Students enter silently according to the daily entrance routine. Today’s assignment includes some of the most commonly missed question-types from the most recent mock interim exam. When we review the answers, I make sure to focus on the third question, about overtime. This item was especially confusing on their unit test due to the wording and the students being unfamiliar with the context. I only changed the name and the numbers in the word problem to allow review of the complexity of the words in the original problem.
Students often make mistakes in problems like these because they simply do not understand the wording or what is happening in the problem. There's information they know, information they don't, and they have to figure out how to put it all together to go in search of the answer. Forming a plan or having problem solving strategies, like the use of the bar models can help with these struggles.
Students worked for 5 minutes to complete these three problems and must enter their answers into Senteo clickers. This allows me to generate instant data for each class. This way, I can prioritize which problems we should review given the constraints of time.
This is yet another example of an irrelevant Do Now that tries to do too much. The reason you see material that is not relevant to comparing distributions is that I was scrambling for time to review topics before the state test. The draw back from this scramble is that students are not reviewing relevant prior skills to address the day’s lesson. On this particular lesson I needed to return to the topic of comparing distributions because my checks for understanding indicated the first lesson did not go well. A better use of time for this Do Now would be to gauge what students were able to remember from the first lesson, rather than spiraling irrelevant material. With 90 minute blocks, it was best to infuse the last 30 minutes of class with spiral material. This is where I moved the original Do Now. The Do Now I used for this lesson this year is attached. It gauges the following skills:
I used the information from this Do Now to group students strategically for the next “Task” section. If I don’t want to have a part III review of this lesson then I have to find ways to use students as teachers. The more they discuss with each other, the better the understanding they will build. Keeping this in mind, I grouped students with range levels of understanding that made sense. In other words, I did not place students who seemed to fully understand the topic (or were close to it) with students who did NOT understand the topic (or very little of it). I don’t want students to have the opportunity to opt out and let the student who understands more do the bulk of the work/thinking. By pairing students who are closer in their level of understanding, I hope to push everyone’s thinking.
Next, students are given a copy of the class notes. They are asked to fill out the heading and the aim, which is displayed on the SmartBoard. We will review measures of central tendency vs. measures of variation and discuss how each of these can help us to compare distributions.
We begin by reviewing what each measure describes:
Next, we identify different measures of center (mean, median, mode) and of variation (range, interquartile range, mean absolute deviation). Once we have identified the purpose of each measure and the different types which can be used, we are ready for an example.
I ask one student to read the work problem at the bottom of the notes. Then, I ask students to take one full minute to read each of the temperatures per city, per month. I ask the question…
… and call on two to three different students to share. Then I ask them to calculate the mean and the range in temperatures per city, given calculators. This must be done independently. After 1 – 2 minutes, we make sure to have the same values and I ask:
The aim in this final discussion is to get students to understand that while the averages are very close in value, the ranges indicate that the temperatures in NYC vary much more than the temperatures in San Francisco. Students should be able to explain in their own words what this means. What does it mean that the temperatures vary more? This is a great way to assess whether or not different students understand the concept of variation within statistics.
Students are given their classwork and placed in groups of 4. I predetermine these groups before the lesson and make sure to mix the kids within each group. There should be at least one student who has proven to understand and explain complex math topics within this group. This student will be the table leader, placed in charge of ensuring everyone in their group is able to explain the complex ideas included in each question on paper and out loud. These students are pulled to the side for about 3 minutes and given Coaching sheets where they must report the progress of each student in their group. They are asked not to give away the answer but instead wait until the student they are helping has stated and defended their own answer.
Students are allowed to work for 15 minutes to complete the class work. As I walk around during this time, I will be assigning each group a question that will need to be put up on the board during the last 5 minutes of this section. Each group must elect one student (NOT the table leader) to explain the solution to their given problem. Two other students will be responsible for putting the work on the board in the last 5 minutes of their “work time”.
In the last 5 minutes of this section the elected student from each group will explain their solutions which should already be on the chalkboard.
It is not unusual for me to reteach this lesson as it presents complex topics within variability. I have also reflected in the past about groups of students like mine this year, which were taught non-common core math and made the switch sometime during their end of their elementary school years. I’ve observed that each year we begin our statistics unit with measures of center, median and mean, students will often ask, “what about range?” At first I wondered why they would ask about range. I realized that before common core, statistics was introduced to students by teaching how to find mean, median, range, and mode. These 4 skills/topics were taught together. Because our approach was more arithmetic, students simply learned how to find these measures without necessarily being taught what they meant. Now, they are expected to understand that measures of center and variability can yield different information and allow us to talk about data sets in different ways.
This year, when I retaught this lesson, I changed the way we completed the task. By grouping students in 4s I realized too many were opting out, allowing the higher achieving students to complete all the work. This task was completed independently and silently this year. Students were given 20 minutes to struggle with their worksheet on their own, writing down questions when they got stuck on a separate sheet of paper. Students were also asked to leave 4 lines blank under each question. As I saw questions written down, I would visit students’ desks to point out where they could find the answer in their notes. If higher students finished early, I would check their answers and then ask them to follow me and “watch how I help a teammate”. They would then be able to walk around the room, helping me to point out the answers to students’ questions in the notes. This created a lot more frustration in students, but it also yielded many good questions that when answered strategically, resulted in deeper understanding of what these measures represented about the data.
After we review the solutions to each problem, students will need to return to their seats for an exit ticket.The problems covered on the exit tickets will review the same skill practiced today: comparing distributions and variability in data sets.
Homework will also be displayed on the SmartBoard. Students will need to turn in their exit ticket at the door, before leaving class.
I changed the Exit ticket this year as well. I wanted to push students to write about this topic. It is not easy. Students have to be pushed to explain their ideas thoroughly though examples and counter-examples. Scoring this exit ticket and giving students another chance to correct any answers encourages the editing process. It normalizes error. I find that students are more likely to discuss math if they are willing to be wrong and want to be convinced through examples. So I ask them to do the same.
I also wanted to focus on comparing data primarily through dot plots. Students should be exposed to various ways of displaying data (histograms, box-and-whisker, stem and leaf, etc), but at the beginning I want students to focus on understanding what the different measures represent and how we can use them to compare data. By using the dot plot in the Exit Ticket, a form they are now familiar with, I am isolating the understanding of the measures of center (mean) and variability (range). I will know students completely understand these concepts if they are able to explain that the mean is a better measure used to compare the data because it is significantly different, as compared to the range which is approximately the same. The range in heights, which is approximately the same, tells us that the boys vary as much as the girls in their heights. The mean for the boys however, is significantly larger than the girls, indicating that they are generally taller.
I use this sample answer to help write feedback on students’ papers for their revision opportunity. Students feel better about making mistakes and developing their understanding when they are given multiple opportunities to write about it. This is how I hope to build fluency in talking about complex statistical concepts.