6-C-2: Higher Level Thinking for the Pythagorean Theorem
Level 0: Concrete, identifies, names, compares, and operates on geometric figures. This level is most comparable to Bloom’s Knowledge and Comprehension level. Students know the basics about geometric figures. They can define the appropriate material at the time, but cannot use ideas associated with the subject to relate them other ideas. They can work on assigned problems and example what they did.
Level 1: Analysis, analyzes figures in terms of their attributes and relationships among attributes and discovers properties and rules through observation. This is closely comparable to Bloom’s Application Level. Students can recognize what methods to use and use the method to solve problems. They can apply ideas and concepts to new situations.
Level 2: Informal deduction, discovers and formulates generalizations about previously learned properties and rules and develops informal arguments to show her or his generalizations to be true. This level is comparable to Bloom’s Analysis level. Students can explain why the solution process works.
Level 3: Deduction, proves theorems deductively and understands the structure of the geometric system. This level is comparable to Bloom’s Synthesis level. Students recognize new problems and develop new tools to solve them. They can create their own plans, models, and/or hypotheses for constructing solutions to problems.
Level 4: Rigor, establishes theorems in different postulational systems and compares and analyzes the systems. This level is comparable to the evaluation level. Students can create a variety of ways to solve the problem and then, based on established criteria, select the solution best suited for the problem. Students will have the ability to appreciate the value of ideas and concepts using appropriate criteria.
How can you use the van Hiele levels to help students learn mathematics?
Since the Van Hiele model is such a natural process in mathematical maturity, the teacher will need to recognize the level the student is currently at. The teacher will need to pay attention to questioning that will help the student understand and make a natural progression from one level to the next level. The teacher will need to understand that not every student will be at the same level, but can use Bloom’s questioning to fulfill all levels so all students can be successful.
Develop additional questions that you could ask students if you were to use this lesson in your classroom.
Illustrate the ways to increase the area, but the perimeter stays the same.
Compare the different perimeters you can have with an area of 16 units². Compare and contrast the area and perimeter.
Create another figure with the same area and perimeter as the original figure.
Create an object with an area of 12 units². What is the greatest possible and least possible perimeter for an object of 12 units²
Access if you can arrange the tiles to form an odd-numbered perimeter. Explain if it’s possible or not possible.