# Lesson 2: How does metacognitive development impact learning in mathematics?

## Objective

Learners will identify real life examples of the major components of the construct of metacognition.

## Case Study

Mrs. B begins teaching a lesson on how to add fractions with like denominators to a small group of 6th grade special education students who are all significantly below grade level in math. After direct instruction including both algorithmic steps (add the numerators and keep the denominator) and conceptual development (Why do we keep the denominator? Did the size of the pieces change?), Mrs. B begins circulating the room. She notices one student is adding the numerators and the denominators (1/5 + 2/5 = 3/10). What should Mrs. B do to help this student?

Look back at what you wrote as a response to the case study scenario in lesson one. Did you include anything that can be related to metacognition? How? Can you think of additional strategies that relate to metacognition? Record your thoughts and save them for later.

## Introduction

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The two major components of metacognition are awareness and regulation. Metacognitive awareness (also referred to as metacognitive knowledge) includes the sub-components of knowledge and experience. Metacognitive knowledge includes an awareness of person, task, and strategy variables (Lai, 2011). An example of metacognitive knowledge regarding person variables is that I know that I do not focus well with the T.V. on while doing my homework. An example of metacognitive knowledge regarding task variables is that I know it will take longer for me to solve a multi-step story problem than a straightforward multiplication problem. An example of metacognitive knowledge regarding strategy variables is that I can answer all straightforward multiplication problems before working on the multi-step story problems. Examples of metacognitive experiences include reading a story problem and realizing that you did not comprehend it fully, and solving a straightforward multiplication problem and realizing that you have to go back and check your work because your answer seems unreasonable.

Metacognitive regulation includes the sub-components of tasks (goals) and actions (strategies). Metacognitive tasks or goals refer to the desired outcome of the cognitive experience. Examples of tasks or goals are solving a multiplication problem, memorizing the order of operations, and learning a new algorithm. Metacognitive actions or strategies refer to behaviors employed to achieve the goals or tasks. In pursuing the goal of solving a multiplication problem a strategy may be splitting the overall problem into smaller problems (i. e. using partial products). A strategy often used to memorize the order of operations is an acronym (PEMDAS) and a strategy used to memorize this acronym is an acrostic (Please Excuse My Dear Aunt Sally). Depending on the task or goal, a wide variety of metacognitive strategies can be employed.

Metacognition requires awareness and regulation. By focusing on the components of each, students can enhance their learning. In mathematics, successful problem solving requires the use of metacognition. In this lesson, you will further explore the construct of metacognition with a specific focus on its role in teaching and learning mathematics. You will read about metacognition and thinking mathematically. You will watch a short video about integrating metacognition in math instruction. You will discuss your thoughts on the integration of metacognition in math instruction. You will identify real life examples of each of the components and sub-components of metacognition. You will create the next section of your quick reference guide by illustrating real life examples of the components and sub-components of metacognition specific to learning mathematics. You will reflect on your own thinking and practice.

Please use the Metacognition in Math Graphic Organizer as you investigate the materials for this lesson.

References

Lai, E. R. (2011). Metacognition: A literature review. Pearson Education, Retrieved from http://www.pearsonassessments.com/hai/images/tmrs/Metacognition_Literature_Review_Final.pdf

## Investigation

1. Read: p. 57 (starting with the section titled "Self-regulation, or monitoring and control")- 81 (first paragraph) of Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics" by Alan H. Schoenfeld http://gse.berkeley.edu/Faculty/AHSchoenfeld/Schoenfeld_MathThinking.pdf
2. Watch: "Metacognition in Math" http://teachertube.com/viewVideo.php?video_id=240808
3. Discuss: How does metacognitive development impact learning in mathematics? What are some real life examples of the use of metacognition in mathematics instruction and learning? Have you been using metacognition in mathematics instruction and learning? Should metacognitive development be integrated with mathematics instruction always? Why or why not? Use the Discussion page for Lesson 2 to discuss your thoughts on these questions. Your initial post should include an answer to these questions which draws on what you have watched and read and connects to your experiences as a student and teacher. Respond to at least two of your classmates' posts.

## Application

You will continue creating a guide for your colleagues on how to integrate metacognitive development in mathematics instruction. To apply what you have learned in Lesson 2, you will create a section for your guide which will illustrate the components and sub-components of metacognition. In this section you must:

• Provide examples of each of the components and sub-components of metacognition as they relate to mathematics.
• Include a case study scenario where metacognitive development in mathematics is necessary (similar to the case study presented in this course).

## Reflection

Use the Discussion page for Lesson 2 to share your reflection on what you have learned and how you have learned it. Your reflection should include:

• what you have learned about metacognition in mathematics.
• how you have learned it.