Overview Realistic Mathematics Education (RME) RME and its challenges Shortcomings RME innovation Netherlands Koeno Gravemeijer Footholds for improvement 1 2 What makes mathematics so difficult? A ‘layman’s view on instruction • How do people learn? • The problem is not in the abstract character of mathematics as such • Layman’s view: By making connections between what is known and what has to be • It is the gap between the abstract learned knowledge of the teachers and the experiential knowledge of the students • Thus: How do people learn mathematics? à Learning Mathematics: making connections with an abstract, formal body of knowledge – Teachers and textbook authors tend to (mis)take their own more abstract mathematical • Problem: Gap between the knowledge of knowledge for an objective body of knowledge the students and the abstract, formal body with which the students can make connections of knowledge • Students cannot make connections with knowledge that is not there for them 3 4 The new mathematical knowledge does not Caveat exist yet • Young children don’t understand the question: • “See” that 4+4=8 “How much is 4+4? & reason that 4+4 equals 8 Even though they know that “4 apples and 4 • Construe resultative counting as a curtaiment of apples makes 8 apples” counting individual objects • Van Hiele Levels: • Construe ‘counting on’ and ‘counting back’ as – Ground level: Number tied to countable objects: “four apples” extensions of resulative counting – Higher level: 4 is associated with number relations: 4 = 2+2 = 3+1 = 5-1 = 8:2 – mathematical object compare Sfard (1991) structural - operational 5 5 1
Van Hiele example: the concept Van Hiele example: the ‘rhombus’ in geometry concept ‘rhombus’ in geometry • Students do not see a square as a rhombus Rhombus as a mathematical object: junction in a network of mathematical relations: square rhombus – Sides are two by two parallel – All sides have equal lengths – Diagonals intersect orthogonal – Facing angles are equal 7 8 Consequences of the common view • We perceive math an independent body of knowledge. – objects such as ‘linear functions’ as can be pointed to and spoken about – being able to talk and reason about these ‘objects’ unproblematically while interacting with others • The body of knowledge only exist in the minds of teachers and textbook authors Ø How can students connect to a body of knowledge that does not exist for them? 9 10 11 12 2
Need for a bottom-up Consequences of the common process view of concept formation • Some people manage to reinvent mathematics even if it is not taught that • The mathematics of the mathematicians way (They may advocate “Learn first, cannot be conveyed by explanations or understand later”) definitions • Most don’t, they learn definitions and algorithms by heart à • Students have to go through a process of concept formation, within which operational – Problems with applications conceptions precede the structural – Problems with understanding conceptions – Math anxiety • expanding common sense (Freudenthal) 13 14 Students have to construct or Students have to construct or reinvent mathematics reinvent mathematics • Constructivism: ‘Students construct their own knowledge’ à (…) students construct their ways of knowing in even the most authoritarian of instructional situations. (Cobb, 1994, p. 4) Ø The question is not, Should the students construct?, but, What it is that we want the students to construct? or: “What do we want mathematics to be?” à People construct their own knowledge Mathematics as a human activity (Freudenthal) 15 16 Students have to construct or reinvent mathematics • Freudenthal: mathematics as a human activity: – Solving problems, – Looking for problems – Organizing subject matter Hans Freudenthal 17 18 3
Freudenthal (1971, 1973) RME THEORY ‘Mathematics as a human activity’ • Generalizing over local instruction theories Mathematics as the activity of mathematicians à domain-specific instruction theory for that involves solving problems, looking for realistic mathematics education, RME, problems, and organizing subject matter; Treffers (1987) – mathematical matter, or • Later, RME theory was recast in terms of – subject matter from reality design heuristics: (mathematizing: organizing subject matter from a – guided reinvention, mathematical point of view) Final stage: axiomatizing (“anti-didactical inversion”) – didactical phenomenology, To engage students in mathematics as an – emergent modeling activity ( Gravemeijer, 1989) Students have to be supported in inventing mathematics à guided reinvention 19 20 Guided Reinvention Addition and subtraction up to 20 • Identify starting points that are experientially Guided Guided reinvention reinvention real for the students • Starting points: informal solution procedures in • Identify the end points of the reinvention route contexts • doubles, five- and ten-referenced number Design heuristics: relations (fingers) 1. Look at the history of mathematics (potential • End points: flexibel use of number relations; conceptual barriers, dead ends, and breakthroughs) derived facts 2. Look for informal solutions that ‘anticipate’ more formal mathematical practices à potential reinvention route 21 22 Derived facts e.g 7+8=... Didactical Phenomenology • Phenomenology of mathematics: Analyze how mathematical ‘thought- 7+3=10 8=7+1 8=5+3 things’ (concepts, procedures, or tools) organize certain phenomena. 7+7=14 7=5+2 • Envision how a task setting may create the need to develop the intended thought thing à starting point for a reinvention process 7+8=5+5+2+ 3 7+8=14+1 7+8=10+5 24 4
Addition and subtraction up to 20 Emergent Modeling: Circumventing the learning paradox Didactical Didactical phenomenology phenomenology Combine, change, compare • External representations do not come with HF, not just structuring and combining sets of intrinsic meaning objects (cardinal) • “Learning paradox” (Bereiter, 1985): • Also counting events, measuring, …(ordinal) How is it possible to learn the symbolizations, • Coordinating cardinality and ordinality: which you need to come to grips with new mathematics, if you have to have mastered this • Problems that are stated cardinally (4 marbles new mathematics to be able to understand those and 3 marbles) are solved ordinally (counting symbolizations? on) 25 26 Emergent modeling Circumventing the learning paradox • Starting point: modeling (situated) strategies Ø The model derives its meaning from the context Emergent Emergent modeling modeling: : it refers to • A dynamic process in which symbolizations • Shift in attention; developing mathematical and meaning co-evolve ó history (Meira) relations • Modeling as a student activity Ø The model starts to derive its meaning from a in service of the learning process framework of mathematical relations; becomes a model for mathematical reasoning “A model of informal mathematical activity becomes a model for more formal mathematical reasoning” 27 28 Addition & subtraction up to 20 “ Model” • Actually a series of sub-models Emergent Modeling Emergent Modeling • From the perspective of the researcher/ • Doubles designer, the series of sub-models • Five & ten referenced constitute an overarching model. • Contexts, coins, ruler, .. double decker bus • This overarching model co-evolves with some new mathematical reality 10 7 3 .. 15 H H H H 5 8 12 .. • Arithmetic rack as a means of support; Ø Imagery: activities with new sub-models scaffolding &reasonig signify earlier activities with earlier sub- models for the students 29 5
Addition & subtraction up to 20 Nashville, with Cobb, Yackel, Whitenack Arithmetic rack as a means of Model scaffolding & communicating • Arithmetic rack , model of ways passengers are seated in the double decker bus • Students realize that 7=5+2 and 8=5+3, and visualize that on the arithmetic rack • Shift to model for reasoning about number relations • They realize that 5+5=10 , or 7+7=14 , or 8+8=16 Mathematics innovation in the Research on mathematics Netherlands innovation in the Netherlands • National surveys PPON • Curriculum innovation in the Netherlands; Little – decline in operations; standard algorithms room for teacher professionalization – improvement in number sense, global arithmetic • Math Wars • Curriculum innovation via textbooks • Research on – Subtraction up to 100 (Kraemer, 2011) • RME innovation in the Netherlands not enacted – Multiplication of fractions (Bruin-Muurling,2010 ) as intended (Gravemeijer et al, 1991) – Algebra (Stiphout, 2011 ) 35 36 6
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