Lesson Analysis: The Four Dimensions
Mathematical Tasks
The worksheet that I did not end up using.
Students were given three tasks:
1.) Identify the number of dots on a dot array card and explain how they saw it.
2.) Generate combinations of five boys and girls using their five frame mats and double-sided counters, both with guidance (“We Do”) and independently (“You Do”).
3.) Write a number sentence that corresponds with the combination of five that they generated.
For the most part, I believe that these tasks supported progress toward my goals. I spent a lot of time thinking through these three tasks and trying to find a perfect balance between placing the cognitive burden on my students and providing the necessary scaffolding. This, in fact, is the basis for my inquiry question for all of Term III. In trying to answer this question I am examining the relationship between the complexity of the skill or task—the “newness” of a procedure or activity, in many instances—and the cognitive demand of the skill or task. As I planned and revised this lesson I realized that nearly all of the procedures or activities were going to be new. My students had never seen a dot array or done a number talk; they had never used a five frame or double-sided counters; they were totally unfamiliar with discourse; and they had not had much practice in writing number sentences. For this reason (and because I had not seen them do enough math to have a great sense of their understanding of these concepts) I decided to reduce the cognitive workload slightly from what I had originally planned. I took out the largest section of independent practice, which was the worksheet.
In hindsight I wish that I had included the worksheet. I think that my lesson fell on the comfortable side of the balance; I think that it could and should have been a bit more challenging. First, I would model creating one combination with my five frame mat and double-sided counters and writing a number sentence on the chart. I would then do the same thing with the help of the students. After that, I would give students the worksheet, have them generate as many combinations as they could on their five frame mats, and instruct them explicitly on how to fill out the worksheet. I would allow them to do this independently but I would also circulate as they did so, checking in with each student individually. Finally, I would bring the group back together and have them each share out a new combination they found and add it to our chart.
1.) Identify the number of dots on a dot array card and explain how they saw it.
2.) Generate combinations of five boys and girls using their five frame mats and double-sided counters, both with guidance (“We Do”) and independently (“You Do”).
3.) Write a number sentence that corresponds with the combination of five that they generated.
For the most part, I believe that these tasks supported progress toward my goals. I spent a lot of time thinking through these three tasks and trying to find a perfect balance between placing the cognitive burden on my students and providing the necessary scaffolding. This, in fact, is the basis for my inquiry question for all of Term III. In trying to answer this question I am examining the relationship between the complexity of the skill or task—the “newness” of a procedure or activity, in many instances—and the cognitive demand of the skill or task. As I planned and revised this lesson I realized that nearly all of the procedures or activities were going to be new. My students had never seen a dot array or done a number talk; they had never used a five frame or double-sided counters; they were totally unfamiliar with discourse; and they had not had much practice in writing number sentences. For this reason (and because I had not seen them do enough math to have a great sense of their understanding of these concepts) I decided to reduce the cognitive workload slightly from what I had originally planned. I took out the largest section of independent practice, which was the worksheet.
In hindsight I wish that I had included the worksheet. I think that my lesson fell on the comfortable side of the balance; I think that it could and should have been a bit more challenging. First, I would model creating one combination with my five frame mat and double-sided counters and writing a number sentence on the chart. I would then do the same thing with the help of the students. After that, I would give students the worksheet, have them generate as many combinations as they could on their five frame mats, and instruct them explicitly on how to fill out the worksheet. I would allow them to do this independently but I would also circulate as they did so, checking in with each student individually. Finally, I would bring the group back together and have them each share out a new combination they found and add it to our chart.
Mathematical Tools
A variety of tools were used in this lesson. Dot array cards were used to get students thinking about different ways of “chunking” numbers. Individual five frame mats and double-sided counters were manipulatives used to represent a real-life situation of five children going to an ice cream store. The chart paper hanging on the white board was a tool that allowed students to organize the representations of 5 students they created with manipulatives into a number sentence. On the chart I intentionally left blank spaces for students to fill in the number of boys/girls in their combination, and I intentionally did not write in the + or = symbols. I believe that this absence of a tool was helpful in that it placed more of the cognitive demand on the student to generate the appropriate symbols for their number sentence.
Overall I believe that these tools supported progress toward my goals. The dot arrays were a great starting point because of their simplicity and because of the students’ familiarity with similar images on dice. Discussion of the different ways of seeing/chunking the dots flowed quite naturally. The individual five frame mats and double-sided counters were helpful because they made concrete the more abstract image of a group of 5 children at an ice cream store.
I do believe that some slight adjustments to these tools could have made progress toward my goals even greater. Specifically, such adjustments could have facilitated a greater understanding of the real-life applications of the combinations of five they generated, thereby ensuring that students did not simply go through the motions of completing the task. For example, I could have put simple pictures of a boy and a girl on either side of each double-sided counter. I also could have changed the scenario to one that is more immediately relevant to my students. The five children given in the problem, for example, could have been sitting at a table at lunch together instead of going to an ice cream store. I have not seen a single ice cream store in the Blankenburg neighborhood, but my students sit at lunch tables every single day.
Overall I believe that these tools supported progress toward my goals. The dot arrays were a great starting point because of their simplicity and because of the students’ familiarity with similar images on dice. Discussion of the different ways of seeing/chunking the dots flowed quite naturally. The individual five frame mats and double-sided counters were helpful because they made concrete the more abstract image of a group of 5 children at an ice cream store.
I do believe that some slight adjustments to these tools could have made progress toward my goals even greater. Specifically, such adjustments could have facilitated a greater understanding of the real-life applications of the combinations of five they generated, thereby ensuring that students did not simply go through the motions of completing the task. For example, I could have put simple pictures of a boy and a girl on either side of each double-sided counter. I also could have changed the scenario to one that is more immediately relevant to my students. The five children given in the problem, for example, could have been sitting at a table at lunch together instead of going to an ice cream store. I have not seen a single ice cream store in the Blankenburg neighborhood, but my students sit at lunch tables every single day.
Mathematical Norms
Stars were put in a five
frame.
Norms were clearly outlined for students at the very beginning of the lesson as I explicitly defined the kind of behavior I expected from them. Students understood the rules: that they must listen to one another quietly, raise silent hands or give silent thumbs up when they wanted to speak, and keep their hands/feet to themselves. I told them that they would earn stars by demonstrating this behavior, and they set a goal for the number of stars they would earn. They seemed very excited about the prospect of getting 5 stars.
My Penn Mentor's observations of my behavior star system.
I believe that the establishment of this norm contributed greatly to the success of my lesson. There were a lot of aspects of this lesson that were new and out of my students’ normal routine. I anticipated, therefore, that behavior challenges might arise. Because my expectations were clear and because I framed them in a way that gave students every reason to believe that they could meet them, a lot of potential behavior problems were avoided. This created an extremely positive and productive learning environment in which students respectfully listened to one another and waited patiently for their turn to speak.
Another norm that I worked to establish was that there was no single “right” answer. This norm was not explicitly stated, but rather was implied through our discussions of multiple ways of “seeing” dot arrays and generating multiple combinations of 5. The establishment of this norm enabled progress toward my goals for two reasons. First, it helped to create an environment in which students felt comfortable sharing their thoughts or answers even if and when they were different from others that had been presented. Second, students heard a variety of solutions to any given problem and were thereby exposed to different kinds of math thinking, which broadened their understanding of part-whole relationships and number composition.
Mathematical Discourse
My pedagogical focus was facilitating mathematical discussion. I believe that that this focus helped me meet my content objective.
With the first task, the dot array, I used discourse to promote the idea that there are multiple ways of ‘seeing’ or chunking numbers, and that all of these ways are “right.” I felt that it was very important for students to grasp this concept during “the hook” portion of my lesson, as they would need to be able to use it later on while creating multiple combinations of five. The use of talk moves was quite effective in facilitating discourse. At one point I asked a student to re-voice, in her own words, what her classmate had just said about how he saw the dot array. This student’s response demonstrated that she had not heard her classmate. She articulated a completely different way—a way which, I believe, was the way that she herself saw it. The video below demonstrates my use of talk moves in this situation to clarify the point.
With the first task, the dot array, I used discourse to promote the idea that there are multiple ways of ‘seeing’ or chunking numbers, and that all of these ways are “right.” I felt that it was very important for students to grasp this concept during “the hook” portion of my lesson, as they would need to be able to use it later on while creating multiple combinations of five. The use of talk moves was quite effective in facilitating discourse. At one point I asked a student to re-voice, in her own words, what her classmate had just said about how he saw the dot array. This student’s response demonstrated that she had not heard her classmate. She articulated a completely different way—a way which, I believe, was the way that she herself saw it. The video below demonstrates my use of talk moves in this situation to clarify the point.
Had I simply accepted the first student’s answer and moved on without asking the second student to re-voice, I would not have realized that she had not heard him. The use of talk moves allowed me to ensure that the second student became aware of the fact that her classmate saw the same thing that she did slightly differently, and was equally correct.
The use of talk moves also helped me ensure that my students explained their reasoning or showed how they came to an understanding. In the following video clip I call on a student to explain why her classmate put the fourth dot in the fourth empty box of the five frame. Her answer does not give me a sense of whether or not she fully understands, so I prompt her for further participation by asking a few different questions, until I am confident that she does understand.
The use of talk moves also helped me ensure that my students explained their reasoning or showed how they came to an understanding. In the following video clip I call on a student to explain why her classmate put the fourth dot in the fourth empty box of the five frame. Her answer does not give me a sense of whether or not she fully understands, so I prompt her for further participation by asking a few different questions, until I am confident that she does understand.
I recognize that the role of discourse could have been stronger. Much of the discourse was between student and teacher. For example, after a student made an assertion and explained his/her reasoning, I often followed up with a question that prompted participation from another student. Students waited to be called on or asked a question before responding to their classmate. This is exactly what I had asked of them. Ideally, however, the conversation would flow more naturally between students and wouldn’t involve me quite as much.
Looking back on it now, perhaps I could have said something ahead of time like, “We learn best by sharing with one another because we all think about and see things differently. When we share the way we think about or see something, other people learn from us. If you agree or disagree with someone else, or if you realize that you did something in the same way or in a different way than someone else, give me a silent thumbs up so that you can share your ideas with the group.” Perhaps explicitly mentioning the value of sharing and discussion would have meant that I would not have had to serve as the intermediary quite as much. Maybe students would have been able to respond more readily to one another without waiting for me to ask questions or call on them.
Looking back on it now, perhaps I could have said something ahead of time like, “We learn best by sharing with one another because we all think about and see things differently. When we share the way we think about or see something, other people learn from us. If you agree or disagree with someone else, or if you realize that you did something in the same way or in a different way than someone else, give me a silent thumbs up so that you can share your ideas with the group.” Perhaps explicitly mentioning the value of sharing and discussion would have meant that I would not have had to serve as the intermediary quite as much. Maybe students would have been able to respond more readily to one another without waiting for me to ask questions or call on them.