Module Three

Textbook Reading

~Describe the stages of graphing experiences that children should encounter and give an example of each stage.

Aubrey, I decided to answer this question as I thought this was an important aspect to understand.

What I found was there are four different stages of the graphing experiences that we must learn about and understand. The first is that of the concrete stage; this stage is constructing graphs that deal with the student at the forefront of the material being graphed. A prime example of this would be asking the question, “How do you get to school?”.  I found it interesting that children should not be used to represent the data collection at this age since they are not able to visually see themselves in the line. I thought that this was interesting to know since I feel most teachers would use the students for the graphing, what do you think?

The next stage is the first part of the overlapping stage because it is called the concrete-pictorial stage. In this stage, the students still need the concrete ideas but now adding an object or picture of an item needed for the graph. The object should remain simple and not give the students an overabundance of options. Something the resonates as a favorite with the students is the best model to use at this stage. The example that stuck out to me was the favorite ice cream, because who doesn’t love ice cream? Aubrey, I was thinking it would be neat to have different colors of cut outs in the shape of ice cream scoops so that the students could place them on a graph to make a graph representation. Would you do this in your class? The next stage is the pictorial-abstract stage, this stage is the transition from using a bar graph of the same item to using different items to mark the number on the graph.  If you are asking them the same question of favorite ice cream, you would then use stickers or tally marks to label the selection. The last stage is the abstract stage during this stage the graphs can represent more. Meaning that instead of having a square to represent each result they can do more of a rectangular bar that represents the whole number. Line graphs are also introduced at this stage, but students need to know that the line graph represents a constant, whereas, the bar graph still represents a category or event. An example of this stage is creating a graph for the number of recycling cans students bring in for a schoolwide project. The representation of a can of soda can be equal to a given amount so that each tally mark doesn’t have to be counted.

Median as a Tool

When looking at the data, it is easy to see that majority of the kindergarten students have not yet lost any teeth, but as we move up in grades the number of teeth lost per students has changed. We still have the zero (0) range for 1^st grade, however, I the next two years the range has changed to two (2). Each grade had a least one student who has lost two (2) teeth. I also noticed that in the 1^st grade the number of teeth lost per student was spread almost equally. The mean and the mode were the same for three grade levels and very close for the 1^st grade.

The mode would tell us that most of the students lost that number of teeth. For Kindergarten we would know that most students did not lose any teeth, 1^st grade most students lost 7 teeth, 2^nd grade most students lost 8 teeth, and for 3^rd grade, most students lost 9 teeth. But we would not be able to tell how many students were asked or the number of teeth students lost.

The median would tell us the mid-point for the number of teeth lost for that grade, we would then be able to compare that number with the other grades and show an increase or decrease in the number of teeth lost. Kindergarten would tell us that 0 was the mid-point, 1^st grade has a 5.5 as a mid-point, the 2^nd grade has 8 as a mid-point and 3^rd grade has 9 as a mid-point.

Using both the median and the range doesn’t tell us much that we can use. Using the median, we will know the mid-point of the line plot, but the range can have extremes and therefore not a reliable method of analyzing the data. Knowing the highest and lowest value will allow for us to know what the extremes are of the data and the median would tell us the middle number, but it still wouldn’t give us an accurate representation of the data.

Designing Data Investigations

Aubrey, I love reading these case studies, we had them last semester too.

So, in Sally’s case, the students noticed that the number of clothespins on the chart did not represent the groupings that the children had created when they separated to count the number in each group. I thought it was neat that the teacher and Sally were being represented in the data collection, they were enablers in the conversation to gather an understanding of the question and how to answer it. By having the guidelines as to what is considered “having milk” a student was able to realize his mistake, Sally instead of accusing the student, simply asked what made him change his answer. Giving the student the chance to explain the reason could also help the other children understand the importance of the question. None of us like to be wrong, so ensuring that the question has clearly defined lines is important.

Nadia’s case study was deep in math processes. The students were set to ask open-ended questions where the person being asked would have not defining measures before the teacher got involved. Knowing what limits need to be put into the question is important, like the group asking how many times have you moved in your life. We are constantly moving from one area to another, room to room, etc. so stating your question with definitive lines is important. I think a better way to ask that question could be, has your family ever moved from one house to another house, whether it be in the same neighborhood, town, or across the country, and lived in that new home for more than 30 days? Do you think that is a clearly defined question? When the students ask the question about playing sports, the teacher challenged them to define what is consider a sport, a student replied with “it’s when you work out” while Ron went from saying it had to be a professional to playing in college, though they were asking 5^th and 6^th graders so they would not be able to collect any data.

When asking a data question, the person asking the question has the result that they want to get out of the question already in their head, so this allows for them to define the question in a way that will allow for the “asker” to receive an answer that fits into the category they are hoping for. If I want to know how many letters are in your name, I would need to specify if I want the first name only or first, middle, and last; so, the question would be better asked as How many letters does your first and last name contain? For Natasha she wants to know how many states a person has purposely visited, meaning that they had a reason to stay and they remain in the state for more than 24 hours. The final question is asking which states have you set foot in, this would give the meaning of entering the state even if just driving through or having a pit-stop for a layover on an airplane. 

What did you think about reading these case studies?

I Scream, You Scream

Wow, I can see this as a goal to be working towards, but I do not agree with it. A four-year-old shouldn’t be worried about trying to pose questions that are defined as we’ve been reading this week so that they can create and represent the data. Even looking at the four stages of progression in graphing make this opening statement seem unreal. How do you feel about this? I think that this would be more suitable for ages 7-9, do you agree?

Explain the importance of recording data in meaningful ways.

If the data collected is not recorded clearly, the information is almost useless. It needs to be clear so that any person can look at the collected data and know what the question was and understand the answers that were given. Depending on the data collected it can be used to show information that could be used for future planning. For example, traffic patterns in front of the school during certain hours of the day. Once the results are recorded the school could use the information to approach the school board for increased traffic measures.

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.

Lower elementary grades- What is your favorite color? How many pets do you have? Are you right-handed or left-handed?

Upper elementary grades- How many family members live with you? What is your favorite T.V. show to watch?