As I planned for Phase 2, I designed my plans based on those that I had formulated at the end of Phase 1. My plans included the following:
Additionally, I moved the lessons to the rug where students sit for calendar time. Instead of sitting at their desks and coming to the board to share their responses, students sat in a horseshoe shape around me. This enabled me to be on the same eye-line as the students as I watched them work on the problems, and it provided us with a more intimate atmosphere for sharing and discussion their solutions.
I also made a very important decision prior to Phase 2 to eliminate the use of written responses. I knew based on the limits of my classroom's schedule that I would only be able to conduct a maximum of eight lessons for Phase 2. This meant that in order to understand what the struggling students needed with respect to CGI, I would do best to eliminate one of the strategies that was most distracting to them. Though some students had used pencil and paper to record their responses to the CGI questions, many students were not using pencil and paper in what adults would consider "useful ways" to help themselves solve the problems. Thus, I chose to have students focus on using the manipulatives to show their work and find solutions to the problems. (See a more in-depth analysis of why I did this in the Results of Phase 1.)
The First Two Lessons: Now What?
The second lesson of Phase 2 was much like the first lesson. Students worked in mixed-level groups doing Joint Result Unknown and Separate Result Unknown problems. Of the fourteen kindergarteners who I did small group work with during the second lesson, eight were able to solve the problems independently. Two students understood that they should first count out the initial amount, and spent most of the lesson trying to count accurately. Four students started by trying to count out the initial amount, but got distracted or did not seem to understand what the next step should be.
Having mixed-level groups seem to have mixed benefits. As I noted, some students could use the manipulatives to help themselves solve the problems. Other students seemed to get lost, and in at least two cases during those first two lessons, they spent part of the lesson making shapes with the unifix cubes rather than counting them.
Mixed-level groups also presented another challenge: while some students could work with high numbers (20 to 30), other students were still practicing numbers less than 20. This meant that some students were still setting up the problems as other students finished solving them. This made sharing solutions more difficult, because not all the students had found the solution yet.
I began to see similarities emerging among the students who struggled with solving the problems. Based on the excitement that I observed, and their overall engagement in the problems, I could tell that the students enjoyed CGI. However, there were six students who seemed distracted, struggled with setting up the problems, or did not seem to understand what was being asked. The student whose work is represented in Sample 5 of Phase 1 (who I will call Daphne) was able to set up the initial amount using manipulatives during Lessons 1 and 2. However, I could see that there was some aspect of the problem-solving process that she was having trouble understanding. In order to consider the needs of the struggling students more deeply, I recorded audio of me and Daphne working together on two problem after the rest of her group had finished Lesson 2. A transcript is below:
Ms. Berger: Are you going to go shopping with your mommy for Valentine’s Day chocolates?
Daphne: Yeah!
Ms. Berger: We’re going to pretend the blocks are chocolates. How many chocolates do you have?
Daphne: [counts them] I have seven.
Ms. Berger: Okay, so you and your mommy went shopping and you bought four more chocolates. How can we show how many you had after your shopping trip using the blocks?
Daphne: We take away.
Ms. Berger: We take away? Why are we going to take away?
Daphne: We’re not going to take away.
Ms. Berger: Okay. You choose. Do you think we’re taking away, or do you think we’re going to add chocolates?
Daphne: We’re gonna add.
Ms. Berger: Okay. Can you show me how to add? You bought four more.
Daphne: [pause as she looks at the blocks]
Ms. Berger: There’s extra blocks over here. [pushing them toward her] What can we do to show how many we had after we bought the cookies?
Daphne: Let’s count. [counts the two groups of blocks separately]
This dialogue is the from the second problem Daphne and I did together. In the first problem we did (a Separate Result Unknown or subtraction type problem), Daphne was able to count out the initial amount and demonstrate "taking away" by removing manipulatives and putting them back in the basket. In the second problem we did, however, Daphne did not seem to understand the relationship between the wording of the problem ("bought more chocolates") and the act of counting out more blocks to add to those that she already had in front of her. I was confused about why Daphne seemed to struggle with this particular problem. Furthermore, as I read the transcripts, I realized that my confusion was causing me to push Daphne towards a solution. While I wasn't sure why Daphne was having difficulty, I understood that she and some of the other students might need more explicit instruction in how to use the manipulatives and how to interpret the action of the problem. (Read more about my insights into Daphne's confusion in the Results of Phase 2.)
Lessons Three Through Six: Making Changes
In order to help the struggling students improve their understanding in the small amount of time that we had together, I resolved to make one group composed of the students who typically had the hardest time demonstrating the problems. With this group (which I will refer to as Group 1), I would use the plans that I had written from Phase 1, but I would emphasize key action clue words such as "more" and "altogether" for addition, and "took away" and "were left" for subtraction. By doing so, I hoped to make more explicit connections between the everyday language that these students used for informal math (such as that they might use at home with their parents, or on the playground with friends), and the problems that we were doing together. I also decided to begin using bear-shaped manipulatives that we had in our classroom in place of the Unifix cubes, based on the fact that students (particularly in Group 1) sometimes lost their focus as they tried to place the Unifix cubes in fixed rows.
Work & Results from the Final Lesson
To the right is an example of a problem that I did during my final lesson with Group 1. My story for this problem was:
Ms. Berger: Gary [a student in the group] found 12 four leaf clovers at recess, and put them in his backpack. He found 3 more at lunchtime, and put those in his backpack. How many four leaf clovers did Gary have in his backpack after lunch?
- Breaking the class into three mixed-level groups in order to observe and discuss strategies more closely.
- Using specific questions to elicit and get a better understanding of student thinking, such as "Can you show me what you were thinking?" and "Can you explain why you did that?"
- Creating problems based only on Join Result Unknown (addition), Separate Result Unknown (subtraction), and Part-Part-Whole (addition with multiple addends), in order to let students become comfortable with those problem types.
- Tracking students' responses in order to gather evidence on the changes in their use of various strategies in response to the problems.
Additionally, I moved the lessons to the rug where students sit for calendar time. Instead of sitting at their desks and coming to the board to share their responses, students sat in a horseshoe shape around me. This enabled me to be on the same eye-line as the students as I watched them work on the problems, and it provided us with a more intimate atmosphere for sharing and discussion their solutions.
I also made a very important decision prior to Phase 2 to eliminate the use of written responses. I knew based on the limits of my classroom's schedule that I would only be able to conduct a maximum of eight lessons for Phase 2. This meant that in order to understand what the struggling students needed with respect to CGI, I would do best to eliminate one of the strategies that was most distracting to them. Though some students had used pencil and paper to record their responses to the CGI questions, many students were not using pencil and paper in what adults would consider "useful ways" to help themselves solve the problems. Thus, I chose to have students focus on using the manipulatives to show their work and find solutions to the problems. (See a more in-depth analysis of why I did this in the Results of Phase 1.)
The First Two Lessons: Now What?
The second lesson of Phase 2 was much like the first lesson. Students worked in mixed-level groups doing Joint Result Unknown and Separate Result Unknown problems. Of the fourteen kindergarteners who I did small group work with during the second lesson, eight were able to solve the problems independently. Two students understood that they should first count out the initial amount, and spent most of the lesson trying to count accurately. Four students started by trying to count out the initial amount, but got distracted or did not seem to understand what the next step should be.
Having mixed-level groups seem to have mixed benefits. As I noted, some students could use the manipulatives to help themselves solve the problems. Other students seemed to get lost, and in at least two cases during those first two lessons, they spent part of the lesson making shapes with the unifix cubes rather than counting them.
Mixed-level groups also presented another challenge: while some students could work with high numbers (20 to 30), other students were still practicing numbers less than 20. This meant that some students were still setting up the problems as other students finished solving them. This made sharing solutions more difficult, because not all the students had found the solution yet.
I began to see similarities emerging among the students who struggled with solving the problems. Based on the excitement that I observed, and their overall engagement in the problems, I could tell that the students enjoyed CGI. However, there were six students who seemed distracted, struggled with setting up the problems, or did not seem to understand what was being asked. The student whose work is represented in Sample 5 of Phase 1 (who I will call Daphne) was able to set up the initial amount using manipulatives during Lessons 1 and 2. However, I could see that there was some aspect of the problem-solving process that she was having trouble understanding. In order to consider the needs of the struggling students more deeply, I recorded audio of me and Daphne working together on two problem after the rest of her group had finished Lesson 2. A transcript is below:
Ms. Berger: Are you going to go shopping with your mommy for Valentine’s Day chocolates?
Daphne: Yeah!
Ms. Berger: We’re going to pretend the blocks are chocolates. How many chocolates do you have?
Daphne: [counts them] I have seven.
Ms. Berger: Okay, so you and your mommy went shopping and you bought four more chocolates. How can we show how many you had after your shopping trip using the blocks?
Daphne: We take away.
Ms. Berger: We take away? Why are we going to take away?
Daphne: We’re not going to take away.
Ms. Berger: Okay. You choose. Do you think we’re taking away, or do you think we’re going to add chocolates?
Daphne: We’re gonna add.
Ms. Berger: Okay. Can you show me how to add? You bought four more.
Daphne: [pause as she looks at the blocks]
Ms. Berger: There’s extra blocks over here. [pushing them toward her] What can we do to show how many we had after we bought the cookies?
Daphne: Let’s count. [counts the two groups of blocks separately]
This dialogue is the from the second problem Daphne and I did together. In the first problem we did (a Separate Result Unknown or subtraction type problem), Daphne was able to count out the initial amount and demonstrate "taking away" by removing manipulatives and putting them back in the basket. In the second problem we did, however, Daphne did not seem to understand the relationship between the wording of the problem ("bought more chocolates") and the act of counting out more blocks to add to those that she already had in front of her. I was confused about why Daphne seemed to struggle with this particular problem. Furthermore, as I read the transcripts, I realized that my confusion was causing me to push Daphne towards a solution. While I wasn't sure why Daphne was having difficulty, I understood that she and some of the other students might need more explicit instruction in how to use the manipulatives and how to interpret the action of the problem. (Read more about my insights into Daphne's confusion in the Results of Phase 2.)
Lessons Three Through Six: Making Changes
In order to help the struggling students improve their understanding in the small amount of time that we had together, I resolved to make one group composed of the students who typically had the hardest time demonstrating the problems. With this group (which I will refer to as Group 1), I would use the plans that I had written from Phase 1, but I would emphasize key action clue words such as "more" and "altogether" for addition, and "took away" and "were left" for subtraction. By doing so, I hoped to make more explicit connections between the everyday language that these students used for informal math (such as that they might use at home with their parents, or on the playground with friends), and the problems that we were doing together. I also decided to begin using bear-shaped manipulatives that we had in our classroom in place of the Unifix cubes, based on the fact that students (particularly in Group 1) sometimes lost their focus as they tried to place the Unifix cubes in fixed rows.
Work & Results from the Final Lesson
To the right is an example of a problem that I did during my final lesson with Group 1. My story for this problem was:
Ms. Berger: Gary [a student in the group] found 12 four leaf clovers at recess, and put them in his backpack. He found 3 more at lunchtime, and put those in his backpack. How many four leaf clovers did Gary have in his backpack after lunch?
Students in the classroom were always excited when they found four leaf clovers in the grass on the playground, or in the garden before school. After telling the story of the problem, I held up a single bear manipulative. "Can we pretend that these are Gary's four leaf clovers?" I did this in order to emphasize that the manipulatives are symbolic of the four-leaf clovers from the story.
To the left is a picture of the manipulatives that a student from Daphne's group was using during our last lesson. (This student was in the act of reaching into a basket to get a twelfth bear as I took the picture.) You can see that she has placed the bears in a group, but has not arranged them in any particular way, except for grouping them (somewhat) by color.
To the left is a picture of the manipulatives that a student from Daphne's group was using during our last lesson. (This student was in the act of reaching into a basket to get a twelfth bear as I took the picture.) You can see that she has placed the bears in a group, but has not arranged them in any particular way, except for grouping them (somewhat) by color.
To the right is the work of another student from Group 1 in response to my second story problem, a Separate Result Unknown or subtraction-type problem. This student has also "clumped" her bears, but her grouping reveals that she understands the action of the story (she has separated the manipulatives that are being subtracted from the original group). I asked this student to share her grouping strategy with the other Group 1 students so that they could observe her work and make the connection between the implied action (separating, or subtraction) and the language of problem.
To the left are the results of a student from Group 2 in response to my story for their first problem (pictured below). This student organized his bears very systematically. He arranged the initial number of cookies in a line, and moved the number I "ate" into the middle, symbolically removing them, or separating them, from the initial group. This student's final number was nineteen because he counted the original amount incorrectly. However, his work reveals that he understands the action of the problem, how to represent it symbolically, and how to group the manipulatives to help himself count them.
To the right is the first problem I used for Groups 2 and 3. (Note that I crossed out the heart representing the nine cookies in order to emphasize that this amount was "going away.") Here is the story of the problem as I dictated it to the students:
Ms. Berger: Miss Berger made enough heart cookies for all the students in the class and Ms. T and Ms. O. But she got hungry and ate nine of them! How many were left after she ate nine?
To the left are pictures of examples of the work of two students from Group 3, also in response to the Separate Result Unknown problem pictured above.
These students were seated next to each other. The student on top chose to arrange his manipulatives into rows of 11. The student next to him also chose to organize his manipulatives into rows. (This student, however, might not have noticed that his rows were not identical.)
I could see that by the sixth lesson of Phase 2 students were becoming more organized and systematic about their use of the manipulatives. Read more about the significance of this in my analysis of Phase 2.
Analysis of Phase 2
As I reflected upon the growth that I saw during the six lessons I implemented during Phase 2, I thought back to the choices that I had made after the first two lessons were complete in order to try to find answers to my questions about how to implement CGI in a way that allowed each student to develop his or her own problem-solving skills independently.
I understood based on my knowledge of my students that there might be some correlation between the counting abilities of the students and their ability to demonstrate or "act out" the story problems. I chose to group the students into homogeneous groups for several reasons, which included the students' abilities to count their manipulatives, their ability to interpret the language of the story problems through the use of the manipulatives, and the pace at which they worked through the problems.
One of the ways teachers try to understand the growth of their students is through the use of assessments and assessment software. One of the assessments we conducted in our classroom was a number recognition test using the educational software ESGI. Students are shown the numbers 0 through 30 in random order and asked to identify them. Unfortunately, the most recent assessments for the students had been conducted just before Thanksgiving 2013, almost two months prior to my six lessons from Phase 2 (January 2014). However, the information I gathered from these assessments was quite interesting.
Based on the information we gathered from the student assessments in November, students from Group 1 could recognize an average of 13 out of 31 numbers. Students from Group 2 could recognize an average of 28 out of 31 numbers. Students from Group 3 could recognize an average of 24 out of 31 numbers. What accounted for these differences?
As I reflected upon the growth that I saw during the six lessons I implemented during Phase 2, I thought back to the choices that I had made after the first two lessons were complete in order to try to find answers to my questions about how to implement CGI in a way that allowed each student to develop his or her own problem-solving skills independently.
I understood based on my knowledge of my students that there might be some correlation between the counting abilities of the students and their ability to demonstrate or "act out" the story problems. I chose to group the students into homogeneous groups for several reasons, which included the students' abilities to count their manipulatives, their ability to interpret the language of the story problems through the use of the manipulatives, and the pace at which they worked through the problems.
One of the ways teachers try to understand the growth of their students is through the use of assessments and assessment software. One of the assessments we conducted in our classroom was a number recognition test using the educational software ESGI. Students are shown the numbers 0 through 30 in random order and asked to identify them. Unfortunately, the most recent assessments for the students had been conducted just before Thanksgiving 2013, almost two months prior to my six lessons from Phase 2 (January 2014). However, the information I gathered from these assessments was quite interesting.
Based on the information we gathered from the student assessments in November, students from Group 1 could recognize an average of 13 out of 31 numbers. Students from Group 2 could recognize an average of 28 out of 31 numbers. Students from Group 3 could recognize an average of 24 out of 31 numbers. What accounted for these differences?
According to the observations of educator and developmental psychologist Herbert Ginsburg, young children who are learning to count have multiple challenges to overcome. "Not only do children [learning to count] have to consider things once and only once," he writes, "they have to assign one and only one number word to each object considered." He concludes that "their mistakes can be traced to certain weaknesses in planning and organization, and to an overreliance on memory." Our classroom assessments try to measure student knowledge by showing them numbers in random order, thus prohibiting the students from relying on their memories of counting order. Counting large numbers and recognizing their equivalents in written form are both skills that take practice and repetition. While development may influence the rate at which these skills are acquired, students need practice working with the numbers with which they are already familiar.
Four of the students in Group 1 could recognize numbers 0 through 11 out of sequence. One of the students was an English Learner who could count from 0 to 6 in English, while another student (a native English speaker) could count from 0 to 5. According to Ginsburg, "For the young child, addition is simply an extension of counting objects." If this is true, then students who struggle to interpret the action words of CGI might simply be overwhelmed by the quantities they are being asked to count. In order to help students understand the action of the problem, it might help to use small quantities (for instance, a Joint Result Unknown problem with a sum of 4) so that students can make the connection between the language and the action of the problem intuitively based on a firm knowledge of those numbers.
While almost all students at the beginning of Phase 2 were attaching their Unifix cubes to each other in long rows in order to count them, putting together and detaching the cubes was sometimes awkward and distracting. Switching to the bear manipulatives eliminated this distraction. Having individual manipulatives (and not long rows of attached cubes) made accurate counting more easy, and it was easier for me to observe which students had effectively "pushed aside" the amount they were subtracting during Separate Result Unknown problems. Ginsburg says of the "pushing aside" strategy, "This strategy is enormously powerful since it reduces the strain on memory." Whether through spontaneous invention or their observation of others, I could see that the students had developed a very efficient strategy for representing the problems and counting the manipulatives.
Ginsburg writes that students are interested in developing new strategies that allow them to count more effectively. I was very interested in the work of the two students whose work I showed above, who had grouped their manipulatives into rows. Though the student who made two groups of 11 did not know how to count by 11s, he could count by 10s. Both these students were familiar with the concept of grouping, and were demonstrating this by constructing rows with their manipulatives. Ginsburg writes, "Children proceed from counting one by one to applying arithmetic operations to groups of elements." With more practice, I believe that these students would learn to group so that they could count rows, rather than counting the manipulatives individually.
Upon reflection, my choice to move our lessons to the rug was an effective one because it allowed me to engage with students directly and it allowed me to observe their work very closely. Though my choice to group the students according to their problem-solving abilities was partly based on making the groups more manageable, I believe that four lessons was not sufficient for me to gain an understanding of whether certain students had improved as a result.
Possible Extensions
Ideally, Phase 2 of my research would incorporate as many differentiated small group lessons as possible. Based on the evidence from my lessons, students need regular practice with the manipulatives to learn how to use them more effectively. Additionally, student absences during my research made consistent observations of student progress more challenging. Extensions to my research would incorporate regular assessment of students through photographs of their work, and, if possible, recordings of students discussing how they perceive the problems and how this is reflected in their grouping of the manipulatives. Lessons would be differentiated within each group in order to focus not only on particular quantities, but specific skills and problem types. Ultimately, students would also be able to design and articulate their own problems for other students to solve.
Four of the students in Group 1 could recognize numbers 0 through 11 out of sequence. One of the students was an English Learner who could count from 0 to 6 in English, while another student (a native English speaker) could count from 0 to 5. According to Ginsburg, "For the young child, addition is simply an extension of counting objects." If this is true, then students who struggle to interpret the action words of CGI might simply be overwhelmed by the quantities they are being asked to count. In order to help students understand the action of the problem, it might help to use small quantities (for instance, a Joint Result Unknown problem with a sum of 4) so that students can make the connection between the language and the action of the problem intuitively based on a firm knowledge of those numbers.
While almost all students at the beginning of Phase 2 were attaching their Unifix cubes to each other in long rows in order to count them, putting together and detaching the cubes was sometimes awkward and distracting. Switching to the bear manipulatives eliminated this distraction. Having individual manipulatives (and not long rows of attached cubes) made accurate counting more easy, and it was easier for me to observe which students had effectively "pushed aside" the amount they were subtracting during Separate Result Unknown problems. Ginsburg says of the "pushing aside" strategy, "This strategy is enormously powerful since it reduces the strain on memory." Whether through spontaneous invention or their observation of others, I could see that the students had developed a very efficient strategy for representing the problems and counting the manipulatives.
Ginsburg writes that students are interested in developing new strategies that allow them to count more effectively. I was very interested in the work of the two students whose work I showed above, who had grouped their manipulatives into rows. Though the student who made two groups of 11 did not know how to count by 11s, he could count by 10s. Both these students were familiar with the concept of grouping, and were demonstrating this by constructing rows with their manipulatives. Ginsburg writes, "Children proceed from counting one by one to applying arithmetic operations to groups of elements." With more practice, I believe that these students would learn to group so that they could count rows, rather than counting the manipulatives individually.
Upon reflection, my choice to move our lessons to the rug was an effective one because it allowed me to engage with students directly and it allowed me to observe their work very closely. Though my choice to group the students according to their problem-solving abilities was partly based on making the groups more manageable, I believe that four lessons was not sufficient for me to gain an understanding of whether certain students had improved as a result.
Possible Extensions
Ideally, Phase 2 of my research would incorporate as many differentiated small group lessons as possible. Based on the evidence from my lessons, students need regular practice with the manipulatives to learn how to use them more effectively. Additionally, student absences during my research made consistent observations of student progress more challenging. Extensions to my research would incorporate regular assessment of students through photographs of their work, and, if possible, recordings of students discussing how they perceive the problems and how this is reflected in their grouping of the manipulatives. Lessons would be differentiated within each group in order to focus not only on particular quantities, but specific skills and problem types. Ultimately, students would also be able to design and articulate their own problems for other students to solve.