transforming my view about reasoning and proof …...journal for research in mathematics education,...
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Transforming my View about Reasoning and
Proof in Geometry • ZULFIYE ZEYBEK
• ENRIQUE GALINDO
• MARK CREAGER
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.
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.
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.
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.
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)
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.
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.
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.
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?
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
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.
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.
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?
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.
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?
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
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.
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.
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?
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?
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
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.
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?
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.
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?
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
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 two-
dimensional figures, describe a sequence that
exhibits the similarity between them.
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
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. 140-154). 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, 31-48.
O Chazan, D. (1993). High school geometry students’ justification for their
views of empirical evidence and mathematical proof. Educational Studies in
Mathematics, 24(4), 359-387.
O Knuth, E. J., Choppin, J. M., & Bieda, K. N. (2009). Examples and beyond.
Mathematics Teaching in the Middle School, 15 (4), 206-211.
O Martin, W. G. & Harel, G. (1989). Proof frames of preservice elementary
teachers. Journal for Research in Mathematics Education. 20 (1), 41-51.
Contact Information
O Zulfiye Zeybek [email protected]
O Enrique Galindo [email protected]
O Mark Creager [email protected]
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