What+do+you+think?

Read p 63 #5 "How much television?" in the text for a context for the work shown below. (This graph is also on p 65 of your text.) One set of students constructed the display below. 1. Comment on what the display tells you about what the students understand and don't understand relative to constructing summarizing displays. 2. What questions would you ask the group to help you get a better sense of their understanding?

By Thursday midnight, respond to each of these questions in the space below the display. To help see different responses from your classmates, put your response in a different color from the previous response.



**What I've done is bolded statements in many of the following responses and provided a comment after the paragraph. I've numbered those comments as well to help you find them.**

Group C does not seem to have an understanding of condensing the data into more manageable forms **__where we can make interpretations based on the original questions asked.__** By looking at their graph, my first resonse would be that 25-30 students watch 120 hours of T.V a week! They should not have summed up the hours of each data interval. I would ask them after they seperated the data into five groups, why did they total the groups data. M.C

1. A good question to always ask when you consider your display.

Group C does not understand what data they were asked to represent. The axes for display b are labeled correctly, however the students conceptual idea of separating the data into five groups differs from how the teacher has asked them to separate the data. This is shown in display a, students then transferred the new data in display a onto display b. They have in fact separated the data into five groups but, the groups represent the sum of portions of the data set. ** I would ask the student how their groups of data will show parents how many hours of TV students in to class watch? ** Depending on responses I would follow up by asking group c to show me on their display how many students watched between 3 and 5 hours of TV? If they can't answer I would ask how could they change their groups to show this?

2. A good question to focus student attention on what you want this display to do. It reminds them of the overall purpose of a graphical display as well.

**It seems that the students do not understand that a bar graph is used to interpret the frequency of each interval-not the sum of that intervals ** data pieces. For example, the bar representing the interval 1-6 should reach a frequency of 10. The students must remember that each piece of data represents the hours of television ONE student reported watching in a week. They need to treat this data as individual cases (or pieces) that should not be added together. Perhaps when the students saw the word //summarize // as something they needed to do with the data they interpreted it is SUMmarize-as in add it all together. Moreover, group C does understand that in creating something that represents the data, they must initially use the given data, "tease it out" (as Dr. B would say) in a way that manages it-ie: putting it in numerical order or creating intervals, and create a graph or table from it. I might ask students questions such as, how did we get the number 1.5 in our data set? Answer being something along the lines of "from Trevor, who only watches and hour and a half of television each week" Reinforcing the idea that each piece of data needs to be treated as an individual when creating a display...not added to a classmates information. I would find it important to tell students that they are on the right track in using some sort of graph to display the information to their parents, **but ask what is most important to show their parents? ** The number of hours the class as a whole watches television, or something that shows the deviation (or shape) of our data set.

3. This is an "oops". Can you spot why? 4. Good focus question on intent of display of the variable under consideration.

The students seem to understand how to organize a graph, by labeling the axes, and deciding on intervals. **But they seem to have misunderstood what the graph should be conveying ** and what data to put into **each bar on the bar graph **. They are adding the hours together, which doesn't show how much television a student watches on average. Each data point (amount of hours of one students tv watching) needs to be plotted individually. For instance, for the interval, they should have put how many students watch 1-6 hours of tv. They have the frequency confused. I would ask the students, who constructed this graph, how many students watched 1-6 hours of TV and how is it represented on the graph. I would also ask what they wanted to show most importantly when making the graph. (JJ)

5. A critical point to notice: they have lost focus on the purpose of the graph. 6. Another "oops" given the data.

**I think that using a bar graph ** is a goog representation but they need to change what goes where on the axis. It looks like they just added up the smaller numbers (0-4hrs.), then the next size up (4.5-8hrs.). I feel that they didn't have a good understanding of how to deviate or graph the information. I would say to put a range of hours on the x axis 10-19hrs., 20-29hrs and so on. For the y axis (is where the frequency should go) I would put the number of students (1-30). I think they were thinking in the right direction but mixed things up, one thing is adding up the hours watched. I would ask them why they added them up like they did and why they didn't group them into similar hours (3+3=6hrs, 5.5 +5.5=11hrs. and then graph that). I would also ask them what other kind of displays would work for this data.

7. "Oops"

==== The students understand how to construct a Bar Graph, ** however I do not think that they understand the data they are using is numerical data and that a numerical display would give them a better representation of the data they are presenting. ** The bar graph groups the data together and they are losing data that is vital to their overall goal. They need to display this so that the parents can see **all** of the **data values** in order to understand how many hours of television **each** student is watching. I would ask the students "How many hours of television did each student watch in the 1-6, 7-12, 13-18...category?" Once they answered the question I would ask "What type of display would be a better representation for this type of data?" SW ==== 8. Relates to my point 5 and what students are not understanding or noticing.

==== ** The graph loses the value of all the individual data. It shows how many students watched tv in the intervals, but you miss out on that specific individual values. The students did a great job on keeping the intervals even with the same range. I would perhaps suggest using a stem and leaf plot to display the individual data. A question to ask students would be what does this graph show on terms of the data? Is there another type of display that might represent this data better? Why did you add up the hours into five groups? ** ====

9. I wonder if we could think of a good question that would help focus the students' attention there without just coming out and saying this. 10. Good to ask and listen to their mathematical thinking. Perhaps they have some appropriate thinking but it just went astray. We won't know unless we ask.

Creating the graph seemed to be no problem for the students. All the axes and labels are correct. What they did not understand was what the teacher was asking them to do. They were able to break the data into five groups, but they took it a step too far and they added the individual data in each group together creating five sums and then graphed the sums as the amount of children who watched television for the amount of time within that interval. What they are saying on the graph i that 15 children watch 1 to 6 hours of television a day when really the data tells us that only 6 children do. To help the students understand more clearly i would pose the question " **How many students are represented** in your data compared to the number represented **within the graph?"** depending on how they responded i would then say "How many children watched television within each interval (1-6, 7-12,...)? [SD] 11. A good question to help the students see a weakness in their graph as they would struggle to match up the number within their graph and the total in their data set.

=== I think the students understand how to graph the data but, have lost focus on what has been asked of them. The axes are labeled correctly. The intervals are even with the range. It seems as if they do not understand the data. I would ask the students, How many hours of television did each student watch in each category. What other display type would you use to record this data === 12. What evidence do we have to show they understand how to graph data?

**First off, the students were asked "how many hours of TV do you watch in a week?".. the words "HOW MANY" is the indication that we are looking for numerical data, or numbers as an answer.** These students used a bar graph to display their data, which by our class standards isn't appropriate for numerical data. Next, the students used # of students as the X axis, and put them into groups. I don't believe that you can group these students into groups,because it isn't significant. Students in group 1-6 could have easily been in group 13-18 for example, and the graph would have been completely different. By adding up the numbers of hours of tv also doesn't show them any information. Knowing that a specific 6 students in the room combined watch only 15 hours while a different 6 students in the room watch a combined 116.5 hours doesn't matter. And like I said by switching which students were in which groups would have altered the combined hours watched per group. Grouping the students is also irrelevant because we aren't comparing how many hours the first 6 students in the room watch vs the last 6 students (25-30), we are comparing how many hours each student watches compared to the entire class. I believe this would have worked if the students would have made a dot plot (with out students grouped). this way each student's amt of hours would have been easily visible and a better way to show the data. I'd first start by asking the students what would happen if they switched the students around and re-added the data, if they would get the same answers. Next I would ask them if they could locate how many hours a specific student watched in a week, if they couldn't I would suggest that this type of display isn't the best for displaying this type of data. (NI)

13. Good to focus on the question asked to gather data to help determine the variable type and from there, an appropriate display.

The students understand how to physically construct a graph (for example they labeled the x-axis and y-axis correctly, as well as titles) However they do not understand how to correctly summarize the information onto a display (this bar graph). The goal was for students to display a summary of how many hours of tv a student watches each week. The data was summarized, but not in a way to make it very clear to compare the results of each individual student. On the x-axis the students gave intervals of the students instead of each individual student. By making intervals of 7, this is too big of a range and does not show/give a specific number of hours being watched (by each student). To see how well students understand I would ask them questions such as, **"From this display, would parents be able to tell how many students watch less then 5 hours of TV a week?"** .. Depending on students response (the answer should be NO), I would ask them how can they better represent that (each individual students) better in this display.. "Is there another type of display that could represent (each individual student) data better?" 14. Similar to #2 in that it points the students in the direction of what can we really read from their display.