Module 3

Hey Jess!

Hope you're having a great week!

Textbook Reading

I am so glad the chapter was scanned into Blackboard, because I am still waiting for my book to come in! This chapter was a great refresher before doing this assignment! Going through the questions, I did not actually solve each problem, but it has been a while since I did this kind of math so it was good to review. What I found very interesting was page 69, with the Data Collection standards listed. It was so interesting to me how intricate the standards were, even from Kindergarten. Expecting students to formulate their own questions at that age is a great introduction to what they will need to do further on in school, but seems really young to start! What do you think?
Question: Explain the three measures of central tendency, and how teachers can help children to better understand these measures.
Central tendency is a way to determine the typical or centered location of the data collected. The three measures used are mean, median, and mode. In simple terms, mean is the average of the data points collected, median is the center, and mode is the most often selected answer. To remember these, I was always taught that median sounds like middle, most sounds like most, and finding the average is the hardest to do, so it is mean! To find the mean of data, add up all of the numerical answers and divide by the number of plots given. To find the median, write out each numerical answer or plot them on a line plot and cross off one at a time from each end until one answer is left in the center. If you have an even number of data, there will be two median numbers. If they are the same number, then that number is the median. If not, average the two numbers to find the median. Lastly, mode is the easiest. Which number appears as an answer most often? This is most easily found on a line plot because it is clear to see which has the highest line. The best way to teach these concepts is by actually doing them as a class! In the classroom, we could collect data as a class and determine the central tendencies of the data collected.

Median as a Tool

I have attached a picture of my line plots. At the bottom of each, I wrote the mode, median, and range. For Kindergarten, I got 0 as the mode, 0 as the median, and 6 as the range. For First Grade, I got 7 as the mode, 5.5 as the median, and 12 as the range. Did you also get 5.5 for the median? Since it is an even number of data, I got the median to fall between 5 and 6, and I averaged them. For Second Grade, I got 8 as the mode, 8 as the median, and 11 as the range. Finally, for Third Grade, I got 9 as the mode, 9 as the median, and 18 as the range. This is the only one that gave me a problem. How did you count the “don’t know” section? I used it to calculate the median, but with or without it the median stays the same as long as that data is put after 19. I feel like it is better to use it without, because “don’t know” count go at the beginning of the data or after 19, and where it is placed, could change the median. For the range, I thought of it as a 20^th option, so I did use it to calculate the range. I’ve never worked with numerical data where one option was not a number at all, so I am not sure if I should have included it. If not, the range would be 17, not 18. As for comparing the sets, the ranges got a lot larger from Kindergarten to Third Grade. In Second, the range is smaller than in First, but that is because the data starts at 2 rather than 0. This concludes that the higher grade levels have more students that have lost more teeth. The modes and medians also increased with each grade level, proving that same point. The biggest jump is from Kindergarten to First Grade, which can let us assume that students generally begin losing teeth around that time. If we only had the mode, for these data plots, we would still get the same ideas I listed above. These are unique because in 3 of the 4 plots, the median and mode are the same. We could not, however, be able to tell the increase in range, how many students were studied, or how large the data got. If we were to just receive the median, I would say the same thing. This data is unique in that way since the median and modes are identical or very similar. With just the median and range, we could say tell almost everything about the data that I listed above, however, we would not get an adequate picture of the data. For example, in Kindergarten, If we knew that the median was 0 and the range was 6, we could almost imagine that every student chose 0 except one, making 6 an outlier. However for First Grade, if we were told that the median is 7 and the range is 12, we could picture the data easier, knowing that 7 is a little over half way between 0 and 12, so there may be a good amount of data along the graph. Yet, with Second Grade, if we were told that the median is 8 and the range is 11, we may picture a good bit of data from 0-7, when in fact there are only 3 plots there. In order to get an adequate picture, we really could not tell from just the median and range. Do you agree? Did I miss a point that could help us get a more adequate picture of this data with just those statistics?

Designing Data Investigations 

Moving onto the data case discussion, in Sally’s case, the students started noticing that their peers were not answering honestly, or at least in the same way they answered with their clothes pins, and it was changing the data. Sally and the teacher are included in the poll, but their roll in this activity is more to guide the conversation and decisions rather than tell the students what is right and wrong. They are encouraging their level of thinking and analyzing, while teaching them ew vocabulary to use in mathematics. These students are learning that sometimes “yes” and “no” are not adequate answers if the question is not properly defined. In mathematics, these questions and categories must be previously defined in order to get accurate data. What if Terri hadn’t brought up her confusion and simply marked “yes?” What if the girl wearing overalls had also put yes? This would have changed the data, whereas when it was defined, Terri put no. As for emotional issues, students are worried about their peers’ honesty. How can they avoid people putting the wrong answer, or choosing one side because more teachers were there? With the “Are you wearing jeans?” question, they were able to decide as a class who was wearing jeans if there was any confusion. No one could lie and not have to prove it. However, while they can define what counts as milk, they do not know who actually had milk with breakfast and they cannot prove it. They are learning that when taking surveys, you may not always get honest answers and that can change your data. That is one of the reasons it is so important to survey a large group of people when needing reliable data.

In Nadia’s case, the students learned just how greatly their results could differ if questions were not properly defined beforehand. Some words, such as “moving” can be interpreted as something specific like moving houses, or something very vague, like physically moving your body. Or with the language question, many people know bits and pieces of a language, but asking them if they speak it fluently is a whole different question. It seems that the students knew what they were trying to ask, but as the teacher asked them questions about their data questions, they decided that their questions were not specific enough to try to guarantee accurate results.

With Andrea’s case, it was made clear how difficult it can be to pose a concise, yet specific question. While you may mean one thing in asking the question, if it is not posed properly, the answers you get may not be what you are looking for at all. With the question “How many people are in your family?” the question needs to be made more specific based on what information you want. If you are trying to find out how many people are in their large extended family, it may be better to ask “How many living relatives do you have?” but many may not know. Otherwise, if they meant immediate family, you could ask “How many people are in your immediate family, such as parents and siblings?” With the question about houses on one street, the question itself is not the issue, but the fact that some people may have really long streets, or not live in a house at all. The people who live on long streets could essentially find out how many houses are on it, but it would be more difficult for them. However, for people who live in an apartment, the question could be clarified as “Or, how many apartments are in your complex?” or on their hallway. Can you think of a better way to word that question that was not talked about in the case study? Something that would be specific and clear? In Natasha and Keith’s question, Natasha is clearly unhappy with the outcome. She wanted a very specific answer, almost like asking how many states they have vacationed to. However, it ended up being “How many states have you set foot in?” when she really did not care about states people spent five minutes in. They possibly could have asked “How many states in the United States have you vacationed to or lived in?” and gotten a more specified answer like she wanted. Even “stayed overnight in” could have helped her, however, some layovers last overnight and she did not seem to care much about those types of visits. How can we as teachers help students who are struggling to verbalize the question they want to ask?

I Scream, You Scream

In the I Scream, You Scream article, the first statement is a bit surprising. As the author explained, expecting students as young as age four to gather and represent data is already a bit to ask, as they may struggle in keeping track of it all. However, the part of this statement that got me was that they need to be able to pose the questions for their data. With the case studies, we just saw how difficult it can be to ask a clear and concise question and ask it in a way that is specific enough to get the answers you want. This is a struggle for even adults, so how do we expect this from a 4-8 year old? This would possibly be a good age to introduce them asking their own questions, but it would probably be a whole class discussion explaining why some questions are not clear enough and helping them clarify. They are probably too young to do that completely on their own.

Explain the importance of recording data in meaningful ways. 

While asking the question is very important as we have seen, it is also necessary to record the data received in a meaningful way. The data results need to be organized and clearly represented to share with others. If the data is all over the place, it is very hard to easily pick out clear answers. For example, when we made the line plots for number of teeth lost in each grade, it was much easier to compare the data and analyze it than it would have been if we only had the initial chart to look at.

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.

Questions that lower grades may ask:

What is your favorite color?

How many siblings do you have?

Do you wear glasses?

Questions that upper grades may ask:

What is your favorite school subject between Math, Science, Language Arts, and Social Studies?

What form of transportation do you use to get to school?