Reflection
I learned a lot from planning, teaching, and reflecting on this lesson, both about how children learn and about the teacher that I am developing into.
How Children Learn
My Penn Mentor wrote, "Not really ready to go beyond on an abstract level."
I believe that one inherently valuable aspect of using discourse in a math lesson is that it exposes children to the different ways that people think and to the idea that there is no one “right” way of doing so. It was my assumption going into this lesson that most students would pick up on new ideas and alternative ways of thinking just by hearing a classmate discuss his/her approach. In this lesson, for example, I hoped that by giving students a chance to articulate how they “saw” the dots in the dot array or how they found a certain combination of five boys and girls, those students who didn’t initially understand the concept of grouping/chunking would pick up on it. I realized, however, that mere exposure to these concepts is not always enough on its own. One of my students articulated a counting strategy for all three dot arrays, even though all four of her peers articulated grouping/chunking strategies. Simply hearing the grouping/chunking strategies did not help her internalize them, at least not from what I saw in a 45-minute lesson. This student may need more practice with, or even perhaps explicit instruction on, grouping/chunking as she continues to develop part-whole understanding.
After teaching this lesson I am still wondering about the extent to which my students understand the real-life application of the combinations of five they generated. I am confident that my students are able to generate multiple combinations of 5. What is unclear to me, however, is how these combinations are relevant to their lives. When one student suggested a combination of 5 boys and 0 girls, for example, I asked the rest of the group whether they thought this was possible. My students’ initial reaction was to say, “No.” It wasn’t until I had them visualize a lunch table that had 5 boys and 0 girls that they changed their minds.
This makes me wonder whether they truly understood the concept or whether they simply had a good grasp of the task at hand and recognized that the double-sided counters could be manipulated in a variety of ways. I have realized that just because I told them that the double-sided counters represent boys and girls going to an ice cream store, I cannot expect that it is automatically relevant to them. As Tate (2005) discusses, real-life examples are only meaningful if they have a true connection to the lives and experiences of the specific students that you are working with. Given that there are no ice cream stores in the neighborhood immediately surrounding Blankenburg, my students have probably not had many experiences taking trips to an ice cream store. While they might be able to imagine what a trip to the ice cream store looks like, therefore, this example may not have been as accessible as I thought it would be. The example of 5 students sitting at a lunch table, on the other hand, seemed to help my students visualize and understand the meaning of 5 girls and 0 boys. I will be more conscious now of the importance of making deliberate choices about the real-life connections I insert into my lessons.
As I’ve been examining through my inquiry question, a great lesson finds a way of balancing the complexity of both content and process skills. Many of the tools, activities and tasks in this lesson were new and unfamiliar to my students, so I worried that they might struggle to understand the content. What I found, however, is that my students were actually quite capable of understanding both the new tools/task and the content. I think that I did a good job of teaching the new tools/tasks by explicitly describing their function, modeling their use, and guiding students through using them. This was a powerful reminder to me that children are capable of learning quite a bit at once as long as it is taught in a way that is clear and intentional.
After teaching this lesson I am still wondering about the extent to which my students understand the real-life application of the combinations of five they generated. I am confident that my students are able to generate multiple combinations of 5. What is unclear to me, however, is how these combinations are relevant to their lives. When one student suggested a combination of 5 boys and 0 girls, for example, I asked the rest of the group whether they thought this was possible. My students’ initial reaction was to say, “No.” It wasn’t until I had them visualize a lunch table that had 5 boys and 0 girls that they changed their minds.
This makes me wonder whether they truly understood the concept or whether they simply had a good grasp of the task at hand and recognized that the double-sided counters could be manipulated in a variety of ways. I have realized that just because I told them that the double-sided counters represent boys and girls going to an ice cream store, I cannot expect that it is automatically relevant to them. As Tate (2005) discusses, real-life examples are only meaningful if they have a true connection to the lives and experiences of the specific students that you are working with. Given that there are no ice cream stores in the neighborhood immediately surrounding Blankenburg, my students have probably not had many experiences taking trips to an ice cream store. While they might be able to imagine what a trip to the ice cream store looks like, therefore, this example may not have been as accessible as I thought it would be. The example of 5 students sitting at a lunch table, on the other hand, seemed to help my students visualize and understand the meaning of 5 girls and 0 boys. I will be more conscious now of the importance of making deliberate choices about the real-life connections I insert into my lessons.
As I’ve been examining through my inquiry question, a great lesson finds a way of balancing the complexity of both content and process skills. Many of the tools, activities and tasks in this lesson were new and unfamiliar to my students, so I worried that they might struggle to understand the content. What I found, however, is that my students were actually quite capable of understanding both the new tools/task and the content. I think that I did a good job of teaching the new tools/tasks by explicitly describing their function, modeling their use, and guiding students through using them. This was a powerful reminder to me that children are capable of learning quite a bit at once as long as it is taught in a way that is clear and intentional.
My Teaching
Overall I was pleased with my ability to facilitate mathematical discourse in my first real attempt at doing so. Because of the norms I set at the beginning of the lesson, students did not talk over each other and instead listened to one another respectfully. They seemed excited to have the opportunity to explain their thinking, and they did so happily when I prompted them to elaborate through the use of talk moves. Moving into next semester, however, there are some areas in which I could stand to improve when it comes to facilitating mathematical discourse. As I mentioned in my analysis, I think that the discussion in my lesson would have benefitted from an intentional explanation of discourse and my student’s participation in it. Had I started the lesson by explaining to students that I wanted them to talk with one another about their ways of seeing the dot arrays or their combinations of five, perhaps they would have been more likely to respond to one another without me first calling on them or prompting them with a question. Next semester I will make a point of including this in an introduction to a lesson.
Something I noticed about myself as I was watching the video was that between my body language, facial expressions, and tone of voice I came across as quite serious. While it is important to convey a level of seriousness and confidence as a teacher, it is also important to remember to have fun with a lesson! I know that when I have taught in the past I have been much more dynamic and enthusiastic. I think that the intense level of preparation that went into my lesson, as well as my focus on asking the most appropriate follow-up questions, managing behavior, and assessing comprehension, distracted me a bit from the larger reason for my even being enrolled in this program to begin with, which is to enjoy the process of learning and teaching with children. I believe that with practice some of my nervousness, seriousness, and intensity will fade, but it is definitely something I want to work actively to improve upon for next semester. Especially when working with younger children I believe that it is important to celebrate student ideas and risk-taking in big, enthusiastic ways that show them that learning is fun. I was extremely impressed with my students’ understanding of the math concepts and their ability to articulate their own understanding and their classmates’. I thought that I conveyed this sense of pride and happiness to them at the time, but as I watched it I realized that it wasn’t very clear. I will keep this in mind moving into next semester.
Finally, I have spent some time thinking about how this lesson relates to my inquiry question. I think I did a pretty good job of placing a high cognitive demand on the students while also supporting them through scaffolding. This allowed my students to work in their “sweet spot,” or challenge zone, for the most part. I recognize, however, that they probably would have been capable of handling even more of a cognitive demand. I wish that I had been able to have them try the worksheet that I had originally prepared but later scratched from my plan. In a similar situation that will undoubtedly arise in the future, I hope to be able and willing to deviate from the lesson I have planned if my students’ needs or abilities are a different level than that for which the lesson is set up. This kind of in-the-moment flexibility comes, I believe, with both preparedness and confidence. This means being able to anticipate a range of possible responses from students, and preparing in such a way that you are ready to go down whichever path their responses lead you. It also means trusting yourself to improvise if their responses are not ones that you anticipated.
Something I noticed about myself as I was watching the video was that between my body language, facial expressions, and tone of voice I came across as quite serious. While it is important to convey a level of seriousness and confidence as a teacher, it is also important to remember to have fun with a lesson! I know that when I have taught in the past I have been much more dynamic and enthusiastic. I think that the intense level of preparation that went into my lesson, as well as my focus on asking the most appropriate follow-up questions, managing behavior, and assessing comprehension, distracted me a bit from the larger reason for my even being enrolled in this program to begin with, which is to enjoy the process of learning and teaching with children. I believe that with practice some of my nervousness, seriousness, and intensity will fade, but it is definitely something I want to work actively to improve upon for next semester. Especially when working with younger children I believe that it is important to celebrate student ideas and risk-taking in big, enthusiastic ways that show them that learning is fun. I was extremely impressed with my students’ understanding of the math concepts and their ability to articulate their own understanding and their classmates’. I thought that I conveyed this sense of pride and happiness to them at the time, but as I watched it I realized that it wasn’t very clear. I will keep this in mind moving into next semester.
Finally, I have spent some time thinking about how this lesson relates to my inquiry question. I think I did a pretty good job of placing a high cognitive demand on the students while also supporting them through scaffolding. This allowed my students to work in their “sweet spot,” or challenge zone, for the most part. I recognize, however, that they probably would have been capable of handling even more of a cognitive demand. I wish that I had been able to have them try the worksheet that I had originally prepared but later scratched from my plan. In a similar situation that will undoubtedly arise in the future, I hope to be able and willing to deviate from the lesson I have planned if my students’ needs or abilities are a different level than that for which the lesson is set up. This kind of in-the-moment flexibility comes, I believe, with both preparedness and confidence. This means being able to anticipate a range of possible responses from students, and preparing in such a way that you are ready to go down whichever path their responses lead you. It also means trusting yourself to improvise if their responses are not ones that you anticipated.