Part one of our series on the Mathematical Practices focuses on why they matter for student learning, and the role high-quality instructional materials play in supporting teachers to incorporate them meaningfully into the classroom.
In this four part series, the EdReports mathematics team explores the Standards for Mathematical Practices and why they are essential for every student to learn and grow.
Our classrooms today do not need to look like the classrooms we were taught in. Students should never be sitting in rows with their textbooks and instructions to solve all the odd numbered problems and check their answers in the back.
Instead, students should engage with mathematics beyond the memorization and regurgitation of facts and figures. They should work collaboratively to solve relevant problems and tasks. Students should be challenged to think critically, to make sense of questions and construct viable arguments, choose appropriate tools and strategies to tackle problems, and learn to justify and convey their thoughts. These are all skills that will serve students in math class and throughout their lives.
Mathematics of the present and the future can be creative and collaborative, can involve student voices and experiences, can look different depending on who you are, and can inspire a generation of leaders and mathematicians who will contribute so much to our changing world.
The 8 Standards for Mathematical Practice emphasize the idea that mathematics is more than calculations. The Practices were developed and adopted as part of college and-career-ready standards in 2009 based on “processes and proficiencies” with “longstanding importance in mathematics education”. However, unlike the content standards, which differ for each grade or course level, the Math Practices are consistent across grades K–12.
By utilizing the Math Practices, students immerse themselves in developing arguments, pursue a problem through multiple strategies, or model those strategies for their classmates whether they’re learning numbers or tackling advanced calculus. This makes the Practices powerful tools in ensuring all students can access the kind of learning that has the ability to transform their futures.
Part one of our series on the Mathematical Practices focuses on why they matter for student learning, and the role high-quality instructional materials play in supporting teachers to incorporate them meaningfully into the classroom. Let’s dive into the first three Practices.
|Example of MP1 in a Kindergarten Classroom|
Students engage in a Number Talk with the teacher. The teacher holds up Dot Cards (images of dots in scattered configurations) and asks “How many do you see?”
Students are encouraged to respond with various numbers, and the teacher records their responses. Students have the option to share different responses than their classmates or agree with another. The teacher then asks students to describe how they “see” the number.
Students write their responses for all to see. The variety of examples may include:
3 + 2 + 3 = 8
3 + 3 + 2 = 8
5 + 3 = 8
2 + 3 + 3 = 8
2 + 1 + 2 + 1 + 2 = 8
Students have the option to share a different strategy for how they reached their conclusion or agree with another student.
Why this Practice is important:
Students engage with MP1 when they have opportunities to analyze and make sense of information in problems and are challenged to employ strategies of their choosing to solve them. Throughout the process, students monitor and evaluate their own progress and determine if their answers make sense. Students reflect on and revise their strategies as needed. As students engage in this Practice, they also begin to participate in critical thinking and conceptual understanding.
In many classrooms, students simply duplicate what the teacher has demonstrated on similar types of problems, which reduces sense making and perseverance. MP1 seeks to dismantle that approach by instilling a doggedness in students to come up with different ways to solve problems. Ultimately, students will have the skills to solve any problem—their approach might just look different from those of their classmates.
|Example of MP2 in a 4th Grade Classroom|
Students engage with word problems involving multiplication of a fraction by a whole number, and are asked to develop both visual fraction models (a visual representation of the problem using: pictures, drawings, number lines, bar diagrams, etc.), and equations (abstract) to represent the problem.
Students can have access to various manipulatives: fraction tiles, fraction cubes, fraction circles, cuisenaire rods, rulers, etc. to aid in creating concrete (physical) representations of the problem using manipulative(s) of their choice.
Why this Practice is important:
As students engage in MP2, they analyze a problem, understand what units are involved, and attend to the meaning of the quantities. For example, can a car be divided in the same way as an apple pie?
Through abstract and quantitative reasoning, students can represent situations symbolically and are able to explain what the numbers or symbols represent. They’ll also be able to understand the relationship between problem scenarios and mathematical representations. These connections help students to see that a word problem, an equation, a table, or a graph can all be different ways of representing the same situation.
Being able to reason abstractly and quantitatively allows students to continue to advance in their mathematics learning. Bringing math into the real lives of students and incorporating local context is part of keeping students engaged and illustrating that mathematics concepts can have real-world applications. Allowing students to showcase what that meaning is ensures skills gained from MP2 can be connected to a variety of applicable situations.
|Example of MP3 in an 8th Grade Classroom|
Students solve one-variable linear equations to determine if they have one solution, infinitely many solutions, or no solution. Individually, students construct an argument based on the structure of the equations.
As a small group, students share their findings and engage with MP3 by:
- Constructing a viable argument (explain their strategies) using verbal or written explanations.
- Critiquing and evaluating their own thinking and the thinking of other students.
- Asking questions to one another (and the teacher) to clarify their understanding.
Why this Practice is important:
As students engage in MP3, they make conjectures and progress logically as they explore the truth of their assumptions. Students analyze problems and are able to recognize and use counterexamples. Communicating and justifying their conclusions, as well as engaging in the arguments of their classmates, are key to this Practice.
When students are constructing viable arguments, sharing them with each other, and offering questions and critiques to make those arguments stronger, they are fully engaged in the learning. I have been in many classrooms where I ask questions and the room is silent. But, as soon as students are the ones leading the discussion, conversation flourishes and their excitement is palpable.
Through this practice, students are also building critical thinking skills for learning in other content areas and life beyond school. In our lives, all of us are challenged to construct arguments or explanations, justify those claims with evidence and reasoning, and refine our arguments or explanations based on the critiques of others.
Quality instructional materials support students to engage in the Math Practices in two primary ways: 1) Through tasks that elicit the Practices and; 2) By providing guidance and resources for teachers. Students are able to share who they are, what they’ve learned and apply that to mathematics concepts. When great materials are in the hands of skilled teachers, students have access to the kind of learning that inspires a love of mathematics and builds the know-how they need for the future.
The Math Practices and their meaningful connection to high-quality standards are central to the EdReports mathematics review tools. Our tool specifically focuses on examining how materials support the intentional development of the Practices as integral to college and career-ready standards. Each Practice is addressed in our review criteria, and educator reviewers evaluate every page of a program and document with evidence how materials support students and teachers to engage in the Practices.