Silver, E. Eostering creativity through instruction rich in mathematical problem solving and problem posing. Silverman, L.
- Nouvelle Révolution commerciale (La) (Sciences Humaines) (French Edition);
- Assemblées générales.
The measurement of giftedness. Shavinina Ed. Smedt, B. Working memory and individual differences in mathematics achievement: A longitudinal study from first grade to second grade. An analysis of constructs within the profession and school realms.
Handbook of intelligence. New York: Cambridge University Press. Swanson, H. Math disabilities: A selective meta-analysis of the literature.
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Review of Educational Research, 76 2 , Thomas, M. Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function. Torrance, E.
Torrance tests of creative thinking. Unsworth, N.
On the division of working memory and long-term memory and their relation to intelligence: A latent variable approach. Acta Psychologica, 1 , Van Hiele, P. Structure and insight: A theory of mathematics education. New York: Academic Press. Wechsler, D. San Antonio, TX: Harcourt. Winner, E. Giftedness: Current theory and research. Mathematical reasoning ability: Its structure, and some aspects of its genetic transmission. Unpublished doctoral dissertation. Hebrew University, Jerusalem. The other day in a second-year course on simulation and modeling, I reached the point where we had to integrate sin X from 0 to 7t.
She then presented me with the phone showing the complete calculation including all of the intermediate steps. I could tell she was wondering why she had worked so hard in first-year calculus, learning to do what her phone could already do better than she could. My colleague asked to see the graph of a challenging function and was surprised that the display on the phone contained just about everything you would ever want to know and also offered the ability to zoom in and out. What does it mean when our students are actively questioning our curriculum because they have a better understanding than we do of the expert tools available to do mathematics?
How can we ask students to master algorithms for long division, the quadratic formula or the integration of rational functions when they can simply speak questions into their phones and get the answers immediately?
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This immediate access to information is the cultural norm for students growing up with the vast online repository of knowledge and skills the internet has become. Given the constant presence of this resource, our students have shifted away from memorizing facts to simply remembering where to find them. Members of this new online culture have no interest in copying portions of the internet into their own brains! As a consequence, whether we like it or not, the current generation is in the process of retiring certain older approaches to knowledge and information like the following: 1. Memorization of simple concrete facts beyond a common cultural core.
Unassisted hand calculation. Machine correctable information such as grammar and spelling. Transmission of information in physical documents or face to face. Recording information by hand. Working with purely textual information that is not interactive or supported by media.
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Understanding of the conceptual underpinnings and applications of a wide variety of mathematical models and the ability to adapt this knowledge to new situations. The ability to assess a technical situation and correctly select the right conceptual framework and associated expert systems. The ability to find and learn specialised mathematical information.
The ability to manipulate, visualize and analyze large data sets. The ability to create and handle computer programs in a variety of contexts to solve mathematical problems and process information. These new skills are a practical necessity in a trans-textual world where information is conveyed by multimodal expert systems capable of simulating intelligent interactivity.
In the presence of this wealth of technology, we have to wonder why the mastery of pencil and paper algorithms is still the primary measure of mathematical achievement in our schools. Our continuing struggle to update our traditional classrooms can be compared with the difficulty Europe had in making the transition from Roman Numerals to the Hindu-Arabic system.
His book was so methodical and understandable that it began to be taken seriously and eventually served as the principal textbook in mathematics for centuries. It only goes to show what can be achieved by a great mathematics teacher with a few good stories about frisky rabbits! Individuals master hand calculation and memorize portions of the knowledge base with no assistance from technology.
Individuals are able to work more independently of technology. A certain amount of hand calculation and memorization helps conceptual understanding. Advantages of the New System Much faster and more accurate. More conceptually clear. Extremely fast and able to handle the complexity of our current culture.
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Ends the need for humans to calculate by hand or memorize large amounts of information. Allows students to spend more time learning concepts and how to apply them. It is trans-textual in the sense that it uses media to transform information into visual and auditory forms Enables large scale projects by groups of people.
Disadvantages of the New System Major investment of time to learn to write and calculate with the new numbers. Loss of conceptual understanding as the knowledge of algorithms for calculations and much of the knowledge base is off-loaded to computers. Individuals are no longer autonomous but must work in conjunction with technology. Loss of the textual and computational culture core that we all had in common. First adopters of the New System Traders around the Mediterranean. Businesses, engineers and young people who grew up immersed in online social networks.
Education system entrenched in centuries of Roman Numerals. Cultural emphasis on creating completely autonomous individuals. Education system entrenched in centuries of hand calculation and memorization. By , TAPSIR will be the norm — the internet will have evolved into a nearly sentient expert system that has completely devoured our current educational culture.
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When skills are no longer needed by our culture, what justification can we offer for retaining them? A standard answer to these questions is that in each educational context there is a common cultural core of skills that we expect students to have that might include the 5-times table, long division, the area of a circle, the quadratic formula, etc. In a future that accommodates both the traditional and technological flavours of mathematics, students might be encouraged to learn two complementary types of knowledge: autonomous knowledge consisting of skills and memorized facts that can be demonstrated without the assistance of technology, and linked knowledge that is contingent upon access to expert systems.
Autonomous knowledge would concentrate on traditional elements of mathematics like theorems and proofs, as well as solving problems by hand. TAPSIR pedagogy is in its infancy and is so different from traditional teaching that most of us, including myself, are nervous about handling its implementation. Using expert systems to explore concepts and perform calculations 2. Visualizing and analyzing large data sets 3. Writing and using computer programs to build models, create simulations and investigate mathematical problems 4.
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Making and testing conjectures. The level of commitment and creative engagement of students in these courses have been remarkable to watch. Students learn VB. NET and even in first year are asked to make conjectures and then test them by writing computer programs. Over the course of the MICA program, they use technology to investigate a wide range of mathematical concepts such as RSA encryption, stochastic models, chaos, the stock market, epidemics, warfare, traffic light synchronization, predator-prey models, and in their third year they learn C-H- and use it to study models based on partial differential equations.
In other courses like calculus, they learn to do certain calculations with MAPLE which frees up time for learning concepts and applications. Students are also expected to work singly and in groups to create several original technology-based projects. We can no longer justify the teaching of ancient algorithms that can now be trivially handled by friendly expert systems that we carry with us at all times.
As mathematics educators, it is our job to define the role of teachers and students within this new paradigm. The revolution is upon us and we have to move quickly to assert and clarify the role of mathematics in the presence of extraordinary technological tools. We want to retain the tremendous accomplishments and culture of mathematics, but we are under siege these days as mathematics departments are closed or cut back around the world. Part of our problem is that mathematics departments have pursued a somewhat isolationist approach to the teaching of mathematics and have not actively pursued relationships with other specialties.
In order for mathematics to continue to thrive, we must continue to demonstrate its relevance to modern times. We can achieve this goal by developing a mathematics curriculum that openly engages with other disciplines and also expands the palette of mathematics graduates to include the mastery of technological tools.