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Van de Walle Chapter: Geometry Levels of Learning

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Task 1

Hmmm…There seems to be a pattern here. We get stuck because we are not presented with problems requiring higher levels of thinking, such as creating theorems, analyzing proofs, writing proofs and reasoning. I would guess that I am at level 1 according to the statistics from a study of high school students,  presented by Usiskin in 1982. Most elementary students are at the level 0, visual classification. Today, preschools are integrating properties of shapes and the statistics may be better for k-1. If this is typical of high school students, at what van Hiele level do most K-5 students fit?  I would think at level 1, because pre-school curriculum is stressing recognizing these shapes in the environment…

Task 2

            In video 1, Mason is asking convergent questions. Convergent questions:

  1. How do you know it’s a triangle?
  2. How do you tell someone what a triangle looks like?
  3. How could you tell someone the description of a square?

The student is functioning on Level 1 because she is basing her selection on appearance alone rather than on reasoning. She cannot describe the attributes of a triangle or square, such as the number of sides, the length of the sides in relationship to one another or the number of points. The evidence of that level is the learner said, “Because it looks like this,” several times, tracing the shape in the air with her finger.

            In Video 2, “How can you tell a triangle?” is a higher-level, convergent question. She changed the orientation and asked, “What if I turn it like this. Is it still a triangle? The learner said, “Cos it has a point and is straight at the bottom.” He is still at level one because he didn’t think the same shape was a triangle when it was turned. Why is “because it was upside down.” She built on that understanding and continued to turn the piece. “Is it were still a triangle?”  He made an adjustment to his schema of classification of properties in space when he said, “Upside down triangle.”

In Video 3, the learner didn’t know what to call an obtuse triangle and said, “Because it’s pointed like this. It’s straight across like this and it’ going diagonal to here.” He didn’t know what shape it is. The teacher asked questions to make him describe properties, “How many sides does it have?” He knew the shape had three sides, but he did not associate properties with the definition of a triangle.

In Video 4, the student was asked an open-ended question which allowed the questioner to discern what concept she knows about the meaning of congruent and similar. The next question allowed the questioner to determine what the learner thought was important about the characteristics of the triangle. Her answers show that the student is at Level 0 because she has grasped the concept that there are classes of shapes. She defines attributes and classifies by properties but does not analyze their relationships.

In Video 5, the teacher asked a level 1 questions that indicates classification by properties and 2 question, analysis, about the relationship (comparison) of the degrees to the angle. The learner is beyond visualization because the same angle is cut within circles of different size.

In Video 6, the interviewer asks the learner to sort and classify level 0. The level 1 question, (how you sorted), indicates level of reasoning, analysis by properties. Naming the shapes and defining is also a level 1 question. He knew the term 3-D but did not yet describe the difference between a square (2-D) and a cube (3-D).

In Video 7, the student sorted and classified by appearance, level 0. He related their shapes according to his experience, his name. He classifies visually and not by property. He indicates experience with tandems as he sees the star in the placement of the triangles. He did not recognize the difference between properties of the triangle, sub classification, level 2.


Written by autumnsfall2

November 11, 2010 at 3:16 pm

Posted in Uncategorized

One Response

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  1. Autumn,
    It was interesting to see the levels you put the different children at and yourself. I perceived myself as a Level 3. I was stumped by the last student in video 7. To me it didn’t seem like he verbalized anything about the shapes he just wrote his name and made a star that didn’t look like a star. Great observation and insight. Thanks, Angela


    November 14, 2010 at 2:29 am

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