Transforming my View about Reasoning and Proof in Geometry ZULFIYE ZEYBEK ENRIQUE GALINDO MARK CREAGER

Transcription

1 Transforming my View about Reasoning and Proof in Geometry ZULFIYE ZEYBEK ENRIQUE GALINDO MARK CREAGER

2 Objectives O To reflect on how to transform our teaching by engaging students in learning proofs in geometry. O To explore and illustrate several ways to support students to learn to conjecture, justify, and develop convincing arguments. O To explore different methods including hands-on strategies, mental experiments, dynamic manipulation on the computer, and building logical arguments.

3 van Hiele Model The theory developed by P. M. van Hiele and D. van Hiele-Geldof suggests that reasoning in geometry develops through a sequence of five levels, beginning with visual identification and ending with rigorous mathematical thinking (Burger & Culpepper, 1993; Burger & Shaughnessy, 1986 ) Today the van Hiele theory has become very influential in the geometry curriculum in the U.S.

4 There are five levels O Visualization (Level 0) O Analysis (Level 1) O Informal Deduction (Level 2) O Deduction (Level 3) O Rigor (Level 4) Levels are sequential and not age dependent.

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6 Level 0 (Visualization): Students reason about basic geometric concepts, such as simple shapes, primarily by means of visual considerations of the concept as a whole without explicit regard to properties of its components. Level 1 (Analysis): Students reason about geometric concepts by means of an informal analysis of component parts and attributes. Necessary properties of the concept are established. Level 2 (Abstraction): Students logically orders the properties of concepts, forms abstract definitions, and can distinguish between the necessity and sufficiency of a set of properties in determining a concept. Level 3 (Deduction): Students reason formally within the context of a mathematical system, complete with undefined terms, axioms, an underlying logical system, definitions, and theorems. Level 4 (Rigor): Students can compare systems based on different axioms and can study various geometries in the absence of concrete models.

7 O Many students accept inductive arguments as valid mathematical proof (Chazan, 1993; Knuth, Chopin, & Bieda, 2009; Martin & Harel, 1989) O One way of helping students develop an understanding about these limitations is to give them tasks in which generalizing from several cases does not lead to a correct generalization ( Knuth, Chopin, & Bieda, 2009)

8 1. Area of Quadrilaterals In this activity we will work with quadrilaterals. We will explore whether In a quadrilateral at least one diagonal cuts the area in half. We will check this conjecture with specific quadrilaterals such as Kites, Squares. We will explore what happen when we do this with different quadrilaterals.

9 1. Area of Quadrilaterals 1. Consider a kite. Does this conjecture hold true for a kite? 2. What if the quadrilateral is a rhombus? 3. What if the quadrilateral is a square? 4. Rectangle? Parallelogram? Work on the exploration and write down points to take away.

10 1. Area of Quadrilaterals 1. Will this conjecture hold true for all quadrilaterals? 2. What if the quadrilateral is a trapezoid? 3. What if the quadrilateral is a generic quadrilateral? Work on the exploration and write down points to take away.

11 Area of Quadrilaterals O Any thoughts about Reasoning and Proof? O Any thoughts about strategies students may use depending on van Hiele Levels? O What are some connections to CCSSM?

12 Area of Quadrilaterals O Reasoning and Proof O Easy to visualize outcome from hands-on activity O Important to realize generalizing from specific cases will not always lead to correct answer O Students may use deductive arguments to justify outcomes

13 Area of Quadrilaterals O Strategies and van Hiele Levels O Different strategies depending on level O Connections to CCSSM? O CCSS.Math.Content.6.G.A.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

14 O Although examples are not sufficient to prove the truth of a statement, they do play an important role in making conjectures. O Presenting students with a variety of tasks in which examples play different roles can help students develop an appreciation and understanding for their use as means of justification.

15 2. Reflecting a Triangle across its Sides In this activity we will work with scalene triangles. We will make a scalene obtuse or acute triangle. We will explore what shape is formed when the triangle is reflected across its sides. What shapes are produced when we do this for different triangles?

16 2. Reflecting a Triangle across its Sides 1. Consider an obtuse or acute scalene triangle. Reflect it across its sides. What shape is formed? 2. What if the triangle is an isosceles? 3. What if the triangle is an equilateral? Work on the exploration and write down points to take away.

17 Reflecting a Triangle across its Sides O Any thoughts about Reasoning and Proof? O Any thoughts about strategies students may use depending on van Hiele Levels? O What are some connections to CCSSM?

18 Reflecting a Triangle across its Sides O Reasoning and Proof O Easy to visualize outcome from hands-on activity O Students may use deductive arguments to justify outcomes

19 Reflecting a Triangle across its Sides O Strategies and van Hiele Levels O Different strategies depending on level O Connections to CCSSM? O CCSS.Math.Content.8.G.A.1 : Verify experimentally the properties of rotations, reflections, and translations. O CCSS.Math.Content.8.G.A.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

20 3. Rotating Triangles In this activity we will rotate shapes and deduce properties of the new shapes formed using knowledge that the images are congruent to the original triangles.

21 3. Rotating Triangles Consider a right triangle. Rotate it 180 degrees through the midpoint of it s hypotenuse. What quadrilateral will be formed? Do you believe the same type of quadrilateral will be formed if other triangles are rotated?

22 Rotating Triangles O Any thoughts about Reasoning and Proof? O Any thoughts about strategies students may use depending on van Hiele Levels? O What are some connections to CCSSM?

23 Rotating Triangles O Reasoning and Proof O Easy to visualize outcome from hands-on activity O Students may use deductive arguments to justify outcomes O Strategies and van Hiele Levels O Different strategies depending on level

24 Rotating Triangles O Connections to CCSSM? O CCSS.Math.Content.8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations: O CCSS.Math.Content.8.G.A.1a Lines are taken to lines, and line segments to line segments of the same length. O CCSS.Math.Content.8.G.A.1b Angles are taken to angles of the same measure. O CCSS.Math.Content.8.G.A.1c Parallel lines are taken to parallel lines.

25 4. Connecting Consecutive Midpoints of the Sides of Quadrilaterals In this activity we will work with quadrilaterals. We will make a shape by joining consecutive midpoints of the sides. We will explore what shape is formed. What shapes are produced when we do this for different quadrilaterals?

26 4. Connecting Consecutive Midpoints of the Sides of Quadrilaterals 1. Consider a generic quadrilateral. Make a shape by joining consecutive midpoints of its sides. What shape is formed? 2. What if the quadrilateral is a rectangle? 3. What if the quadrilateral is a kite? 4. Rhombus? Square? Work on the exploration and write down points to take away.

27 Connecting Consecutive Midpoints of the Sides of Quadrilaterals O Any thoughts about Reasoning and Proof? O Any thoughts about strategies students may use depending on van Hiele Levels? O What are some connections to CCSSM?

28 Connecting Consecutive Midpoints of the Sides of Quadrilaterals O Reasoning and Proof O Easy to visualize outcome from hands-on activity O Students may use deductive arguments to justify outcomes

29 Connecting Consecutive Midpoints of the Sides of Quadrilaterals O Strategies and van Hiele Levels O Different strategies depending on level O Connections to CCSSM? O CCSS.Math.Content.7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. O CCSS.Math.Content.8.G.A. Given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.

30 Questions and Discussion Points to take away about: O Reasoning and Proof O Van Hiele Levels O Connections to CCSSM O Standards for Mathematical Practices

31 References O Burger, W. F., & Culpepper, B. (1993, February). Restructuring geometry. In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (p ). Indianapolis, IN: MacMillan Reference Books. O Burger, W., & Shaughnessy, M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17, O O O Chazan, D. (1993). High school geometry students justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), Knuth, E. J., Choppin, J. M., & Bieda, K. N. (2009). Examples and beyond. Mathematics Teaching in the Middle School, 15 (4), Martin, W. G. & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education. 20 (1),

32 Contact Information O Zulfiye Zeybek zzeybek@indiana.edu O Enrique Galindo egalindo@indiana.edu O Mark Creager macreage@indiana.edu

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