Context The structure for our study lesson is “Let the Chalk Do the Talk,” in which students work silently in groups to explore new concepts and make conjectures and generalizations. In their groups, they communicate through writing and pictures, asking questions of each other and the teacher, who is circulating around the room and offering questions or suggestions to groups in order to advance their mathematical thinking. As students conclude their work on the task, they are able to work cooperatively on an additional challenge problem that requires the students to build upon the knowledge that they have just acquired and apply it to a related (yet with key differences) situation. Afterwards, we debrief the task as a class, asking groups to share specific observations that the teacher noticed on their papers with the class, with a particular focus on showing differing strategies in a scaffolded manner. We have selected this structure for our study lesson with the goal of developing our students’ ability to accurately utilize mathematical language, as well as our own skill with asking questions to advance mathematical understanding and generate discussion without lowering cognitive demand. Often, in our dialogue with students, we inadvertently (or sometimes intentionally) give hints that indicate a direction for students to pursue in the task, which effectively lowers cognitive demand by specifying a pathway. High cognitive demand tasks require students to utilize diverse kinds of thought and skill, which in turn produces the most significant and valuable outcomes for student learning. Through this lesson study, we have hopes to gain a deeper understanding of the ways in which we lower cognitive demand and methods of questioning and guiding students that minimize the diminishing effects on cognitive demand. Initial Response
I taught this lesson to three classes, and the class which was observed was noticeably the least enthusiastic and engaged. In this group, there is a significant range of how much the students are willing to work and cooperate with me. In particular, there are a few students who seem to have recognized that although I am the only one who directly instructs them, my mentor teacher has more authority. As a result, when she leaves the room (as she did in during this lesson), this group begins to question me and what I am asking them to do far more than I expect they would have had she been present. That being said, the majority of the class was engaged and contributing to their group’s progress through the task, though a few students continued to cooperate less than I would have liked. In enacting the lesson, I did suggest a few strategies (drawing pictures and making a table), as well as overtly hinting that the amount of fencing was the perimeter (as opposed to the area) of the rectangle. Hinting that the fencing was the perimeter was in response to a misconception in an earlier teaching of the lesson that several students had struggled to move past -- after several hints they were still wanting 24 to be the area rather than the perimeter -- so I had decided that the suggestion would be beneficial and serve more to focus their thinking than to explicitly specify a pathway. I considered that suggesting strategies for how to show their thoughts (drawings, table) might influence how students went about the task, but it would not specify a mathematical pathway so it would not diminish the cognitive demand. As I explained the task, I had several students who immediately objected to the premise of communicating only through writing. One student in particular required an enormous amount of my energy in attempting to focus him on the task and create as many entry points as possible. As a result, I was less able to push other students to work with their groups and cooperatively solve the problem. However, in examining the papers after the task, I happily observed that each group had at least two or three of its members participating in drawing the rectangles, which was the most fundamental portion of the task. I would have liked to see the students more equally participating across the whole task, but I was reassured that many students had grasped the mathematical big ideas of the lesson after seeing how each had engaged in the task. Revisions to Lesson One simple revision which we will be making in our future enactments of the lesson is to better sequence instructions, such as having students select their color for writing after all instructions are given so that the “twistables” (or whatever writing implement we use) are not a distraction to students. On a related note, we will also ask them to put these away before beginning the post-task discussion and wrap-up. This is for two purposes: firstly, to prevent them from being a distraction, and secondly, to ensure that they are not adding to their work during the discussion in order to allow us to better assess their thinking during the task (as we can assess their thinking during the discussion by examining the video to see what they said/how they said it). We are also revising our lesson to clarify instructions and ensure that students are focused on what they should be doing and how to respond to encountered difficulties. Specifically, we will redirect questions onto the class, “What do you think you should do if…?”, and ask the class to repeat back key points from the directions. I found that several students had similar questions, and I could have done a better job of ensuring that everyone understood the first or second time around had I done this. A new instruction which we will be adding is to remind students that their group should work together on every question (not each working on separate questions) and that they may get up and move around their table to be able to better contribute. Depending on the skill level of the class, we may also read the questions from the assignment aloud, but with caution to avoid lowering the cognitive demand by suggesting pathways. I found that a few students were a little confused by the context and might have benefited from this additional processing time. In order to facilitate the sharing of ideas from a wide range of students (especially those who are disinclined to volunteer), we will be asking groups as they are completing the task to share specific ideas in the discussion or to allow us to show their method on the projector. As we would like to show a variety of strategies, we may also prompt groups who complete the task quickly to look for additional strategies or ways to organize their work. Surprisingly, although I suggested that students draw a table (which they have done many times before), no students in the observed hour did this or used any other obvious organizational system. I noticed that many students required some prodding to push them to see the generalization and to explain how to apply that knowledge once they found it. I think that a reason for this was the way the questions were formatted, where we asked students to do a few things within the same question. Although the intent of the activity is for students to develop their own thinking, I think that some of them would have benefited from a little more guidance by means of the question formatting (e.g. break down the questions into smaller parts, more explicitly specify what constitutes an acceptable answer to that question, etc.). In response to my prior observations of these students, I had split the third and fifth questions into two parts, which I believe helped students to understand which questions they were being asked to answer. However, I think explicitly stating what we are expecting to see in their answer (e.g. “at least five possibilities,” “and an explanation of your work,” etc.) without specifying what pathway to take (in order to preserve cognitive demand) would be beneficial and enhance student outcomes without lowering the cognitive demand. Influence on Future Practice One place where I felt that I struggled with this lesson was to “hook” disengaged students and help them find entry into the task. Specifically, one student immediately reacted to finding out that there would be no talking in the task by loudly announcing “I can’t do that!”, and seemed to shut down for the remainder of the class. Throughout the time that students were working on the task, I frequently attempted to create entry points for this student and encourage him to try something even if he was unsure that it was correct. This has been a recurring problem with a few students, though this was the most significant resistance that I have received. I have been working more individually with a few of these students to help raise confidence in their ability to complete tasks and build relationships, though it has been a slow process and one which I plan to continue. One new strategy which I plan to take is to help these students create specific goals to focus on when they are having a bad day, focusing them on a specific skill/concept for the day, since I have noticed that these “bad days” can snowball as they become increasingly overwhelmed. Although asking them to focus on a single skill/concept for the class is asking them to do less than their peers and they will need to catch up, it is better than them being overwhelmed and giving up immediately. I have also recently noticed (particularly during the observed lesson) that a few students seem to view me as significantly less of an authority figure than my mentor teacher, though she hardly ever interacts with the class beyond assisting me in helping students during individual work time and reminding students of expectations. In response to this issue, I am planning to set up an arrangement with the students which asks students to select who they would like to sit with, explain what they will do to be successful in these new seats, and suggest fair consequences if they are unable to be productive, responsible and respectful in their new seats. The students seem to respond well to me discussing this issue as a trade, where I will allow them to sit with friends as long as they reciprocate with being actively engaged in whatever I am asking of them. This experimental seating arrangement will soon be implemented, and I will be actively observing how students are working together, with specific attention to the following questions:
Goals for Future Lesson Study In this round of the lesson study, I was the acting teacher, so I look forward to being able to look at the lesson from a different perspective. Specifically, I am excited to be able to focus on specific facets of student engagement and teacher questioning which I was less able to observe from my perspective of being too fully immersed in the actual teaching of the lesson to truly analyze what was happening as it happened. For the group, my goals are that we are able to better predict student responses and misconceptions in order to better develop questions to advance their thinking from wherever it stands. I would like for us also to establish means to engage students who are struggling with the premise of the task (not speaking) or with being actively engaged in the task. Personally, I would like to hone my skills in developing questions that advance student thinking and help them progress through the task without lowering cognitive demand by being overly generous with hints or suggested problem-solving pathways. In planning each successive lesson, I would like to improve my own ability to both plan questions in advance, but I hope also to more naturally create questions on the spot in response to actual student thinking. In gaining more experience in the classroom, I believe that my skills in predicting student thinking and understanding how to move it forward without lowering cognitive demand are improving and will continue to do so, but I think that the cooperative planning and observational aspects will enhance my learning of these skills. My goal for observing the second round of lesson study is to evaluate my own practice and identify practices employed by my co-teachers which I could apply in my own classroom. Specifically, I would like to examine the techniques which they use to address two major issues which I encountered in the enactment of my lesson: eliciting more equal participation and responding to resistance from students. Although I had at least half of the members in each group contributing to the drawings (and all contributed to some degree), participation overall was noticeably uneven, typically with one or two members dominating the work effort. I believe that these two issues are intertwined, in that the students who were resistant from the beginning were also those who were less engaged throughout the task. I would like to examine the strategies (beyond responding consistently to these issues, which is also a skill upon which I need to improve) that my co-teachers employ to address these issues and work to apply these strategies in my own practice.
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Lesson StudyA project in which I partner with fellow teacher interns to collaboratively develop a high cognitive demand lesson over the course of a year. We implemented our lesson in multiple grade levels at Owosso Middle School and Grand Blanc East High School. Archives
April 2016
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