ASSESSMENT AND CORRECTION MATHEMATICS EDUCATION:

ECED 4251

Rosalind Duplechain, PhD

University of West Georgia

College of Education

Geometry and Measurement

Module 11

Basic Structure of PPt

• Lecture (slides 3-10)– How the D&C Process works with Geometry and Measurement– Geometric Thinking and Measurement

• Application (slide 11)– See textbook for more examples of error patterns associated

with geometry and measurement.

• Other related ideas (slides12-13)– What you should student know?– Other ideas related to correction

• Homework - (See Course Calendar).

The D&C Process: Four Sub-processes

Correcting 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

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.

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

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.

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.”

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

Unit Conversions of Measurement

• Some common measurement relationships:– 1 Gallon = 4 quarts– 1 Foot = 12 inches– 1 Quart = 4 cups– 1 Yard = 3 feet

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

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.

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

Homework

• See Course Calendar.