Geometry: An Introduction

1. Geometry: An Introduction Sept 11 and 13, 2007

2. 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

3. 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

4. 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

5. 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?

6. 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?

7. 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

8. Level 3: Informal Deduction (Abstraction) • MDL’s- “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 2 3 triangle with one right angle a right triangle = 1 2 Write an MDL for a rectangle and for a parallelogram.

9. 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

10. 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.

11. 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”

12. van Hiele Levels and Cognitively-appropriate Manipulative Levels

13. Let’s guess…. • What percentage of entering high school geometry students are Level

14. Data-driven Instruction: An Introduction September 11 and 13, 2007

15. 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

16. Basic Facts

17. Computational work

18. Procedures with Connections

19. Doing Mathematics

20. Various Types of Assessment • Norm-referenced • Criterion-referenced • Performance-based

21. 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

22. 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?

23. 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, >300 = advanced, 200-299 = proficient • Can all students be proficient? • Examples • State assessments • North Carolina End of Grade (EOG) test

24. Creating Assignments • Criterion-referenced Suggestions for making this PWC or DM?

25. 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

26. 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?

27. 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

28. 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?

29. 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 • Let’s look at some examples…