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?