Implement tasks that promote reasoning and problem solving

Teacher Actions

Student Actions

• Motivating students’ learning of mathematics through opportunities for exploring and solving problems that build on and extend their current mathematical understanding.

• Selecting tasks that provide multiple entry points through the use of varied tools and representations. 

• Posing tasks on a regular basis that require a high level of cognitive demand. 

• Supporting students in exploring tasks without taking over student thinking. 

• Encouraging students to use varied approaches and strategies to make sense of and solve tasks.

• Persevering in exploring and reasoning through tasks.

• Taking responsibility for making sense of tasks by drawing on and making connections with their prior understanding and Ideas. 

• Using tools and representations as needed to support their thinking and problem solving. 

• Accepting and expecting that their classmates will use a variety of solution approaches and that they will discuss and justify their strategies to one another.

Equitable Teaching

• Engage students in tasks that provide multiple pathways for success and require reasoning, problem solving, and modeling, thus enhancing each student’s mathematical identity and sense of agency.

• Engage students in tasks that are culturally relevant.

• Engage students in tasks that allow them to draw on their funds of knowledge (i.e., the resources that students bring to the classroom including their home, cultural, and language experiences).

Dig in more!

Why Check it Out

Room to Grow Podcast:  Balancing Conceptual Understanding and Procedural Fluency

In this episode, Joanie and Curtis consider the balance between conceptual understanding and procedural fluency in mathematics teaching and learning. Acknowledging that this idea is like a pendulum that has swung back and forth over several decades, they unpack the meanings of these terms and discuss the impact and implications on student learning.

Article:Supporting High Level Tasks and Talk Through Lesson Study

(NCTM Mathematics Teacher:  Teaching & Learning PK - 12:  Need NCTM Membership to Access)

A cross-grade level team participates in lesson study to develop skills in engaging students in deep mathematical thinking and high-level mathematical discourse.

Fostering Math Practices: Decide and Defend Tasks

This website provides worked examples that can be used to connect procedural fluency with conceptual understanding.

Colorado Department of Education Resources for Tasks

Provides a great summary of the importance of tasks and the actions of teachers and students in relation to this effective teaching practice

Illustrative Mathematics Tasks

A database, sorted by grade level/strand, of rich tasks with which to engage student learning, discourse and conceptual understanding.

From Principles to Actions

Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solutions.

Research on mathematical tasks yielded these major findings:

1. Not all tasks provide the same opportunities for student thinking and learning.

2. Student learning is greatest in classrooms where the tasks consistently encourage high-level student thinking and reasoning and less in classrooms where the tasks are routinely procedural in nature

Build procedural fluency from conceptual understanding

Teacher Actions

Student Actions

• Providing students with opportunities to use their own reasoning strategies and methods for solving problems.

• Asking students to discuss and explain why the procedures that they are using work to solve particular problems.

• Connecting student-generated strategies and methods to more efficient procedures as appropriate.

• Using visual models to support students’ understanding of general methods.

• Providing students with opportunities for distributed practice of procedures.

• Making sure that they understand and can explain the mathematical basis for the procedures that they are using.

• Demonstrating flexible use of strategies and methods while reflecting on which procedures seem to work best for specific types of problems.

• Determining whether specific approaches generalize to a broad class of problems.

• Striving to use procedures appropriately and efficiently.

Equitable Teaching

• Connect conceptual understanding with procedural fluency to help students make sense of the mathematics and develop a positive disposition toward mathematics.

• Connect conceptual understanding with procedural fluency to reduce mathematical anxiety and position students as mathematical knowers and doers.

• Connect conceptual understanding with procedural fluency to provide students with a wider range of options for entering a task and building mathematical meaning.

Dig in more!

Why Check it Out

Progression of K-5 Math Thinking Videos

These videos are a great resource to develop an understanding of the progression of math within a learning trajectory.

Nix the Trick

This website supports a learning focus on the big ideas in mathematics rather than the common tricks that hide the mathematics for students.

Relational Thinking Prompts

These relational thinking prompts from Pam Harris help students focus on the structure of numbers and how they are related to make equivalent expressions.  The goal is to remove the compulsion to calculate without building reasoning first!