During the Needs Assessment phase of my Action Research, I observed a need for more open-ended, problem-based instruction among my students. Though math instruction in our classroom was diverse and interactive, it was generally conducted as an entire class. Instruction did not incorporate time for small-group instruction, or regular challenges for the students using independent problem-solving. I wanted to learn more about research-supported methods of math instruction, and how to utilize those methods to make our classroom's math curriculum more balanced. I also wanted to learn more about how to create a system of math instruction that could fulfill the intellectual needs of students who would seen be facing the rigors of the Common Core.
The Challenges of the Common Core
Throughout September and October, I attended at least three meetings with my master teachers meant to outline the process of adjusting to the Common Core. I was particularly struck by the comments of teachers from higher grades. "How are we supposed to teach our students to explain things like algebra?" one of the sixth grade teachers from another school said, with a discernible tone of frustration in her voice. An article from The Atlantic magazine summarizes the issue:
'Another recent article explains, "This curriculum puts an emphasis on critical thinking, rather than memorization, and collaborative learning." In other words, instead of simply teaching multiplication tables, schools are adopting "an 'inquiry method' of learning, in which children are supposed to discover the knowledge for themselves."'
As I read through the Common Core requirements for both kindergarten and first grade, I was struck by the section marked 'Mathematical Practices.' According to their description, the Mathematical Practices section of the Common Core is an attempt to create standards for the thought processes students should develop while learning the subject content (National Governors Association Center for Best Practices, 2010, p. 6-7).
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Use appropriate tools strategically.
These standards made it very clear to me: my kindergarten students will no longer simply be asked to recall and apply basic math facts. They will need to learn strategies in order to "persevere in solving" complex, multi-step problems. They will need to use reasoning skills in order to construct arguments that provide reasoning for their strategies. In order to do this, they will also need to be familiar with the different kinds of tools they can use to find answers in order to construct effective explanations (National Governors Association Center for Best Practices, 2010, p. 7).
I was overwhelmed by the considerations. How might I incorporate ways of teaching mathematics that also address the standards of the Common Core?
Math Concept Acquisition and Instruction
As I mentioned in my introduction, one of the sources I turned to for research into strong classroom math instruction was a book by educational researchers Craig Hart, Diane Burts, and Rosalind Charlesworth. Their book, Integrated Curriculum and Developmentally Appropriate Practice: Birth to Age Eight, summarizes the work of influential developmental psychologists such as Jean Piaget and Lev Vygotsky. It also discusses the results of multiple research studies about effective instruction in elementary school classrooms. One of the strongest points the authors emphasize is the effectiveness of allowing students to build communication skills by sharing and discussing their solutions to various problems, working independently or in small groups (Hart, et al, 1997, p. 63). Additionally, in a work from 1995, Rosalind Charlesworth describes three ways that children typically acquire mathematical concepts.
- Naturalistic, in which children construct knowledge based exploration and interaction with the environment
- Informal, in which children engage in naturalistic activities, during which adults might offer suggestions or ask questions
- Structured, in which children are involved in preplanned activities designed by the teacher
"Provided with all three types of experiences," Charlesworth writes, "children encounter a variety of instruction from free exploration to teacher-structured activities." Our classroom provided students with both naturalistic and structured math instruction, but lacked opportunities for informal instruction in which they could share and discuss strategies to problems they could engage with directly.
Cognitively Guided Instruction
It was during a professional development event at our school that I was introduced to Cognitively Guided Instruction, or CGI. CGI is a research-based philosophy developed over the course of twelve years by educators Thomas Carpenter, Elizabeth Fennema, and their colleagues at the Wisconsin Center for Education Research. Editor for Education Week Debra Viadero states about CGI,
"A central idea of [Cognitively Guided Instruction] is that children do not come to school as blank slates, but that they already have certain intuitive understandings about math and can use a variety of fairly predictable strategies to solve problems."
I was intrigued by this statement. "Do kindergarteners really already possess problem-solving skills?" I wondered.
In the first book written about CGI, Children's Mathematics: Cognitively Guided Instruction, developer Dr. Thomas Carpenter maintains that if teachers can understand and articulate the strategies that their students use to solve mathematical problems, they can better guide their students in ways that will help them build on what they already know (p. 4). I learned during my research that CGI addresses the four Mathematical Practices of the Common Core in important--and even essential--ways.
CGI, Mathematical Practices, & The Common Core
I gradually realized during my research that many of the challenges of the Common Core--particularly those associated with its Mathematical Practices--could be met by the incorporation of Cognitively Guided Instruction into the curriculum.
1. Make sense of problems and persevere in solving them.
Drs. Carpenter and Fennema wrote in an article from 1996, "[Within CGI] the teacher is not perceived as the source of knowledge and does not provide ready-made explanations and representations." CGI makes use of contextualized problems and encourages teachers to present the problems as opportunities for children to find solutions in their own ways. Students might use their own unique strategies or find ways to adapt the strategies of others. Debra Viadero wrote in her article from 1995, "Rather than plug numbers into formulas on work sheets, [students] must come up with their own formats and solutions." In order for my kindergarten students to become independent problem solvers, they must be ready to use their own strategies and persevere in approaching problems from a number of different perspectives.
2. Reason abstractly and quantitatively.
In a study conducted about CGI's long-term effects on teaching instruction, one teacher spoke of CGI, stating
"[It's] just a way of reaching the child through more of a hands-on approach, using manipulatives, relating abstract concepts to concrete concepts that they can make a relationship to, so that they learn strategies to solve word problems better."
Though these teacher's observations don't necessarily encompass all of the researched benefits of CGI, she mentions one very important advantage of CGI's problem-based approach to mathematics. Rather than simply practicing rote learning of numbers (for example, memorizing the numbers one through ten), students must make use of numbers through one-to-one correspondence (Carpenter, 1999, p. 2).
One of the findings from the research and development of CGI is that though students can use strategies in unique and unexpected ways, their strategies typically evolve along a predictable path. The earliest strategy children typically use while attempting to solve CGI problems is known as the Direct Modeling Strategy (Carpenter, 1999, p. 15). Students use objects (for instance, unifix cubes) in order to represent the quantities described in the problem. Over time, students begin to use Counting Strategies, which are more efficient and abstract compared to Direct Modeling (Carpenter, 1999, p. 21). Dr. Carpenter summarizes, "Counting Strategies are more efficient and abstract than modeling with physical objects. In applying these strategies, a child recognizes that it is not necessary to physically construct and count the sets described in a problem" (Carpenter, 1996, p. 7). CGI not only provides students with opportunities to reason abstractly and quantitatively, it necessitates that they do!
I gradually realized during my research that many of the challenges of the Common Core--particularly those associated with its Mathematical Practices--could be met by the incorporation of Cognitively Guided Instruction into the curriculum.
1. Make sense of problems and persevere in solving them.
Drs. Carpenter and Fennema wrote in an article from 1996, "[Within CGI] the teacher is not perceived as the source of knowledge and does not provide ready-made explanations and representations." CGI makes use of contextualized problems and encourages teachers to present the problems as opportunities for children to find solutions in their own ways. Students might use their own unique strategies or find ways to adapt the strategies of others. Debra Viadero wrote in her article from 1995, "Rather than plug numbers into formulas on work sheets, [students] must come up with their own formats and solutions." In order for my kindergarten students to become independent problem solvers, they must be ready to use their own strategies and persevere in approaching problems from a number of different perspectives.
2. Reason abstractly and quantitatively.
In a study conducted about CGI's long-term effects on teaching instruction, one teacher spoke of CGI, stating
"[It's] just a way of reaching the child through more of a hands-on approach, using manipulatives, relating abstract concepts to concrete concepts that they can make a relationship to, so that they learn strategies to solve word problems better."
Though these teacher's observations don't necessarily encompass all of the researched benefits of CGI, she mentions one very important advantage of CGI's problem-based approach to mathematics. Rather than simply practicing rote learning of numbers (for example, memorizing the numbers one through ten), students must make use of numbers through one-to-one correspondence (Carpenter, 1999, p. 2).
One of the findings from the research and development of CGI is that though students can use strategies in unique and unexpected ways, their strategies typically evolve along a predictable path. The earliest strategy children typically use while attempting to solve CGI problems is known as the Direct Modeling Strategy (Carpenter, 1999, p. 15). Students use objects (for instance, unifix cubes) in order to represent the quantities described in the problem. Over time, students begin to use Counting Strategies, which are more efficient and abstract compared to Direct Modeling (Carpenter, 1999, p. 21). Dr. Carpenter summarizes, "Counting Strategies are more efficient and abstract than modeling with physical objects. In applying these strategies, a child recognizes that it is not necessary to physically construct and count the sets described in a problem" (Carpenter, 1996, p. 7). CGI not only provides students with opportunities to reason abstractly and quantitatively, it necessitates that they do!
3. Construct viable arguments and critique the reasoning of others.
I learned in preparing for my Action Research that one of the primary goals of Cognitively Guided Instruction is to provide teachers with the framework and the tools that they will need to access and assess the mathematical understanding of their students. In their article from 1996, Drs. Carpenter and Fennema write,
"In CGI, the emphasis is on what children can do rather than on what they cannot do. This leads to a very different approach to dealing with errors than an approach in which the goal is to identify students' misconceptions in order to fix them. For CGI teachers the goal is to work back from errors to find out what valid conceptions students do have so that instruction can help students build on their existing knowledge."
In order to understand which concepts students can already grasp, teachers must use carefully chosen phrases to elucidate their students' knowledge and gain greater understanding of their strategies. Simultaneously, students must try to provide knowledge of their own thought processes. Drs. Nancy Knapp and Penelope Peterson, two education researchers who conducted a study examining the long-term affects of CGI on classroom instruction, remarked about one of the teachers implementing CGI that "she rarely verified or corrected answers, instead encouraging students to work out the correct answers together by presenting and evaluating mathematical arguments." It is my goal that students will be able to present and evaluating each others' reasoning in a way which builds on their collective knowledge.
4. Use appropriate tools strategically.
In order to better understand what the writers of the Common Core meant specifically by "tools," I read a more detailed explanation of their meaning. They write that students successful in mathematics must consider and utilize the available tools when solving various math problems. These tools, they write, "might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software" (National Governors Association Center for Best Practices, 2010, p. 7). Though kindergarteners don't yet have a need for protractors or spreadsheets, it is important that they learn to utilize the tools at their disposal, including their fingers, and pencil and paper. Descriptions from the teachers' implementations of CGI from the study conducted by Knapp and Peterson describe the diverse ways in which students used manipulatives or counters from their classrooms in addition to counting on fingers and creating drawings on their own papers. "We would go back and forth," one of the teachers states, "and we would usually go back to some type of modeling [with manipulatives or drawings]" (Knapp, 1995, p. 48). Within CGI, students are encouraged to make use of classroom tools to devise their own strategies.
Having researched Cognitively Guided Instruction and begun to feel confident about its abilities to fulfill the needs of the Common Core and encourage the independent problem-solving skills of young students, I now had to consider how to implement it and measure its impact in my classroom.