Title: Geometry: An Introduction
1Geometry An Introduction
2van 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
3Level 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
4Level 1 Visualization
- Shape Hunts
- Around the school find as many different
two-dimensional shapes as possible - Draw pictures, take photos with a digital camera
5Level 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?
6Level 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?
7Level 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
8Level 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.
9Level 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
10Level 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.
11Beyond 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
12van 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
13Lets guess.
- What percentage of entering high school geometry
students are Level
14Data-driven Instruction An Introduction
- September 11 and 13, 2007
15How 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
16Basic Facts
17Computational work
18Procedures with Connections
19Doing Mathematics
20Various Types of Assessment
- Norm-referenced
- Criterion-referenced
- Performance-based
21Various 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
22Creating Assessments
- Norm-referenced
- National Assessment of Educational Progress (NAEP)
What would you classify this task as? How could
you make this PWC or DM?
23Variance 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
24Creating Assignments
Suggestions for making this PWC or DM?
25Various 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
26Creating 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?
27Assessment 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
28Can 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?
29Assessing 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