## Archive for the ‘**Uncategorized**’ Category

## Head Start Cut 5 Percent

CBS reports on the federal budget effect on Head Start, the children who benefit from it’s services, and the mothers who work in their communities.

http://m.cbsnews.com/fullstory.rbml?catid=57589532&feed_id=1&videofeed=37

## 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?

## 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.

## Foundation for understanding algebra

Reading the article “Children’s Understanding of Equality: Foundation of Algebra, by Falkner, Levl and Carpenter in *Teaching Mathematics*, I was surprised that children see the equal sign as an operation rather than a relationship. I recall hearing the words “is equal to” or “the same as” when reading math problems. Also, I was surprised by that 84 percent of the 145 sixth graders who did not correctly compute 8 + 4 = ? + 5. I found that the small sample of one of three fourth-graders in my clinical class got the addition problem wrong. Two of three got the multiplication problem 9 x 9 = ? + 63 correct; but they did not make the connection that 63 is a multiple of 9. They just finished factoring. The third child is working on the tens number system, and number sense. We can help children build number sense by given the equations context and working with a manipulative like a set of 100s blocks.

## Addressing Student Errors

The student errors in multiplying by double-digit multiplicands and multipliers were not basic fact errors. The student knew that 2*4 = 8 and 3*1 = 3. They did not understand the form of columns representing decimal places in relationship to multiplication. They dutifully followed the rules they had learned in addition. The top-to-bottom procedure related to their experience with columns in adding double-digit numbers. I imagine the student thinking, “Why would anyone suddenly cross over to the ones column to multiply 3 * 4 in “32 * 14 = 38?” I assessed a fourth-grader who obviously had been taught the term “cross over” to multiply this type of problem. She dutifully used the term “cross over” as she placed the products in locations only an artist would appreciate. An array breaks the double-digit numbers into parts that can be kept in the proper decimal columns. The transitional strategy shows the student where numbers originate and their proper decimal place in the product. The relationships of the numbers become clearer to the students with visible scaffolding.