Math Practices Video Series: Mathematical Practice #2

We are continuing with our video series on The Standards for Mathematical Practice. Last week, we reviewed Mathematical Practice #1, and today we are sharing our video for Mathematical Practice #2.

There are multiple parts to each practice. The parts help students develop the habit of mind that is the main practice. Remember that the practices are defined as ways to help students become mathematically proficient. As we look at each practice, think of ways we can help students to take ownership of these practices.

In the second video, students are learning how to determine whether or not a situation is fair. The Essential Question asks: “How can proportions help you decide if things are fair?”

Observe how the teacher begins with a demonstration so that students can develop their understanding about fairness. What questions does she ask? Students are seeing the relationships between different quantities and are able to discover the meaning of those relationships. When they begin to work on problems in their groups, they will be able to use these strategies, thereby building their proficiency.

Mathematical Practice #2: Reason abstractly and quantitatively.

• Mathematically proficient students make sense of quantities and their relationships in problem situations.

• Mathematically proficient students bring two complementary abilities to bear on problems involving quantitative relationships:

- the ability to decontextualize – to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents

- the ability to contextualize – to pause as needed during the manipulation process in order to probe into the referents for the symbols involved

• Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

As you look at your classroom, you probably see students with varying degrees of expertise in this practice. Our job, as educators, is to help students develop a habit of mind that helps them naturally think before they begin to reason abstractly and quantitatively. Ask yourself:

Do you give students enough time to understand the relationship between the different quantities in the problem?

Are students able to visually represent the relationship between the two quantities?

Can the students explain the visual representation to demonstrate an understanding of the problem?

As students take ownership of their learning and develop expertise using the mathematical practices, the content standards (knowledge, skills and understandings, procedural skills and fluency, and application and problem solving) will make sense, allowing students to achieve success in mathematics.

Math Practices Video Series: Mathematical Practice #1

We are continuing with our video series on The Standards for Mathematical Practice. Last week, we introduced you to the series and today we are sharing our video for Mathematical Practice #1.

There are multiple parts to each practice. The parts help students develop the habit of mind that is the main practice. Remember that the practices are defined as ways to help students become mathematically proficient. As we look at each practice, think of ways we can help students to take ownership of these practices.

In the first video, students are learning how to read and solve a word problem. The Essential Questions asks, “How does rewriting a word problem help you solve the word problem?”

Observe how the teacher prompts students to make sense of the problem. What questions does she ask? Notice that the students are making sense of the problem and planning a solution pathway. When they begin to work on problems in their groups, they will be able to use these strategies in building their proficiency.

Mathematical Practice #1: Make sense of problems and persevere in solving them.

• Mathematically proficient (MP) students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.

• MP students analyze givens, constraints, relationships, and goals.

• MP students consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.

• MP students monitor and evaluate their progress and change course if necessary.

• Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need.

• MP students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.

• Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem.

• MP students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”

• MP students understand the approaches of others to solving complex problems and identify correspondences between different approaches.

As you look at your classroom, you probably see students with varying degrees of expertise in this practice. Our job, as educators, is to help students develop a habit of mind that helps them naturally think before they begin, make sense of what they are doing and persevere in their work.

Ask yourself:

• Do you give students enough time to ask themselves the meaning of the problem?

• Are students aware that there may be more than one entry point to a solution?

• Do they monitor their own progress?

As students take ownership of their learning and develop expertise using the mathematical practices, the content standards (knowledge, skills and understandings, procedural skills and fluency, application and problem solving) that are applied in the classroom will become externalized, thereby allowing students to grasp and achieve success in mathematics.

Teacher Comment: Ocala, FL

Lisa Goldsmith, Math Specialist, visited classrooms in Ocala, Florida where she facilitated activities from the Big Ideas Math program. Here is a comment from one of the teacher’s classes:

“I want to thank you for coming to my class this week. The lesson went well for the group in period 5. One student told me that she understood radius better after the hands on activity.”

In this particular class, Lisa facilitated Activity 6-3 from the sixth grade book. Like all the activities, this gives the students an opportunity to discover the mathematics, discuss the mathematics with their peers and understand the why and how of the mathematics.

Mathematical Practices Video Series: Introduction

Happy New Year to you all! In 2012, we have lots of goodies in store for you. The Big Ideas Math blog is back and ready to give you the content you’re looking for.

To begin the new year, we will be posting videos and descriptions for each of the Standards of Mathematical Practice.  Today, we’re sharing some background information about the standards.

Common Core State Standards Initiative

The Common Core State Standards Initiative states as a mission statement:

“The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.”

In addition to well defined grade level standards, the framers of the Common Core State Standards also developed Standards for Mathematical Practice.

“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).”

The Big Ideas Math program was developed, from the ground up, using the 8 Mathematical Practices as the foundation for learning.

During the next month we will blog about each practice. We hope that you will join in the discussion as together we work to help all students reach their potential!