1. ASSESSMENT AND CORRECTION MATHEMATICS EDUCATION: ECED 4251 Rosalind Duplechain, PhD University of West Georgia College of Education Geometry and Measurement Module 11

2. Basic Structure of PPt Lecture (slides 3-10)Lecture –How the D&C Process works with Geometry and MeasurementD&C Process –Geometric Thinking and MeasurementGeometric Thinking Measurement Application (slide 11)Application –See textbook for more examples of error patterns associated with geometry and measurement. Other related ideas (slides12-13)Other related ideas –What you should student know? –Other ideas related to correction Homework - (See Course Calendar).Homework

3. The D&C Process: Four Sub-processes DIAGNOSE errors in pre-data GOAL: Find out exactly what student is doing to get problem wrong so error can be corrected. CORRECT errors GOAL: Use specified correction steps to correct problems that are wrong so student can be successful. EVALUATE correction strategy GOAL: to determine if errors have been corrected by administering a post-test. REFLECT on post-test score GOAL: to determine next step.

4. CorrectingCorrecting Algorithm Errors… Conceptual Only – Using manipulatives/drawings only, show and talk aloud while solving problem. Emphasize ideas related to student’s error. Repeat until student can do alone. Teacher Guided Experiences Intermediate – Using manipulatives/drawings, show and talk aloud while solving problem. Also, teach and show that the algorithm is a step-by-step record of what is being done with manipulatives. Emphasize ideas related to student’s error. Repeat until student can do alone. Teacher Guided Experiences Procedural Only – Using only the algorithm, show and talk aloud while solving problem. Emphasize ideas related to student’s error. Repeat until student can do alone. Teacher Guided Experiences Independent Practice (procedural) – Provide problems for student to solve alone, using only the algorithms. Once practice is completed, teacher checks work. If work earns <85%, teacher repeats correction cycle beginning on either the intermediate level or the procedural only level. Student-only practice Teacher feedback

5. 5 Geometric Thinking… Pierre and Dina van Hiele, a Dutch husband-and-wife team of mathematicians investigated and described how children develop an understanding of Euclidean forms for many years. They concluded that children pass through five stages of geometric understanding (Van de Walle et al., 2010, pp. 400 - 404), irrespective of age (p. 404): –Stage 0: Visualization –Stage 1: Analysis –Stage 2: Informal Deduction –Stage 3: Deduction –Stage 4: Rigor “Most students in Pre-K through grade 8 will fall within the” first three stages (p. 404). –This suggests that Pre-K to 5, our certification range, would need to focus on the first two stages: Stage 0 and Stage 1.

6. Geometric Thinking van Hiele – Levels of Geometric Thinking –Level 0: Visualization –Level 1: Analysis –Level 2: Informal Deduction/Abstraction –Level 3: Deduction –Level 4: Rigor For specific information: See Van de Walle (2010), pp. 399-404 Teaching and implications, pp. 404 - 433 Google van Hiele or Geometric Thinking

7. Geometric Thinking… Level 0: Visualization (Van de Walle, 2007, pp. 413- 414) –Recognize, sort, and classify shapes based on global visual characteristics, appearances. “A square is a square because it looks like a square.” “If you turn a square and make a diamond, it’s not a square anymore.” –Because appearance is dominant at this level, appearances can overpower properties of a shape.

8. Geometric Thinking Level 1: Analysis (Van de Walle, 2007, p. 414) –Recognize, sort, and classify shapes based on their properties (number of sides/faces and edges and the size of angles). “A square is a square because it has four equal sides and four equal angles.” “This is a right triangle because it has three sides and three angles and one of those angles is a right angle.” Because an understanding of how properties of shapes relate is lacking, each property is understood in isolation of other properties. –“A square is not a rectangle.”squarerectangle

9. Measurement Area - “The measure of a bounded region on a plane or on the surface of a solid” (Webster 1996, p. 72). –Bounded region = inside Perimeter - “The outer boundary of a figure or area; circumference” (Webster, 1996, p. 1004). –Outer boundary = outline

10. Unit Conversions of MeasurementMeasurement Some common measurement relationships: –1 Gallon = 4 quarts –1 Foot = 12 inches –1 Quart = 4 cups –1 Yard = 3 feet

11. Application Let’s apply what we’ve learned today about the D&C Process to violations of algorithms, and in particular to Geometry and Measurement. –Martha –Oliver –Denny –Margaret

12. What Should Student Know? Determining what a student should know about solving these types of problems is very similar to analyzing student work for their errors. Work each problem on the pretest and compare student’s work (step by step) and answer to your work (step by step) and answer. –For any problem that is wrong, ask yourself: What exactly is student doing to get this work (step by step) and this answer? Making this kind of comparison enables you to do two things: –1) Develop a checklist for these types of problems. Then you can use this checklist to help you diagnose future errors with these types of problems. –2) Tells you exactly what the student is doing to get the problem wrong. Then you can devise a strategy for correcting his/her error.

13. Other ideas related to correction For numerous activities that can be turned into learning center activities or that can be tweaked to fit into the correction process discussed in this course, refer to Van de Walle (2010), pp. 404-433. –Level 0 Thinkers –Level 1 Thinkers –Level 2 Thinkers

14. Homework See Course Calendar.