Module 5
Generating Meaning
This article defines mean, median, mode, and range, and lists struggles that students often have with these topics. It is a great article to help educators effectively teach and define these terms and their uses. Discussing how one number can provide information on the data set and how knowing the mean or average gives you a glimpse at the data helps us to define these things to our students. One thing that really stuck out to me was when the article said, "One word cannot describe everything about a person, but the word can provide a verbal snapshot of information about the person. The number that we find for an average provides a glimpse of the facts about a data set." That is an exemplary way to define this term to younger students. The article discussed how students struggle with understanding what the mean actually means. They understand the process to find the mean, but not what significance it has or why it is important. When I was in elementary school, there was a huge amount of time spent on teaching us how to find the mean of a data set, but I do not remember ever talking about why. Like the article, we would complete data collecting activities about the students in our class to connect us with the data and make it meaningful to us, but when it came to finding the mean, there was no real explanation given behind why we were doing it. It was not until 6th grade that I remember understanding why the mean, or average, was actually so significant. We did a data collecting project, but before we began, we went through samples of averages. There were examples of sports teams, homes in America, and so many things that we could relate to. We would talk about what information we were given when we knew the average, and what we still did not know. This really helped me to understand what was given when we found the mean of a data set.
Jess, what concept do you think you will struggle teaching the most? Mean, median, mode, or range?
This article defines mean, median, mode, and range, and lists struggles that students often have with these topics. It is a great article to help educators effectively teach and define these terms and their uses. Discussing how one number can provide information on the data set and how knowing the mean or average gives you a glimpse at the data helps us to define these things to our students. One thing that really stuck out to me was when the article said, "One word cannot describe everything about a person, but the word can provide a verbal snapshot of information about the person. The number that we find for an average provides a glimpse of the facts about a data set." That is an exemplary way to define this term to younger students. The article discussed how students struggle with understanding what the mean actually means. They understand the process to find the mean, but not what significance it has or why it is important. When I was in elementary school, there was a huge amount of time spent on teaching us how to find the mean of a data set, but I do not remember ever talking about why. Like the article, we would complete data collecting activities about the students in our class to connect us with the data and make it meaningful to us, but when it came to finding the mean, there was no real explanation given behind why we were doing it. It was not until 6th grade that I remember understanding why the mean, or average, was actually so significant. We did a data collecting project, but before we began, we went through samples of averages. There were examples of sports teams, homes in America, and so many things that we could relate to. We would talk about what information we were given when we knew the average, and what we still did not know. This really helped me to understand what was given when we found the mean of a data set.
Jess, what concept do you think you will struggle teaching the most? Mean, median, mode, or range?
Working with the Mean
Share your answers to the peanut problem.
The other two bags could have 7 and 8 peanuts, or 5 and 10.
How did you use the cubes to figure out the problems?
I used the cubes to make towers of the amount of peanuts that would be in each bag. The first tower had 5 cubes, the next had 7, then 8, then 9, and finally 12. I worked with the cubes to make even towers of 8. To do this, I took 4 off of 12 and 1 off of 9. I added 1 to the tower of 7 cubes and 3 to the tower of 5 cubes. This left me with one cube left over to start my next tower, meaning that to make 7 equal cubes of 8, I needed 15 more cubes. I could make this by adding 7 cubes to my one left over, and making a new tower of 8. This gave me my answers of 7 and 8. I figured out this way that any combination of 15 would work to get a mean of 8.
How does this model help demonstrate what the mean represents?
This model showed that the mean represents what number the data set could be split evenly into. On average, there would be 8 peanuts per bag of 7 bags, even though not every bag actually had 8.
How did you use the line plot to figure out the problem?
I used the line plot in a very similar way, mostly to check my work. I crossed off Xs and added them to lines to ensure that what I did with the cubes made sense.
How does this model help demonstrate what the mean represents?
I feel that the line plot tells us the same as the cubes did, but in an easier, quicker manner. Some may prefer working with the cubes because it will be hands-on, but I found the line plot to be easier and less frustrating.
What does an average tell us about the whole data set?
The other two bags could have 7 and 8 peanuts, or 5 and 10.
How did you use the cubes to figure out the problems?
I used the cubes to make towers of the amount of peanuts that would be in each bag. The first tower had 5 cubes, the next had 7, then 8, then 9, and finally 12. I worked with the cubes to make even towers of 8. To do this, I took 4 off of 12 and 1 off of 9. I added 1 to the tower of 7 cubes and 3 to the tower of 5 cubes. This left me with one cube left over to start my next tower, meaning that to make 7 equal cubes of 8, I needed 15 more cubes. I could make this by adding 7 cubes to my one left over, and making a new tower of 8. This gave me my answers of 7 and 8. I figured out this way that any combination of 15 would work to get a mean of 8.
How does this model help demonstrate what the mean represents?
This model showed that the mean represents what number the data set could be split evenly into. On average, there would be 8 peanuts per bag of 7 bags, even though not every bag actually had 8.
How did you use the line plot to figure out the problem?
I used the line plot in a very similar way, mostly to check my work. I crossed off Xs and added them to lines to ensure that what I did with the cubes made sense.
How does this model help demonstrate what the mean represents?
I feel that the line plot tells us the same as the cubes did, but in an easier, quicker manner. Some may prefer working with the cubes because it will be hands-on, but I found the line plot to be easier and less frustrating.
What does an average tell us about the whole data set?
An average tells us what number the data can be evenly split into. It is a round-about estimate of what is in each bag using the total number of bags and peanuts, without telling us how many is in each bag. It essentially tells us that if we were to have 8 peanuts in each bag, there will be an even number in all 7 bags and the total of all bags would still equal 56 peanuts.
Do you think this way is most efficient? I found it pretty quick and easy, but I had to think it through to understand how to initially go about it. Could I have done this in an easier way? Did you also think it was easier using the line plot?
How Much Taller?
In order to compare two sets of data quantitatively, it’s necessary to find a way to summarize each data set. In the videotape you just saw and in Maura’s case 27, students are comparing the heights of two groups. Make a list of at least five different measures or features of the data students use to compare. Cite an example from the video or from case 27 for each item and write a sentence for each to describe the mathematical ideas students are developing.
1. Using the mode was offered as a way to compare the data of first and fourth graders. Jason in the video, although he said average, offered that they could use the mode of the 1st grade data, which was 53, to figure out how tall the typical first grader was.
2. Using the median was offered as a way to compare. Grady in the video said that 51 or 52 inches could work, since the data had an even number, the median would fall between 51 and 52. In Lydia's case, Erin also offered to use the median.
3. Using their own height, knowing if they were taller or shorter than most, they could guess the average height. In the video, Samantha knew that she was on the taller side of the class data, so she proposed that they use her height and subtract an inch or two.
4. Comparing the lowest and highest numbers of each grade. In Maura's case, Emma compares the ranges to determine which class has the bigger range, and using that to compare them.
5. Comparing the sum of all the data in each group. In Maura's case, Leah suggests that they add up all of the data in each grade and compare the totals. She is starting to understand the average, but not quite there.
In Lydia’s, Phoebe’s, and Maura’s cases, look for statements from children that suggest they have a beginning idea of the mean as an average. Find two or three examples of student ideas that you think are foundational to an understanding of the mean and explain why you think these ideas are important.
As mentioned above, in Maura's case, Laura begins to understand the mean when she suggests that they add up the totals of each grade. This is the first step in finding the mean.
In Lydia's case, Tyler realizes that a lower group of numbers leads to a lower average and a higher group of numbers leads to a higher average.
In Phoebe’s and Nadia’s cases, look for ideas that might get in the way as students begin to develop an understanding of the mean as average. Select at least two student ideas and explain why you see them as problematic.
1. Using the mode was offered as a way to compare the data of first and fourth graders. Jason in the video, although he said average, offered that they could use the mode of the 1st grade data, which was 53, to figure out how tall the typical first grader was.
2. Using the median was offered as a way to compare. Grady in the video said that 51 or 52 inches could work, since the data had an even number, the median would fall between 51 and 52. In Lydia's case, Erin also offered to use the median.
3. Using their own height, knowing if they were taller or shorter than most, they could guess the average height. In the video, Samantha knew that she was on the taller side of the class data, so she proposed that they use her height and subtract an inch or two.
4. Comparing the lowest and highest numbers of each grade. In Maura's case, Emma compares the ranges to determine which class has the bigger range, and using that to compare them.
5. Comparing the sum of all the data in each group. In Maura's case, Leah suggests that they add up all of the data in each grade and compare the totals. She is starting to understand the average, but not quite there.
In Lydia’s, Phoebe’s, and Maura’s cases, look for statements from children that suggest they have a beginning idea of the mean as an average. Find two or three examples of student ideas that you think are foundational to an understanding of the mean and explain why you think these ideas are important.
As mentioned above, in Maura's case, Laura begins to understand the mean when she suggests that they add up the totals of each grade. This is the first step in finding the mean.
In Lydia's case, Tyler realizes that a lower group of numbers leads to a lower average and a higher group of numbers leads to a higher average.
In Phoebe’s and Nadia’s cases, look for ideas that might get in the way as students begin to develop an understanding of the mean as average. Select at least two student ideas and explain why you see them as problematic.
In Nadia's case, James seems to think that the average is the mode. He thinks that the mean is the number that occurs most frequently. This could cause some confusion when he needs to know the difference between mode and mean and how to find them both.
In Nadia's case again, Laurel is confused because she thinks the mean needs to be rounded. It is a bit difficult to explain to students why a mean can be something that is not a whole number when the answers themselves need to be whole numbers, and this may confuse her in the future.
In Nadia's case again, Laurel is confused because she thinks the mean needs to be rounded. It is a bit difficult to explain to students why a mean can be something that is not a whole number when the answers themselves need to be whole numbers, and this may confuse her in the future.
Find examples of averages in a daily newspaper, from the sports page, or any page. Then describe what these averages “mean” – their significance, implications within the context of the story, and so forth.
Because of my love for dogs, I decided to look up the average number of pets in an American household. This was mentioned in our article reading and it peaked my interest. In a chart online, I found the following information:
Average number of households owning dogs: 43,346,000
Average number of households owning cats: 36,117,000
Average number of dogs per household: 1.6
Average number of cats per household: 2.1
Total number of dogs in the United States: 69,926,000
Total number of cats in the United States: 74,059,000
First, I wondered how they determined what they used as their total numbers. They are clearly rounded to the nearest thousandth, and I do not know how they could know an exact total number for each animal. But I figured out that they essentially divided the total number of dogs by the average number of households owning dogs to get their number of 1.6 dogs per household, and they did the same with cats. These averages show us that if every dog-owning household in America had the same number of dogs, they would each have 1.6 dogs. However, there is no such thing as owning one sixth of a dog, which shows us that some houses have more than this number and some have less. No one has exactly 1.6 dogs, but that is about how many they would have if the number of owned dogs was split evenly.
Was that a good way to explain it, or does it sound too complex? I feel like we as college students can understand that, but do you think our students would?
Because of my love for dogs, I decided to look up the average number of pets in an American household. This was mentioned in our article reading and it peaked my interest. In a chart online, I found the following information:
Average number of households owning dogs: 43,346,000
Average number of households owning cats: 36,117,000
Average number of dogs per household: 1.6
Average number of cats per household: 2.1
Total number of dogs in the United States: 69,926,000
Total number of cats in the United States: 74,059,000
First, I wondered how they determined what they used as their total numbers. They are clearly rounded to the nearest thousandth, and I do not know how they could know an exact total number for each animal. But I figured out that they essentially divided the total number of dogs by the average number of households owning dogs to get their number of 1.6 dogs per household, and they did the same with cats. These averages show us that if every dog-owning household in America had the same number of dogs, they would each have 1.6 dogs. However, there is no such thing as owning one sixth of a dog, which shows us that some houses have more than this number and some have less. No one has exactly 1.6 dogs, but that is about how many they would have if the number of owned dogs was split evenly.
Was that a good way to explain it, or does it sound too complex? I feel like we as college students can understand that, but do you think our students would?
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 searched the annual mean wage of teachers in Georgia, since that is hopefully where we will end up. I found a great site, TeachingDegree.org, where you can enter your degree level, program, and state and it will tell you the averages. I found that the average starting salary for a certified teacher in Georgia is $38,925. The annual mean of all elementary school teachers in Georgia is $53,750. Different counties have different averages and teachers receive increases based on their level of education and time spent teaching. This essentially tells me that I can start at somewhere around $39K a year in Georgia and if I earn higher degrees, I could end up making around $54K in my career. However, it does not tell me which county I could get paid higher in, or if working in a private or public school would increase or lower these numbers. This is information that I can search for myself, but it is not given just by these averages. I also do not know the range of the salaries, or the highest or lowest paid teaching job.
I searched the annual mean wage of teachers in Georgia, since that is hopefully where we will end up. I found a great site, TeachingDegree.org, where you can enter your degree level, program, and state and it will tell you the averages. I found that the average starting salary for a certified teacher in Georgia is $38,925. The annual mean of all elementary school teachers in Georgia is $53,750. Different counties have different averages and teachers receive increases based on their level of education and time spent teaching. This essentially tells me that I can start at somewhere around $39K a year in Georgia and if I earn higher degrees, I could end up making around $54K in my career. However, it does not tell me which county I could get paid higher in, or if working in a private or public school would increase or lower these numbers. This is information that I can search for myself, but it is not given just by these averages. I also do not know the range of the salaries, or the highest or lowest paid teaching job.
Hey Megan! Awesome post. I loved you insight into the article. I think that is one of the great things about these discussions and getting to see someone else's views and help build on what I had already picked up. I think the hardest to teach will be mean. As we saw in the article sometimes it can be difficult for students to really grasp what average means and not just be able to do it without understanding it fully. I did also think the activity was easier with the line plot. I think you did a really great job of explaining. It made me realize more about the activity than I initially had. This is another concept I think is important to keep in mind as a teacher, we may so quickly go through a problem without putting enough thought it into. I really do think we can always learn a lot from our students as well! I think students would be able to understand. I think some students may need additional explanations or information, but that comes with any topic or lesson! Somethings that come easy to you may be more challenging for me or vice versa, since we are all different learners. But I definitely think the way you explained it was thorough and detailed! :)
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