## literature review

Thinking is a requirement for learning, and instead of doing the thinking for students and modeling math problems I feel that students need time to feel empowered to be problem solvers. When students are given opportunities to figure things out for themselves they learn how to reason and understand why certain rules apply in math, be discovering these patterns and relationships. Young mathematicians need ample opportunities to practice making sense of everyday math through documenting and communicating their thinking and reasoning.

Cognitively Guided Instruction (CGI) is based on research of

Teachers who use CGI in the classroom pose real-world problems which align with a student’s zone of proximal development (ZPD) (Vygotsky, 1978). The problem should be accessible to students, but should require the them to push their thinking, and build on their prior knowledge to solve the problem, thus extending their mathematical thinking. Vygotskian theory states that students construct knowledge through their independent problem solving, and then build upon this knowledge through social interactions when they team up with their peers. CGI capitalizes on ‘social constructivism’ and this collaborative exchange between students, as it helps children “co-construct knowledge by assuming the mutuality of such interactions” (Inoue, 2012, p. 97).

Teachers are learners along with the students. Teachers study students’ thinking and problem solving strategies and help to guide them to discover the rules, patterns, and relationships within a given math problem. Helping students to be flexible thinkers and problem solvers takes careful planning and time learning about the students’ thinking as they work. When teachers study their students they “construct” how their students are learning, and can implement methods are the most successful for individual students and the class as a whole. “The methods that work best, as identified from the synthesis of meta-analyses, lead to a very active, direct high sense of agency, in the learning and teaching process. Such teaching leads to higher levels of learning, autonomy, and self-regulation on behalf of the learner (whether student or teacher)” (Hattie, 2008).

John Hattie (2008) a researcher in education, states in ‘exemplary schools’ there is an, “emphasis on the engagement of students in the learning process, teachers articulating strategies of instruction and paying attention to learning theories, and the school building as an infrastructure to support such instruction.” The daily monitoring of how students are learning is a vital part of this type of math program. To

Strategic ‘warm ups’ help students to make sense of their math thinking and trust their instinctive knowledge about numbers and mathematical concepts. Five to ten minutes in the beginning of a math lesson is typically set aside to prime student’s thinking and articulation of their thinking. Teachers often relate the warm-up to an area in math that students are struggling with, nudging them to discover efficient strategies. The warm-up can also be used to reinforce a skill or introduce new mathematical concepts. When a teacher facilitates the discussion of student’s mental math strategies, they are “helping students develop and maintain flexibility in thinking about numbers” (Parrish, 2011, p. 206). The rich conversation about what students notice about numbers or shapes creates a community of mathematicians that respect a variety of strategies and perspectives. Boaler writes, “Number talks are the best pedagogical method I know for developing number sense and helping students see the flexible and conceptual nature of math” (2016, p. 47). This is an effective way to begin to prepare young mathematicians for a challenging math problem.

Once a math problem is posed to the class, the teacher helps students ‘unpack’ what the problem is asking by allowing students share their questions about the problem, or restate what is being asked. Once students understand what is being asked of them, they go to their desks armed with their own strategies and begin to solve the problem. “Thinking happens mostly in our heads, invisible to others—effective thinkers make that thinking visible, meaning they externalize their thoughts through speaking, drawing, and writing. They can direct and improve those thoughts,” write Ritchhart and Perkins (2008, p. 58). Students make their thinking visual first as they start to solve problems in a way that makes sense to them. Whether it is a picture, numbers, dots or lines, it is a beginning document that evidences their thinking. Later, this visual work is referenced when sharing their math problem solving with their colleagues, and students begin to co-construct meaning socially with their peers. When students see evidence that there is a variety of ways to come to an answer, they are seeing the relationships of ideas and concepts. Teachers are actively questioning and studying the student’s conversations and work, noting the variety of approaches.

Throughout this type of math learning, teachers are also offering feedback. Feedback can serve as a form of instruction and is shown to have a powerful effect on student achievement. Hattie defines useful feedback as addressing: ‘How it’s going?’, ‘Where they are going?’, and ‘How to get to the next place?’ with the learner. This was found by analyzing the effect size of types of feedback. The most effective aspects of feedback included cues and/or reinforcement to the learner, process-oriented feedback (Hattie, 2012). Clarity, timing, and targeting the purpose and the desired learning goal is important when teachers offer feedback. Personal feedback, as in ‘good job’ or ‘I like that’ is not effective, but helping learners notice where they are and where they are going in the learning process is valuable. Teachers often point students to see what the next steps would be by naming where they are in the learning process, letting them discover correct path, and solve the problem themselves. For example, if a student was direct modeling and drawing 823 cherries and stopped after several circles, crossed it out, and used a counting and adding strategy to attack the problem and wrote ‘800+20+3’ the teacher might offer feedback on that process, “I see you were recording the individual cherries and revised your work to represent them in numbers. You accurately recorded this part of your thinking. What are you planning to do now?”

Hattie’s findings also revealed, “even students’ self-feedback and setting specific goals are effective. They have dramatic effects on the development of self-efficacy, which in turn affects the choice of difficulty of goals. Feedback without goal setting is less effective, and goal setting without feedback is ineffective,” (Hattie, 1999, p. 11). “How to get to the next place” or the actionable item in feedback is the critical component.

All this feeds into helping students reflect and set goals for their learning. Having high expectations for all students is important, and engaging students in goal setting and learning how to take action to achieve them. “A student's own predictions of their performance should not be the barriers to exceeding them, as they are for too many students” (Hattie, 2008, p. 31). On the journey to deep understanding, one needs time to reflect on their learning, and learning goals. “An effective way for students to become knowledgeable about ideas they are learning is to provide some class time for reflection,” (Boaler, 2016, p. 150). Reflection reinforces concepts learned and gives information to the teacher to continue the learning cycle. Focusing on ones thinking—metacognition, is an essential component of math.

Research on children’s learning mathematics supports the idea of honoring students’ thinking by allowing them the time to construct, reflect, and document their thinking. When teachers take time to listen to and examine students’ work, teachers can see the path their students are taking on the journey to constructing meaning and making sense of their mathematical thinking. The goal of my curriculum will be to help students document their mathematical thought processes throughout a typical math routine: warm up, problem solving, collaboration, and reflection. In doing so, I hope to arm them with the tools they need to be powerful problem solvers independently and in their math community. I want celebrate their visible learning with them. I hope in the process of recording their work, they are able to recall information and become more accurate independent problem solvers.

RQ:

Subquestion:

**Cognitively Guided Instruction (CGI)**Cognitively Guided Instruction (CGI) is based on research of

*how*children learn and what students understand to inform what is being taught. Students solve math problems construct their understanding of concepts over time, and creating a deep understanding. “The teachers discuss CGI as a philosophy, a way of thinking about the teaching and learning of mathematics, not as a recipe, a prescription, or a limited set of knowledge. CGI teachers engage in sense-making around children's thinking” (Franke & Kazemi, 2001). Cognitively guided instruction gives students a chance to try a variety of strategies to help them understand conceptually and procedurally*why*these strategies work with the support of their teacher and their peers.Teachers who use CGI in the classroom pose real-world problems which align with a student’s zone of proximal development (ZPD) (Vygotsky, 1978). The problem should be accessible to students, but should require the them to push their thinking, and build on their prior knowledge to solve the problem, thus extending their mathematical thinking. Vygotskian theory states that students construct knowledge through their independent problem solving, and then build upon this knowledge through social interactions when they team up with their peers. CGI capitalizes on ‘social constructivism’ and this collaborative exchange between students, as it helps children “co-construct knowledge by assuming the mutuality of such interactions” (Inoue, 2012, p. 97).

Teachers are learners along with the students. Teachers study students’ thinking and problem solving strategies and help to guide them to discover the rules, patterns, and relationships within a given math problem. Helping students to be flexible thinkers and problem solvers takes careful planning and time learning about the students’ thinking as they work. When teachers study their students they “construct” how their students are learning, and can implement methods are the most successful for individual students and the class as a whole. “The methods that work best, as identified from the synthesis of meta-analyses, lead to a very active, direct high sense of agency, in the learning and teaching process. Such teaching leads to higher levels of learning, autonomy, and self-regulation on behalf of the learner (whether student or teacher)” (Hattie, 2008).

John Hattie (2008) a researcher in education, states in ‘exemplary schools’ there is an, “emphasis on the engagement of students in the learning process, teachers articulating strategies of instruction and paying attention to learning theories, and the school building as an infrastructure to support such instruction.” The daily monitoring of how students are learning is a vital part of this type of math program. To

*cognitively guide*math instruction, teachers choose appropriate, high-cognitive demand mathematical tasks for students based on the knowledge of their students understanding of their previous work, and in turn students engage in math that is rigorous enough for them without becoming frustrating. This strategic approach enhances both teacher and student efficacy as it affords both parties to follow the learning that is happening day-to-day.**Math Routines**Strategic ‘warm ups’ help students to make sense of their math thinking and trust their instinctive knowledge about numbers and mathematical concepts. Five to ten minutes in the beginning of a math lesson is typically set aside to prime student’s thinking and articulation of their thinking. Teachers often relate the warm-up to an area in math that students are struggling with, nudging them to discover efficient strategies. The warm-up can also be used to reinforce a skill or introduce new mathematical concepts. When a teacher facilitates the discussion of student’s mental math strategies, they are “helping students develop and maintain flexibility in thinking about numbers” (Parrish, 2011, p. 206). The rich conversation about what students notice about numbers or shapes creates a community of mathematicians that respect a variety of strategies and perspectives. Boaler writes, “Number talks are the best pedagogical method I know for developing number sense and helping students see the flexible and conceptual nature of math” (2016, p. 47). This is an effective way to begin to prepare young mathematicians for a challenging math problem.

Once a math problem is posed to the class, the teacher helps students ‘unpack’ what the problem is asking by allowing students share their questions about the problem, or restate what is being asked. Once students understand what is being asked of them, they go to their desks armed with their own strategies and begin to solve the problem. “Thinking happens mostly in our heads, invisible to others—effective thinkers make that thinking visible, meaning they externalize their thoughts through speaking, drawing, and writing. They can direct and improve those thoughts,” write Ritchhart and Perkins (2008, p. 58). Students make their thinking visual first as they start to solve problems in a way that makes sense to them. Whether it is a picture, numbers, dots or lines, it is a beginning document that evidences their thinking. Later, this visual work is referenced when sharing their math problem solving with their colleagues, and students begin to co-construct meaning socially with their peers. When students see evidence that there is a variety of ways to come to an answer, they are seeing the relationships of ideas and concepts. Teachers are actively questioning and studying the student’s conversations and work, noting the variety of approaches.

Throughout this type of math learning, teachers are also offering feedback. Feedback can serve as a form of instruction and is shown to have a powerful effect on student achievement. Hattie defines useful feedback as addressing: ‘How it’s going?’, ‘Where they are going?’, and ‘How to get to the next place?’ with the learner. This was found by analyzing the effect size of types of feedback. The most effective aspects of feedback included cues and/or reinforcement to the learner, process-oriented feedback (Hattie, 2012). Clarity, timing, and targeting the purpose and the desired learning goal is important when teachers offer feedback. Personal feedback, as in ‘good job’ or ‘I like that’ is not effective, but helping learners notice where they are and where they are going in the learning process is valuable. Teachers often point students to see what the next steps would be by naming where they are in the learning process, letting them discover correct path, and solve the problem themselves. For example, if a student was direct modeling and drawing 823 cherries and stopped after several circles, crossed it out, and used a counting and adding strategy to attack the problem and wrote ‘800+20+3’ the teacher might offer feedback on that process, “I see you were recording the individual cherries and revised your work to represent them in numbers. You accurately recorded this part of your thinking. What are you planning to do now?”

Hattie’s findings also revealed, “even students’ self-feedback and setting specific goals are effective. They have dramatic effects on the development of self-efficacy, which in turn affects the choice of difficulty of goals. Feedback without goal setting is less effective, and goal setting without feedback is ineffective,” (Hattie, 1999, p. 11). “How to get to the next place” or the actionable item in feedback is the critical component.

All this feeds into helping students reflect and set goals for their learning. Having high expectations for all students is important, and engaging students in goal setting and learning how to take action to achieve them. “A student's own predictions of their performance should not be the barriers to exceeding them, as they are for too many students” (Hattie, 2008, p. 31). On the journey to deep understanding, one needs time to reflect on their learning, and learning goals. “An effective way for students to become knowledgeable about ideas they are learning is to provide some class time for reflection,” (Boaler, 2016, p. 150). Reflection reinforces concepts learned and gives information to the teacher to continue the learning cycle. Focusing on ones thinking—metacognition, is an essential component of math.

**Summary**Research on children’s learning mathematics supports the idea of honoring students’ thinking by allowing them the time to construct, reflect, and document their thinking. When teachers take time to listen to and examine students’ work, teachers can see the path their students are taking on the journey to constructing meaning and making sense of their mathematical thinking. The goal of my curriculum will be to help students document their mathematical thought processes throughout a typical math routine: warm up, problem solving, collaboration, and reflection. In doing so, I hope to arm them with the tools they need to be powerful problem solvers independently and in their math community. I want celebrate their visible learning with them. I hope in the process of recording their work, they are able to recall information and become more accurate independent problem solvers.

RQ:

*How does having students document their mathematical thinking visually help them achieve understanding and master the content?*Subquestion:

*How does having students record all aspects of their mathematical thinking in a math lesson inform my teaching?***References:**

Boaler, J. (2016). Mathematical Mindsets: Unleashing Students' Potential Through Creative

Math, Inspiring Messages, and Innovative Teaching. San Francisco, CA: Jossey-Bass &

Pfeiffer Imprints.

Franke, M. L., & Kazemi, E. (2001). Learning to Teach Mathematics: Focus on Student

Thinking.

*Theory into Practice,*

*40*(2), 102-109. doi:10.1207/s15430421tip4002_4

Hattie, J. (1999).

*Influences on Student Learning*. Inaugural Lecture: Professor of Education in

Lecture presented at University of Auckland, Auckland. Retrieved February 02, 2017,

from https://www.researchgate.net/publication/237248564_

Influences_on_Student_Learning

Hattie, J. (2008). Visible Learning: A Synthesis of Over 800 Meta-Analyses Relating to Achievement (Kindle Locations 928-930). Taylor and Francis. Kindle Edition.

Inoue, N. (2012).

*Mirrors of the Mind: Introduction to Mindful Ways of Thinking*

*Education.*New York, NY: Peter Lang.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn

mathematics. Washington, DC: National Academy Press.

Parrish, S. D. (2011). Number Talks Build Numerical Reasoning.

*Teaching Children*

*Mathematics,*

*18*(3), 198. doi:10.5951/teacchilmath.18.3.0198

Ritchhart, R. & Perkins, D. N. (2008). Making Thinking Visible. Educational Leadership, 65

(5), 57-61.

Vygotsky, L. S. (1978).

*Mind in society: The development of higher psychological processes*

Cambridge, Mass.: Harvard University Press.