knowledge for teaching geometry - fostering geometric thinking：教学知识的几何几何思维的培养

A Habits of Mind Framework Supports Knowledge for Teaching Geometry,Daniel HeckHorizon Research, Inc.Chapel Hill, North Carolina,Rachel Wing & Mark DriscollEducation Development CenterNewton, Massachusetts,Research reported here is support by the National Science Foundation under Grant ESI-0353409,Workshop Overview,Geometric Habits of Mind framework (G-HOMs)Knowledge for Teaching GeometryHow G-HOMs and Fostering Geometric Thinking (FGT) materials support 2 types of Knowledge for Teaching Geometry Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry Questions,Workshop Overview,Fostering Geometric ThinkingGeometric Habits of Mind framework (G-HOMs)Knowledge for Teaching GeometryHow G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry How FGT field test addresses impact of G-HOM-based materials on 2 types of Knowledge for Teaching Geometry Questions,Fostering Geometric Thinking (FGT),Identify productive ways of thinking in geometry (G-HOMs)Create professional development materials based on G-HOMs40 hours 20 two-hour sessionsGroup-study materialsStructured Exploration Process (Kelemanik et al. 1997) Stage 1: Doing mathematics Stage 2: Reflecting on the mathematicsStage 3: Collecting student workStage 4: Analyzing student workStage 5: Reflecting on students thinkingG-HOMs framework is a lens for analysis,FGT is support by the National Science Foundation under Grant ESI-0353409,Fostering Geometric Thinking (FGT),Field TestResearch Questions: In what ways does the use of professional development materials that target students geometric ways of thinking Q1: increase teachers content knowledge and understanding of student thinking in geometry and measurement? Q2: affect instructional practice in geometry? Knowledge for Teaching Geometry,Workshop Overview,Geometric Habits of Mind framework (G-HOMs)Knowledge for Teaching GeometryHow G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry Questions,Geometric Habits of Mind (G-HOMs),Seeking Relationships Checking Effects of Transformations Generalizing Geometric IdeasBalancing Exploration with Deduction,Seeking Relationship,Actively looking for relationships within and between geometric figures, in one, two, and three dimensions. Relationships can be between/among: figures whole figures and their parts concepts,Which two make the best pair?,Internal Questions:“How are these figures alike?” “In how many ways are they alike?” “How are these figures different?” “What would I have to do to this object to make it like that object?”,Which two make the best pair?,Seeking Relationships Checking Effects of Transformations Generalizing Geometric IdeasBalancing Exploration with Deduction,Geometric Habits of Mind (G-HOMs),Checking Effects of Transformations,Analyzing which attributes of a figure remain invariant and which change when the figure is transformed in some way.Attributes of the figure that may be affected by transformations include: Orientation Location Area, perimeter and volume Side lengths and ratio of side lengths Angles,A squares diagonals always intersect at 90-degree angles. Is this true of a rhombus?,Internal Questions:What changes? Why? What stays the same? Why?”,A squares diagonals always intersect at 90-degree angles. Is this true of a rhombus?,Seeking Relationships Checking Effects of Transformations Generalizing Geometric IdeasBalancing Exploration with Deduction,Geometric Habits of Mind (G-HOMs),Generalizing Geometric Ideas,Wanting to understand the always and the every related to geometric concepts and procedures.Generalizing progresses through stages: Conjecturing about the “always” & “every” Testing the conjecture Drawing a conclusion about the conjecture Making a convincing argument,Connecting the midpoints of this quadrilateral created a parallelogram. What other quadrilaterals will this work for?,Internal Questions:Does this happen in every case? Why would this happen in every case? Can I think of examples when this is not true? Would this apply in other dimensions?,Connecting the midpoints of this quadrilateral created a parallelogram. What other quadrilaterals will this work for?,Seeking Relationships Checking Effects of Transformations Generalizing Geometric IdeasBalancing Exploration with Deduction,Geometric Habits of Mind (G-HOMs),Balancing Exploration with Deduction,An iterative and cumulative process that alternates between: exploring structured by one or more explicit limitation/restriction and taking stock of what is being learned through the exploration,Is it possible to draw a quadrilateral that has exactly 2 right angles and no parallel lines?,Internal Questions:“What happens if?”“What did that action tell me?”,Workshop Overview,Geometric Habits of Mind framework (G-HOMs)Knowledge for Teaching GeometryHow G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry How FGT field test addresses impact of G-HOM-based materials on 2 types of Knowledge for Teaching Geometry Questions,Knowledge for Teaching Geometry,Influenced byOur work developing the FGT materials and measures for the field testPrevious work on Connected Geometry and Fostering Algebraic Thinking ToolkitVan Hiele (1959; 1986) model of cognitive development in geometryBall et al.s (2001; 2003) work on Content Knowledge for Teaching Mathematics,Knowledge for Teaching Geometry,Knowledge for Teaching Geometry,Knowledge for Teaching Geometry,Workshop Overview,Geometric Habits of Mind framework (G-HOMs)Knowledge for Teaching GeometryHow G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry Questions,Knowledge of productive ways of thinking in geometry,FGT G-HOMs frameworkLanguage for thinking and talking about geometric thinking,“If we could find the area of all the pieces and then take the square root”,Balancing Exploration with Deduction,Knowledge of productive ways of thinking in geometry,FGT G-HOMs frameworkLanguage for thinking and talking about geometric thinkingIncrease use of G-HOMs in problem solving,Perimeter of A = Perimeter of B,A B,Perimeter of A = Perimeter of B = Perimeter of C,C,“Will the perimeter always be the same no matter how many mounds?”,“Why is this happening?”,Knowledge for Teaching Geometry,Knowledge of how to promote productive, and potentially productive, ways of geometric thinking in students.,FGT G-HOMs internal questions and emphasis on potentialQuestion students in a way that encourages G-HOMS,Tangrams and Seeking Relationships,S: If you take the whole thingthere are 7 different parts of this (rectangle theyve built). Some of them may be different shapes, different sizes, but theyre all equal to 1 part of this shape.T: So, would you say then that theyre all 1/7th of the shape?S: Yeahwell, nokind of cause like this (points to large triangle) could be 1/7th or like if you had a greater denominator it could be greater than 1/7th cause its bigger than everything else.T: So theyre not the same size pieces, but there are 7 pieces.S: Yeah. And theyre all 1/7th of this thing, even though some are smaller than the others.T: So this (small triangle) is 1/7th and this (large triangle) is 1/7th?S: Yeah, because like this (small triangle) has to be at least 1/7th because you dont really go into decimals and that stuff.,Knowledge of how to promote productive, and potentially productive, ways of geometric thinking in students.,FGT G-HOMs internal questions and emphasis on potentialQuestion students in a way that encourages G-HOMS Capitalize on potential,“Bs area is bigger”,A B,“Area of A = p x 3 x 3; Area of B = 2 x 6 x 6”,A B,But“Bs area is bigger”,A B,Workshop Overview,Geometric Habits of Mind framework (G-HOMs)Knowledge for Teaching GeometryHow G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry Questions,Knowledge for Teaching Geometry in the Context of FGT Materials,Explore FGT problem: Dissecting ShapesReflect on G-HOMs ability to capture productive ways of thinking in geometryVideo of students working on Dissecting ShapesDiscuss how the G-HOMs framework affected how you analyzed students thinking in the videoHow FGT would use this artifact with teachers to promote pr