Module 14

Textbook

Use the Writing to Learn questions on page 26 to focus your reading. Include at least two of your responses to the questions within your blogs.
Explain what it means to measure something. Does your explanation work equally well for length, area, weight, volume, and time?
To measure something means that you are going to give a value to describe an object. This object can be anything from a cardboard box to a sound.
I think that this definition can work with length, area, weight, volume, and time, but it may need a bit more description of each.

Four reasons wer offered for using nonstandards units instead of standrds units in intructional activities. Which of these seem most imortant to you and why?
I believe the reason "when standard units are used, estimation helps develop familiarity with the unit." is most important.

Circumference and Diameter 

Questions:
Describe Ms. Scrivner's techniques for letting students explore the relationship between circumference and diameter. What other techniques could you use?
The students drew a circle on the carpet with chalk. I find this very cool, I have not seen that before. The students place a pencil in the middle of the circle and use a string to create a circle. The string was the radius. The students demonstrated holding a pumpkin in their arms. That would be the circumference. They break down the work circumference. Circle and circumference have circ-.
They are going to measure the circumference and diameter of different items in the classroom.

In essence, students in this lesson were learning about the ratio of the circumference to the diameter. Compare how students in this class are learning with how you learned when you were in school.
In school, when studying diameter and circumference of circles, I remember measuring things given to us by a teacher and also working directly with formulas. This activity allows the students to explore more.
Do you rememeber any specific activities you got to do as a student with circumference?

How did Ms. Scrivner have students develop ownership in the mathematical task in this lesson?
The students got to choose what was being measured.

How can student's understanding be assessed with this task?
With a group discussion and with their completed data sheet.

For Further Consideration…

We have explored numerous areas throughout this semester. Pick five ideas that you will later use in your classroom.

1. Tangrams and the activities
2. Mira. I thought this was fun to explore.
3. Early data collection activities (graphs) for younger grades. Maybe a data question as morning work.
4. Types of graphs and when to use them.
5. Hands on activities and technology. we have been able to explore various topics with online activities and may be very beneficial for students.

What was your favorite area this semester?

Module 13

Measurement Misconceptions

After you’ve viewed the talk-over PowerPoint and read the vignette, respond to the following questions in your Blog. 

Why do you think the students are having difficulty? 
I think the students are having difficulty reading the ruler. They do not understand that the 0 is the start of the measurement. When counting, students start with the number one and they are trying to do the same with the ruler.

What misunderstandings are they demonstrating? 
They are misunderstanding how to compare the units with the attributes of the object being measured. Like the student who measure 41 inches. He knew that was wrong because he "thought the pencil was shorter".

Have you witnessed any students experiencing some of these same difficulties?
I have not witnessed this yet with students.

What types of activities could you implement that would help these children?
If I had students that were struggling with this concept, I would start the day with a Measuring Morning work. Give the student a chance to get familiar and perfect measuring one day at a time.

Have you seen a student struggle with measuring? If so, in what way? Were you able to work with teh child?

Angles Video and Case Studies

Discuss how the children in the video view angles. What ideas make sense and what ideas need further development?

One child described angles as "when two lines meet and it is the space between them".  The students were able to show an angle using their arms. One student mentioned how the angle is at the opening of a cross, but an angle is considered at the small end as well as the larger opening. Another student mentioned that when you put your two hands together, an angle is created. She then uses her classmate as an example. The pushes his hands closer together and explains that moving these hands together creates a smaller space and that is a smaller angle as well.
One thing the students need to have a further discussion about is what makes an angle. The student drew on a board a curved line and a straight line creating a point. She said this creates an angle. Her classmates also agreed. I think the students need to go through and further discuss why this is not considered an angle and what can create an angle.

Respond to the following questions after you read the case studies: 
1. In Nadia’s case 14 (lines 151-158), Martha talks about a triangle as having two angles. What might she be thinking? 

I think Nadia is showing one angle which uses the bottom and one side. Instead of doing the same for the top angle, the student split that angle to distinguish the making of the second angles.

Have you come across a student that had a similar understanding of angles?

2. Also in Nadia’s case (lines 159-161), Alana talks about slanted lines as being “at an angle.” What is the connection between Alana’s comments and the mathematical idea of angle? 

This one is a little difficult. A line represents 180 degrees, but I think the student is more referring to the slant line as one of the lines that form an angle.

3. In Lucy’s case 15 (line 251), Ron suggests that a certain angle “can be both less than 90˚ and more than 90˚.” Explain what he is thinking.

I think that Ron is showing an understanding that during the action, the skater started at one angle, then moves to another. On paper, if we showed the action on paper, the skater could show both 120 and 60 degrees next to each other on a line.

4. In case 13, Dolores has included the journal writing of Chad, Cindy, Nancy, Crissy, and Chelsea. Consider the children one at a time, explaining what you see in their writing about angles. Determine both what each child understands about angle and what ideas you would want that child to consider next. 

Chad mentions how you have a small angle with two pencils together and a larger angle when you put your arms together. I think the thing that chad needs to consider is that the lines that are making the angle are different sizes, but the angles could be the same.
Cindy mentions how angles can be in any direction, but doesn't explain what an angle is. I would want Cindy needs to learn more about what an angle is, and how it they are formed. She doesn't show much understanding what an angle is.
Nancy shows a great understanding with little words. She understands that there are a variety of angles, obtuse and acute. I would move on with the student and show the student various angles that looks like the are different in size.
Crissy also has a great understanding of angles. She mentions acute, right and obtuse angles. She doesn't quite have a great description of an acute angle.

5. In Sandra’s case 16 (line 318), Casey says of the pattern blocks, “They all look the same to me.” What is he thinking? What is it that Casey figures out as the case continues? 

Based on the images provided in the case study, I am wondering if the student is confusing the linear line as a 90 degree angle since that's whats in common of the three shapes.
Casey made the connection that a 90 degree angle is still a 90 degree angle not matter the size of the object creating it.

What do you think causes students to have confusion about angles when comparing the same sized angles from different objects or shapes?

How Wedge you Teach?

What ideas will you take from this article into your classroom? Was there anything surprising about what you read that made you change your thinking about children’s understanding of calculating an angle? What possible misconceptions might children have about angles and what misconceptions did you have about angles?

When the students began performing the activity, it amazed me that the students were able to use prior knowledge of angle measurement to help determine an angle. For example, one student said “The whole circle is 360 degrees; so if you cut it in half, that straight line would be 180 degrees. If you cut it into fourths, … each angle would be 90 degrees.” This is great use of their prior knowledge. I remember as a child, one thing that I did not understand was that a circle was an angle (360 degrees). That was a hard concept for me to grasp because I assumed angles were created by two straight lines that made a point. This could be something stdents struggle with when learning about angles.

Exploring Angles with Pattern Blocks

Post your responses to the questions posed within this task. 

Was this activity difficult for you? If so, which part?

For Further Discussion

As adults we use standard measurements almost without thought. Nonetheless, nonstandard measurements also play a role in speech and behavior. We ask for a “pinch” of salt or a small slice of dessert. We promise to be somewhere shortly or to serve a “dollop” of whipped cream. In some situations we might even prefer nonstandard measurements – for example, in cooking, building, or decorating. Do you or someone you know use nonstandard measurements on a regular basis or even instead of a standard system? Give some examples and discuss why a nonstandard measurement might be preferred in some cases. 

My grandmother to this day uses nonstandard measurements when cooking. My grandmother prefers this because she always said they never made appropriate measuring cups for the amounts she needed. So she always said measurements like spoonful, handful, a pinch, a tad bit, or even a "good" amount of something. It was measurements that she was comfortable and familiar with using.

Module 12

Coordinate Grids

List the websites that you explored and discuss which ones you would use in your own classroom? Discuss the advantages and disadvantages of using online programs in the classroom? 

http://www.shodor.org/interactivate/activities/GeneralCoordinates/
http://www.beaconlearningcenter.com/weblessons/GridGraph/default.htm
http://www.shodor.org/interactivate/activities/MazeGame/

I liked the maze game as an activity to do in the classroom. All the websites were fun and incorporated points on a grid different ways. The disadvantage of the maze game is you can select any points on the grid to make a maze. You are not having to locate specific points. The beacon learning center game does require you to figure out points 1-5. The sight however is not very kid friendly.

Did you find any websites that you like? What were they?

Miras, Reflections, and the Kaleidoscopes

After you’ve completed all the activities, take a picture of the boy on the swing set showing how you were able to successfully use the mira. Have you ever used a Mira before? Did you find any part of this problematic? How did this build on your understanding of transformations? Discuss the ideas from the article that you will take with you into the classroom.

This was my first mira activity. I had never used one before. I had trouble getting the mira just right to make an accurate reflection. These activities were very fun. It helped me better understand transformations by having to perform them. In the article they mention that for students to have a better understanding they must learn more than the definitions. I think that the activities discussed in the power point and article can give the students a visual representation of the word reflection.

Was the mira hard for you to do? Have you done anything like this?

Case Studies

Respond to the following questions in your Blog:

1. What ideas about measurement do the children in Barbara’s class (case 12) bring to school before they are taught about it?

The students were discussing the size of the box and emphasized on the box being big. They then started comparing themselves to the box. The pointed out the box was bigger than certain students.  The students were asked how they could measure the box at school, and the students began using objects in the class to measure the box. They tried bins, a chair, and books.

2. Many children struggle with the idea that the larger the unit, the fewer the number of units needed to cover a length. Go through the cases by Rosemarie (case 13) and Dolores (case 14) to identify how different children are making sense of this issue.

In case 14 Dolores, the students were given the scenario of two boys measuring a room. One got 30 steps and one got 43 steps. Students had varying conclusions like one miscounted, one took longer steps, and one had bigger feet. All the students had different understandings. on case 13 Rosemarie, the students wanted to measure the distance using hand and feet. They made the connection that they could not use hand and feet. They must use things the same size to get the right measurement.


3. In Dolores’s case, line 245, Chelsea notices that Tyler and Crissy both measure the width of the basketball court as 62 “kid feet.” Why didn’t everybody measure the width as 62 kid feet? What discrepancy is Chelsea noticing? What is Henry noticing? How are their observations related to the issue that arises in Sandra’s seventh-grade class (case 17)?

Not everyone had 62 because of the varying shoe size and mistakes made. One student mentioned when comparing the measurements that some of the measurements were the same or close together. Later Chelsea made the conclusion that people with bug feet take fewer step.

How would you demonstrate to the students that the different lengths mean the unit they are using are different sizes?

4. The children in cases by Mabel (case 15) and Josie (case 16) are working out the use of standard tools for measuring length. Specifically, the children in both classes discuss how to place the tool and how to read the number of units. What do the students have to say about these two issues? What do they understand about measuring with accuracy and precision?

They discuss using a ruler to measure objects. If the object is longer than the ruler, you have to use several rulers. If you do not have a more rulers, then you can out your finger and move the ruler. The students also discuss that when using your finger as a place holder, it can alter your length a little because of your finger. If you have to do this several time, it may make a big difference in length.

5. By comparing the cases from second, third, fourth, and seventh grades to Barbara’s kindergarten (case 12), can we identify ideas that, by the older grades, are understood by the children and no longer warrant discussion. What are some issues that still lie ahead for Barbara’s students to sort out?

The older group discussed measurements using various terms and were able to understand how to measure objects with different types of units. The younger group needs to work on their vocabulary and how to describe objects when comparing sizes.

For further discussion

A fellow teacher says that he cannot start to teach any geometry until the students know all the terms and definitions and that his fifth graders just cannot learn them. What misconceptions about teaching geometry does this teacher hold?

This teacher is relying on the fact that students need to know definitions. You shouldn't strictly rely on the students vocabulary knowledge. In the article, the author mentions "Geometry is more than definitions; it is about describing relationships and reasoning".

Did you have a different misconception? What do you recommend when teaching geometry? Definitions?

Now that you’ve had some time to explore the world of geometry, how has your view of the key ideas of geometry that you want your students to work though changed? 

My view has changed. In school, I remember learning definitions. It wasn't until later grades was I required to show what each term meant. I think that the best way to learn and understand geometry, you have to show students examples while teaching the definitions. Working with just definitions and minimal examples can be confusing, especially for lower grades.

Module 11

Pentonimo Activities

http://www.mathsphere.co.uk/fun/pents/pents.html
This is one I found. You are given all the pentonimoes and must fill in a rectangle with them. It caused a lot of frustration because you can't flip them and place them perfectly together. It was very hard. I had to hit the help button.

http://pentomino-puzzle.toogame.com/play
This game is very similar, but you can use some of the pentonimoes more than once. You can also turn the shapes and make adjustments.

For the powerpoint, I completed the activities just fine.

Did you find any interesting websites you would use in your future classroom?

Pentonimo Narrow Passage

16 spaces 

Was this difficult for you? How many spaces was your path? Did you have any strategies?

Tessellating T-Shirts

Summarize the article and then describe how the article furthered your understanding of transformational geometry?
It is expected for students k-2 should work with putting together and taking apart two- and three-dimensional shapes. Grades 3-5 should be transforming shapes. Many teachers believe these concepts should be taught at a higher level. The best way to start introducing transformational geometry is by creating translating tessellations. When creating the shirts, the students need to use problem solving strategies. Students are more successful with large shapes.

What does it mean to tessellate?
Creating a repetitive connecting pattern.

Look online for different examples of tessellations and share what you’ve found.

I love the tshirt idea to try in my future classroom. How would you allow the students to create their tessellations? Would you provide shapes or allow them to create it?

Tangram Discoveries

Post pictures of the shapes you created and then share your responses to the following questions:

triangle, square

Trapezoid, square

Parallelogram

  Which polygon has the greatest perimeter? …the least perimeter? How do you know? (You must describe how you know this to be true.)
Greatest perimeter: trapezoid, parallelogram, an triangle,
Least perimeter: square
I gave my own lengths of sides and determined a value for each.
 Which polygon has the greatest area? …the least area? How do you know?
The area is the same for each shape since they are using the same three shapes all the same size.

How did you figure these out?

Ordering Rectangles

Share your answers to questions 1-5 from this activity in your blog. Spend some time discussing your thoughts with your blog partner.
1. My first thought was C had to have the smallest perimeter, but when I really examined them, C and D seem pretty close.
AGEFBDC Largest to Smallest
2. Smallest Area is c. Largest is G.
GAEFBDC Largest to Smallest
3. (A D E) B G C F Small to Large
I did horrible on this one. I am surprised how far off I was.
4. C D B A E F G Small to large
I did a lot better on are than perimeter. I only have a few displaced.
5. I need to remember that when looking at perimeter and area, area is the amount of space the shape takes up and perimeter is the outside length. Area and perimeter are different and can have different answers when comparing the two.

Was this difficult for you? In what way?

For Further Discussion

Multicultural mathematics offer rich opportunities for studying geometry. Research the art forms of Native Americans and various ethnic groups such as Mexican or African Americans. What kinds of symmetry or geometric designs are used in their rugs, baskets, pottery, or jewelry? Discuss ways you might use your discoveries to create multicultural learning experiences.

I see a lot of triangles, squares, and circles used for design.
I would make a connection of how various cultures used geometric shapes for design and use different images as examples. Then the students can create their own masterpiece using certain geometric shapes.

Module 10

Nets Activity

How could you use a similar activity with students in the classroom? Were you able to complete the activity without too much frustration? What are some anticipated issues while doing this activity with students? 

I thought thus activity was fun. I did not find it frustrating at all and found each x spot correctly. I think that this required a different level of thinking, and would be great to try in the classroom. I would set up different groups and have various pentominoes, hexominoes, and tetrominoes and ask the students to find various shapes.  I think that some issues would be that the students would be tempted to just create and fold the paper and not try to think about which ones could create the shape.

Did you find this activity hard?

Textbook Reading

All the questions on page 60 will aid you in your reading. You are encouraged to include your answers to all of the questions on your blog but you must answer number 4 and at least one more

Spatial Readings/Building Plans

In your blogs, talk about these three experiences and answer the required questions: 
 Did you find any of these activities challenging? If so, what about the activity made it challenging?

I did not find this activity challenging. I remember doing something similar when I was in elementary school. I always found it fun to do.

 Why is it important that students become proficient at spatial visualization? 

It is important because we need spatial visualization in our everyday lives. When students see things out in the real world, like merging traffic lanes, a student must understand how to interpret this in order to follow traffic correctly.
Do you know of any example of spatial visualization in the real world?

 At what grade level do you believe students are ready for visual/spatial activities 

I think it should be taught at an early age like third grade. I think that they have the ability to understand what it is and how it can affect us.

 How can we help students become more proficient in this area? 

I think just giving the students time to practice with spatial visualizations, whether with blocks, cubes, on the computer, etc. The more the student is exposed to it, the better they will master spacial visualization

Tangrams

Begin this section by attaching a picture of the tangram pieces that you created with your paper. Throughout this module, you are asked to respond to a variety of questions. Choose at least two of the questions from this module to include in your blog. You are encouraged to choose areas that are causing confusion as your blog partner may help you make the connections that you need for your understanding. 

Problem A3:
a. I moved the triangle on the far right to the left by rotating it the shape to the right side.
b. I rotated the top triangle to the right and you can slide the triangle to the left to the right side of the triangle we just moved.
c. I flipped the triangle to the left. Then slide the triangle down.

Problem b2: Start with a right triangle. Dissect the triangle so that you can rearrange the pieces to form a rectangle.

I first cut the triangle down the middle (b to the middle of c and a), Then I had two triangles. I cut those two triangles in half. I then used the four triangles to make two separate squares and then placed then together.

Did you find this activity challenging? I struggled with some more than others. Which were easy or difficult for you.

For Further Discussion

Informal recreational geometry is an important type of geometry in many childhood games and toys. Visit a toy store (or go to an online store) and make an inventory of early childhood toys and games that use geometric concepts. Discuss ways these materials might be used to teach the big ideas of early childhood geometry. 

Perfection is a game that definitely teaches the students about geometry. The children are required to match the game piece to the appropriate hole. I think this helps the children recognize different attributes and characteristics when observing and working with different geometric shapes.
I also found the shapes box for young children. The kids must place the toys back in the bx through the appropriate hole. There are triangles, squares, stars, rectangles, and circles. 

Did you see similar games? Did you use any games like these growing up or for your children?

Module 9

Quick Images Video

When I first viewed the shape, I thought it was a crescent shape with a small circle in the middle. When the students began their discussion, they compared to the shape to things it reminded then of. Different answers they gave were the shape was a moon, it was like the letter C, it looked like part of jet ski, and looked like a part of a circle. 

Was you first thought different than the students? If so, how?

Shapes and Geometric Definitions

First, write about your own thinking. 

What are your definitions for these geometric terms? 
Triangle- A three sided shape
Square- Four sided shape with all equal sides
Rectangle- Four sided shape with the two different paralleled side lengths. The first two are short, the other sides are longer. 
Parallelogram- A shape with two sets of parallel sides.

Did you have a hard time coming up with definitions? Were there any harder than others?

What is the difference between a definition and a list of properties or attributes? 
A definition defines the shape, and attributes are used within the definition to describe the shape.

What is the purpose of a definition? 
A definition explains a word. It can also break down the word.

Did you think differently?

Then, examine this set of issues through the eyes of the students. 

What specific issues do the students need to consider in order to make sense of definitions for triangle, square, rectangle, and parallelogram? 
I think students need to familiarize with the shapes and various attributes. If a definition mentions 90 degree angles, the student must be familiar with what that is and look like to know what shape is begin referred to.

What process do the students go through as they learn to apply their definitions? 
They learned how to build their own definitions of each shape first. They described each shape by its attributes and tried to be specific to prevent confusion.

Looking beyond the specific geometric content of this set of definitions, how do children develop a sense of the purpose of definition? 
They honestly are leaning how to use and interpret a definition. This can not only help them in a geometric sense, but can also help them in other subjects and everyday things.

Respond to the following questions:

Follow the thinking of Susannah throughout Andrea's case 19. What does she understand about triangles? What is she grappling with? What ideas or questions does she contribute to the class discussion? What does she figure out by the end of the case? 
The students came up with a few facts:
Triangles have three straight lines. 
If they have wavy or round sides, it is not considered a triangle. 
A triangle must have three points.
The students debated these different facts. They gave various examples why it could be true based on the few triangles they were looking at. All had three points, straight sides, and sides of different lengths. The students also discusses that no matter how you change the position or stretch the shape, it is still the same shape. The student compared himself to the topic.

Now go back and follow the thinking of Evan throughout Andrea's case 19. What does he understand about triangles? What is he grappling with? What ideas or questions does he contribute to the class discussions? What does he figure out by the end of the case? 
Evan understood that if you have a shape and you stretch it, or change its position (rotate or flip), it is still the same shape. Evan compared himself. He said if you flipped him him, or stretched him, he would still be the same person. He even wrote that you can turn a triangle in any direction and still be a triangle.

Consider Natalie’s case 20. What are the students learning about squares and rectangles? What do they still need to figure out? Refer to specific examples from the case to illustrate your ideas. 
They are learning the difference between a square and rectangle. When they listed definitions of both, they have same definitions. They need to make the definition more specific.

Also in Natalie’s case 20, after line 250, the students are working to define the term square. Their conversation is as much about what a definition should be as it is about the particular term square. What does their discussion make clear about definitions? In particular, consider Roberto’s definition (“four sides, four corners, four angles, and it’s a square”) and the other children’s responses in the lines that follow. 
The realized that they need to specify various attributes and facts about shapes in the definition to decrease confusion. If you don't, then some of the definitions sound like the same shape.

In Dolores's case 18 (lines 25-43) and in Andrea's case 19 (lines 162-168), students are talking about what it feels like to make sense of a new idea. Describe their conversations. Refer to specific portions of the text in your discussion. What is your reaction to their comments? 
The students in case 18 mention how a "regular" triangle might feel ",ore like a triangle" than others. This lead me to believe that since students are initially introduced a "regular" triangle, the students need to learn that there are variations to shapes. The initial shape introduced is not the only shape. They definitely are trying to work on their definitions and different attributes of each shape.
Was there a particular students in the cases you followed closely? I enjoyed reading and following Ethan.

Reflect on what you just read and discuss how this will impact what you will do in your future classroom
I really liked the open discussions of the class. The students building on one another's contributions and giving great examples. This is something I would like to do in my classroom. It seemed to have a great outcome.

Math Activity with Color Tiles

When making these shapes, I built off of previously made shapes. I continued to create shapes until I felt I had made them all, and I had. I created the entire set. When I named them, I noticed I had one category a shape short, so I had to recount. I found which one I had miscounted. I forgot to count two sides of one shape.

Did you enjoy this activity? Did you find it difficult? I enjoyed the activity. Would you do this in a classroom?

For Further Discussion

If I were to describe my home, it is made up of various geometric shapes. My walls and floors different shaped quadrilaterals, my floors are hardwood with quadrilaterals, and frames are quadrilaterals. I live in a duplex that creates a dodecagon with our patios sticking out. It is crazy how geometry is everywhere in our lives.

Module 8

Key ideas in Geometry

The key ideas that I would like the students to work on is being able to identify various shapes. They must know the names of the shapes and to also list the different distinctive characteristics of the shapes as well.

Van Hiele Levels and Polygon Properties

I enjoyed doing that activity. I got all three shapes correct. You definitely need to know vocabulary to get the correct shapes.

Did you enjoy this activity? Where there questions that confused you? Did the terminology make it difficult?

Knowing that different students grasp the concept and knowledge of shapes at different levels, I would have to approach geometry instruction differently than I initially thought. We need to focus on characteristics of shapes and discuss vocabulary. We need to help student move from the visualization level to the rigor level.

I would rate myself a level two overall. I think I am very well with properties of shapes and comparing shapes. It has been a while since I have dealt with various shapes and needing to provide proof using appropriate definitions.

How do you feel about geometry? Did you struggle in high school?

Thinking about Triangles

Tricycle

Triathlon

Trident

Trio

This really made me think. Did you find this difficult?

Is it possible to make a three-sided polygon that is not a triangle?

No. Any shape with three sides, no matter the length is a triangle.

Is it possible for a triangle to have two right angles?

No, you can only have one right angle in a three sided shape to have each side connect.

How many different right triangles can be made on the geoboard?

I made 12 different right triangles. I’m unsure where the other two are.

Did you get 14? What strategy did you use to get your right triangles?

How many different types of triangles can you find? I made 5 triangles.

Equilateral, scalene, acute, obtuse, and isosceles.

Where you able to find these? Which wasa most challenging? Did you repeat any shapes by accident?

Module 7

Textbook Reading

After reading the pages in the textbook, answer the following questions:
 What are the advantages of having students conduct experiments even before they attempt to figure out a theoretical probability?

The students are able to focus on enhancing their problem solving skills. The students can also practice and learn how to collect data appropriately and accurately.

Did you think of other advantages?

 Go to the Illuminations website and explore one of the virtual experiments. List the title and describe the experiment that you explored and discuss the advantages and disadvantages of virtual experiments. (http://illuminations.nctm.org/ and use the search site: probability and check interactives)

Adjustable Spinner.
For this experiment, you spin the spinner and record which color you land on. There are 6 different colors (the number can change) on the spinner and data chart is already provided. The theoretical % is given for each color. When the number of each color is recorded, the experimental percent is given as well.
Advantages: Students get to see how to collect data from a spinner. The students can see changes when you change different factor (number of colors on the spinner)
Disadvantages: The students aren't physically spinning the spinner or filling out the chart. The percentages automatically change once the student hits the spin button.

Was your activity similar? Did you encounter similar advantages and disadvantages?

A Whale of A Tale

Did you have any problems with you chart? What there one category harder to answer than the other?

Dice Toss

• Ms. Kincaid wanted the students to make predictions about their experiment on the basis of mathematical probability. Discuss preconceptions that students exhibited about tossing dice even after discussing the mathematical probability. Discuss the instructional implications of dealing with these preconceptions. 

The students predicted the dice will roll the sum of 7 more because it has the most sum combinations with two dice. They also discussed 12 as the number to least likely to occur.

• Were these students too young to discuss mathematical probability? What evidence did you observe that leads you to believe that students did or did not grasp the difference between mathematical probability and experimental probability? At what age should probability be discussed?

These students are not too young to discuss mathematical probability. At the beginning the students were able to make the appropriate predictions before completing the experiment. Yes, they made mistakes, like exceeding the number of rolls required. They did however catch their mistakes and had to retrace their steps. One student does seem to not understand that rolling a sum of two is just as likely as rolling a 12. In their data, they had more ones rolled, and assumed that a one is more likely to show up than sixes.

• The teacher asked the students, “What can you say about the data we collected as a group?” and “What can you say mathematically?” How did the phrasing of these two questions affect the students’ reasoning? 

With the first question, the students describes what the graph looked like. The student described the graph to look like a rocket ship. Once she asked what it looks like mathematically, the student was able to inform the teacher that the sum of seven was rolled most by the group.

• Why did Ms. Kincaid require each group of students to roll the dice thirty-six times? What are the advantages and disadvantages of rolling this number of times? 

This is to control the number each group rolled and to have more accurate data. If different numbers are rolled, then the results will be skewed. Advantages is that the students have more opportunities to roll and will ave more data to collect. I think a disadvantage is that it is a lot of data. Many students may be disorganized when recording and trying to keep track of all the rolls and numbers.

• Comment on the collaboration among the students as they conducted the experiment. Give evidence that students either worked together as a group or worked as individuals. 

They worked as groups. I liked the students taking turns for each roll and telling the student assigned to recording data their number. This gave each group member a roll.

• Why do you think Ms. Kincaid assigned roles to each group member? What effect did this practice have on the students? How does assigning roles facilitate collaboration among the group members? 

I think she did this to make sure everyone had a roll and an opportunity to learn. If it was only two students working in the group, then the other two students would not learn as much. They may have a harder time understanding what is going on and how they received their data. Also assigning the roles means that to complete the experiment, each team member must communicate together to be successful.

I like the idea of assigning roles for groups. Do you? If so, what kind of roles would you have assigned in an experiment such as this one?

• Describe the types of questions that Ms. Kincaid asked the students in the individual groups. How did this questioning further student understanding and learning? 

Ms. Kincaid had to ask a group all the possible numbers you could roll. This question was to help the student who was in charge of data collection. She was helping steer the girl in the right direction on which numbers she needed on her sheet before beginning the data collection process.
In another group, Ms. Kincaid asked the students "is this what you expected would happen?"

Are there any question you would additionally ask as a future teacher?

• Why did Ms. Kincaid let each group decide how to record the data rather than giving groups a recording sheet that was already organized? When would it be appropriate to give students an organized recording sheet? Discuss the advantages and disadvantages of allowing students to create their own recording plans.

I think she wanted the students to figure out for themselves how to effectively record data. This will help students learn from their mistakes. I think if there is data you need recorded a certain way when coming together as a group, that would be the time to give the students a specific recording data sheet. Advantages are the students will be independent as learn how to record their own data without instruction. A disadvantage would be making mistakes or using the wrong kind of data plans and can cause the student to have to restart their data collection.

Did you have similar advantages and disadvantages? If not, what were they?

For Further Consideration

Knowing what you now know about probability concepts in the elementary school, how will you ensure that your students have the background to be successful with these concepts in middle school?
To ensure that students are prepared for middle school, I would make sure to continually work with the students on probability. We can have a weekly probability exercise in the mornings as morning work. We can work on our vocabulary and also our data collection skills. In the textbook, it mentions how in middle school there is more emphasis on theoretical values. We can work on those concepts and work on our predictions.

Module 6

Box Plots

In the PowerPoint you were shown a graph comparing two classes of data which is posted below. Discuss which class you’d like to teach a follow-up lesson to and state why. Make sure to support your defense using reference to the five finger summary which an estimate can be determined by analyzing the Box Plot.

I would reteach class A. Class A has a smaller range and seems to have more students that have done poorly. Class B does have the lowest number, but it also has the highest. There may only have been one student who scored badly which causes the plot to be very low. We definitely know that no students in Class  scored above a 60 according to the maximum number.

Which class did you choose? Was it difficult for you to select a class, if so, why?


a. Make a box and whisker plot for the data in your class and draw it under the German class’s plot using the same scale. (Upload a picture of your graph to your blog!) 

b. Suggest three good questions that you could ask your class in making comparisons between the two plots. Answer each of your questions. (When posing questions, you want students to think beyond the obvious. Your goal is higher order thinking.)  

1. Which plot had heavier weights of trash.
2. If I changed the Maximum number on graph two, would the data change dramatically?
3. How do we know which plot had more data?

Common Core Standards

• Write down two “first impressions” you have about the data standards
• Two first impressions I have are the information given is easy to navigate through and very organized, and it is intimidating. It seems like a lot of information that needs to be covered in the classroom. It makes me nervous.

How did you feel about data standards? Did you feel differently? Did you look at a specific grade?

• How do the concepts progress through the grades? Need to provide specific examples to support what you see are the progressions. 
• They begin teaching the fundamentals in the lower grade and then build on top of the previously taught knowledge. For example, Kindergarten Geometry requirements are to identify and describe squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres. In first grade geometry, students must compare shapes and identify attributes. They work with two-dimensional and three dimensional shapes.
• How do the concepts change and increase in rigor and complexity for the students? Again, provide specific examples. 
• The students are required to use higher level thinking. As stated before, it is like building blocks and the students build off of their prior knowledge. Continuing from first grade Geometry, students in the second grade start to partition shapes. They are learning how to divide these shapes into smaller shapes ad equal parts.

Now look at both the Common Core and NCTM standards to respond to the following questions:

• Does the Common Core Standards align with what NCTM states students should be able to know and do within the different grade level bands? (Note that NCTM is structured in grade level bands versus individual grade levels.) 
• When looking at common core and NCTM standards, I see some similarities, but the NCTM standards has more advanced requirements.

Do you think the NCTM standards asks for more?

• Give examples of which standards align as well as examples of what is missing from the Common Core but is emphasized in the NCTM standards and vice versa.
• For K-2 in common core, the work their way up with data collection. They should count the number of objects in a category, then learn to organize, represent, and interpret data, and finally measure data and create graphs. According to NCTM K-2 should formulate questions, use appropriate statistical methods to analyze data, then develop and evaluate inference and predictions based in data.

Curriculum Resources

Grade 1: Would You Rather? Activity 1
The students practice taking surveys. They will be figuring our how to sort and display the data to the class.  Question: Suppose you could be an eagle or a whale for a day. Which one would you rather be?

• When using this activity, what mathematical ideas would you want your students to work through? 
• Data collection, comparing numbers (interpreting data)
•  How would you work to bring that mathematics out? 
• I would ask students questions to get them thinking. Why do the charts look different? What does it mean? 
• How would you modify the lesson to make it more accessible or more challenging for your students? 
• I would add more choices to choose from, or expand our survey and have more people.
• What questions might you ask the students as you watch them work? 
• What do these numbers mean? Which animal has more people? What is he difference between the two? 
• What might you learn about their understanding by listening to them or by observing them?
• I am hoping to learn that they understand one graph is larger than the other because more people chose one over the other. I am hoping to see an understanding of data collection through their charts.
• How do the concepts taught in this lesson align to the Common Core Standards? 
• It lines up with the measurement and Data domain. Students are required to organize, represent, and interpret their data collected.

Module 5

Generating Meaning

This article is about how to introduce mode, mean, median, and range through various lessons. Start with range, and work your way through the categories. Let the students identify these categories and allow them to create the definitions for each. This helps the students develop a basic understanding.

I don't recall the process of going through mode, median, mean, and range like this one. I do remember reading about them in a textbook and we had a worksheet with various number sets that we were told to complete. 

I was shown the procedure. My teacher stuck to textbooks and worksheets. We were told what it is, how to find it, and then we were sent to do it ourselves.

Did you have a similar experience, or did your teacher help you create your own understanding of each feature?

Working with the Mean

6 and 10
4 and 10

For the cubes, I lined up my cubes from smallest to largest and mad my cubes of 8 lie in the middle. I began plugging in numbers that were skipped and tried to keep 8 in the center.
Did you use your cubes differently?

For the line plot, I drew it out and plugged all my numbers. The line had a steady incline so I plugged in missing numbers on the line and numbers before and after the line.

For this data set, I believe that the average gave us the middle number. We had no reoccurring numbers and the average would be similar to the median.

How much Taller? Video and case studies 

1.
The students measured their own heights in inches (282). 
The students my have known that to compare data, they must be the same units.

The students used two line plots to compare data (292 and figure 41).
The students created two data plots to compare the grades. They are using critical thinking skills by visually comparing the two.

The students notice range on their plots (312-314).
The students at different points in the case mention range. They were using range as the deciding factor on which grade was actually taller. They were noticing which had a higher range, where the range started and an overlap in the two grades as well.

The students notice a clump (328).
Students notice a clump on the graph. They are slowly beginning to make notice of mode without realizing.

The students notice mode (367).
The students took into consideration that mode could be the factor to base their reasoning off of. "The clump" is higher on the fourth grade graph than the third grade graph.

Did you find different features? If so, what were they?
2. 
One thing mentioned in case 27 (Maura), on line 453-455, Leah mentions that they could add up all the numbers (heights) to determine which is largest. I think this is a great first step in helping the students determine mean. 

The student Ilsa (Nadia's case) wanted to find the middle number but wasn't sure how. Her thoughts were to add or subtract, but she couldn't quite figure out how. She had a great thought process that with the data that was collected, we needed the middle number.

3. 
In case 26 (Phoebe), a small group of students took the average of their small group to determine the average height of their class. One student mentioned how they need to measure everyone because "what if the people in your group were different sizes from everyone else in the class?" (151-155). he made the connection that this is not the proper way to find the mean, but they are off to  good start. The student student is headed in the right direction, but sometimes taking a small sample to determine the whole can be problematic and can cause misinformation.

Annual salary is often a touchy subject for teachers whose low pay and high workloads are axiomatic. Search the virtual archives of a newspaper in an area where you would like to teach. Look for data about averages and entry-level salaries as well as information about pay scales and increases. Evaluate the data. What does it tell you? What doesn’t it tell you?I had to go to several different places to find the salaries. 

The annual salary for Texas teachers starting pay is $28,080. If you look for pay by location, they give you the average made. In my hometown, the average salary is $53,640. They do not mention the starting pay, so this number is very misleading when you are beginning your teaching career. They do not list the range where the average was taken.

Where do you plan on teaching? Are the number what you expected?

Find examples of averages in a daily newspaper, from the sports page, or any page. Then describe what these averages “mean” – their significance, implications within the context of the story, and so forth.

In a sport article about Dallas Mavericks Player Dirk Nowitzki, it mentions how he averages about 23.9 points per game. This means that when Dirk plays his games, he typically scores around 23-24 points per game. This is not a guarantee because the range may vary, he could score lower or higher than the average. However, comparing the numbers from all his games (which is a lot), he does however typically stay in the same range.

Module 4

Lost Tooth Video

• First video segment 

The teacher may ask the students to look at their data and what are some differences between these graphs. She can guide them to the direction to notice the different ranges. The students did a great job comparing their own life situations when they were talking about why the ranges were so different. The students made the comment that people lose teeth at different times. Everyone is different. They also had a change of data, from their initial survey. Someone had lost a tooth over night and that had to be put into consideration, which caused a change in their range.

What do you think about how the teacher asked the students questions? Do you think they could have been worded differently or were they just right?

• Second video segment - Did the children notice what you consider to be important features of the data? Are there features that they didn't notice? 

The students noticed range on their chart, mode (but they did not specifically call it mode, and they also mentioned clusters that they had on the chart. The students did not notice median, but that wasn't not something I expected them to. They did a very good job.

• Final video segment - Consider the same questions above.

In the final section, the third grade survey had a "I don't know" category. One of the students mention how they could have lost teeth previously and my have forgotten. I think that was an interesting inference.



After you view the video:

• Why might the teacher ask students to think about differences in the range at each grade level?

I would line all the graphs up and ask students what they notice about the range. I would guide their discussion and maybe select specific grades for the students to discuss.

• What insight do you get into children's thinking as they talked about why the ranges would be different? 

I noticed two things. One student noticed that the younger grades lost less teeth (kindergarten had 10 lose 0 teeth), but they also mentioned how age may not be a factor because they had a kindergartner lose a lot more teeth than predicted. They mention how everyone is different and my lose different amount of teeth at different times.

• Did the children notice what you consider to be important features of the data? Are  there features that they didn't notice? 

I think this class has mastered range and mode very well. They pointed out the range on several charts and they always mentioned which category had the most students. I did not notice the students mention median in any way.

Were you ever required to survey other classes like this? If so, what was your data question.

Stem and Leaf Plots

Describe anything new that you learned from this article. If you didn’t learn anything new, describe the ideas that were confirmed. How will you use stem-and-leaf plots in your future classroom?

They mention how they can set up the stem an leaf plot differently. They have two categories for the tens place value (so two 20 columns). The first one takes the ones 0-4 and the second one 5-9. I don't remember ever doing this to a stem and leaf plot and found it very interesting. It is definitely a way to keep organized. I think that there is different ways we can use stem and leaf plots in the classroom. One way is to have various weekly activities for the students. I could have a permanent stem and leaf plot on the board and change the question each week. (When is your birthday, how many pets do you have, how many pairs of pants, how many red cars are on your street)

Did you do stem and leaf plots in school? If you remember, what data did you collect? Did you enjoy doing them?

What is the difference between a bar graph and a histogram?  

A bar graph is used to compare results collected with bars (each bar represents one numerical value). A histogram is a type of bar graph, but instead of one value, it can many different numbers in the data set. The histogram compares data that range in intervals. Another difference is the bars on a bar graph are spaced out, while the bars on a histogram are side-by-side.

Find an example of a line graph and share on your blog. Describe the data used in the graph and why the line graph is an appropriate representation. 

In this graph, a class compared the number of magazine sales between Monday and Friday. This graph shows a gradual increase in sales, and a sudden decrease on Friday. They had the lowest sales on Friday, and the most on Thursday. This kind of graph is a great way to compare changes in data in a numerical sense. For example, this graph is great comparing sales, temperature, attendance (like people in crowds), etc.

How would you use a line graph in the classroom?