is new and yet to be explored. • How do concepts developed in cognitive research, in visual design studies, and in the analysis of mathematics curriculum considerations can help formulate terms that may prove a useful starting point for the R&D of interactive mathematics textbooks? 5
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Student’s Math Text
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Electronic Library for the teacher-:
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Math text written as an encyclopedia http://mathworld.wolfram.com
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Interactive Definitions
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Resources • How do concepts developed in cognitive research, in visual design studies, and in the analysis of mathematics curriculum considerations can help formulate terms that may prove a useful starting point for the R&D of interactive mathematics textbooks? 10
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The Function Web Book is based on: • Student exploration and conjecturing • Curricular emphasis on “big ideas” and support for teacher understanding of them • Responsiveness to student ideas and questions • Use of technology developed specifically for educational settings, which emphasizes visual and qualitative thinking • Use by students of all levels 11
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Multi Modal Text
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Software Tools & Explorations Knowing with tools • Knowledge with tools involves both the intuitions used and developed with the tool that are connected to the formalities of the knowledge system involved • The design of the artifact as a didactic object (Yerushalmy 1999, Schwartz 1995, Thompson 2003). • Instrumental genesis: the actions students perform together with the knowledge they construct are shaped by the artifacts they use (Artigue, 2002; Mariotti, 2002; Rabardel & Bourmaud 2003 )
Transfer, Abstraction, Generalization • Design pre-planed abstractions (or actions of transfer) but learn to attend to what can be assumed as transfer in the students‟ work. • Identify processes that occur with the tool and could be viewed as “tool situated abstraction.” 13
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Text & Explorations • There is a tension between “construction of meaning” and printed single-ordered text • Writing to the reader often cause to over simplification of the text • Exploration tasks are often either not understood or over guided
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Cognitive research on Diagrams • Representational function, serving as a redundant source of information that overlaps or repeats what is already included in the text, providing students with a second opportunity for learning or checking text comprehension. • Organizational function, visualization of the macrostructure and of the relations within the text, often suggesting aspects that are not apparent through text processing. • Interpretational function, whereby the picture illustrates difficult parts of the text using analogies or visual metaphors. • Transformational function, a mnemonic role used to help improve the memory • Decorative function that addresses the appearance of the text and is usually assumed to be external to the content
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Multimodality and Images semiotic functions • Presentational (also called ideational meta-function by Halliday (1985) and representational by Kress & van Leeuwen (1996)); • Orientational (or interactive function) • Organizational (also called textual meta-function or compositional ) These functions raise the following questions regarding semiotic resources of visual images used in text: What a diagram is representing? How does a diagram interact with the viewer? How do the elements of text and visuals combine to deliver a specific style or message ? 16
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Thinking Math with Diagrams • Most current definitions of conceptual understanding of mathematics include the argument that emergence of meaning is both: dependent on the capability to think with, and can be demonstrates as interpreting and linking among multiple representations • The discourse of investigation (or the guided inquiry) attempts to organize the practice of teaching and learning along tasks that involve active processes of transforming experiences with different representations. Textbooks and tasks groups of writers are seeking ways to use visualization to achieve clarity and focus while maintaining the 'inquiry' tone of the text • At the same time diagrams, graphs and illustrations is often a source of processes that 'narrow down' student„s images • Often diagrams do not turn to be a problem-solving tool because the mathematical structure of the problem situation is not sufficiently recognized in the diagram. 17
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Functions of Interactive Diagrams Neat and Sketchy • Interactive diagrams should not be assumed transparent; they require work and must be changed not merely viewed — the tools needed to do so being an integral part of the diagram and the task. • Reading diagrams means operating the given components and tools (numeric data, history of construction) as the means of exploring the sketch. • interactive diagrams, if designed to appear explicit but at the same time subtle, could function both as a neat diagram and as a good sketch that presents rough information, generates ideas, and maintains a non-authoritative tone. • While paper sketches are often not sufficiently informative and even misleading (the lack of detail precluding answers to questions raised by the sketchy appearance), • Interactive diagrams can function both as sketches and as diagrams in the sense that they can reveal their details. 18
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Multi Modal Interactive reading An example
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The major capabilities that the design of interactive diagrams is based upon • Each visual drawn element as well as the paper or more generally, the digital surface on which it is drawn are digital objects. • Drawings are mathematical procedures, each subject to algorithmic manipulation. • images are reproducible and can be animated and rescaled. • The reproduction can be done in a way that leaves traces to provide a temporal dimension to the diagram by preserving the history and the origin of the image.
Thus, the design of the diagram helps the reader attend to two different layers, each with its own variability and the tools that control it: the screen that provides different ‘papers’ to draw on and the drawn objects . 20
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What are the instructional affordances of such design? • Unlike software applications, interactive diagrams are structures already designed around a pre-constructed example. • Each diagram appears with a specific example that is interesting enough by itself, but the reader is called upon to change it. • The pre-constructed example attempts to present a complete picture without any unnecessary details that might obscure the idea of the presentation.
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Rendering random initial example • The diagram appears differently at different times for different users. • Randomness is constrained and the random example is limited to a well-defined setting (action or object). • Random examples are often chosen for activities that introduce new concepts as part of activity definition.
piecewise functions 22
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Specific interactive example • In Burning calories the diagram presents graphic and numeric data specific to but not provided in the text. • The specific presentation changes according to reader input, so the diagram not only provides the missing data but also a way to personalize the data that is given. 23
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Generic example • constant acceleration • It represents a situation that could have been part of the given task but it is not given in the text and it does not necessarily complement the text. • It is generic in the sense that it is an example that calls for change as specified in the task and it allows the reader to see what the diagram can generate. • Generic – dictionary definition: relating or applied to or descriptive of all members, available for common use. • The idea that the example is only an initial statement and is a subject for change using a well-defined structure eases somewhat the complexity of choosing an initial generic example. 24
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Genres of Compositions • Complementary diagrams, the major role of which is to mentor the beginning of thinking about the problem • Narrative diagrams that assume the role of providing a story line • Elaborating diagrams that assume the role of facilitating explorations.
Within each category I examine the specific logic and affordances of representation and interaction offered by the diagram. 25
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Complementary Diagrams • Provides some representation of the objects dealt with in the task mostly at the intuitive level and has an expository role. • Emphasize on perceptions rather than on tools to think with 26
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Narrating Diagrams • Prompt actions that are predominantly narratives of guided inquiries, designed to invite participation in devising a strategy for solution either by means of a construction of a specific mathematical model or with tools for solving the problem. • Designed to be the principal delivery channel of the activity‟s message • Used as a way of balancing openended exploration with the specific content and objectives of the task • Ask the reader to take action that supports the construction of the principal ideas of the task. 28
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Elaborating Diagrams • Elaborating diagrams are designed to support explorations by inviting participation in a setting that fosters the elaboration of ideas or actions intended to widen the focus or even distance the reader from the original task • The conceptual image that will emerge depends to a large extent on what attracts the reader in this diagram, so the definitions may vary. Average velocity can be the average slope of the position graph; the constant speed at which one should move to start and end at the same time as the accelerated motion; or the average size of the step of the linearly changing step sizes. • The text refers explicitly to the diagram and invites the reader to take as many views and representations as possible, and to create and describe motion situations with the diagram and explain the options it offers. 29
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How can the design of interactive book of this sort contribute to the attempt to change the traditional interaction with mathematics texts? • I presented here my designer‟s points of view on a structure that seems powerful and sometimes surprising and on function that relies on creativity beyond attempting to be practical • Design is related to the instructional values brought to it • In a text that attempts to maintain and make explicit an inquiring tone, creating relations between the reader and the author is an important value and a major task. • Studies about use of software tools suggest that designed tools are instrumented by students in a way that makes the mathematics meaningful to them so designed environment is an important instructional tool only if it is open and flexible to various modes of uses and users.
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Widening authors‟ options • The design of the text and of the diagrams is oriented toward an active reader; although it presents the point of view of the designer, it also requires the user to act and change the given diagram based on personal relevance • The dynamic nature of the diagrams and the fact that the structure of the task is embedded in the tool by means of a limited set of commands and a pre-constructed example allows authors to reduce the amount of written instructions • Helps to enjoin the reader to experience the processes to be learned. • interactive diagrams increase the opportunity for authors to write structured, but not predetermined, text supporting meaningful learning of mathematical content 31
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Terms for developers to think with •
We simply cannot limit our explorations of possible future literacies to extrapolations of what we think we understand about literacy today. (Changing Minds diSessa 2000 p.22) A pragmatic result of this work is the definition of a framework or terminology for developers seeking ways to create curricular materials that would support teachers in unpacking knowledge for their students
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Helps inject order and consistency in the language, syntax, and appearance of the diagrams and the tasks we have devised
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The methods offered to readers to create data in the diagram were at first borrowed from the design of software tools. It became evident that new considerations should lead the design
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The framework presented in this study proposes a possible set of terms that may be useful in the communication between developers of new tools in an interactive environment
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Researchers‟ discourse • Support the study of learning and teaching situated in such environments Since any construction of knowledge is situated in some setting, it is necessary to identify the practices associated with this type of design • How would the reading proceed in an environment that combines non-hierarchical text with visual components? • How are we to analyze the ways in which learners make meaning by linking diagrams and text? • Which type of diagram will become a direct problem-solving tool and what would be the different modes in which it might be used? • Will tools and surface capabilities come and shape the language of mathematics? 33
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