Bringing CGI to my Classroom
In order to implement lessons in Cognitively Guided Instruction regularly in our classroom, I sought to find a time that would not interrupt regular math lessons. My master teachers agreed that every Monday through Thursday for approximately two and a half weeks, I would lead the entire class in one or two short math lessons (of no more than a half-hour total) as the children returned from lunch at 1:00. The after-lunch schedule is most often a story followed by an hour of "Center Choice Time." (During Center Choice Time, students can choose from a number of play-oriented activities, including play-dough projects, building with blocks, or drawing.) Though this was not an ideal time of the day to feature new academic material, I hoped that having an open time separate from the rest of the academic content of the day would help the students feel more comfortable adjusting to a new routine. However, I also understood that I would need to keep the lessons short, as many of the students might not yet have the stamina to complete complex math problems during what is usually structured playtime.
Understanding the Problem Types
After setting aside a time for instruction, I set about designing a way to monitor student understanding of the problems I would be presenting. Within Cognitively Guided Instruction, there are approximately eleven different subtraction and addition problem types that fall into four classes (Carpenter, 1999, p. 12). In order to ease students into the language and strategies associated with CGI, I decided to focus my research on four problem types, with an emphasis on those problems which most clearly mimic the types of math problems the students had been focusing on during their regular math instruction.
The two major problem types I wanted to focus on are known as Join Result Unknown (a type of addition problem in which students must find the sum), and Separate Result Unknown (a subtraction problem in which students must find the difference). However, in order to challenge the students, I chose to include one Join Change Unknown problems as well as two Part-Part-Whole problems. Join Change Unknown problems are similar to Join Result, but students must find the missing addend rather than the missing sum. In a Part-Part-Whole problem, students must divide a final amount into several equal, smaller amounts. Though the problems themselves would emphasize slightly different approaches, ultimately, according to Carpenter and Fennema,
"The distinctions among problem types are reflected in children's solution processes. For the most basic strategies children use physical objects (counters) or fingers to directly model the action or relationships described in each problem. Over time, children's strategies become more abstract and efficient."
Using this information, I created a graph that reflects the major kinds of strategies that children usually attempt in Cognitively Guided Instruction. I created a column for students who used Direct Drawings, a type of "direct modeling" strategy that also includes using fingers and unifix cubes to model the action of the problem. I also understood that some children might have already progressed to Counting Strategies. These children understand numbers in a more abstract way, and can begin counting forward from the first addend represented in a problem (Carpenter, 1999, p. 7). (For example, in a problem such as four plus four, these students would begin counting on from four to reach eight, rather than drawing two groups of four or using unifix cubes to find the total.) Finally, some students might employ an entirely abstract strategy in which they use knowledge of numbers and quantities to complete problems mentally using Number Facts. One of the teachers interviewed in the study conducted by Knapp & Peterson found that students who regularly used number facts also often wrote their own mathematical equations (Knapp, 1995, p. 53). For this reason, I decided to include a column labeled 'Equation.'
In order to implement lessons in Cognitively Guided Instruction regularly in our classroom, I sought to find a time that would not interrupt regular math lessons. My master teachers agreed that every Monday through Thursday for approximately two and a half weeks, I would lead the entire class in one or two short math lessons (of no more than a half-hour total) as the children returned from lunch at 1:00. The after-lunch schedule is most often a story followed by an hour of "Center Choice Time." (During Center Choice Time, students can choose from a number of play-oriented activities, including play-dough projects, building with blocks, or drawing.) Though this was not an ideal time of the day to feature new academic material, I hoped that having an open time separate from the rest of the academic content of the day would help the students feel more comfortable adjusting to a new routine. However, I also understood that I would need to keep the lessons short, as many of the students might not yet have the stamina to complete complex math problems during what is usually structured playtime.
Understanding the Problem Types
After setting aside a time for instruction, I set about designing a way to monitor student understanding of the problems I would be presenting. Within Cognitively Guided Instruction, there are approximately eleven different subtraction and addition problem types that fall into four classes (Carpenter, 1999, p. 12). In order to ease students into the language and strategies associated with CGI, I decided to focus my research on four problem types, with an emphasis on those problems which most clearly mimic the types of math problems the students had been focusing on during their regular math instruction.
The two major problem types I wanted to focus on are known as Join Result Unknown (a type of addition problem in which students must find the sum), and Separate Result Unknown (a subtraction problem in which students must find the difference). However, in order to challenge the students, I chose to include one Join Change Unknown problems as well as two Part-Part-Whole problems. Join Change Unknown problems are similar to Join Result, but students must find the missing addend rather than the missing sum. In a Part-Part-Whole problem, students must divide a final amount into several equal, smaller amounts. Though the problems themselves would emphasize slightly different approaches, ultimately, according to Carpenter and Fennema,
"The distinctions among problem types are reflected in children's solution processes. For the most basic strategies children use physical objects (counters) or fingers to directly model the action or relationships described in each problem. Over time, children's strategies become more abstract and efficient."
Using this information, I created a graph that reflects the major kinds of strategies that children usually attempt in Cognitively Guided Instruction. I created a column for students who used Direct Drawings, a type of "direct modeling" strategy that also includes using fingers and unifix cubes to model the action of the problem. I also understood that some children might have already progressed to Counting Strategies. These children understand numbers in a more abstract way, and can begin counting forward from the first addend represented in a problem (Carpenter, 1999, p. 7). (For example, in a problem such as four plus four, these students would begin counting on from four to reach eight, rather than drawing two groups of four or using unifix cubes to find the total.) Finally, some students might employ an entirely abstract strategy in which they use knowledge of numbers and quantities to complete problems mentally using Number Facts. One of the teachers interviewed in the study conducted by Knapp & Peterson found that students who regularly used number facts also often wrote their own mathematical equations (Knapp, 1995, p. 53). For this reason, I decided to include a column labeled 'Equation.'
Monitoring Behavior & Involvement
To monitor student involvement, I created an Observation Guide. In order to make it clear at a glance which type of CGI problem and which quantities I was using, I created a space in which I could write "JRU" (e.g., for Joint Result Unknown) and the specific phrasing I had used for the problem. Each student had his or her own box in which I could jot quick notes with regard to involvement. If a student began to play with the unifix cubes (rather than using them as a counting manipulative) or otherwise became distracted, I would redirect them and write RD in the graph. If a student disengaged from the lesson and began talking to his or her neighbor, I would try to redirect them and would write DP in the graph (for 'did not participate'). If a student wanted to participate and raised his or her hand to share a strategy, I would write RH.
Developing the Lessons
In researching CGI problems and problem types, I discovered that texts associated with CGI do not necessarily provide any detailed descriptions of how to write the problems themselves. Instead, the articles and books I read explained that CGI is meant to function as a framework within which teachers may develop their own effective interpretations of the problem types (Carpenter, 1996, p. 4). Therefore, I decided to create my own CGI problems based on a number of pieces of evidence.
I derived the phrasing of my problems from the example problems supplied by the books and articles written by Drs. Carpenter and Fennema. Joint Result Unknown problems are worded in order to emphasize the "joining" of two distinct amounts. Similarly, Separate Result Unknown problems are worded in order to emphasize the "separation" of two distinct amounts. An example of a Separate Result Unknown problem from Children's Mathematics: Cognitively Guided Instruction is written as follows:
There were 8 seals playing. 3 seals swam away. How many seals were still playing?
One important factor I also wished to consider was that two of my students were English Language Learners. In order to scaffold the academic language acquisition process for these pre-reading students, I sought to create interesting visuals which would emphasize the "joining" or "separating" of distinct amounts.
I chose the visuals for my lessons based on a number of factors. The most successful teachers from the study of the long-term implementation of CGI "used contextualized problems," according to Drs. Knapp and Peterson. In order to create context, I printed and colored a graphic of JiJi the Penguin, the mascot of ST Math. I did the same with a cartoon drawing of an owl, who I intended to introduce as a cousin of our class mascot, a stuffed Great Horned owl named Ozzie.
The introduction of JiJi the Penguin and "Ollie" the Owl served a number of purposes. The fact that they are bright and colorful adds visual interest for the students, and also creates continuity within the curriculum. Simultaneously, I hoped that by using JiJi as a main character within each problem, I would encourage students to think of both CGI and ST Math as lessons in perseverance and problem-solving.
The next challenge I had to consider was how challenging I wanted the problems themselves to be. I had already decided that I would focus mostly on the easiest types of problems--that is, Joint Result and Separate Result Unknown, but I had not yet decided on which quantities to use. Then, I came upon the following quote from Carpenter and Fennema's first article about CGI:
"As long as children can count, they can solve problems involving two-digit numbers."
Based on this, I decided to use a variety of single and double-digit numbers, with a result of no more than 16. While the majority of the class can recognize and write the corresponding amounts for numbers of 20 or more in sequence, as of mid-November, 6 kindergarten students are still working on counting to 18. Additionally, about half the class needs additional practice with the names and quantities for 12 and 13. (See the chart printed below of the students' most recent assessments.)
"As long as children can count, they can solve problems involving two-digit numbers."
Based on this, I decided to use a variety of single and double-digit numbers, with a result of no more than 16. While the majority of the class can recognize and write the corresponding amounts for numbers of 20 or more in sequence, as of mid-November, 6 kindergarten students are still working on counting to 18. Additionally, about half the class needs additional practice with the names and quantities for 12 and 13. (See the chart printed below of the students' most recent assessments.)
Below are samples of the types of problems I will introduce to the students:
Sharing (and Understanding) the Strategies
In order to assess and track students' strategies, I will observe them during lessons and will take notes based on their worksheets after the lessons. I will also use the Observation Guide to track which students are using the manipulatives to solve the problems. After the majority of students have finished solving the problems, I will call four or five to the front of the class in order to discuss the strategy that they used in order to solve the problem. In order to help students understand that there are many potential strategies to solving a problem, I will use their self-reported strategies to explain the concept of a strategy, and will use the terminology when we discuss the problems, in order to help students understand the importance of experimentation in problem-solving.
Tracking Student Reactions to CGI
In order to keep track of students' attitudes towards the CGI lessons, I will have them each complete a Leichert scale in lieu of an exit slip in response to a prompt that I will give them at the end of each lesson. I will review with them that each "Garfield" shows a different reaction, and that they should circle the one which most closely resembles how they feel in response to my prompt.
- If JiJi threw 3 snowballs, and Ollie threw 2, how many snowballs did they throw altogether?
- If Ollie puts 7 presents under the Christmas tree, and JiJi unwraps 3 of them, how many are still left under the tree?
- When JiJi fills this jar with gum balls, it can hold 12 gum balls. It has 6 gum balls in it right now. How many more does he need to add to fill it up?
- If JiJi and Ollie catch 6 fish, how many do they each get to eat?
Sharing (and Understanding) the Strategies
In order to assess and track students' strategies, I will observe them during lessons and will take notes based on their worksheets after the lessons. I will also use the Observation Guide to track which students are using the manipulatives to solve the problems. After the majority of students have finished solving the problems, I will call four or five to the front of the class in order to discuss the strategy that they used in order to solve the problem. In order to help students understand that there are many potential strategies to solving a problem, I will use their self-reported strategies to explain the concept of a strategy, and will use the terminology when we discuss the problems, in order to help students understand the importance of experimentation in problem-solving.
Tracking Student Reactions to CGI
In order to keep track of students' attitudes towards the CGI lessons, I will have them each complete a Leichert scale in lieu of an exit slip in response to a prompt that I will give them at the end of each lesson. I will review with them that each "Garfield" shows a different reaction, and that they should circle the one which most closely resembles how they feel in response to my prompt.
Estimated Timeline
Phase One
November 18-20: Joint Result Unknown, Part-Part-Whole, Join Change Unknown
November 25-26: Join Change Unknown, Separate Result Unknown, Part-Part-Whole
December 2-5: Join Change Unknown (x3), Separate Result Unknown (x2)
Reflection
December 9-12: Reflection and development of Phase Two.
December 16-18, January 6-9, January 13-16: Implementation of Phase Two.