Lesson Study Cycle 3​
Problem of practice: How do we support students in their ability to create and interpret graphs in mathematics?
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Research Theme: How do we empower students to create and interpret visual representations of data to better understand and act in their world?
How do we empower students to become data literate by interpreting visual representations like graphs in order to better understand and act in their world?
Street Data
Focal Student Interview Summaries
Focal Student 1
The first focal student was chosen because of his solitude and lack of eagerness to work with other in class. He was able to pass math the first semester with a strong push of revising work at the end of the semester but did not engage regularly in class. He feels like people see him and acknowledge his presence but do not know how he is doing internally. He had a hard time thinking of a time he felt belonging in school but eventually talked about his last semester when his class was doing a video project, “I was the video editor. I had experience previously. It felt good to actually know what I was doing. It becomes easy.” He is unsure of what his future looks like as he would like to take a year or two to find what he enjoys doing but said "mom wants me to go to college and dad wants me to join the air force.” He is confident outside of the classroom and has a strong belief in his ability to be successful in life. He does not like collaboration because it forces him to change his approach depending on who he is working with.
Focal Student 2
The second focal student was chosen due to her quiet nature and challenges to complete and turn in work regularly during the first semester of math class. She often keeps to herself during class time and is reluctant to engage with her group. In our interview she said “I can be super quiet or very energetic depending on how close I am to them. I am hyper around my closest friends and described as shy by people who do not know me because I am quiet.” She feels like people don’t know a lot about her because she does not share a lot with the classmates that she does not feel close to. Her closest friends go to San Diego High and she transferred to HTH for her mom. She felt belonging in her biology class this year because she had already taken biology in freshman year and had done well. She had experience and felt some expertise and also liked working with oysters. She plays the role of big sister to her sister and brother and often takes care of them. She feels that challenges help us develop as people and has felt successful each time she has persevered through a challenge in her past . She feels like most aspects of her life are challenging but is determined to persist until she feels successful.
Shared Math Task
Students were given the pie graphs on the left from the New York Times What's Going on in this Graph page without the source. We had a whole class discussion about their thoughts, agreements, and disagreements with the data. It was then revealed that the data came from fourth graders and we came up with our own ideas. There was high engagement in the class discussion with some unusual status being revealed. Students, not necessarily the regular contributors to math discussions, were able to influence their class mates with each class creating a different list biggest problems. Both focal students, stated that they did not believe they were learning things in school to address the problems listed as FS 1 did not know if adults are doing a good job and stated "I feel as if we need to learn more about how to live our lives as adults" while FS 2 disagreed that adults are doing a good job and mentioned "Students should be be aware of real world situations that are impacting peoples lives everyday." This activity demonstrated that data collected from the real world and seen as directly impactful could stimulate lots of engagement in the math classroom.
My students' responses:
PDSA Cycles
Cycle 1 - Notice/Wonder with Relevant Graphs
Students provided noticing, wonderings, and feelings about the graphs above independently in a google form. We then had a group discussion about the graph. I had 28 out of 88 students (32%) correctly identify an ordered pair on the graph when asked to do so for the kid with the highest income rank. Students were anecdotally successful in identifying the general trends of the graph and identifying systemic inequalities highlighted by the graph. FS 1 did not answer the ordered pair question and FS 2 was unsuccessful in answering accurately although she was close. Neither focal student was able to explain the meaning of the ordered pair with words. Once again there was a lively class discussion when students were presented the graphs. The discussion provided opportunities for students to share who are often reluctant to share in math class although neither focal student shared out loud with the class.
Cycle 2 - Notice/Wonder with Relevant Graphs
Students were given the graph seen in the student responses below in a Desmos classroom activity where they were asked to annotate using the digital tools provided. Students provided what they noticed and wondered about the graphs before answering the following closed questions:
1. What does it mean if a country is close to the x-axis of this graph?
2. What does it mean if a country is close to the y-axis of this graph?
3. Which country is located at (24, 39)?
After students answered these questions individually, we discussed as a group and the majority of students in the room were engaged in the discussion with more volunteers than average. Students were able to identify the difference between countries above and below the y=x line after we defined what that line represented. After students answered these questions individually, we discussed as a group and the majority of students in the room were engaged in the discussion with more volunteers than average.
44/91 (48%) answered question 1 correctly
39/91 (43%) answered question 2 correctly
68/91 (75%) answered question 3 correctly
Neither focal student participated in the whole class discussion but they did provide the annotations on the graphs below:
Focal Student 1
This student interpreted the line of y=x as separating younger generational problems from older generational problems. He was able to answer question 3 correctly.
Focal Student 2
This student correctly observed that respondents older than 40 years old were more likely to have agreement percentages less than 50%. She also label the x and y axes correctly but was unable to locate the country at the ordered pair (24,39) despite annotating the graph. This represents more engagement than she displayed on most classroom activities throughout the school year.
Summary of Process
The use of informational and relevant data presented through graphs promoted lively class discussions and engagement. All students were willing and able to answer questions about what they notice, wonder, and feel about the given graphs individually including annotating on the graphs themselves. This led to more students participating in the whole class discussion and different status dynamics developing within the room. I would like to continue these strategies and look for ways to then leverage them to create the same amount of engagement when reviewing content that is less directly applicable to students' lives. For these reasons, we designed our lesson study around the presentation of relevant data and the annotation of graphs showing that data.
Lesson Plan
For a periodic function that models a relationship between time and minimum daily temperature in Pala, students will be able to interpret and annotate key features including relative extrema, periodicity, and midline to support students in understanding how rising temperatures connect to the environmental injustices affecting the Pala Band of Luiseño Indians.
Before the Lesson
Before this lesson students constructed an understanding of the shapes of the sine and cosine functions through their understanding of the unit circle. They mapped the x or y coordinate of a point on the unit circle individually versus the angle of rotation counterclockwise from 3 o'clock to form the cosine and sine graphs respectively. They then defined the important aspects of their new wave functions such as amplitude, midline, maxima, and minima. These understandings would form the foundation of content knowledge they could use to investigate the new periodic phenomenon presented in this lesson.
Lesson Strategy
The lesson begins with a short video clip of a Pala elder describing a mural at the Pala Mission. He states that the mural has snowcapped mountains because at the time of its painting it was typical for snow to fall on the surrounding mountain ranges where it no longer does. Despite the changing environment, he goes on to explain that the making of acorn meal remains an important aspect of their culture.​​
Students then are shown on the projector screen a generic graph showing the daily high, average, and daily low monthly average temperatures over the course of the year. They write down their noticings and wonderings independently in their notes before sharing out in a whole class discussion. The teacher leverages this graph to reiterate the definitions of maxima, minima, and midline.​​
Students are then given a graph showing average daily low temperature from a 6 year timespan. Both the timespan and location of the data is intentionally withheld from the students to create dissonance in their discussions as they consider the extrema and midline for their graph.
It is revealed that the graphs are from three different decades and all represent temperatures from the Pala Reservation.The three decades are 1900-1910, 1950-1960, and 2010-2020. There is then a whole class discussion revolving around the exponential growth of temperatures visible from the graphed data and the implications of those changes on the people living there.
Reflection
By employing graphing annotation strategies while students were making sense of the periodic nature of sine functions, students engaged deeply with the climate change data of the Pala band of Luiseño and Cupeño Indians. Students in the lesson study were writing independently on their graphs for the entirety of a ten minute work period. They even asked for another five minutes to continue adding their thoughts (some of which can be seen below). Some classroom structures and routines could have been employed to push this engagement toward peer-to-peer and whole class discussion to build more communal understanding but the relevance of the data to the students was evident.
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The biggest area of growth for our lesson was dropping the anchor at the end of the lesson to connect their understanding of the periodic phenomenon back to the impacts of climate change on the Pala people. To be a true social justice lesson, there is an action to be taken by the students that was not present at the conclusion of ours. This action could have been a policy recommendation or push to make this data public to the people being impacted by these changes. This type of conclusions is powerful and requires planning that was not possible on our timeline presenting a weakness in the lesson study process as presented in this program.
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Despite this limitation of the efficacy of our lesson, we found that using annotation strategies on graphs of relevant data engaged students in independent work with those graphs as well as whole class discussions around those graphs. This led to a greater diversity of voices sharing in my classroom as well as a building confidence in all students around graph literacy. This confidence resulted in greater fluency of identifying aspects of the graphs with most students in my classes including focal student 2. The next step in this research would be to learn strategies to leverage these high engagement activities to carry over to content that is less easily identified by students as relevant to their lives.
