Module Seven

Textbook Reading

What are the advantages of having students conduct experiments before they attempt to figure out a theoretical probability?

·        It is all about extending the understanding of the concepts. Using objects that the students are not familiar with help the children have an unbiased experiment. The understanding of the experiment part of probability is important as it allows for the children to predict what could happen next, and see if they were correct. Aubrey-do you agree, or did you get a different answer?

List the title and describe the experiment that you explored and discuss the advantages and disadvantages of virtual experiments.

·        Adjustable Spinner

o   In this virtual experiment you are conducting a probability of what color the spinner will land on with a certain number of spins. The advantages are that you can increase the sectors to a 12 different colors and do any number of spins between 1 and 100 to get accurate data collection. This would allow the students to see how the number of colors selected and the number of spins could change the results.  However, the disadvantage is that it is a computer program, and we really don’t know how random the spins really are. I also think that because it is not all that interactive the students could get bored rather quick.

Aubrey, do you think using the computer or “virtual experiment” is good?

A Whale of a Tale

Dice Toss

1.     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.

·        Most of the students were able to already understand probability due to the previous work they had done with coin flip activity. The students were able to come up with all of the possible results that could happen with two dice, but the teacher had to remind them a few times that the total had to include both dice. Some students

2.     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?

·        No I don’t believe they were too young to discuss mathematical probability, they are 4^th graders and this teacher is allowing them to have an early understanding of what it means to develop probability. The students were able to give the proper definition of both mathematical and experimental probability without help from the teacher. They also understand that because more combinations of dice could make the number 7 that it had a higher chance of being rolled. We should start discussing probability as soon as possible, as the book mentions, even talking about the chance of rain is early stages of probability.

3.     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?

·        When the question was asked about the data, the first group she went to expressed what they saw on the graph, and how it looked like a “rocket” but once she changed to mathematically what do you see they knew that she wanted the number answer using math vocabulary.

4.     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?

·        The total number of possible combinations that could be rolled was 36, so you would want the experiment to have at least that many rolls to ensure you get the maximum outcomes for possibilities. However, with the way some students recorded the data or preformed the experiment they had too many rolls for their group. Instead of them restarting they tried to erase the last answers they remembered.

5.     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.

·        It seemed like most all of the groups worked together, they all had input of how they could collect the data, and they all had a job. It was interesting to see how the different groups decided to do the dice rolling, some rotated with one turn at a time while others did all 9 rolls at once. It was an important task to keep count and even when the one group over rolled, they worked together to recall their past rolls and erase the data.

6.     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?

·        Giving each student a job in the group allowed for them to not only work together but to also hold each other accountable. For example the group that rolled to many times, the person that was supposed to be counting got carried away and caused an error in the data. It also ensures that each member is an active part of the group and no one member is taken control of the experiment.

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

·        The questions Ms. Kincaid asks of the individual groups allows the students to make connections to what they predicted would happen and what actually happened. This allowed the students to gain a deeper understanding of probability and how the number of outcomes relates to the probability that it could show up more often. They were able to relate personal experiences as well to this experiment.

8.     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.

·        This allowed the groups to create the way they thought would best work to organize the results. As Ms. Kincaid mentions, one student’s way was unorganized and she had to go back and reorganize the data so that they could understand what they had collected. Ms. Kincaid showed examples to the students of ways they previously used to collect data, they could have used this as an idea. It would be better for the lower-elementary age to have a recording sheet or the first time the idea is presented so that they students can see how to record data. Allowing them to create their own allows for the students to learn from possible mistakes it also lets them decide which way works best for them. The downfall is the unorganized collection that could result in inaccurate responses or if time is an issue.

Aubrey, I would using this as a lesson on probability, would you? If so, would you alter any part of it?

For Further Consideration…

I will be sure to introduce the ideas of probability early on, since I would like to teach kindergarten, I would ask my students questions about the weather. If it is cloudy, what is the chance of rain today? Introducing this concept early on will allow for the students to understand that probability is about experimenting with the given information and finding out how often it can happen. The experiments in the book seemed like they could be fun beginner lessons to create an understanding and a way to use the natural language of the children (Higgins, 2014).

I am interested in reading how you will ensure a background knowledge. 

Module Six

Box Plots

Hi Aubrey, I think I finally figured out the box-and-whiskers graph and how they work! So I have decided that I would want to teach a follow-up lesson to Class A. I chose this class because the median score was about 43 or 44 percent, which isn’t great but I felt that most of the students were on the same page as far as their understanding goes. Class B had extremes on both sides that could result in some students being utterly bored with if they had to sit through another lesson about the same thing. I would hope that having a follow-up lesson about the same information would help class A improve the test scores. Do you agree or disagree with my reasons?

Question 1. What kind of comparisons statements can you make of our class and the German class?
~The minimum value for both is about the same at 32 pounds, and that’s about it.
Question 2. Do we or do the Germans produce more trash daily? Support your answer with evidence from the box plot.
~Our class produces more trash daily; the box plot shows that our median value alone is roughly 30 pounds higher.
Question 3.  Does the class size play a difference in the results of the box plots? Why or Why not?
~No, the German class was already larger than our American classroom, and we are already producing more trash than the 42 members of the German class.
Any other questions that you think would have promoted more higher-level thinking?

Common Core Standards

My first impressions are that it seems pretty intense when reading it, almost like wait-what do I have to teach the students...my second impression was that in itself it seems pretty specific on how and what they want of the teaching standards per grade level. I tried to look at the whole of the criteria to get a better understanding, but I stuck with the file that was created.
I am interested to read what your impressions were!

Progression

Kindergarten is classifying object with a category limit of 10.
First grade is organizing and interpreting data and questioning the data collected.
Second Grade is measuring the length and representing the data on a line plot and also creating bar graphs and now having up to four categories.
Third grade is now drawing scaled graphs to represent the data and several categories; the graphs can represent a number of items per square.
Fourth grade is making a line plot with fractional numbers and solving problems that use the line plot as a guideline.
Fifth grade is making the line plot as well, and solve problems but with a different level of difficulty.
Sixth grade is learning about statistics and probability and using data collection as a way to solve statistical problems. They also continue to use the line plot but introduce other graphs, and summarize the data sets.
I’ve listed out the changes that I see from grade level to grade level, some of the ideas are the same, and they build upon each other as the year’s progress. The NCTM is a more detailed description of what the grade levels should be focusing on. 
Do you agree with how I wrote the progression out?

Common Core and NCTM

The earlier grades of K-2nd I think the standards align together quite well. But once the third-fifth grade is read, I feel like they are not all that similar. The NCTM Standards in third through fifth grade is asking students to design investigations allowing the students to formulate questions, it brings in the terms median and comparing the representations of the data, then it says the students should develop and evaluate inferences and prediction. I don’t see anything of the same detail in the CCSS. Both the CCSS and the NCTM have base instructions for the statistical process. They deal with classifying objects, understanding the data, and interpreting the results; all of which are important to the statistic process.
Do you think differently? 

Curriculum Resources

Grade 1 Would You Rather?

The lesson focuses on the question “Would You Rather be an Eagle or a Whale?” and the students learn to not only ask the question of each other but also how to make the representation. They also give reasons why they’d like to be one animal or the other. They then take what they learned from this activity and are given a new question that they are responsible for collecting data from the class.
Learning how to make representations of the given data is essential since there are various ways in while the students can accurately and inaccurately calculate the information. It says that you would have the students work in pairs to collect the data on the new question they were asked, I would have each person in the partnership collect data using a different type of representation. One could use tally marks while the others draw a picture to represent each classmate's answer. Having the students check the data that the other person gathered to ensure they recorded the same results would also be helpful. Finally deciding which way the pair is going to show the data representation, they will have to determine what collection method they liked better or how they could accurately display the results they came up with.
To make this more challenging I would have the pairs come up with their question to ask the class, they would have to ensure that the question is one that follows the “this-or-that” answers that we are currently learning. The question would need to be approved before they can begin the data collection.

Questions that could be asked of the students as they collect the data could be:
What results do you predict to get?
How do their representations align with the number of students in the class?
How are you going to collect your data effectively now that you have a question?
Did your results match your prediction?
At the discussion point, I would ask the students to describe the data that they collect.

As a result of listening to the students and observing the work that they are doing, you will be able to see which students have grasped the concept of data collection methods. Another thing we could learn is their ability to work with a partner; I would try to pair them up with different classmates each time we had activities by using the names on a stick method.
This lesson aligns with the standards because it is asking the students to collect and represent data into categories, they are formulating and asking questions, and then describe the data they collected. It follows both standards, but I think with the modification of the student formulating the questions is more so following the NCTM Standards.

Would you use this lesson plan in your classroom? If so, what changes or modifications would you make?

Module Five

Generating Meaning

 This article is about teaching the students the how to understand the true meaning of the mean, mode, and median without giving them the terminology at the very beginning. They are being asked to look for patterns, develop their meaning of the mean, mode, and median, and then understand the representation of the data set. They focus on finding the meanings of the words and then how to find them. I don't remember learning these specifically in elementary school, so I cannot truly recall how it was taught. I know when I took MAT-143 at Coastal Carolina Community College over the summer these were introduced and the teacher more so explained it. Because it was an online class it was up to us to learn and understand it, which I did from the textbook readings. 

Working with the Mean

I got the following possible answers: The first set would be 5, 7, 7, 8, 8, 9, 12

        The second set would be 5, 6, 7, 8, 9, 9, 12

~Honestly at first when you messaged me I hadn’t even looked at the assignment yet….then I did and I was like ummmm what?!?

But as I played around with the cubes I knew I had to have the average number of 8 “peanuts” per bag. I made the 5 bags I had into 8 “peanuts” as much as I could and then added pieces until I reached 8 “peanuts” per bag. Then I separated the original “peanuts” back out to each bag and played with the remaining “peanuts”. The remaining number of “peanuts” I had to place on the representation was 15, so I could have selected any combination of numbers that totaled 15. 1 and 14, 2 and 13, 3 and 12, 4 and 11, 5 and 10, and the chosen ones 6 and 9 and 7 and 8.

Because the mean is the average, using the cubes helped me average out what I had and then allowed me too visually see the numbers and what I could create with them.

The line plot….well first I think I made it incorrectly so I went back and tried again. After watching the video on the Annenberg site it made this whole process a little clearer. I made the bottom represent the bags and then color coordinated the marks, as I moved them around I crossed them off and placed the color in the new row. This again allows us to visually see the numbers on the paper to create the answer of the numbered pair for the remaining 2 bags.

The average tells us the equal shares in the distribution, which can then tell us the total number of items we have.

What do you think Aubrey? Do the pictures help, sorry it is sideways?

How Much Taller

In the video Dede answers the question of “how much taller is a fourth grader”. She says 10 inches because that is the difference between the tallest 4^th grader and the tallest 1^st grader. 

Grady answers “how tall the typical 1^st grader is” by organizing the data collected in order from smallest to largest and finding the median.

While Jason uses the mode to answer the question because he selected 53 inches as the most common height.

In Case 27, Zia is using the range to decide “if one group is taller than the other”, she recognized that the 4^th grade range is 52 to 62, while the 3^rd grade range is 51 to 54.

 Abby also mentions her use of the line plot, she states that the 4^th grade line plot starts an inch after the 3^rd grades plot. There are 5 students in 3^rd grade that are 51 inches, but no 4^th graders are that height.

In Lydia’s case, Erin has the idea of the mean, she wants to use the number in the middle and is trying to explain it to her partner. Erin seems to know that she needs the middle value but is at the beginning stage of explaining why. She is understanding that the 2 numbers are important but not sure how to use that information yet.

In Phoebe’s case, Trudy and Javier are both very close to understanding the average, but at first they only see it as a number between the calculations. They took the height of everyone in their group and divided which gave them their own group average. Though the teacher has to “pull” the information out of Trudy at the end she is able to understand that they essentially took numbers off the taller members and put them on the shorter members. This is important to grasp because to me it is the foundation of understanding what it means to find the average. The total number stays the same but the individual numbers inside can change. Does that make sense to you Aubrey?

In Nadia’s case, the first group with James, Jordan and Laurel all decided that the average was 13 not 13.2 because you can’t have .2 of something. This is problematic because while you can’t have a fraction of a letter you can still have a fraction. In the Annenberg video is shows to represent it in a fraction so it would be better understood if the answer was 13 3/20.

Another problematic student idea I found was Linnea in Phoebe’s case, she is misunderstanding the definition of the word average. Average is not the most common and this will cause her to have errors in her data set. Learning and understanding the difference between mode, median, and range is very important.

Aubrey, did you find other problematic areas or do you agree with mine?

I think the children for the most part understand how to reach the answer of the mean but not really what it means, or how to explain mean in words. I can relate, I have to stop and think about the difference between them when working with these concepts.

Do some reading and thinking about the concept of the average or mean and its application in schools through the bell curve. What does the mean suggest in terms of grades and achievement? Why is the concept represented with a bell curve? What are the implications for grading on the curve? Is it fair? Why or why not?

Ahh, the bell curve. I can remember never liking this method, as a student who prides herself on studying and getting good grades, I found it unfair that a teacher would give a better grade to another student because some did better than others. Those extreme outliers either way could potential hurt the grades of all the students. But while reading an article on grading on a curve I have come to realize that if the teacher uses the results in different ways it would be beneficial to all students, like for example if most students got certain questions right more points could be rewarded but I think my favorite way would be to round up to 100% and then add the same percentage to each students grade. The biggest downfall I see is that if a student does badly on a test and the teacher grades on a curve they are still being rewarded for not knowing the answers…but I also understand that some people, like me, have severe test anxiety and the curve could help ease the stress. Aubrey did you ever have a teacher use a curve? I had a college professor who did, but I don't think he did a curve a certain way. 

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 want to teach with DODEA so I decided to look at the pay scale for them. They have all of the pay charts available online so you can see the past few years as well.

https://www.dodea.edu/Offices/HR/salary/index.cfm

This allowed for me to see the pay increase percentage over the years. For example the starting pay for 2016-2007 for a Bachelor’s Degree and Step 0 was $46,981 the same starting level for this year has a pay rate of $48,390. The information provided shows a 3% increase per year. However, these charts do not take into account the taxes for the state that you work in or the cost-of-living in the area. The data I found does also show me the increase for each step, but I don’t know where that comes into play, and the amount if I have different times of service and degrees.

I decided to look at the pay rate for North Carolina and there is a significant pay difference between DODEA and North Carolina teachers, I hope I can get back into the DODEA system.

Aubrey, are you planning on teaching on or off base? Are you staying here in North Carolina?

Module Four

Lost Teeth Video

I believe that the purpose for questioning the students about the range for the different grade levels is, so they can share what ideas and thoughts they took from collecting the data the previous day. They can gain an understanding of the differences between the grade levels in reference to the number of teeth lost in each grade. Knowing that the Kindergarten classes have probably lost the fewest number of teeth they know that the range will be smaller, whereas, for the older grades the ranges will be farther apart. I think the students did well in trying to explain the features of the data, they had some trouble with wording but that is common for students. They were able to explain the difference between their class and the other class. I liked that they were surprised that students in the lower class had already 12 teeth, but that they made a connection that they might have had those teeth removed by the dentist. I think it was higher level thinking that the children thought this was a possibility, what about you? For the third-grade class a new category was entered into the discussion, and I really liked the explaining of the student “they lost the teeth so long ago they can’t remember.” Again, I think the students did well explaining the graph they made, but I feel like they were a little confused on the “clump” on the graph. The representation of the information on the graph was a little difficult for the students to understand due to the difference in size and the placement of the data on the graph.

Stem and Leaf Plots

Within the first few sentences I was surprised at what I was reading, having children in grade school currently I have not seen this idea of statistics being taught at all. I also don’t recall hearing about the stem-and-leaf plots prior to this article and module. Do you have any experience with this? I found it interesting that the idea behind the stem-and-leaf plot if you will be that of number placement. As I read the article and noticed the information about the “other uses” I realized that while in Barcelona the bus charts were listed the same way as the example. I had no issue reading the bus schedule while in Barcelona, but had no idea that it was considered a stem-and-leaf plot, pretty cool huh? I think including this graphing tool will be beneficial to the students as it allows them to see another way to represent the data.

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

Histograms are a way to graphically represent a given frequency. Histograms are similar to bar graphs but the histogram groups the numbers into ranges. Bar graphs are more useful when the data is categorical, and space should be left between the bars unlike in the histogram. I wasn't aware of the differences between the bar graph and the histogram until reading about it, but I found the gaping or no gaping to be most interesting, have you noticed the difference before?

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.

The data that is being represented in this line graph is the temperature changes in Seattle in March of 2012. It has given us the high and low for 14 days so that it can be graphed. This graph is showing the change in the temperature over the 2 week span of temperature collection and is being used in an appropriate manner.