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?

Module Two

Categorical Data Sort and Analysis

When I saw the list of names, I thought of the variety that was given. Some of the names I hadn't heard of before so it was interesting to look them up and see who are they and what I thought would be a conversation that I could have with them. I ended up doing about 5 or 6 different sorting's and one of them I chose to do was names I knew versus having to look them up. I knew of 17 of the names (counting Jesus Christ twice) and 6 of the names I was not familiar with. 

The sorting that I want to share with you is: 

Who is this person?

Activist	Actress	Artist	Political	Musician	Novelist	Computer Programmer	Religious	Royalty
Rosa Parks	Lauren Graham	Antoni Gaudi	John F. Kennedy	Drake	Jane Austen	Bill Gates	Jesus Christ	Queen Victoria
Haregewoin Teferra	Oprah Winfrey		Barack Obama	Paulo Coelho	Malala Yousafzai	Mark Zuckerberg	Jesus Christ	Queen Elizabeth I
Malala Yousafzai			Michelle Obama		Paulo Coelho	Aaron Swartz		King Leonidas
			Maxine Waters		Oprah Winfrey
			Nelson Mandela		Michelle Obama
					Mark Twain

Sorry the chart wouldn't line up any better.

If we look at the way I ultimately decided to sort the names, some of us thought about influential people of the past and present in relation to political people. Though one could classify musicians as a type of artist, I chose to keep Gaudi separate from the others. While I didn't personally think of Jesus Christ when I first read the question, I found it interesting that only two classmates thought of Him. Another thing I noticed was that a few of the names could go into different categories, and I wasn't really sure if that was okay or if it "messes" this data graph up. What do you think Aubrey?

To further this data, one could ask a more concise question such as, What one political person would you have a conversation with, dead or alive? Or What one activist? 

I did have some names that I wasn't sure how to put on my list, what I ended up doing was doing a little more research on the person to see how I could maybe rename or classify a group I already had listed. I had difficulty with where to put Queen Elizabeth, Queen Victoria, and King Leonidas, but I came with with Royalty. I originally had Mark Zuckerberg and Bill Gates as Social Media, but realized that that was not really a good fit, when I thought about what they did in creating a program I was able to also place Aaron Swartz in the group.

Students in a classroom would be able to share similar responses, I think that this activity could really show the realm of understanding that the student may have in terms of people that influence them enough to want to have a conversation. 

Aubrey, do you think I could have narrowed down my categories to have a few less columns? Do think it was okay to put different people into more than one grouping? 

Article- Statistics in the Elementary Grades

I don't remember much from my elementary years, but after reading the article I feel like we did the shoe thing at least once growing up. As I was reading the article I was surprised at the many different ways one could interpret the questions and the results. For the shoe data, it suggests "pushing the students' thinking with extension questions" for example is type of shoe related to age? I don't know if I would have thought of that question as a child, but the question is valid and it could change the data collection based on the grade level. 

For the soccer goal problem, analyzing the data was different. I don't recall hearing the fair, or equal share that is talked about in the article. Aubrey, have you heard of this before? I have not had any data experiences in my field experience classrooms that I have been in, and I would really like to see this thinking in action, and whether or not the teacher can actually engage the students in the understanding of "difference in distributions of data." Aubrey have you had the chance to see data collection in any field experience classes? 

How Many Pockets? Video

As I was watching this video I noticed that first the teacher just asked a simple question, "How many pockets do you have?" At first she didn't explain, she gave them time to count what they had on and what they considered a pocket. After a few seconds the teacher again asked the students a question, but this time it was for more of a clarification of what was considered a pocket. I noticed that once she made this comment that several of the students began to double check and recount the pockets that they had. As the students started to review the data that they had collected they were able to make connections between the number of students that had a select number of pockets, however, some students struggled with the wording concepts and needed help from classmates. I felt that the teacher was active enough in the conversation to allow the students to build upon what they had learned, she was also able to help the students create a higher level of thinking and exploring. The one thing I wasn't happy with is that the teacher mentions predictions, and we did not hear what the predictions were. 

But the students are working on learning about numerical data collection and how to process the information they gathered. Aubrey, I think the students had some confusion on counting the number of pockets they had and if they had reported it to the teacher, what do you think? 

When reviewing the data collected, I thought the teacher did well on keeping the students on the same idea to help explain the results that they saw. The teacher also made sure that the students used the proper wording of "some" instead of "most". 

Exploring Questions

When dealing with a statistical process we must first formulate a question. The question should be answered with data that can be used for an array of other information once gathered. The next step is to collect the data, this can be done in a variety of different ways depending on the question asked. The third step is to analyze the data, this is the process that separates the information they gathered into the categories that they belong with. The final step is interpreting the results, here you would summarize the data and consider other questions that could be answered based on the results.  For example, we could ask inside a classroom: What is your favorite food to eat for lunch?

Observing a group of people was difficult, but I did while waiting in line at the grocery store. So many different ways to categorize them came to mind. Did they have a purse? How did they pay for their purchase? How many items did they have in their cart? And since it is a grocery store, was it an individual shopping or a entire family? 

Any of these ways can have interesting results, looking at how they paid for the groceries can lead to more informative data collection. The grocery store could use that data to decide whether or not more self checkout machines would be useful or if having a cash only line could benefit customers. 

I never thought of all the different ways in which one could categorize a single experience but the ways are unlimited. 

About Me!

Hello Hello! My name is Tracie LaFauvre and we currently live in Jacksonville, NC. However, I am originally from Florida, and I have lived in California, Maryland, and Okinawa, Japan. I have been married for 16 years to my amazing husband Chris and we have 3 wonderful children together; Kaitlyn, Jonathan, and Stephanie. I don't have much spare time when school is in session, but we are a family of soccer players. I love to read and I enjoy cooking when I have the time. Something unique, my half sister is the same age as my older daughter. They were born 9 months apart and are like best friends when they are together. 

My family during our winter wonderland!

Math is...well after last semester I learned that math is a lot more than I thought it was. Math is having an understanding of how numbers and symbols work together to create solutions. 

When it comes to learning mathematics, I feel...better this semester than I did last semester. I know that I will learn valuable knowledge this semester from your class.

When it comes to teaching mathematics, I feel...still a little scared. I have to learn to trust myself and what I know to share with others and be able to actually explain how to do it not just keep showing them over and over again.

Elementary school mathematics means...stressful and confusing times for the students if you as the teacher do not know how to teach what needs to be taught. 

Being good at mathematics means...having the ability to understand what you were just taught by repeating or explaining the ways to solve the problem in your own words. 

What do "good" mathematics teachers do to help their students learn mathematics...they ensure that the students understand what they were taught by engaging the students in activities that allow them to demonstrate their knowledge. They allow for the students to come up with solutions and explain how they solved the problem. The teacher also understands that not all students learn the "way" and must show different methods that all students can learn from. 

I decided to have a conversation with Antoni Gaudí. Many of you know that I recently visited Barcelona Spain for the first time in November with my son. We were able to tour Park Güell, the Sagrada Família, Casa Milá, and Casa Batlló. The design of his buildings were so unique and I learned so much but I would love to ask him more about his creations. His buildings do not have straight lines, they bend and curve and replicate nature. Interesting fact about the Sagrada Família, the inside was created to represent trees of the jungle. I would love to hear more about his inspirations and what he thought about when Park Güell didn't turn out to be exactly what he thought it would. I would like to know what he thinks about the fact that he has influenced most of Barcelona's building design even today. 

Inside view with the sunlight coming in.

Here you can really see the formation of trees in the canopy. 

Outside