Outline

=​Outline=

Algebra
 * Historical
 * concepts were gained in the mathimatical world with mistrust and took many years to develop, yet we expect students to be able to accept them quickly. (Sfard 1995 p.16 & 34)
 * Mention something about history of .... (Kaput p. 47)
 * Kaput 22 Historian quote
 * Generalized Arithmetic
 * ( Sfard 1995 p.18) Generality is the characteristic that separates arithmatic from algerbra (Kaput p.21)
 * Mathematical discourse and mathematical objects create eachother this prcoess occurs at bothe the individual and classroom level as well as historically (Kaput 21)
 * Whenever thinking about arithmetic in a general way, this is where the switch is made to algebra. p. 18 sfard
 * Operational symbolizim is a main feature of algebraic reasoning.p 18 sfard
 * "Algebraic reaosnng shares dual aspects with most mathematics and that is generalizing and expressing generalizations and that is about using special systems of symbols to reaosn with." (Kaput 21)
 * (Bringing out the algebraic.... Preface)
 * arithmetic is taught first before algebra, and in pre algebra, algebra and arithmetic are taught side by side.
 * Equals sign related to process vs object ( equals/ makes) (also in kaput p 13)
 * arithmetic is part of algebra (a proper subset) they should teach them together
 * to consider arith as a part of alg encoruages us to view isolate examples and topics as instances as more abstract ideas and concepts (xiii)
 * (Kaput p.13 ) Transitions between algrbra and arithmatic
 * strand 1, primary role into ealgebra generalizing arith oprerations and their properties and reasoning about more general realtionships and their forms (heart of alg) (p 12)
 * strand 2, generalizng and moving toward idea of function (p 13)
 * Strand 3, modeling expressing patterns and regularities in situations and also modeling involves generalizing from solutions to single asnwers from arithmetic word problems????.....p14


 * Process vs. Object
 * (Kaput p.11ish)
 * two faces maybe viewed as dynamic processes or static objects or int he words of Sfard 1991 as an operational or instructional duality. (Graham & Thomas 2000 p.266)
 * Three levels of understanding (Heid and Blume)
 * Connections to Geometry
 * Sfard 1995 p.22-23
 * Symbolization
 * Kaput p.20, 9, 25
 * "The heart of algebraic reasoning is comprised of complex symbolization processes that serve purposful generalizations and reasoning with generalizations" p.9
 * Having the symbols allows us to express and reason with generalty and further symbolization p.20
 * Winshield analogy p.25
 * use symbolization to help form/build on ideas p.21
 * Two kinds of algebraic symbolization (Kaput p.44)
 * Generalizing and action lifting
 * "These kinds of historical phenomena remind us that the process of symbolization in school mathmatics has a deeply transformative effect on what is known" p.47
 * first concepts are met through process later they are able to view that symbolism as encapsulating the product of the process (object). (Graham and Thomas 2000 p.26 7)
 * first concepts are met through process later they are able to view that symbolism as encapsulating the product of the process (object). (Graham and Thomas 2000 p.26 7)

Concept image/Concept Definition related to Kaputs model
 * Evolution between worlds
 * representing world and represented world
 * Concept image affecting definitionn (& vice Versa)
 * (Tall p.1-4)
 * concept image consists of cognative structure in the mind that is associated with a given concept
 * in this process the concept is given a symbol or a name which enables it to be comunicated and aids in mental manipulation
 * The concept image can be formed and built up over the years through expierences
 * concept definition generates its own concept image
 * concept images can be biult from informal usage but can be changed after a formal definittion is given
 * concept definition is the form of words used to specify a concept, it is dynamic and may vary from the formal definition
 * Symbols affect world and perception of it
 * (Kaput Model p.30) using symbolization to form representations and generalizations
 * expierences produce raw reprsentations that in turn form a new way of thinking. The new way of thinking causes new symbols and representations which in turn form a newer way of thinking, and the process continues.
 * Learning Spiral (Kutzler p.60)
 * There are three stages to gaining new concepts; expirementation, exactification, and application.
 * Expirement with examples which is the first step to forming conjectures or patterns about a concept
 * Exactification is a deductive process so proving the conjuctures that were found in expirementation
 * Application, is a production where by using the theroms proven new patteren and examples are formed to be expieremented with

Use of Scaffolding
 * Relate to past knowledge or concepts
 * Build from primitive knowledge (onion model) (Pirie & Kieren, 1994)
 * Structure of lesson-Ausebel's advanced organizer (organization to promote learning-moving from one level to another)
 * Technology as tool to build (Kutzler, 2003)
 * (Hied & Blume p.80) Being able to use certain functions as a leaning expierence and not to overgeneralize.
 * build concepts and symbolizations (building understanding)
 * K. Hied chapter 2
 * Appropriate use of technology (not as a crutch)
 * Kutzler p.56 car analogy
 * Goldenberg p.13
 * Moving on to higher level skills by using CAS to compensate for lower level skills (Kutzler p.67)
 * (Leah McCoy p.2) Three roles of technology tutor, tool, and tutee
 * (Leah McCoy p.2) Three roles of technology tutor, tool, and tutee

Immediate Feedback
 * Simultaneous Change (repeated experiences)
 * (Brasell p.393)
 * Allows students to correct misconceptions and to make the "abstract" concrete (Hale 2000)
 * Beichner
 * impact of MBL
 * Student control of situation
 * Beichner 1990 p.812
 * The ability to make changes and to instantly see the effect is vital to the efficacy of mcb kinetic labs, due to both the viual and kinestetic senses.
 * Expirementation is a vital part of learning (Learning spiral-Kutzler)
 * Without the expiermentation phase of the learning spiral there is nothing to build the deduction and production off of. It becomes meaningless

Teacher Use of Technology to Support Learning (If and how the techer uses the tool Kutzler p.63 )
 * Use of open-ended question (question type) (Kutzler, 2003, p. 63)
 * "A lack of 'expressing' activity seems to inhibit the students from moving beyond their previous image." (Pirie and Kieren, 1994, p. 76)
 * Reflection on practice
 * (Lapp, 2000, p. 508) Class and group discussion, not only does this deepen understanding, but it also prevents groups from "converging on misconceptions".
 * make sure not to give examples/ use technology in a way that helps confirm misconceptions (Lapp......)
 * when using tech it is nessasary to repair misconceptions that may develop (Hale 2000)
 * (Hied & Blume 2008 p.95)
 * (Pirie and Kieren) to "express" and reflect what you are learning in order to reach and understanding
 * "The belie that learing occurs through students deliberate action and thier reflection on that action suggests that stuidents should generate instances of a pattern and also reflect on the meaning of those results." (Heid chapter 2 p.41)
 * Group work- mathematical insight comes mostly from disscussion with peers first (M.K. Hied & G.W. Blume 2008 p.73)
 * Group work also helps to solidify expression of different representations (McCoy, Baker & Little 1996)
 * Use of cognitive conflict to motivate "sense-making"
 * Hale p.417 repair misconceptions
 * Tall p.2-3 cognitive conflict factors