Module 2 Blog

 Textbook Reading

       I learned a lot of great information from the text reading this week. I thought it was interesting that the chapter opened up with sample statistics questions appropriate for eighth grade students. I will admit that this was a great refresher for me, since it’s been a while since I’ve even looked at this level of math. Considering my past struggles with math, I can certainly see how students could have a difficult time understanding these problems. I like how the book explains that when teaching students about collecting and organizing data, students must have a reason for collecting and understanding data. They need to have a question that initiates the collection of data—a question that might arise from a brainstorming session or simply from a recent discussion.

               As for recording/representing the data collected, the book mentions several ways this can be done. I gained a lot from reading about these different ways and learned that children learn to read data graphs are at different levels: the concrete stage, concrete-pictorial stage, pictorial-abstract stage, and the abstract stage. This was very interesting and helpful! It makes sense that children need concrete, tangible objects to interpret the data and build up to a stage where dots or lines would suffice (abstract stage). Other methods of graphing data were explained in the text like circle/pie graphs, line plots, histograms, etc. These are all things I can remember learning and using in my early years as a student.

        It was also interesting reading about idea of interpreting data—also known as statistics. I have never taken a statistics class before but I was surprised to see that a lot of the terms and methods described were the same ones I have learned in elementary school. For example, I am very familiar with the terns mean, median, and mode and have had a lot of experience with describing data (descriptive statistics). However, thanks to this chapter, I was introduced to a variety of new ways to help children understand these ideas.

         The text shares that probability is another method used to interpret data. I have to say that out of the things I learned in elementary math, probability was one of the hardest things for me to grasp. I was great to read up on the different perspectives and lesson/activity ideas given in the chapter.

Data and Representation—Annenberg Module

               The Annenberg lessons always provide a lot of information! This session certainly revealed a lot about data representation. The first couple of parts explained the importance of variation in data that we collect. Because of the variations, it makes sense to provide a variety of descriptions of the data. I can understand why we had to do this with the “Well-known person” survey last week. There may have been a lot of names and potential categories but it was easy to make different statements about them.

               The session also went into detail about line plots and frequency tables and how they represent the same data in different ways. It was interesting to see that, depending on the type of question, it is often easier to use a line plot than a frequency table. After working through the different “raisin” problems, I would agree that it was easier to answer the questions about individual counts using the line plots. On the other hand, the questions about a certain range of raisins were easier to solve using the frequency table. I would ask my blog partner: Do you agree with this idea? What were you experiences when completing these problems?

               I will say that I was a little confused by the idea of cumulative frequency tables. I understand the concept but had problems figuring out some of the questions. For example, in problem C5 c, the answer to the question “How many boxes have between 26 and 28 raisins, inclusively (i.e., including 26 and 28)?” is 10. After looking at the table, I can’t figure out how to come up with that answer. However, it wasn’t until the following section that it made sense. I would ask my blog partner (or anyone else): Did you have any problems when solving these questions? I will say that some of the questions were easy to solve because all I had to do was look at the table and the answers were plainly given. Moving on to Session 2, Part C, I honestly felt like the idea of cumulative frequency tables became more complicated. It just may be that I need more practice with the idea since I have never worked with cumulative frequency before.

               Part D of the session, which talked about the idea of using the median, was a lot easier for me to understand. I actually liked how the lesson explained how you are able to find the median of a set of numbers on a line plot. I didn’t know that could even be done. It was simple to just number the dots and count up to the median. However, when going over the part about finding the median on a cumulative frequency table, I was confused again. Was this part easy or hard for you to figure out?

               I’m glad to say that Part E went a little better for me because it talked about something I am already familiar with—bar graphs. I thought it was very interesting to see bar graphs in a different way using the idea of relative frequency. I will say that this was a little hard to grasp but like the other ideas mentioned in this Annenberg session, it’s something that I will have to review and practice.

 

 

Median as a Tool in Data Description Activity

 

 

 

 

1.      Describe and compare the four sets of data. Are the four grades similar? Different?

     The line plot for Kindergarten shows that a majority of the students surveyed have yet to lose any teeth. Only one student reported already losing 6 teeth. For the First graders, we see that more students reported losing teeth. The amount of teeth lost for this grade ranges from 0 to 12 whereas the range for Kindergarten was 0-6, so this was a big difference. There wasn’t that big of a difference for the range for 2^nd grade because the student who lost the most teeth only lost 13. However, we see that a majority of the students surveyed in Grade 2 (8) had lost 8 teeth. Every student in this grade lost at least two teeth, something we don’t see in the line plots for Kindergarten and 1^st grade. I noticed an interesting range take place here. Other than one student losing 2 teeth, the other children lost between 7 and 13 teeth. We see a major jump for Third grade because, unlike students in K-2^nd grade, some lost up to 19 teeth. For these 3^rd graders, 5 of them reported having lost 9 teeth while the others said they lost more or less. Similar to the 2^nd grade line plot, we see that the all of the 3^rd graders lost at least 2 teeth. It is interesting to note that two students were unsure of how many they had lost.

2.      If just the mode for each grade were reported to you what would that tell you about the data? What wouldn’t you be able to tell?

Modes for each grade

Kindergarten:  0
Grade 1: 7
Grade 2: 8
Grade 3: 9

      If we only looked at these number, we would get the idea that these are the amount of teeth these students should be losing in a given grade or that a majority of the students have lost. Many people might conclude that not many other students lost more or less teeth. However, this is not necessarily the case. Looking at the plot for Kindergarten, we see that 10 of the 17 students hadn’t lost teeth yet. With just the mode, we wouldn’t be able to tell the common number of teeth lost by other students. The same goes for the plot for Grade 1. With a mode of 7, we get the idea that most of the students lost 7 teeth. This again, is not accurate. With just this number, we don’t get to see the large range the plot shows because only 4 reported having lost 7 teeth. These 4 out of a total 20 students doesn’t seem to speak for a majority of the class. We see a steady range throughout. With just the mode for 1^st grade, you wouldn’t be able to tell that more students lost between 0-6 and 8-9 teeth than they did 7.

       The same can be said about the data collected from the 2^nd grade class. The mode alone might suggest that a majority of the students lost 8 teeth.  Without other information, we wouldn’t be able to tell that other students lost more and even less than 8 teeth. Again, the line plot shows that 1 student in this grade lost only 2 teeth, which is a big difference.

     If we only had the mode for 3^rd grade, you could certainly tell that a higher number of students (5) reported losing 9 teeth than any other amount. This may be true but solely looking at this number wouldn’t reveal that the 18 other students surveyed reported losing from the entire 2-19 tooth range. With just the mode, we are not able to see that more than half of the students surveyed had lost more or less than 9 teeth. Moreover, we don’t see that 2 of the students did not provide a number for they were unsure of how many teeth they lost. It could be that if these two students gave a number, the mode would have been different. Just a thought.
     

3.      Find the median number of teeth lost for each grade. If just the median for each grade were reported to you, how would that help compare the grade levels?

Median Levels for Each Grade:

Kindergarten: 0
Grade 1:  5.5
Grade 2:  8
Grade 3:  9

Our text explains that the term: median, means “in the middle” (p. 66). In order to find the middle number, you must place the data in order from least to greatest and then find the middle value.

    If we were solely using these numbers to compare grade levels, we could see a major increase of lost teeth as we move up each grade. For example, we can look at the number for Kindergarten (0) and 1^st grade (5.5) and see that not a lot of Kindergartners have lost teeth but will likely lose one or more teeth when they reach 1^st grade. Zero to 5.5 is a big jump so I think this also indicates that there will be a range of lost teeth in 1^st grade. From the medians of 1^st and 2^nd grade, we see another jump, perhaps suggesting that the children continue to steadily lose teeth. From 2^nd to 3^rd grads medians, there is not a significant change in numbers, so this might mean that children finish losing their teeth around this grade level/age range.

4.      If just the median and the range were reported to you, what could you say about the data? If the lowest value, the highest value, and the median for each grade were reported, what could you say about the data? Would these statistics give you an adequate picture of the data? What wouldn’t you know?

 

When determining the range of the data for each grade level, you subtract the lowest value from the highest level. Below, I have provided these ranges and the medians from the previous section:

Kindergarten: 6-0= Range-6   Median 0
Grade 1: 12-0 = 12 Range-12 Median 5.5
Grade 2: 13-2 = 11 Range 11 Median 8
Grade 3: 19-2 = 17 Range 17 Median 9

With the ranges, I think it’s easier to see an almost steady increase in teeth lost as you move up a grade level. I also think that if we only had these numbers, it would make it easier to determine the average number of teeth lost by the students. For example, the range for Kindergarten indicates that no students lost more than 6 teeth. With a middle number of 0, it shows that there were a lot of students who had lost 0 teeth, yet some had lost a certain amount. One might conclude that Kindergarten students may not be ready to lose teeth. However, I do think that analyzing the range can give us an inaccurate representation of the true data. When looking at the numbers for 1^st grade, we see a much higher median value out of a range of 12 teeth lost. What we don’t see is that only one child reported losing 12 while others lost less. If we were given the highest and lowest points, we know that students did lose a given amount of teeth but we don’t know how many students. The points in between these values are important in knowing and determining an accurate pattern. I think that having only the median and range to draw conclusions would throw the data off because only one child lost so many teeth (i.e.: 1 child in 3^rd grade lost 19 teeth) while most of the other children lost less. It does not provide a fair depiction.

 

Designing the Data Investigations Case Studies

      I will say that since taking EDN 322, I have always gotten a lot out of reading and reflecting on the case studies about teachers’ observations in their math classes. It really helps to see things from a different perspective and explore the different ways teachers have engaged students into meaningful discussions about math.

     The first scenario (Case 4) presented some great ideas about students’ observations of a set of data. I felt like the students were very engaged in the lesson and were very passionate about providing and collecting accurate data.  When Eddie and Jean Pierre caused a standstill in the data collection and analysis, it was interesting to recognize how emotional and involved the students were about fixing the problem. I liked how that the teacher and the case study writer (Sally) acted as mediators instead of teachers in this situation. They simply asked a lot of questions to guide the students to the appropriate steps to take. I think that these students learned the importance of checking your data to make sure you have an accurate count. Moreover, I think Eddie’s confusion of the initial question brought up an interesting point. Perhaps that by the end of the session, the students began to see that maybe they should have worded the question a little differently. It may have confused other students without them even knowing, possibly causing them to obtain inaccurate results. As the “asker” you have a specific question that you want answered in a specific way. I think the children in this case knew what they wanted and showed a desire to obtain the answers.

      I thought the teacher in Case #5 asked a lot of really great questions to help her students develop appropriate data questions. Many of the students learned that they had to be very specific in the way they asked their questions. Asking a question such as “How many sports have you played” is very broad. Moreover, the people being asked this question need an idea of what counts as a sport. The students that worked on the question about speaking more than one language found that revising their question was necessary because a lot of people can speak other languages, just not fluently—a major piece of information they were looking for. They learned to revise their question to gather data they wanted. The group of students that talked about questioning people about how many times they moved realized that they had to reword their question to yield appropriate results. What counts as a “move”? was the topic of discussion here. In an excerpt of her journal, I like how Michelle notes that it’s possible to gather inaccurate data “because the person might not take the question in the way the asker meant it”.

    In Case #6, I was glad to see more students grasping the importance of wording a question to gather appropriate data. I immediately noticed how engaged the students were in their ideas for questions. The “how many people in your family?” question led to an interesting discussion of what really qualifies as a “family member”. People being asked this question may count extended family members and in-laws as their family while others may not.  It was great to see these young students understand that their question needed to be reworded to collect the data they wanted. The same goes for the “how many houses on your street?” question. It was interesting to read about the problem Keith and Natasha had about asking people about how many states they had visited.  Evidently, there is a discrepancy in such a question because they did not specify what a “visit” really means. I can see Natasha’s thinking and can understand her frustration in attempting to put what she wants to know in the form of a question.

TCM Article: I Scream, You Scream

       I really enjoyed reading this article about the young students conducting a survey about their classmate’s favorite ice cream flavors. I will say that the opening comment: “That students should be able to pose questions, gather data, and represent that data in graphs seems reasonable enough – until you consider that these are the stated expectations for children just four to eight years old!” was very interesting! It’s not unusual for most people to think about graphs, data collection, and seeing answers to questions and think about a high school statistics project or research completed by college students. It makes sense that these students would know about this process but not many people think about small children doing it. It may be surprising to see young students collecting, displaying, and analyzing data but I think this opening statement implies that they are perfectly capable of completing these tasks. Moreover, I think young children have a desire to gather data—they just don’t see it as math or statistics. Children are naturally curious. They want to know everything so with them in lower grade levels, it is the perfect time to introduce them to the idea of data collection. The students described in the article seemed to enjoy completing their project and learned a lot about their peers in the process.

 

Other Questions to Consider:

Explain the importance of recording data in meaningful ways.  

    As we read in the case studies, we see how important it is to appropriately word a question you are going to ask. Moreover, it is very important to record data in ways that make sense and in ways that will help accurately display the information collected. Our text does a great job of explaining how children learn to view data in certain stages, so it’s important to make sure you are displaying the data in their appropriate stage(s). Also, I think it’s interesting to point out that the way in which you display your data depends on the type of question you ask.

The purpose of data analysis or statistics is to answer questions. Give some examples of questions that children in the lower elementary grades might want to answer by collecting data. Also give some examples for the upper elementary grades

       I think it’s safe to say that students in lower grades have a lot of questions they would love to have answered. They like to ask questions about things they enjoy or find interesting. I think about how well the students from ice cream survey article enjoyed the project. Thinking along these lines, perhaps students would enjoy collecting data by asking questions like:

·        What is your favorite color?

·        After reading two different books, which one was your favorite?

·        What is your favorite food?

       For upper elementary students, questions could begin to ask questions that have numerical values that can be displayed on many types of graphs. Questions such as:

·        How many siblings do you have?

·        How many coins do you have in your pocket?