Core Decisions of Lesson Design
What
In this lesson students will begin to develop an understanding of number composition and part-whole relationships through the use of dot arrays and five frames. Specifically, it is my goal that students understand how to decompose and compose the number 5 so as to be able to identify the six different combinations of 5. Their understanding of 1:1 correspondence and addition will be reinforced and built upon in the process. According to Chapin and Johnson (2006), a child’s understanding of quantity is extended beyond counting and cardinality when he/she is able to consider numbers as parts of a whole. Chapin and Johnson state that being able to “see” small numbers without counting them--using imagery to make sense of a quantity—is an important step in developing part-whole understanding. Part-whole understanding, then, leads to a better understanding of number concepts in general, and specifically of place value. A strong conceptual understanding of composition and decomposition of numbers, then, is an absolutely essential foundation for future learning of the Base-10 system and place value.
Math is not taught consistently in my classroom on the days I’m there--some days my teacher gets to it, and many days she doesn’t. However, most recently my students have been learning how to use tally marks. My lesson, therefore, will be somewhat of a shift from the content of the curriculum.
I anticipate that some students may find parts of my lesson confusing or challenging. They have not been exposed to either dot arrays or five frames so far this year. Some may, therefore, have trouble seeing small groups of numbers the first few times I show them an a dot array and may want to instead count each individual dot. Additionally, because they seem to be used to the idea that there is usually one correct answer to any given math problem, I anticipate that some of them will have trouble coming up with multiple combinations of 5. This brings me back to my pedagogical focus of facilitating mathematical discussion and my process goal of having students practice explaining and communicating their reasoning. I am hoping that as students share their strategies for solving problems they will learn together how many different ways there are to both approach a problem and arrive at an answer.
Math is not taught consistently in my classroom on the days I’m there--some days my teacher gets to it, and many days she doesn’t. However, most recently my students have been learning how to use tally marks. My lesson, therefore, will be somewhat of a shift from the content of the curriculum.
I anticipate that some students may find parts of my lesson confusing or challenging. They have not been exposed to either dot arrays or five frames so far this year. Some may, therefore, have trouble seeing small groups of numbers the first few times I show them an a dot array and may want to instead count each individual dot. Additionally, because they seem to be used to the idea that there is usually one correct answer to any given math problem, I anticipate that some of them will have trouble coming up with multiple combinations of 5. This brings me back to my pedagogical focus of facilitating mathematical discussion and my process goal of having students practice explaining and communicating their reasoning. I am hoping that as students share their strategies for solving problems they will learn together how many different ways there are to both approach a problem and arrive at an answer.
How
Students will investigate the part-whole relationship first through whole-class discussion of the use of dot arrays and five frames as tools for organizing numbers, both individual and whole-class exploration of five frames, and finally through whole-class discussion of combinations that can be made out of 5.
Tasks
1.) Students will participate in a dot array number talk.
2.) Students will engage in a class discussion about the use of dot arrays and five frames to organize and group numbers.
3.) Students will use their five frame and double-sided counters to find combinations of girls and boys that total 5 children in all.
4.) Students will share their combinations of 5 by articulating a number sentence.
5.) Students will engage in a whole-class discussion about their observations of the combinations of 5.
Discourse
Students will participate in whole-group sharing and discussion at various points throughout the lesson. My first goal is that they understand the norms around discourse so that it can be a productive tool. My second goal is that they use discourse as a tool to listen to their classmates’ strategies to solving problems in order to strengthen their own mathematical understanding.
Tools
Whiteboard, chart paper, dot arrays (3, 4, 5), 5 moveable dot cut-outs (made from construction paper ahead of time) taped onto a piece of white paper, one individual five frame per student, 5 double-sided counters per student.
Norms
Norms will need to be discussed and established ahead of time since they are not used to whole-class discussion/discourse. Students will need to know to raise their hand before speaking, to listen respectfully and actively during discussions, and to be quiet during independent practice time.
Tasks
1.) Students will participate in a dot array number talk.
2.) Students will engage in a class discussion about the use of dot arrays and five frames to organize and group numbers.
3.) Students will use their five frame and double-sided counters to find combinations of girls and boys that total 5 children in all.
4.) Students will share their combinations of 5 by articulating a number sentence.
5.) Students will engage in a whole-class discussion about their observations of the combinations of 5.
Discourse
Students will participate in whole-group sharing and discussion at various points throughout the lesson. My first goal is that they understand the norms around discourse so that it can be a productive tool. My second goal is that they use discourse as a tool to listen to their classmates’ strategies to solving problems in order to strengthen their own mathematical understanding.
Tools
Whiteboard, chart paper, dot arrays (3, 4, 5), 5 moveable dot cut-outs (made from construction paper ahead of time) taped onto a piece of white paper, one individual five frame per student, 5 double-sided counters per student.
Norms
Norms will need to be discussed and established ahead of time since they are not used to whole-class discussion/discourse. Students will need to know to raise their hand before speaking, to listen respectfully and actively during discussions, and to be quiet during independent practice time.
Why
I chose to create this lesson on my own as opposed to altering a unit from Everyday Math. This is in part because of the inconsistency with which math is taught in my classroom. Additionally, when it is taught it is often through the use of worksheets that deal more with procedures for learning math than the deep, conceptual understanding that I am aiming for. Based on my observations, it seems that students are memorizing the strategies that they are given more than anything else. Many of them do not understand number composition for the numbers 1-10, which, as mentioned before, is an absolutely necessary conceptual foundation.
There are other things that I observed about my students’ understanding that informed my decisions. First, when I taught a small-group lesson for my 'Teaching Diverse Learners' class I noticed that some of them still relied on counting the individual dots on dice—they did not recognize the numbers at a glance. This tells me that they need more practice with and exposure to grouping. Second, I noticed that while they learned how to fill in a five frame quickly, they had trouble identifying the empty boxes as indicators of how many more they needed to get to five. This tells me that the empty boxes were too abstract—they still depend on concrete tools to solve problems, a fact that is not surprising given their stage in development.
Finally, I believe strongly in the idea that learning works best when done in a communal setting. I don’t think that lessons should always be either exclusively teacher-led or exclusively independent practice. I think that children like sharing their approaches and hearing others’. Furthermore, implicit in all of our lessons should be a focus on teaching social skills such as listening, communicating, and waiting to be heard. I want to give my students a chance to learn these skills, which is why I have chosen to focus my pedagogical practice on facilitating mathematical discussion.
There are other things that I observed about my students’ understanding that informed my decisions. First, when I taught a small-group lesson for my 'Teaching Diverse Learners' class I noticed that some of them still relied on counting the individual dots on dice—they did not recognize the numbers at a glance. This tells me that they need more practice with and exposure to grouping. Second, I noticed that while they learned how to fill in a five frame quickly, they had trouble identifying the empty boxes as indicators of how many more they needed to get to five. This tells me that the empty boxes were too abstract—they still depend on concrete tools to solve problems, a fact that is not surprising given their stage in development.
Finally, I believe strongly in the idea that learning works best when done in a communal setting. I don’t think that lessons should always be either exclusively teacher-led or exclusively independent practice. I think that children like sharing their approaches and hearing others’. Furthermore, implicit in all of our lessons should be a focus on teaching social skills such as listening, communicating, and waiting to be heard. I want to give my students a chance to learn these skills, which is why I have chosen to focus my pedagogical practice on facilitating mathematical discussion.
Pedagogical Focus
Throughout the course of this lesson I will focus on the pedagogical practice of facilitating mathematical discussion. I chose this focus for a number of reasons. First, there is very little mathematical discussion that takes place in my class currently. I have not yet seen anything that I would consider true discourse, nor have I seen my teacher use many of Chapin et. al’s (2003) “talk moves.” In my own experience as a student of math, now and throughout my many years of schooling, I have been in many classrooms where discourse is used to advance understanding. I therefore believe strongly in the benefits of learning through discourse, so I would like to introduce it to some of my students and get a chance to practice it myself.