Understanding Oklahoma's Mathematical Actions and Processes (MAPs)

Oklahoma’s Academic Standards for Mathematics Content Standards should be taught through the lens of the Mathematical Actions and Processes (MAPs). Therefore, working to appropriately embed these process standards throughout each lesson is fundamental to helping students gain mastery of mathematical concepts and skills.

Introduction

The Oklahoma Academic Standards for Mathematics are developed around four main content strands with the content progressing throughout PK-12 with seven process standards that are consistent throughout each grade. The goal is for students to grow in their ability to problem solve, communicate, and reason about mathematics in ways that will help them become both college and career ready.

Shown below is the complete text of the Mathematical Actions and Processes (MAPs). However, we believe the MAPs can be made more impactful if the text is interpreted in terms of the behaviors one should expect to observe as students engage with each MAP, therefore, the bulleted list which follows each MAP is intended to restate the text in terms of student behaviors. Ideally, students would view the bulleted statements as the success criteria by which they could self-assess their ability to fully problem solve, communicate, and reason with mathematics on a regular basis.

MAP 1: Develop a Deep and Flexible Conceptual Understanding

Demonstrate a deep and flexible conceptual understanding of mathematical concepts, operations, and relations while making mathematical and real-world connections. Students will develop an understanding of how and when to apply and use the mathematics they know to solve problems.

Mathematically literate students will:

• Make sense of quantities and relationships in problem situations.
• Flexibly move between the context and abstract representations of problem situations.
  • Remove the context from a problem situation.
  • Produce a symbolic representation of the problem.
  • Manipulate the symbolic representation and put it back in context.
• Express connections between concepts and representations.
• Understand and correctly use the units involved in the problem situation.
• Flexibly use properties of operations.

To incorporate MAP 1 effectively in the classroom, teachers can help students by providing opportunities to use manipulatives when investigating concepts. They can guide students from concrete to pictorial to abstract representations as understanding progresses; and expect students to give meaning to all quantities in a task. Additionally, teachers should give students ample opportunities to see how various representations are useful in different situations.

MAP 2: Develop Accurate and Appropriate Procedural Fluency

Learn efficient procedures and algorithms for computations and repeated processes based on a strong sense of numbers. Develop fluency in addition, subtraction, multiplication, and division of numbers and expressions. Students will generate a sophisticated understanding of the development and application of algorithms and procedures.

Mathematically literate students will:

• Select efficient and appropriate methods for solving problems within the given context.
• Maintain flexibility and accuracy while performing procedures and mental calculations.
• Complete tasks accurately and with confidence.
• Adapt procedures to apply them to a new context.
• Use feedback to improve efficiency when performing calculations.

To incorporate MAP 2 effectively in the classroom, teachers can help students by providing the flexibility to solve problems by selecting a procedure that allows them to solve efficiently and accurately. Teachers should provide ample opportunities for students to practice efficient and generalizable methods. It is most impactful if teachers ask students to reflect on the method they used and determine if a more efficient method could have been used.

MAP 3: Develop Strategies for Problem Solving

Analyze the parts of complex mathematical tasks and identify entry points to begin the search for a solution. Students will select from a variety of problem-solving strategies and use corresponding multiple representations (verbal, physical, symbolic, pictorial, graphical, tabular) when appropriate. They will pursue solutions to various tasks from real-world situations and applications that are often interdisciplinary in nature. They will find methods to verify their answers in context and will always question the reasonableness of solutions.

Mathematically literate students will:

• Analyze the problem in a way that makes sense given the task.
• Create a plan for solving the problem.
• Continually ask themselves, “Does this make sense?” as they work through a task.
• Modify their methods as necessary when solving a challenging task.
• Make sense of and explain connections among mathematical ideas and various representations.
• Stay engaged and maintain a positive mindset when working to solve problems.
• Verify their solutions, as well as understand solutions presented by others.

To incorporate MAP 3 effectively in the classroom, teachers can help students by cultivating a community of growth mindset learners. They can foster perseverance in students by choosing tasks that are interesting and challenging, involving meaningful mathematics. Teachers should look to present problems that allow for multiple strategies and multiple solutions. Importantly, teachers should recognize students’ efforts when solving challenging problems.

MAP 4: Develop Mathematical Reasoning

Explore and communicate a variety of reasoning strategies to think through problems. Students will apply their logic to critique the thinking and strategies of others to develop and evaluate mathematical arguments, including making arguments and counterarguments and making connections to other contexts.

Mathematically literate students will:

• Construct possible arguments based on stated assumptions, definitions, and previously established results.
• Use counterexamples appropriately.
• Analyze the arguments of others and ask probing questions to clarify or improve the arguments.
• Compare the effectiveness and efficiency of two or more plausible arguments.
• Recognize errors and suggest how to correct those errors.
• Justify results by explaining mathematical ideas, vocabulary, and methods effectively.

To incorporate MAP 4 effectively in the classroom, teachers should establish a culture in which students ask questions of the teacher and their peers, and error is an opportunity for learning.

They should select, sequence, and present student work to advance and deepen the understanding of correct and increasingly efficient methods. Additionally, teachers should help students develop their ability to justify methods and compare their responses to their peers’ responses.

MAP 5: Develop a Productive Mathematical Disposition

Hold the belief that mathematics is sensible, useful, and worthwhile. Students will develop the habit of looking for and making use of patterns and mathematical structures. They will persevere and become resilient, effective problem solvers.

Mathematically literate students will:

• Recognize patterns, structures, and relationships within quantities, processes, and expressions, and across topics.
• Use properties and operations to make sense of problems.
• Use patterns, structures, multiple representations, and relationships to identify an effective and efficient solution path.
• Decompose a complex problem into manageable parts.
• Use patterns and repeated reasoning to solve complex problems.

To incorporate MAP 5 effectively in the classroom, teachers should encourage students to look for structure, not simply to apply a rule or structure given by the teacher. This means encouraging students to notice key features, such as identifying characteristics of shapes or noticing whether the order in which you add numbers changes the sum. Patterning activities also support attention to structure. Teachers can ask young children to identify the part of a pattern that repeats over and over and can ask older children to figure out a rule for predicting a new instance in a growing pattern or function table.

As with MAP 3, teachers can foster perseverance in students by choosing tasks that are interesting and challenging, involving meaningful mathematics.

MAP 6: Develop the Ability to Make Conjectures, Model, and Generalize

Make predictions, conjectures, and draw conclusions throughout the problem-solving process based on patterns and the repeated structures in mathematics. Students will create, identify, and extend patterns as a strategy for solving and making sense of problems.

Mathematically literate students will:

• Use recognition of repeated reasoning to help identify and understand procedural shortcuts.
• Apply prior knowledge to independently model, understand, and represent real-world problems.
• Identify repeated reasoning in calculations or processes to make generalizations.
• Make predictions and conjectures based on observed patterns and repeated structures and continually evaluate the reasonableness of the intermediate results, comparing to the initial prediction/conjecture.
• Identify, analyze, and draw conclusions about quantities using tools such as diagrams, two-way tables, graphs, flowcharts, and formulas.
• Check to see if an answer makes sense within the context of a situation, improving/revising the model where needed.

To incorporate MAP 6 effectively in the classroom, teachers should present opportunities to reveal patterns and repetition in thinking, so students can make a generalization or identify a rule. They should help students connect new tasks to prior concepts to extend the learning of a mathematical concept. Additionally, teachers should ask for predictions about solutions at midpoints throughout the solution process.

MAP 7: Develop the Ability to Communicate Mathematically

Students will discuss, write, read, interpret, and translate ideas and concepts mathematically. As they progress, students’ ability to communicate mathematically will include increased use of mathematical language and terms and the analysis of mathematical definitions.

Mathematically literate students will:

• Communicate their understanding precisely to others using proper mathematical language.
• Use clear and precise definitions in discussion with others and in their own reasoning.
• Calculate accurately and efficiently, and express answers to the appropriate degree of precision.
• Accurately communicate the meaning of units and clearly and accurately label diagrams.

To incorporate MAP 7 effectively in the classroom, teachers should consistently model the use of precise mathematics language and symbols and expect their students to do the same. They should ask students to identify symbols, quantities, and units in a clear manner. Teachers should also set expectations as to how precise the solution needs to be and help students understand when estimates are appropriate for the situation.

What's the Big Idea?

It is important to embed these MAPs into lessons every day. Teachers should look for ways to integrate appropriate MAPs in authentic ways to deepen students’ understanding of the mathematics content standards. Ultimately, the goal should be to engage students in rich, high-level mathematical tasks that support the approaches, practices, and habits of mind which are called for within these standards.