Math Practices Video Series: Mathematical Practice #8

Our video series on the Mathematical Practices is wrapping up today as we share our video for Mathematical Practice #8. If you’ve missed any of this video series, you can catch up with our previous posts:

Mathematical Practice #1: Make sense of problems and persevere in solving them.
Mathematical Practice #2: Reason abstractly and quantitatively.
Mathematical Practice #3: Construct viable arguments and critique the reasoning of others.
Mathematical Practice #4: Model with mathematics.
Mathematical Practice #5: Use appropriate tools strategically.
Mathematical Practice #6: Attend to precision.

Mathematical Practice #7: Look for and make use of structure.

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 eighth video, students are learning how to describe an equation. The Essential Question asks: “How do you describe the equation y=mx+b?”

Notice how the teacher probes the students rather than supplying the students with answers. What questions does she ask? 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, thereby building their proficiency.

Mathematical Practice #8: Look for and express regularity in repeated reasoning.

• Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

• By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3.

• As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details.

• Mathematically proficient students continually evaluate the reasonableness of their intermediate results.

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 your students notice patterns in mathematics?

Are your students able to use patterns to formulate a solution?

Do they evaluate the reasonableness of their solution?

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 #7

Our video series on the Standards for Mathematical Practice is coming to a close, but we still have two videos left to share! Today we are sharing our video for Mathematical Practice #7. If you’ve missed any of this video series, you can catch up with our previous posts:

Mathematical Practice #1: Make sense of problems and persevere in solving them.
Mathematical Practice #2: Reason abstractly and quantitatively.
Mathematical Practice #3: Construct viable arguments and critique the reasoning of others.
Mathematical Practice #4: Model with mathematics.
Mathematical Practice #5: Use appropriate tools strategically.
Mathematical Practice #6: Attend to precision.

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 seventh video, students are learning how to break apart a composite figure into its pieces to calculate surface area. The Essential Question asks: “How can you find the surface area of composite figures?”

Observe how the teacher activates the students’ previous knowledge. What questions does he ask? 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, thereby building their proficiency.

Mathematical Practice #7: Look for and make use of structure.

• Mathematically proficient students look closely to discern a pattern or structure.

• Mathematically proficient students can see complicated things as single objects or as being composed of several 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, make sense of what they are doing and persevere in their work. Ask yourself:

Can your students break composite figures into the appropriate pieces?

Are students able to make conjectures and define a strategy to solve problems?

Do they realize composite figures exist in everyday life?

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 #6

Today in our video series on the Standards for Mathematical Practice, Larry Dorf presents Mathematical Practice #6. If you’ve missed any of this video series, you can catch up with our previous posts:

Mathematical Practice #1: Make sense of problems and persevere in solving them.
Mathematical Practice #2: Reason abstractly and quantitatively.
Mathematical Practice #3: Construct viable arguments and critique the reasoning of others.
Mathematical Practice #4: Model with mathematics.
Mathematical Practice #5: Use appropriate tools strategically.

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 sixth video, students are learning how to use rates in real life situations. The Essential Question asks: “How can you use rates to describe changes in real life problems?”

Notice how the students are given time to discuss and write their own word problem. Notice also how the teacher questions and discusses the comparison of rates. Students are discussing situations with which they can relate and with which they will be able to explore. When they begin to work on problems in their groups, they will be able to use these strategies, thereby building their proficiency.

Mathematical Practice #6: Attend to precision.

• Mathematically proficient (MP) students try to communicate precisely to others.

• MP students try to use clear definitions in discussion with others and in their own reasoning.

• MP students state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.

• MP students are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.

• MP students calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context.

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 so that they can mathematically communicate clearly.

Ask yourself:

Do you promote discussion between your students?

Are students able to accurately express their answers?

Are your students able to justify their reasoning with one another?

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 #5

We are continuing with Mathematical Practice #5 in our video series on the Standards for Mathematical Practice. If you’ve missed any of this video series, you can catch up with our previous posts:

Mathematical Practice #1: Make sense of problems and persevere in solving them.
Mathematical Practice #2: Reason abstractly and quantitatively.
Mathematical Practice #3: Construct viable arguments and critique the reasoning of others.
Mathematical Practice #4: Model with mathematics.

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 fifth video, students are learning how to determine the sum of integers. The Essential Question asks: “Are the sum of two integers positive, negative or zero and how can you tell?”

Observe how the teacher immediately makes a real life connection for the students giving meaning to the mathematics.  What questions does she ask?  Students are using multiple representations to display the mathematics.  When they begin to work on problems in their groups, they will be able to use these strategies, thereby building their proficiency.

Mathematical Practice #5. Use appropriate tools strategically.

• Mathematically proficient students consider the available tools when solving a mathematical problem. These tools 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.

• Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

• Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

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 your students use manipulatives to assist them in making sense of a problem?

Are students aware that there may be more than one way to find a solution?

Are students given time to discover the rules of integers rather than just being told?

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 #4

Have you been following our video series on the Standards for Mathematical Practice? If not, you can catch up with our posts on Mathematical Practice #1, Mathematical Practice #2, and Mathematical Practice #3. Today we are discussing Mathematical Practice #4: Model with mathematics.

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 fourth video, students are learning how to use data displays to make decisions about real life situations. The Essential Question asks: “How can you display data in a way that helps you make decisions?”

Observe how the teacher makes a connection to real life, thus providing meaning to the mathematics being taught. Notice how she validates each student’s choice for graphing data but asks the student to justify whether or not his/her decision makes sense. When students begin to work on problems in their groups, they will be able to use these strategies, thereby building their proficiency.

Mathematical Practice #4: Model with mathematics.

• Mathematically proficient (MP) students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

• MP students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.

• MP students are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas.

• MP students can analyze relationships mathematically to draw conclusions.

• MP students routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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 model mathematics. Ask yourself:

Do you give students the opportunity to discuss the connections between math and everyday life?

Are students comfortable making suggestions regardless of their accuracy?

Do students understand the importance of data displays?

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.