Piper's Lesson Study Notes

Lesson Study Notes (9/29/09)

Lesson Action

Introduction

Most of my notes in this section catalog the exchange between the class and the teacher. Note: not all of the exchanges of the lesson were recorded on here, hence some of the gaps later on in the lesson when the teacher seems to “suddenly jump” to the journal. There was conversation before this and more conversation about the math journal, but I did not have the chance to record it.

T: “What does area mean to you?”

• “Space”
• “area….like”
• “like this is my area, space”
• “It’s like squares”

T: “What does perimeter mean to you?”

• “measure the outside of an object”
• “the outline of the notebook”
• “perimeter is the outline of an object”

T: “Where else have you seen area and perimeter in the real world?”

• “if you’re getting enough aluminum foil to cover food before you put it in the oven”
• Tile a floor, need to know how big tiles are
• “perimeter is by the edge of the desk. Area is inside it”
• “on a construction site”

T: (Noted the importance of knowing area and perimeter of irregular shapes)

T: “How would we measure perimeter (of the rectangle on the ELMO) with pencils?”

• “With a ruler?”

T: “We can’t because we don’t have one.”

• “We can put pencils on the shape”

T: (puts one pencil on the top of the rectangle line) “What’s wrong with this picture?”

• “The pencil’s too long.”

T: “What do we do?”

• “Cut it.”

T: “How about estimating?”

• (Voiced assent)

T: “What about area, how could we find it, with these hexagons?”

• (incoherent student response)

T: “So we’re going to fill the shape.”

• “You’re not filling all the spaces!”

T: “What are the problems?”

• “Spaces”

T: “What do we have to do?”

• “Estimate”

T: “How many do we know for sure?”

• 5

T: “How many can we estimate?”

• 2 and ½

T: “So the answer is 7 and ½ what? Pencils?”

• No! Hexagons!

T: (instructs class to open their math journals and set them up for the activity)

**Students are AMAZED by the ELMO**

T: “Was it (the measurements with pencils and hexagons) accurate?”

• No, because we had to estimate.

T: “Why did we estimate area?”

• Because there were hexagons that went over the shape and open spaces

T: (Gets ready for the transition into work time) “Should we hear any talking at this time?”

• “Yeah, about math.”

T: “Right.”

During

For this section, I first looked around the room and focused on which manipulatives were picked first. Then, I keyed in on five students in particular, whom I assigned either a number or letter to, within my notes.

Initial reaction: First manipulatives chosen: tape measure, ruler, paperclips.

Student Q

• Used paperclips to get the leaf’s perimeter, and added up the number of half paperclips.
• Said this was not accurate, because “I had to estimate them because they’re either too big or too small.”
• Then, he used triangles for area measurement.
• “I’m measuring the inside with the triangles.”
• “There’s not enough triangles, I’m gunna use squares.”

Student 4

• (a fourth grader)
• “The ruler is too hard to use”
• Then, he went onto using the triangles. He first traced the triangles and then he counted them.
• Student did uncover a problem with measuring the stem “because you can’t trace the triangle on the stem, it doesn’t work.”
• The student then went onto explain his tracing and counting method to another student.

Student 3

• “It (the ruler) is too hard to use.”
• This student was very frustrated with the activity. She was also very unmotivated and off-task constantly.
• Student 3 is a fourth grader

Student 5

• She was done with the first two measures of area and perimeter very quickly, and she moved onto measuring area and perimeter with a third method.
• As she used the circles to measure area, she overlapped many of them.
• She noted that the circles were “not accurate, they’re too big, because their fatter than the stem.”
• She then went onto measure the perimeter by using the lengths of the felt squares.
• Additionally, she used the triangles and squares to measure both perimeter and area.

Student 7

• Unbent the paperclips to wrap them around the lead and measure perimeter.

Post-Activity Class Discussion

In this section, I made sure to note, what I felt were the most important exchanges between students and teacher in the discussion. Again, this is not the whole discussion, just pieces of it.

T: We’re going to share how some people got their area and perimeter measures. “If you think something happened that you don’t agree with, then feel free to speak up, respectfully.”

• Quiet student commented on her method to wrap the string around the leaf (mostly incoherent from where I was sitting)

T: “Did anyone find a good way to find area?”

• “I traced the leaf on grid paper and I counted the squares.”

T: “Squares to find perimeter? How did you do that? Was it accurate?”

• “I lined the squares on the perimeter. No, because they were overlapping.”

T: (student name)?

• “ I hooked the paperclips together and measured around the perimeter”

T: “Was that accurate?”

• (incoherent response)

T: “Anyone else use paperclips to find area, or perimeter? Were they accurate?”

• I did, and it wasn’t accurate.

Post-Lesson Debriefing Session

I took notes on our initial ideas of improvements that can be made to the lesson, which we discussed in the staff lounge.

• More description of using units that are non-standard
• More scaffolding needed for the meaning of “accurate”
• More time should be allowed
• Maybe the lesson could be broken down into two days, the first focusing on perimeter, and the second area allowing them access to the same manipulatives on both days.
• Maybe add preassessment before the lesson so we can see the growth of knowledge as a result of their exploration.

Readings

Area
~Area is defined as the amount of space covered by the inside boundries of a two demensional figure and the formula is A=LXW

~Finding the area of irregular shapes
~Creative ways to come up with answers
~Explore own ways to find area of irregular shapes on grid paper (come up with their own strategies). Then share in pair share or classroom discussion, or they could write in their math journals as writing across the curriculum.

~Might be beneficial to discuss what units of measure are used in area. To piggy back that idea, what does a squared centimeter look like, etc. Then they could explore areas and their appropriate units of measure (EX. Not going to measure a soccer field in squared inches)

~Trace their hand on grid paper then find the area of their hand.

~In everything that they are doing they should reflect in their math journals

Which Measure is It
~Can two shapes have same area but different perimeters? Explain?

~To show understanding students need to construct visual representations of figures, generate word problemss, and justify properties.

Quick Lesson Idea

Hey guys!

I just had a quick lesson idea, wondering if anyone else might like it. Basically the kids at mine, Brad's, and Andy's school are OBSESSED with four-square. Maybe after we do the initial explanation/definition section about perimeter and area (if that's what we decide to do) we can take them outside and have them find the perimeter and area of the 4-square game board? Just an idea for an application of area and perimeter that may engage the students.

Piper

Brad's Readings

By The Unit or Square Unit

Intro:

• This article poses a very interesting question to the students. They ask "For the school carnival, sponsors will pay for advertising their products on signs. How can the school make the most money: by charging the sponsors by the square unit of area or by measuring the perimeter of the signs and charging by the linear unit?"

Why Area and Perimeter are Difficult?

• Students put borders on things or cover something with paint. However, the true meaning of common terms are not understood by students.
• Students struggle to grasp the non constant relationship between perimeter and area.

Class 1

• Presented the question above to students with examples
• Examples all had area of 100, but perimeters varied
• Students also talked about units used to measure different things correctly

Class 2

• Measured different objects irregular and regular to compare perimeter and area numbers
• Students realized the concept of objects more like a square have less perimeter

Class 3

• Much the same as the other two classes
• Proposed the question of how to find the area and perimeter of a shoe print accurately.

Importance of Lessons

• The longer the shape the greater the perimeter
• The area can be the same but the perimeter different
• The more balanced the sides the bigger the area.

Summary Quotes

"Some procedural knowledge is always necessary to do mathematics efficiently, and short ways of computing perimeter and area by adding or multiplying are worth knowing. However, children also need to develop conceptual knowledge of the fundamental properties of these topics."

"Some educators might argue that discovering the relationship between perimeter and area is an extension activity - a pursuit for more advanced argue that such relationships are essential to mathematics. All children should have the opportunity to work with problems for which the answer is not immediately evident..."

Why Do Some Children Have Trouble Learning Measurement Concepts?

This article was mainly written about 1st graders:

• It found that 1st graders can learn a great deal about measurement, though Piaget's theories don't suggest so.
• Most benefit comes from concrete instructions and evaluation.
• Experience, experience, experience... is most beneficial

Integrating Measurement Across the Curriculum

A big thing from this article I took was the fact that measurement is in every subject and tested for. However, the other subjects have some sort of instruction and measurement is just expected to be understood. The article recommends the following which will probably benefit us the most.

• Students should have varied experience with measurement tools
• Integrating across subjects
• Students should communicate measure processes and results
• take advantage of real world connections
• take advantage of special occasions ex. olympics

Piper's Readings

Break the Boundaries

Introduction:

• Area has always been a topic that has been tough for students to deal with.
• Many students only have a superficial understanding of area
• past instruction has only focused on formula and not learning the actual concept of area.
• Students need experience with:

1. partitioning a region with a two-dimensional unit of measure
2. iterating a unit to cover the region without space or overlap
3. exploring conversation of area
4. structuring an array

The Professional Learning Team (PLT)

• group of elementary teachers team up to do a lesson study on area
• wanted to improve upon students' understanding from a recent lesson on area
• used the lesson study format to design a study

Previous lesson

• called "Shapes, Halves, and Symmetry"
• used various activities to have student cover shapes with a unit of area
• they covered shapes with unit blocks
• teachers felt students didn't have enough time to communicate or engage in hands-on activities
• teachers want students to communicate and justify their solutions

The New Plan

• Teachers devised a new plan to supplement the previous lesson.
• This time, students would use triangle units to cover shapes and they would count the number of triangles used to obtain the area.
• This allowed students to partition a complex shape into known shapes.
• communication was also key

Implementation

• The lesson involved finding all the ways the students found the area of the shapes
• Students had to explain their solutions
• Teachers found that some students initially struggled with the activity because they would count 2 triangle that formed a block as 1, instead of 2. This was later remedied
• This lesson allowed students to recognize using a common unit when computing area.

Taking it further

• Teachers can offer challenges to more advanced students by either having them find the area of a shape they create, or have them figure out if it's easier to measure with bigger or smaller units.

Learning About Area by Working with Building Plans
Introduction

• Students have an over-reliance on using the area formula
• teachers wanted to see how they could help students see the connection between area and direct measurement.
• Teachers wanted students to 1)use benchmarks to estimate area and (2) calculate area from building sketches.
• The Area Teaching was divided into 3 Lessons.

Lesson 1 (Introduction to Area Measurement)

• Students asked what to use perimeter for, versus area.
• students given a square foot linoleum tile and asked to measure its area using a smaller tile unit.
• Methods and answers were compared.
• Students then took linoleum tiles and were asked to measure area in parts of the school.
• The goal of this section was to teach area that they don't have to "fill in" all the space with their tile, but instead, they create rows and columns and can multiple their numbers.

Lesson 2

• Students went to two nearby neighborhood houses and estimated their area measurements.
• They also had to determine a good benchmark (or unit) to use to measure.

Lesson 3

• Students were given floor plans for real houses and asked to determine the area of the living space in the house.
• This often involved students having to add or subtract odd ends. In other words, they weren't finding the area for simple square houses.
• Students worked on this for one half hour and presented their findings on the overhead

Results

• Many students benefited from this lesson with their understanding of area as determined by pre and post-assessments.
• Many students still struggled with determining area from irregular shapes.
• Other ideas for area units include: emphasizing benchmarks for area, designing building plans, and extension to rational numbers.
  

Suzie's Readings

Here's a summary of my articles... as well as some quotes I took from the articles that I thought spoke for themselves.

Developing Spatial Sense through Area Measurement
Foundations of Measure-
1. units of measure should be adapted to the objects of measure- meaning that we measure length with a unit that has length and we measure area with either an object that has area or we find a way to adapt length and width to make area.
2. units of measure should be identical- we can't measure with inches and then switch halfway to centimeters; we should also use conventional measurements (like measuring lumber with feet not paper clips)
3. measurement involves iteraton- things need to be measured by end to end, not overlapping or leaving gaps
4. a scale has a zero point- like the brokn ruler, yet 8 1/2 would be reset to zero to start measuring
5. measurement is characterized by additivity- areas of figures can be adapted (cut or imagined) to fit into another figure;
6. measuring area is based on space filling- no gaps or overlaps


Using Representation to Explore Perimeter and Area
"The problem is that children o not know how tto choose the correct operation or sequence of operations to solve the problem."
"Students often confuse perimeter and area because the topics are learned only as sets of procedures."
"Children also have difficulty determining when to write an answer in inches, and when to write an answer in square inches, because they simply do not know what the number they have obtained represents."
"If we want children to be successful in problem solving , we must give them many opportunities to solve problems in meaningful contexts."
"They also need opportunities to talk and write about these ideas as they begin to construct an understanding of these and other closely related mathematical concepts."

For thursday

Also, what else do we need to have done for Thursday? We will have time to write the rough draft in class. Should we have anything prepared? I feel a little like Suzie and I are out of the loop since the rest of you see each other and are able to discuss things so let us know. Thanks guys!