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.

Math Practices Video Series: Mathematical Practice #3

We are continuing with our video series on The Standards for Mathematical Practice. So far, we’ve looked at Mathematical Practice #1 and Mathematical Practice #2, and today we are sharing our video for Mathematical Practice #3.

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 third video, students are learning how to determine whether or not figures are similar to each other. The Essential Question asks: “What information do you need to know to find the dimensions of a figure that is similar to another figure?”

Observe how the teacher engages the students in demonstrating similar figures. Notice how she incorporates previous knowledge and mathematical terminology as she instructs students to form similar right triangles.  Students are justifying their conjectures and providing valid arguments.  When they begin to work on problems in their groups, they will be able to use these strategies, thereby building their proficiency.

Mathematical Practice #3: Construct viable arguments and critique the reasoning of others.

• Mathematically proficient (MP) students understand and use stated assumptions, definitions, and previously established results in constructing arguments.

• MP students make conjectures and build a logical progression of statements to explore the truth of their conjectures.

• MP students are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.

• MP students justify their conclusions, communicate them to others, and respond to the arguments of others.

• MP students reason inductively about data, making plausible arguments that take into account the context from which the data arose.

• MP students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

• Students learn to determine domains to which an argument applies. They can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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 construct viable arguments and critique others. Ask yourself:

Do you give students opportunities to rely on previous knowledge?

Are students given enough time to explain their thought processes?

Do they validate the conclusions of others?

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.