Students learn best and retain what they learn better by solving problems. Often, mathematics instruction is shaped by the belief that if teachers tell students how to solve problems and then students practice, students will learn how to do mathematics.

Decades of research tells us that the traditional model of instruction is flawed. Traditional instructional methods may get short-term results with procedural skills, but students tend to forget the procedural skills and do not develop problem solving skills, deep conceptual understanding, or a mental framework for how ideas fit together. They also don’t develop strategies for tackling non-routine problems, including a propensity for engaging in productive struggle to make sense of problems and persevere in solving them.

In order to learn mathematics, students should spend time in math class doing mathematics.

“Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving.”³

Students should take an active role, both individually and in groups, to see what they can figure out before having things explained to them or being told what to do. Teachers play a critical role in mediating student learning, but that role looks different than simply showing, telling, and correcting. The teacher’s role is

1. to ensure students understand the context and what is being asked,

2. ask questions to advance students’ thinking in productive ways,

3. help students share their work and understand others’ work through orchestrating productive discussions, and

4. synthesize the learning with students at the end of activities and lessons.

MP1: I Can Make Sense of Problems and Persevere in Solving Them

• I can ask questions to make sure I understand the problem.

• I can say the problem in my own words.

• I can keep working when things aren’t going well and try again.

• I can show at least one attempt to figure out or solve the problem.

• I can check that my solution makes sense.

MP2: I Can Reason Abstractly and Quantitatively

• I can think about and show numbers in many ways.

• I can identify the things that can be counted in a problem.

• I can think about what the numbers in a problem mean and how to use them to solve the problem.

• I can make connections between real-world situations and objects, diagrams, numbers, expressions, or equations.

MP3: I Can Construct Viable Arguments and Critique the Reasoning of Others

• I can explain or show my reasoning in a way that makes sense to others.

• I can listen to and read the work of others and offer feedback to help clarify or improve the work.

• I can come up with an idea and explain whether that idea is true.

MP4: I Can Model with Mathematics

• I can wonder about what mathematics is involved in a situation.

• I can come up with mathematical questions that can be asked about a situation.

• I can identify what questions can be answered based on data I have.

• I can identify information I need to know and don’t need to know to answer a question.

• I can collect data or explain how it could be collected.

• I can model a situation using a representation such as a drawing, equation, line plot, picture graph, bar graph, or a building made of blocks.

• I can think about the real-world implications of my model.

MP5: I Can Use Appropriate Tools Strategically

• I can choose a tool that will help me make sense of a problem. These tools might include counters, base-ten blocks, tiles, a protractor, ruler, patty paper, graph, table, or external resources.

• I can use tools to help explain my thinking.

• I know how to use a variety of math tools to solve a problem.

MP6: I Can Attend to Precision

• I can use units or labels appropriately.

• I can communicate my reasoning using mathematical vocabulary and symbols.

• I can explain carefully so that others understand my thinking.

• I can decide if an answer makes sense for a problem.

MP7: I Can Look for and Make Use of Structure

• I can identify connections between problems I have already solved and new problems.

• I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.

• I can make connections between multiple mathematical representations.

• I can make use of patterns to help me solve a problem.

MP8: I Can Look for and Express Regularity in Repeated Reasoning

• I can identify and describe patterns and things that repeat.

• I can notice what changes and what stays the same when working with shapes, diagrams, or finding the value of expressions.

• I can use patterns to come up with a general rule.