Implementing CGI: The First Three Days
Until I began my AR lessons on November 18, after-lunch time was a very laid-back period of the day. Academic work generally took place from the beginning of school to break time (8:00-10:30), and from the end of break until the beginning of lunch (10:30-12:10). Students would read a story from 1:00 to 1:15, play ST Math on the iPads from 1:15 to 1:30, and then dismiss to Center Choice/creative play time from 1:30 until 2:15. After lunch time was a fairly relaxed, open-ended period of the day.
Until I began my AR lessons on November 18, after-lunch time was a very laid-back period of the day. Academic work generally took place from the beginning of school to break time (8:00-10:30), and from the end of break until the beginning of lunch (10:30-12:10). Students would read a story from 1:00 to 1:15, play ST Math on the iPads from 1:15 to 1:30, and then dismiss to Center Choice/creative play time from 1:30 until 2:15. After lunch time was a fairly relaxed, open-ended period of the day.
On the first day of the new lessons, I began by placing my cutout of JiJi on the board. "Who's this?" I asked the class excitedly. "JiJi!" they replied. I wanted to begin my lessons by attaching great enthusiasm to the characters in order to get the students excited.
"What do we do in ST Math?" I then asked the students. I called on Rachel, one of the most insightful and vocal students, who replied "We play games with JiJi." I then asked her what she might do if she couldn't figure out a game in the program. "We ask the teacher," she answered.
"You could ask a teacher," I answered her, "but what else could you do?" I called on several students, who answered that they would either ask a teacher, or ask a friend. I confirmed to the students that asking the teacher and asking a friend are both strategies for solving problems in the game. "Strategy is a word that means 'a way of figuring something out,'" I explained.
The first three days of my implementation of CGI became a kind of framework for the kinds of challenges that I encountered in the rest of my Action Research. For this reason, I will spend time reflecting on what I learned in those lessons and how it impacted my long-term goals for the project.
"What do we do in ST Math?" I then asked the students. I called on Rachel, one of the most insightful and vocal students, who replied "We play games with JiJi." I then asked her what she might do if she couldn't figure out a game in the program. "We ask the teacher," she answered.
"You could ask a teacher," I answered her, "but what else could you do?" I called on several students, who answered that they would either ask a teacher, or ask a friend. I confirmed to the students that asking the teacher and asking a friend are both strategies for solving problems in the game. "Strategy is a word that means 'a way of figuring something out,'" I explained.
The first three days of my implementation of CGI became a kind of framework for the kinds of challenges that I encountered in the rest of my Action Research. For this reason, I will spend time reflecting on what I learned in those lessons and how it impacted my long-term goals for the project.
Day One
The problem I chose for the first day of CGI was a Part-Part-Whole problem. One of the greatest difficulties I had in developing a visual representation of the CGI word problems was how to make the visuals support the word problem without making the problem itself look like a number sentence. Though I had consulted a number of different examples of CGI problems, many were designed for students who could already read. For my first problem, I had drawn and placed magnets on six drawings of fish, which I placed on the board between the cutouts of JiJi and Ollie.
"JiJi and Ollie went fishing. Together, they caught six fish. How many fish did they each get to eat?"
None of the students used the manipulatives to solve the problem, and several students began talking to their neighbors without seeming to understand what they were being asked to do. Many of the students wrote down the correct answer very quickly.
After I observed that the majority of students had solved the problem, I called one of the students to stand next to me at the board, and asked her "How did you get your answer?" "I looked at the board and I saw the two rows of three," she answered. Each student who attempted the problem got it correct. (Two students did not seem to understand what the objective of the new lesson was, and drew pictures of the fish on their worksheets.) I called on two more students who all shared their answers with the class. My questions for the students were very simple:
Though I thought about introducing the manipulatives as an additional strategy, I decided based on the waning attention spans of the students to wait until the next day to introduce them as an additional strategy.
The problem I chose for the first day of CGI was a Part-Part-Whole problem. One of the greatest difficulties I had in developing a visual representation of the CGI word problems was how to make the visuals support the word problem without making the problem itself look like a number sentence. Though I had consulted a number of different examples of CGI problems, many were designed for students who could already read. For my first problem, I had drawn and placed magnets on six drawings of fish, which I placed on the board between the cutouts of JiJi and Ollie.
"JiJi and Ollie went fishing. Together, they caught six fish. How many fish did they each get to eat?"
None of the students used the manipulatives to solve the problem, and several students began talking to their neighbors without seeming to understand what they were being asked to do. Many of the students wrote down the correct answer very quickly.
After I observed that the majority of students had solved the problem, I called one of the students to stand next to me at the board, and asked her "How did you get your answer?" "I looked at the board and I saw the two rows of three," she answered. Each student who attempted the problem got it correct. (Two students did not seem to understand what the objective of the new lesson was, and drew pictures of the fish on their worksheets.) I called on two more students who all shared their answers with the class. My questions for the students were very simple:
- What answer did you get?
- How did you get your answer?
Though I thought about introducing the manipulatives as an additional strategy, I decided based on the waning attention spans of the students to wait until the next day to introduce them as an additional strategy.
Day Two
After considering the first day's problem and how easy it had been for my students, I decided to challenge them a bit more in the hopes that they would begin using more strategies, and that I might get more time to watch them work as they considered the problem. I also decided to begin the lesson by giving a quick lesson in how to use the unifix manipulatives. I then read out the day's problem:
JiJi, Ollie, and some bird friends went on a field trip. Before the bus left, the driver counted eight legs on the bus. How many bird friends went on the field trip with JiJi and Ollie?
After placing Ollie and JiJi on the board and writing "2" above both of them to indicate the number of legs they each had, I read the problem out to the students, and finished by writing an "8" above the cutout of the bus.
Though I had written the problem down and read it exactly as it was printed, I suddenly became concerned that the problem would be too difficult for the students. I wondered if these visual suggestions were enough to indicate that there was a missing quantity that I wanted to the students to find. Rather than trusting that the students could use their own strategies to try to solve the problem, I tried to "back up" and make the problem easier for them.
In trying (and failing) to help the students understand the question that I had written, I learned a valuable lesson: I should be using my lessons as an opportunity to learn about the students' existing problem-solving skills. Instead, I was using the lessons as a top-down assessment, asking students about their answers and trying to spot the students who couldn't solve the problems.
Day Two was also challenging because I was still learning about which types of CGI problems would be challenging for a diverse group of young learners. Though the numbers and quantities I used in my problem were appropriate for my class, the concepts behind the problem may have been too complex at this point in the students' work with Cognitively Guided Instruction. I had thought of my lesson as a relatively straightforward Part-Part-Whole style of problem, but it was in fact a combination of two types of CGI problems.
In addition to what I learned from this lesson, I decided to focus on developing other aspects of my Action Research. I had designed my Behavior Tracking graph to take notes on which students were talking to their neighbors. However, I also wanted students to share their strategies. I decided to make the following changes:
After considering the first day's problem and how easy it had been for my students, I decided to challenge them a bit more in the hopes that they would begin using more strategies, and that I might get more time to watch them work as they considered the problem. I also decided to begin the lesson by giving a quick lesson in how to use the unifix manipulatives. I then read out the day's problem:
JiJi, Ollie, and some bird friends went on a field trip. Before the bus left, the driver counted eight legs on the bus. How many bird friends went on the field trip with JiJi and Ollie?
After placing Ollie and JiJi on the board and writing "2" above both of them to indicate the number of legs they each had, I read the problem out to the students, and finished by writing an "8" above the cutout of the bus.
Though I had written the problem down and read it exactly as it was printed, I suddenly became concerned that the problem would be too difficult for the students. I wondered if these visual suggestions were enough to indicate that there was a missing quantity that I wanted to the students to find. Rather than trusting that the students could use their own strategies to try to solve the problem, I tried to "back up" and make the problem easier for them.
In trying (and failing) to help the students understand the question that I had written, I learned a valuable lesson: I should be using my lessons as an opportunity to learn about the students' existing problem-solving skills. Instead, I was using the lessons as a top-down assessment, asking students about their answers and trying to spot the students who couldn't solve the problems.
Day Two was also challenging because I was still learning about which types of CGI problems would be challenging for a diverse group of young learners. Though the numbers and quantities I used in my problem were appropriate for my class, the concepts behind the problem may have been too complex at this point in the students' work with Cognitively Guided Instruction. I had thought of my lesson as a relatively straightforward Part-Part-Whole style of problem, but it was in fact a combination of two types of CGI problems.
In addition to what I learned from this lesson, I decided to focus on developing other aspects of my Action Research. I had designed my Behavior Tracking graph to take notes on which students were talking to their neighbors. However, I also wanted students to share their strategies. I decided to make the following changes:
- I would use the Behavior Tracking graph to take notes on which students were counting using fingers or manipulatives, rather than which ones were talking to their neighbors.
- I would not ask students what their answers were. Instead, I would concentrate on asking them about the strategies that they had used.
Day 3
After looking over student responses, contemplating my notes from the past two days, and re-reading chapters of Children's Mathematics, I decided to pick a more straightforward problem for my third CGI lesson. I also introduced a third character known as Jamie the Giraffe. Before reading the problem, I explained to the class that even though I would like them to circle their final answer, I wouldn't be asking them what their answers were when we shared our strategies.
Ollie, JiJi, and Jamie were very hungry. They each ate 4 ice cream cones. How many ice cream cones did they eat altogether?
After reading the problem, I drew "4" above each character. I then turned to the class and observed them as they worked. At least four pairs of students whispered to their neighbors about the correct answer, and shared their drawings of their work with each other. Two students were off-task, and drawing pictures of JiJi and the other characters. Three students were using the manipulatives, though one of them was using them to trace pictures of squares.
When almost all the students had written down their answers, I called on the first student volunteer to share strategies. This student was the same student who had been using the manipulatives to make drawings on his worksheet; his work can be seen below in Sample #4. Our conversation about his strategies was as follows:
Miss Berger: Which strategies did you use to solve the problem?
Student 4: I brainstormed it.
Miss Berger: You brainstormed it. Like, you used your brain? That's excellent. Kindergarten friends, Student 4 used his brain to solve the problem. That's an awesome strategy!
Though Student 4 had not found the right answer, I felt much more confident placing emphasis on the students' strategies rather than the answer itself. I then called on Student 1, whose work can be seen in Sample 1:
Miss Berger: Which strategies did you use to solve the problem?
Student 1: Um, I thought about it, and I thought the answer was 8, but then my dad told me that 4 plus 4 plus 4 is 12.
Miss Berger: Your dad told you that the right answer was 12?
Student 1: Yeah.
Miss Berger: Kindergarten friends, Student 1 did something awesome. He thought about the answer, and then he thought about it again to make sure his answer was right. He was double-checking.
In my conversation with Student 1, I realized that I would have to be very careful about how I asked students which strategies they used. I wanted to draw attention to their logic processes, but not necessarily to the result itself. (I assumed that Student 1's response of "my dad told me" was his perception of his inner voice. Three different students responded the same way in the course of my AR.)
After looking over student responses, contemplating my notes from the past two days, and re-reading chapters of Children's Mathematics, I decided to pick a more straightforward problem for my third CGI lesson. I also introduced a third character known as Jamie the Giraffe. Before reading the problem, I explained to the class that even though I would like them to circle their final answer, I wouldn't be asking them what their answers were when we shared our strategies.
Ollie, JiJi, and Jamie were very hungry. They each ate 4 ice cream cones. How many ice cream cones did they eat altogether?
After reading the problem, I drew "4" above each character. I then turned to the class and observed them as they worked. At least four pairs of students whispered to their neighbors about the correct answer, and shared their drawings of their work with each other. Two students were off-task, and drawing pictures of JiJi and the other characters. Three students were using the manipulatives, though one of them was using them to trace pictures of squares.
When almost all the students had written down their answers, I called on the first student volunteer to share strategies. This student was the same student who had been using the manipulatives to make drawings on his worksheet; his work can be seen below in Sample #4. Our conversation about his strategies was as follows:
Miss Berger: Which strategies did you use to solve the problem?
Student 4: I brainstormed it.
Miss Berger: You brainstormed it. Like, you used your brain? That's excellent. Kindergarten friends, Student 4 used his brain to solve the problem. That's an awesome strategy!
Though Student 4 had not found the right answer, I felt much more confident placing emphasis on the students' strategies rather than the answer itself. I then called on Student 1, whose work can be seen in Sample 1:
Miss Berger: Which strategies did you use to solve the problem?
Student 1: Um, I thought about it, and I thought the answer was 8, but then my dad told me that 4 plus 4 plus 4 is 12.
Miss Berger: Your dad told you that the right answer was 12?
Student 1: Yeah.
Miss Berger: Kindergarten friends, Student 1 did something awesome. He thought about the answer, and then he thought about it again to make sure his answer was right. He was double-checking.
In my conversation with Student 1, I realized that I would have to be very careful about how I asked students which strategies they used. I wanted to draw attention to their logic processes, but not necessarily to the result itself. (I assumed that Student 1's response of "my dad told me" was his perception of his inner voice. Three different students responded the same way in the course of my AR.)
Next I called on the student whose work can be seen below in Sample 2.
Miss Berger: So Student 2, which strategies did you use to figure it out?
Student 2: I counted on my fingers.
Miss Berger: You counted using your fingers? That's a great strategy to use.
In my dialogue with the students, I praised them for their ability to problem-solve and completely disregarded what their answers were. In doing so I hoped to make the use of strategies the focus of the lessons.
The student work samples below also illustrated another challenge of CGI. There were many students who I did not get to speak to during the lesson. How could I interpret their work in order to learn which strategies they were using?
Miss Berger: So Student 2, which strategies did you use to figure it out?
Student 2: I counted on my fingers.
Miss Berger: You counted using your fingers? That's a great strategy to use.
In my dialogue with the students, I praised them for their ability to problem-solve and completely disregarded what their answers were. In doing so I hoped to make the use of strategies the focus of the lessons.
The student work samples below also illustrated another challenge of CGI. There were many students who I did not get to speak to during the lesson. How could I interpret their work in order to learn which strategies they were using?
Analyzing Student Samples
The student from Sample 1 reported that "his dad" told him what the correct answer to the problem was. Based on the fact that there were no other numbers written on his worksheet, and the fact that he did not state that he had used his fingers or the unifix cubes, I believe that Student 1 was using Derived Number Facts to help himself solve the problem. Three of the other children at his table also produced the correct answer, however, so it is possible that the student from Sample 1 copied his answer from a neighbor. However, Student 1 can also count confidently by 2's and 3's, so I believe he most likely counted by 4' mentally.
The student in Sample 2 reported to me that he counted out the solution on his fingers. I am not sure whether or not he was using the numbers on his worksheet as a visual anchor for his counting, or whether he was simply copying the numbers off of the board. It is possible that he was using both strategies simultaneously, as I observed other students doing who also wrote down the numbers from the board.
Based on the observations of teachers who use CGI regularly in their classrooms, students who write corresponding number sentences also tend to rely on Derived Number Facts to find solutions. Like the student from Sample 1, Student 3 can also count by 2's and 3's. I did not observe her using unifix cubes, but it is possible that she, like Student 2, used a blended strategy of Derived Number Facts and finger-counting.
Student 4 reported that he "brainstormed" in order to find his answer. I am not sure based on his written answer where Student 4 got "stuck," but it is possible this student did not realize that all three quantities in the problem needed to be added together to find the final answer.
The student from Sample 5 had to be redirected consistently during the first three CGI lessons. When I tried to rephrase the problem for her during Problem #3, she smiled at me playfully and guessed at what she thought the answer was. This student also typically has trouble focusing much earlier in the day compared with other students, and her motor skills are not quite as refined as some of her peers. However, I hope to help her understand the objective of the new lessons by working with her more closely.
Results for Problems 4 - 7 and Further Reflections
The rest of my Phase One CGI problems were Join Result Unknown, Separate Result Unknown, or Part-Part-Whole (Whole Unknown). 100% of the students who attempted Problems 5 - 7 got the correct answer. I began to see students utilizing the unifix cubes in order to find the solution to the problems, and students were able to state confidently which strategy they had used to find the answer. The strategies began to fall into predictable categories, including:
The student from Sample 1 reported that "his dad" told him what the correct answer to the problem was. Based on the fact that there were no other numbers written on his worksheet, and the fact that he did not state that he had used his fingers or the unifix cubes, I believe that Student 1 was using Derived Number Facts to help himself solve the problem. Three of the other children at his table also produced the correct answer, however, so it is possible that the student from Sample 1 copied his answer from a neighbor. However, Student 1 can also count confidently by 2's and 3's, so I believe he most likely counted by 4' mentally.
The student in Sample 2 reported to me that he counted out the solution on his fingers. I am not sure whether or not he was using the numbers on his worksheet as a visual anchor for his counting, or whether he was simply copying the numbers off of the board. It is possible that he was using both strategies simultaneously, as I observed other students doing who also wrote down the numbers from the board.
Based on the observations of teachers who use CGI regularly in their classrooms, students who write corresponding number sentences also tend to rely on Derived Number Facts to find solutions. Like the student from Sample 1, Student 3 can also count by 2's and 3's. I did not observe her using unifix cubes, but it is possible that she, like Student 2, used a blended strategy of Derived Number Facts and finger-counting.
Student 4 reported that he "brainstormed" in order to find his answer. I am not sure based on his written answer where Student 4 got "stuck," but it is possible this student did not realize that all three quantities in the problem needed to be added together to find the final answer.
The student from Sample 5 had to be redirected consistently during the first three CGI lessons. When I tried to rephrase the problem for her during Problem #3, she smiled at me playfully and guessed at what she thought the answer was. This student also typically has trouble focusing much earlier in the day compared with other students, and her motor skills are not quite as refined as some of her peers. However, I hope to help her understand the objective of the new lessons by working with her more closely.
Results for Problems 4 - 7 and Further Reflections
The rest of my Phase One CGI problems were Join Result Unknown, Separate Result Unknown, or Part-Part-Whole (Whole Unknown). 100% of the students who attempted Problems 5 - 7 got the correct answer. I began to see students utilizing the unifix cubes in order to find the solution to the problems, and students were able to state confidently which strategy they had used to find the answer. The strategies began to fall into predictable categories, including:
- "I used my brain" (Derived Number Facts)
- "I used my fingers to count" (Counting Strategy)
- "I used the unifix cubes" (Direct Modeling)
The results that I collected from Garfield Likert scales that I used to assess the students' attitudes were in some ways surprising. I had predicted that the kindergarteners would consistently select either the "happiest Garfield" (a score of 4) or the "unhappy/angry Garfield" (a score of 1). However, the mean score across all my assessments was approximately a 3, or 77%. During all of Phase 1, there were only five responses of the "unhappy/angry Garfield," four of which came from the same student.
When I compared the individual results of the Garfield Likert scale against whether or not the student had found the correct answer, I could find no relationship between the two. However, when I plotted the averages for the responses for each problem on a graph, I noticed that the averages for the last 3 problems in the series were beginning to fall. 100% of the students who attempted to find an answer for problems 5, 6, and 7 found the correct answer. (This can be compared with Problem 4, in which only 60% of the students found the correct answer, but the average Garfield score was 83%.) Though I am not sure whether the Garfield Likert scale is a reliable way of assessing whether the student feels more or less confident about his or her own performance on the day's problem, I believe it is possible that students were indicating that the problems were becoming too easy, and that they were becoming bored.
In order to refine my answers to my original research questions, I will take the following steps in Phase 2:
- Rather than conduct my lessons as an entire class, I will break the class into three smaller, mixed-level groups in order to observe and discuss their strategies more closely.
- I will challenge the students to provide evidence of their thought processes by using questions such as "Can you show me what you were thinking?," "Can you please explain how you counted?," or "Can you show me what your brain was telling you?"
- I will continue to use Join Result Unknown, Separate Result Unknown, and Part-Part-Whole (Whole Unknown) problems using numbers up to 20.
- I will track students' responses to the problems in order to gather evidence on the individual changes in their use of various strategies.
In Phase 2, I hope to answer the following questions:
- How does Cognitively Guided Instruction affect the development of student's problem-solving strategies?
- How will students' abilities to explain their thought processes change?
- How will the small group structure affect the ways in which students share their strategies?