Geometry: An Introduction

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Geometry An Introduction

• Sept 11 and 13, 2007

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van Hiele Levels

• Sequential development (like Piaget)
• Experience-based (not-like Piaget)
• Geometric experience advancement
• Both instruction and language should be
  developmentally appropriate
• Students typically learn algorithms/definitions
  without experiencing the concept

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Level 1 Visualization

• Sorting and classifying
• Examples
• Triangles vs. non-triangles
• Shapes with all straight sides vs others
• Constructions with
• Geoboards- Virtual geoboard
• Tessellations with pattern blocks
• Pentominoes

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Level 1 Visualization

• Shape Hunts
• Around the school find as many different
  two-dimensional shapes as possible
• Draw pictures, take photos with a digital camera

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Level 1 Visualization

• Location bingo
• Develops students sense of direction
• Every placement is dependent on the previous
  placement

• Put a square under the circle.
• Put a triangle to the right of the square.
• Put a kite above the triangle.
• Fill in the rest of your grid with shapes from
  auto shapes on the drawing tool bar.
• What would clues be for the rest of the shapes?

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Level 2 Analysis

• Shapes are a collection of properties
• Sorting by properties of 2-D or 3-D shapes
• Right triangles vs non-right triangles
• Quadrilateral sort

Squares
Rhombus
Rectangles
Trapezoids
Quadrilaterals
Kites
Parallelogram
What similarities do you notice??? What
differences?
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Level 2 Analysis

• Sorting by characteristics of 3-D shapes
• Faces, vertices, edges
• Shapes with 6 faces, 8 vertices and 12 edges

Shapes with a circular base
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Level 3 Informal Deduction (Abstraction)

• MDLs- minimal defining lists
• Minimal if anything is removed the definition is
  incorrect
• Defining- any object with this definition must be
  that shape

a square quadrilateral with 4 right angles and 4
congruent sides
1
3
2
triangle
with one right angle
a right triangle
1
2
Write an MDL for a rectangle and for a
parallelogram.
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Level 3 Informal Deduction (Abstraction)

• Shape decomposition (Draw lines on the shapes
  using the Drawing toolbar)
• Start with an isosceles triangle
• Make two shapes that have 7 total sides
• Make three shapes that have 11 total sides
• Start with a regular hexagon
• Make two shapes that have 8 total sides
• Make two shapes that have 9 total sides
• Start with a square
• Make three triangles- two of the three need to be
  congruent

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Level 3 Informal Deduction (Abstraction)

• Answer each question with
• always, sometimes or never
• Triangles have one right angle.
• Squares are rectangles.
• Quadrilaterals are rectangles.
• Parallelograms have a right angle.
• Trapezoids have a right angle.

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Beyond Level Three

• Level Four- Deduction
• Students can work through proofs, understand
  axioms, theories and definitions
• Level Five- Rigor
• Fluent at proofs, sophisticated geometry- proofs
  through the contrapositive

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van Hiele Levels and Cognitively-appropriate
Manipulative Levels
van Hiele level Manipulative level Examples
Level 1- Visualization Concrete Tangrams Soma cubes Pattern blocks Power solids Geoboards
Level 2- Analysis Representational Square color tiles for area and perimeter Paper representations
Level 3- Informal deduction Representational Symbolic Paper representations
Level 4 and beyond Symbolic Paper representations Geometric proofs
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Lets guess.

• What percentage of entering high school geometry
  students are Level

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Data-driven Instruction An Introduction

• September 11 and 13, 2007

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How would you grade

• A) 30 basic facts problems
• B) 10 computational problems involving the use of
  an algorithm
• C) A PWC task that involves computation and
  drawing a picture
• D) A Doing Mathematics task that requires
  identifying an approach, finding a solution, and
  explaining the approach and solution

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Basic Facts
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Computational work
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Procedures with Connections
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Doing Mathematics
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Various Types of Assessment

• Norm-referenced
• Criterion-referenced
• Performance-based

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Various Types of Assessment

• Norm-referenced
• Purpose
• Compare students performance to other students
• Format
• Multiple choice
• Scoring
• Compared to other students scores (percentiles)
• 99th percentile scored better than 99 of other
  students
• Can all students score in the 99th percentile?
• Examples
• Iowa Test of Basic Skills
• Stanford 9 Test

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Creating Assessments

• Norm-referenced
• National Assessment of Educational Progress (NAEP)

What would you classify this task as? How could
you make this PWC or DM?
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Variance Types of Assessments

• Criterion-referenced
• Purpose
• Measures students mastery of standards and
  criteria
• Format
• Typically multiple choice exams
• Scoring
• Compared to expected score
• E.g., 400 possible points, gt300 advanced,
  200-299 proficient
• Can all students be proficient?
• Examples
• State assessments
• North Carolina End of Grade (EOG) test

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Creating Assignments

• Criterion-referenced

Suggestions for making this PWC or DM?
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Various Types of Assessment

• Performance-based
• Purpose
• Assess students ability to perform or complete
  tasks related to concepts and skills
• Format
• Tasks- multiple-choice, short answer, multi-part
• Scoring
• Rubric based
• Scores are compared pre-test, post-test
  benchmarks
• Can all students score above the benchmark?
• Examples
• Illinois State Mathematics Exam
• Balanced Assessment in Mathematics- link

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Creating Assessments

• Performance-based
• Merging between higher-order thinking skills and
  content
• Actions- analyzing, evaluating, explaining,
  synthesizing
• North Carolina levels of thinking- link
• How do these levels of thinking align to
  constructivist beliefs about teaching and
  learning?

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Assessment Questions

• Multiple choice
• Question (stem)
• Answer choices
• Typically three to four choices
• In numerical or alphabetical order
• Wrong answers are plausible (common errors)
• Multiple-Multiple Choice
• Question
• Answer choices
• More than one choice can be correct

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Can multiple choice questions

• Assess students higher-order thinking skills?
• Why or why not?
• How can you compose multiple choice questions
  that extend beyond recall of basic knowledge?

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Assessing Students Work

• A brief introduction to rubrics
• Within each task select
• Components of the task
• For each component criteria of an exemplar answer
• Determine point values for each component
• Lets look at some examples