## Archive for **November 2010**

## Lift the lesson out of the textbook and into the learner’s life

## Van de Walle Chapter: Geometry Levels of Learning

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:

- How do you know it’s a triangle?
- How do you tell someone what a triangle looks like?
- 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.

## 4th-grade math

**Direct instruction: my way or the highway**

I am observing a fourth-grade class, in the southeastern part of the county. The students are working on multiplying double digit multiplicands. The lessons start with reviews and then direct instruction. Guided practice follows. Then students work alone. The teacher uses an electronic board to explicitly model her strategy and expects students to follow her methods exactly. Last week, she said that she would take off points for skipping a step in her method. The strategies offered make sense and would work if they could be remembered, the first step in Bloom’s Revised Taxonomy. However, the students will be in trouble if they forget that strategy at the end of the year.

**Building experience with concrete objects**

About three bins of manipulative pieces sit above the lockers. I have seen one bin opened for a rainy day recess inside. The child with an IEP ventured inside and brought out some connecting blocks. I didn’t investigate whether he was solving math problems. I opened another bin looking for pieces for my lesson and they were still sealed in bags with the shipping statement on top. I’m only there twice a week; so I missed the array review for multiplication. Arrays with algorithm were posted on the wall.

**A problem was posed.**

Four students each had some colored pencils. Each student had 3 red and 2 blue pencils. Show 2 different ways to found out the total number of colored pencils. One student drew an array and then counted the squares one by one.

The problem reminded me of one of Dr. Piel’s DMI tasks to determine whether children have progressed from pre-operational to concrete stage of behavior, that indicates whether they have reached the stage of maturation to learn math. So I asked the teacher if I could take a quick poll. “Are there more red pencils or are there more pencils?” About a third of the class of 29 raised their indicating there were more red pencils. Their answer begs the question, “Are we expecting too much too soon of our students?” Rather, is the state requiring developmentally appropriate curriculum?

## Fraction Models

“Fraction Models” representation on the Illuminations website provides number sense through visual experiences with irrational fractions. The site also allows children to see that different representations of fractions may be written with different numbers and operations and still mean the same proportion of the whole. The experience with equivalent fractions will build schema for algebraic equations that require reducing rational numbers.

The ability to change the squares to show area, length, and width removes the mystery of irrational numbers. Children may see that larger numerators simply means complete wholes and part of a whole. At this time, fourth-grade class is working on multiplication of tens, hundreds and thousands. The wide range option on scale would allow children to explore the concept of decimal places. Seeing the area representations change in proportion to the number line slide would allow my children who are still counting on their fingers to multiply make a connection to skip counting. Playing with the pie models would allow my children to build visual understanding of simple rational fractions. The screen is somewhat busy. I predict that some guidance on looking at different parts of the screen to make comparisons would be required. Also, it was fun to enlarge the pie with the scale and not look at the corresponding percentage and decimals algorithms. (I have trouble focusing sometimes also.)

Our training has taught us that playing with models such as this is an excellent way to build experience before introducing new concepts. The experience also would allow children to make discoveries themselves, without having to teach them directly. We may reinforce their ideas through explanation and scaffold their understanding with questions. As I’ve already mentioned, I also see opportunity for reviewing concepts and clearing errors in thinking with the models.

## Equivalent fractions

Okay, maybe it’s late and I shouldn’t admit this; but the Illuminations Equivalent Fractions game was hard!. When I tried to do the math I’d get it wrong. When I let the picture guide me, then I got it right. I guess that is the idea. The concept is that fractions can have different denominators and number of parts, nominators, and still represent the same proportion of the whole. The game builds number sense of fractions. Students may visualize the parts to the whole and then see that the different numbers correspond to the same fraction. Errors in thinking, such as larger numbers mean larger pieces, may are be disproven with a visual representation that is then associated with the symbolic equivalent fractions on the right. The game alow so may help children see how factors can be used to reduce fractions to the lowest common denominator. The game would build upon an understanding of fairly dividing parts of a whole, like a sheet cake. Good night.