The Philosophy of Mathematics in ? Philosophy of MiC The design principles forMathematics in Context

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  • Philosophy of MiC

    The design principles for Mathematics in Contextare derived from Realistic Mathematics Education(RME). Since 1971, researchers at the FreudenthalInstitute have developed this theoretical approachtowards the learning and teaching of mathematics.RME incorporates views on what mathematics is,how students learn mathematics, and how mathe-matics should be taught. The principles that under-lie this approach are strongly influenced by HansFreudenthals concept of mathematics as a humanactivity. From the RME perspective, students areseen as reinventors, with teachers guiding andmaking conscious to students the mathematiza-tion of reality. The process of guided reinvention issupported by student engagement in problemsolving, a collective as well as an individual activity,in which whole-class discussions centering onconjecture, explanation, and justification play acrucial role. The instructional guidance of teachersin this process is critical, as teachers graduallyintroduce and negotiate with the students themeanings and use of conventional mathematicalterms, signs, and symbols.

    The Realistic Mathematics Education philosophytransfers into a design approach for MiC with thefollowing components:

    Developing instruction based in experientially real contexts

    The starting point of any instructional sequenceshould involve situations that are experientiallyreal to students so that they can immediatelyengage in personally meaningful mathematicalactivity. Such problems often involve everyday life settings or fictitious scenarios, although mathematics itself can also serve as a context ofinterest. With experience, parts of mathematicsbecome experientially real to students. Such activities should reflect either real phenomena fromwhich mathematics has developed historically oractual situationsand phenomenawhere furtherinterpretation,study, and analysisrequire the use ofmathematics.

    Designing structured sets of instructionalactivities that reflect and work towardimportant mathematical goals

    A second tenet of RME is that the starting pointshould also be justifiable in terms of the potentialend point of a learning sequence. To accomplishthis, the domain needs to be well mapped. Thisinvolves identifying the key features and resourcesof the domain that are important for students tofind, discover, use, or even invent for themselves,and then relating them via long learning lines. Thesituations that serve as starting points for a domainare critical and should continue to function as paradigm cases that involve rich imagery and, thus,anchor students increasingly abstract activity. Thestudents initially informal mathematical activityshould constitute a basis from which they canabstract and construct increasingly sophisticatedmathematical conceptions.

    Designing opportunities to build connections between content strandsthrough solving problems that reflectthese interconnections

    The third tenet of RME is based on the observationthat real phenomena, in which mathematicalstructures and concepts manifest themselves, leadto interconnections within and between contentstrands as well as connections to other disciplines(for example, biological sciences, physics, sociology,and so on). Although the maps developed for eachof the four MiC content strands contain uniqueterms, signs, and symbols as well as an extendedlearning line for the strand, instruction in actualclassrooms inevitably involves the intertwining of these strands. Problems can often be viewedfrom multiple viewpoints. For example, a geometricpattern can be expressed using numbers relation-ships or algebraically.

    Recognizing and making use of students priorconceptions and representations (models) to support the development of more formal mathe-matics (i.e., progressive formalization), RMEsfourth tenet is that instructional sequences shouldinvolve activities in which students reveal and createmodels of their informal mathematical activity.

    The Philosophy ofMathematics in Context

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  • Philosophy of MiC

    RMEs heuristic for laying out long learning lines forstudents involves a conjecture about the role thatemergent models play in the students learning,namely that students models of their informalmathematical activity can evolve into models forincreasingly abstract mathematical reasoning.Gravemeijer explained this bottom-up progressionin terms of four levels of progressive mathematiza-tion (see Figure 1).

    At the initial situational level, the expectation isthat students develop interpretations, representa-tions, and strategies appropriate for engaging witha particular problem context. At this level, studentscreate and elaborate symbolic models of theirinformal mathematical activity, such as drawings,diagrams, tables, and informal notations.

    At the referential level, students create informalmodels of the problem situation. Such models-ofcontain the collective descriptions, concepts, procedures, and strategies that refer to concrete or paradigmatic situations. At the general level,as a result of generalization, exploration, andreflection, students are expected to mathematizetheir informal modeling activity and begin to focuson interpretations and solutions independent ofsituation-specific imagery. Models at this level areconsidered models-for and are used as a basis forreasoning and reflection. The formal level involvesreasoning with conventional symbols and is nolonger dependent on models-for.

    Figure 1.

    Thus, for students, a model is first constituted, forexample, as a context-specific model of a situation,then generalized across situations. In this process,the model changes in character and becomes anentity in itself, functioning eventually as a basis for mathematical reasoning on a formal level. Thedevelopment of ways of symbolizing problem situations and the transition from informal to formalnotations are important aspects of the selection of problem contexts, the relationships betweencontexts, and instructional assumptions.

    The consequences are that the introduction ofnumbers, number sentences, standard algorithms,terms, signs and symbols, and the rules of use informal mathematics should involve a process ofsocial negotiation in a manner similar to thosefrom which the notations and rules were derived.In the process of reinventing formal semiotics, students are participating in the human mathe-matics process of progressive formalization.

    The design of activities to promote pedagogical strategies that support students collective investigation of reality

    RMEs fifth tenet is that in classrooms the learningprocess can be effective only when it occurs withinthe context of interactive instruction. Students areexpected to explain and justify their solutions, towork to understand other students solutions, toagree and disagree openly, to question alternatives,to reflect on what they have discussed and learned,and so on. Creating a classroom that fosters suchinteraction involves a shift in the teachers rolefrom dictating the prescribed knowledge via a routine instructional sequence, to orchestratingactivities situated in learning trajectories whereinideas emerge and develop in the social setting ofthe classroom. To promote the interaction of student representations and strategies, teachersmust also create discourse communities that support and encourage student conjectures, modeling, re-modeling, and argumentation.

    Philosophy of MiC Teacher Implementation Guide 59

    situational

    referential

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    formal

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  • Philosophy of MiC

    The development of an assessment system that monitors both group and individual student progress

    The teachers role in the RME instructional processinvolves capitalizing on students reasoning andcontinually introducing and negotiating with stu-dents the emergence of shared terms, symbols,rules, and strategies, with an eye to encouragingstudents to reflect on what they learn. Students are seen as reinventors, with the teacher guidingand making conscious to students the mathemati-zation of reality, with symbolizations emerging anddeveloping meaning in the social situation of theclassroom. Students are encouraged to communi-cate their knowledge, either verbally, in writing, orthrough some other means like pictures, diagrams,or models to other students and the teacher.Central to this interactive classroom is the devel-opment of students abilities to use mathematicalargumentation to support their own conjectures.

    A Summary of Implications of RealisticMathematics Education for MiC

    The real-world contexts support and motivatelearning. Mathematics is a tool to help studentsmake sense of their world. Mathematics in Contextuses real-life situations as a starting point forlearning; these situations illustrate the variety ofways in which students can use mathematics.Models help students learn mathematics at dif-ferent levels of abstraction. The ratio tables, doublenumber lines, chance ladders, percent bars, andother models in Mathematics in Context allow students to solve problems at different levels ofabstraction. The models also serve as mediatingtools between the concrete world of real-life problems and the abstract world of mathematicalknowledge.

    Students reinvent significant mathematics.Instruction builds on students own knowledge ofand experiences with mathematics. They serve as a foundation for developing deeper understandingthrough interaction with other students and theteacher. The teachers role is to help students makeconnections and synthesize what they have learned.

    Interaction is essential for learning mathematics.Interaction between teacher and student, studentand student, and teacher and teacher is an integralpart of creating mathematical knowledge. Theproblems posed in Mathematics in Context providea natural way for students to interact before, during,and after finding a solution.

    Multiple strategies are important. Most problemscan be solved with more than one strategy.Mathematics in Context recognizes that studentscome to each unit with prior knowledge andencourages students to solve problems in theirown wayby using their own strategies at theirown level of sophistication. It is the teachersresponsibility to orchestrate class discussions toreveal the variety of strategies students are using.Students enrich their understanding of mathematicsand increase their ability to select appropriateproblem-solving strategies by comparing and analyzing their own and other students strategies.

    Students should not move quickly to the abstract.In Mathematics in Context, it is preferable that students use informal strategies that they under-stand rather than formal procedures that they donot understand. It is important to allow students toexperience and explore concrete mathematics for aslong as they need to. Level 1 units offer studentsan assortment of informal methods for solvingproblems. Students are given the opportunity tomove to more formal strategies in Levels 2 and 3.

    Mastery develops over the course of the curriculum. Because mastery develops over time,teachers should not expect students to mastermathematical concepts after a single section oreven after a single unit. Each unit is connected to the other units in the curriculum. Through a spiraling of the content and contexts, importantmathematical ideas are revisited throughout thecurriculum so that students can deepen theirunderstandingand master theideas over time.

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    x

    y

    San FranciscoAirport

    Source: Rand McNally.

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  • Goals of MiC

    Goals of MiC Teacher Implementation Guide 61

    The Vision for Mathematics in Context

    Pedagogically, Mathematics in Context is designedto support the National Council of Teachers ofMathematics (NCTM) vision of mathematics education as expressed in the Principles andStandards for School Mathematics (NCTM, 2000).It consists of mathematical tasks and questionsdesigned to stimulate mathematical thinking andto promote discussion among students. Studentsare expected to:

    explore mathematical relationships;

    develop and explain their own reasoning andstrategies for solving problems;

    use problem-solving tools appropriately; and

    listen to, understand, and value each othersstrategies.

    The NCTM Principles and Standards state thatbecause mathematical understanding is related topersonal experiences in solving problems in thereal world, school mathematics is enhanced whenit is embedded in realistic contexts that are meaningful to students. Mathematics in Contextprovides these contexts as well as problems thatactively engage students in mathematics.

    NCTM also suggests that students need ampleopportunities to solve realistic problems usingstrategies that make sense to them. Students mustrecognize, understand, and extract the mathemat-ical relationships embedded in a broad range of situations. They need to know how to representquantitative and spatial relationships and how touse the language of mathematics to express theserelationships. They must know how and when touse technology. Effective problem-solving alsorequires the ability to predict and interpret results.Mathematics in Context is a connected curriculumthat requires students to deepen their understandingof significant mathematics through integratedactivities across units and grades.

    The History of Mathematics in Context

    The Mathematics in Context curriculum projectwas initially funded in 1991 by the NationalScience Foundation to develop a comprehensivemathematics curriculum for the middle grades.

    Collaborating on this project were the researchand development teams from the National Centerfor Research in Mathematical Sciences Education(NCRSME) at the University of Wisconsin, Madison,and the Freudenthal Institute (FI) at the Universityof Utrecht in the Netherlands. Thomas A. Romberg,Project Director, had become familiar with thework of the Freudenthal Institute while chairingthe writing team of the NCTMs Curriculum andEvaluation Standards (1989). He recognized that the mathematics curriculum used in theNetherlands was consistent with the vision of theStandards and that it could serve as a model for amiddle grades curriculum for the United States.

    The development of the curriculum units, teachersguides, and supporting materials took six years(19911997). An international advisory committeeprepared a blueprint document to guide the dev-elopment of the curricular materials. Then theFreudenthal Institute staff under the direction ofJan de Lange prepared initial drafts of individualunits based upon the blueprint. Researchers at theUniversity of WisconsinMadison modified thelanguage and problem contexts in these units tomake them appropriate for students and teachersin the United States. Pilot versions of the individualunits were tested in middle schools in Wisconsin.Revised field-test versions of the units were createdfrom feedback received during the initial pilot. Therevised versions were then used in several statesand Puerto Rico. Data from the field-test sites wereused to inform revisions to student books andteachers guides before commercial publication.

    The current revision of Mathematics in Context(20032005) was partially funded by the NationalScience Foundation. The revision team solicitedfeedback from experienced MiC teachers. Thisinformation was used in conjunction with theNCTM 2000 Principles and Standards to shape the changes reflected in the 2006 edition. The staffof the Freudenthal Institute prepared drafts ofeach revised unit similar to the first edition.The revision team at the University of WisconsinMadison then prepared the drafts for field-test, ifnecessary, and for submission to the publisher.Extensive teachers guide notes were contributedby experienced MiC teachers.

    Goals of MiC

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