High Tech High Graduate School of Education
LESSON STUDY CYCLE 1
A Focus on Mathematical Discussion
OUR LESSON STUDY TEAM
Bailey- Bailey teaches third grade. She was the host teacher for our lesson, she invited our team into her classroom to observe how her third graders approached the lesson that we planned.
Rebecca - Becca teaches Science to grades K-5. Becca's knowledge of CGI Math and her devotion to the research brought so much into the process.
Sandy- Sandy teaches kindergarten at Montessori charter school. Sandy's calm demeanor and thought provoking questions contributed so much to the process of this lesson study.
Lillyette (Me)- I teach grades 1-3 at a Montessori charter school.
THE RESEARCH- ANNOTATED BIBLIOGRAPHY
Research is an integral part of the lesson study process. In the beginning of our research and planning, our team was focused on student thinking. I read various articles that focused on getting students to show their mathematical thinking and participate in effective discourse. We also interviewed other educators to get their thoughts and ideas on carrying out effective mathematical lessons and student discourse. As we got deeper into the planning of the lesson, we turned toward articles that had more to do with mathematical thinking. We hoped that the students in our lesson would see the connection between addition and subtraction. To learn more about student thinking, discourse, and mathematical thinking, scroll down to view my annotated bibliography.
THE LESSON
In planning the lesson, our host teacher kept us updated about how her students had been performing in their math discourse, the different strategies her students were currently using to solve math problems, and the strategies and language that she hoped to hear by the day of our planned lesson. Bailey planned to present a CGI Math problem to her class. The problem type was a JSU (join start unknown), something the class had worked on before, but this would be the first time that the problem involved a three digit number.
Below is the research question that guided our math lesson, our theory of action, and goals we hoped to achieve the day of the lesson.
Our Research Question - How can we help students use academic language in regards to mathematical discussion?
Theory of Action - If we as teachers build upon students’ academic language, then students will be able to describe their math thinking, resulting in math agency.
Content Goal - Students will understand that a JSU (join start unknown) can be done by counting up or counting down.
Equity Goal- Students will listen to, value, and build off of each other’s ideas.
For a detailed look at our lesson plan as well as the document we used from day one of our lesson study, click on the links below.
STUDENT THINKING
Our host teacher has a class of 24 third grade students. As I watched her students sit at the rug listening to the problem that they were about to solve, I could tell that this experience was routine for them. They eagerly listened to the problem, "Some children were seated in the commons. 79 children joined them. Now there are 192 children in the commons. How many children were in the commons at first?" Many of them raised their hands to retell the problem. As I observed the students working, I did not see one student that stopped trying to solve the problem.
FOCUS STUDENTS
FOCUS STUDENT 1
We expected to see FS1 direct model and/or draw pictures of cubes and ten and ones blocks. We wanted to see FS1 grappling with invented algorithms/turning his direct modeling into invented algorithms.
Outcome: FS1 used direct modeling, but he also clearly labeled each ten that he drew. He was also able to come up with a clear number sentence to match his strategy. He was confident in his explanation when he participated in the discourse.
FOCUS STUDENT 2
We expected that FS2 might use an invented algorithm. We thought he would chunk “nice” numbers (For example adding by 20, 30, or 40). FS2 might explain how he turned the chunks in to “nice” numbers. FS2 might explain how he incremented bigger numbers to save time. From FS2’s past work we also believed he could still use standard algorithm to solve the JSU problem.
Outcome: FS2 immediately went to standard algorithm. In his second strategy he used an invented algorithm that did not make much sense. He changes his starting number to a completely different number, from 79 to 72, to make it even. This resulted in having to subtract 7 to arrive at the actual answer. It seemed as if FS2 used an invented algorithm to fulfill a requirement rather than to engage in and help him understand the problem.
FOCUS STUDENT 3
We believed that FS3 would draw the numbers on the paper that are written in the problem. We anticipated that FS3 would not attempt to solve unless helped by a peer or adult. We expected to hear FS3 talking about the numbers they built in terms on 10s and 1s.
Outcome: To the left is what FS3 was able to complete. He was unable to accurately solve the problem. In future math lessons, this student needs one on one support, as well as direct instruction in counting on. FS3 did not have full access to the problem that he was being asked to solve.
FOCUS STUDENT 4
We expcted FS4 to start with direct modeling. We expected that after direct modeling, she might move onto an invented algorithm, but refer back to the direct modeling to check her answer. This student was expected to try an invented algorithm because of pre-lessons that were building her up to be confident by making the jump from direct modeling to invented algorithm. We expected to hear FS4 say, “I am going to add 10 every time”, “I am going to take away 10 every time”, “I am adding/taking away 10 because it is a ‘nice number’”.
Outcome: To the left is the invented algorithm that FS4 used. This strategy was used on the back of her original strategy which was direct modeling as expected. FS4 was stuck at 102, she could not mentally subtract the 10. The host teacher gave her a 100 chart and she was able to carry out her strategy of counting down, with ease.
REFLECTING ON PARTICIPATING IN LESSON STUDY
As this cycle of lesson study came to a close, I could not help but feel that our lesson study was "incomplete", or "not good enough". I found myself going back to our theory of action, as well as our lesson plan, and goals. I wondered if they all truly aligned. While any lesson study will have guiding theories and expected outcomes, what actually happens on the day of the lesson can be changed by so many factors. What is learned from observing the lesson might not even match what our hopes were. Regardless, there are so many takeaways and revelations about student thinking and teacher moves. This first go at lesson study is a rough draft for future lesson study cycles. For a deeper reflection on my own research leading up to the lesson and a reflection on what I learned from participating in this first lesson study cyle, click below.