Florida’s B.E.S.T. Standards for Mathematics should be taught through the lens of the MTRs. Therefore, working to appropriately embed these standards throughout each lesson is fundamental to helping students gain mastery of mathematical content.
Florida’s B.E.S.T. standards are certainly written in clear language, therefore incorporating them into the mathematics classroom should be no problem … right? Well, maybe not. It turns out that while the clarity of the standards and communicating the desired outcomes is a good first step in designing mathematics instruction, it is equally important to describe the success criteria by which we can assess whether students are making progress toward the desired outcomes.
Beneath each MTR are bulleted descriptions of behaviors one should expect to observe as students engage with the specific MTR. Ideally, students would view these statements as the success criteria by which they could self-assess their ability to fully reason with mathematics on a regular basis. These statements are followed by Clarifications, intended to guide teachers in ways they can integrate the MTR Standards with the B.E.S.T. mathematical content standards.
Lastly, a description of what success for each MTR might look like in terms of its complexity is given at the surface, deep, and transfer levels – according to the Structure of the Observed Learning Outcome (SOLO) Taxonomy (Biggs and Collis, 1982).
MTR 1 (MA.K12.MTR.1.1): Actively participate in effortful learning both individually and collectively.
Mathematicians who participate in effortful learning both individually and with others:
- Analyze the problem in a way that makes sense given the task.
- Ask questions that will help with solving the task.
- Build perseverance by modifying methods as needed while solving a challenging task.
- Stay engaged and maintain a positive mindset when working to solve tasks.
- Help and support each other when attempting a new method or approach.
Teachers who encourage students to participate actively in effortful learning both individually and with others:
- Cultivate a community of growth mindset learners.
- Foster perseverance in students by choosing tasks that are challenging.
- Develop students’ ability to analyze and problem solve.
- Recognize students’ effort when solving challenging problems.
Surface Success Criteria: Stay with a challenging problem for more than one attempt.
Deep Success Criteria: Try several approaches in finding a solution and only seek hints if stuck.
Transfer Success Criteria: Struggle with various attempts over time and learn from previous solution attempts.
MTR 2 (MA.K12.MTR.2.1): Demonstrate understanding by representing problems in multiple ways.
Mathematicians who demonstrate understanding by representing problems in multiple ways:
- Build understanding through modeling and using manipulatives.
- Represent solutions to problems in multiple ways using objects, drawings, tables, graphs, and equations.
- Progress from modeling problems with objects and drawings to using algorithms and equations.
- Express connections between concepts and representations.
- Choose a representation based on the given context or purpose.
Teachers who encourage students to demonstrate understanding by representing problems in multiple ways:
- Help students make connections between concepts and representations.
- Provide opportunities for students to use manipulatives when investigating concepts.
- Guide students from concrete to pictorial to abstract representations as understanding progresses.
- Show students that various representations can have different purposes and can be useful in different situations.
Surface Success Criteria: Use models to represent and solve a problem and translate the solution to mathematical symbols.
Deep Success Criteria: Use models and symbols to represent and solve a problem, and accurately explain the solution representations.
Transfer Success Criteria: Use of a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem.
MTR 3 (MA.K12.MTR.3.1): Complete tasks with mathematical fluency.
Mathematicians who complete tasks with mathematical fluency:
- 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.
Teachers who encourage students to complete tasks with mathematical fluency:
- Provide students with the flexibility to solve problems by selecting a procedure that allows them to solve efficiently and accurately.
- Offer multiple opportunities for students to practice efficient and generalizable methods.
- Provide opportunities for students to reflect on the method they used and determine if a more efficient method could have been used.
Surface Success Criteria: Use manipulatives, models, discussions, and estimation to explore.
Deep Success Criteria: Find reliable and accurate methods that are generalizable.
Transfer Success Criteria: Use reliable strategies to accurately find solutions to a problem and reflect on efficiency of method.
MTR 4 (MA.K12.MTR.4.1): Engage in discussions that reflect on the mathematical thinking of self and others.
Mathematicians who engage in discussions that reflect on the mathematical thinking of self and others:
- Communicate mathematical ideas, vocabulary, and methods effectively.
- Analyze the mathematical thinking of others.
- Compare the efficiency of a method to those expressed by others.
- Recognize errors and suggest how to correctly solve the task.
- Justify results by explaining methods and processes.
- Construct possible arguments based on evidence.
Teachers who encourage students to engage in discussions that reflect on the mathematical thinking of self and others:
- Establish a culture in which students ask questions of the teacher and their peers, and error is an opportunity for learning.
- Create opportunities for students to discuss their thinking with peers.
- Select, sequence and present student work to advance and deepen understanding of correct and increasingly efficient methods.
- Develop students’ ability to justify methods and compare their responses to the responses of their peers.
Surface Success Criteria: Explain thinking for the solution found.
Deep Success Criteria: Explain their own thinking and the thinking of others with accurate vocabulary.
Transfer Success Criteria: Justify and explain, with accurate language and vocabulary, why their solution is correct.
MTR 5 (MA.K12.MTR.5.1): Use patterns and structure to help understand and connect mathematical concepts.
Mathematicians who use patterns and structure to help understand and connect mathematical concepts:
- Focus on relevant details within a problem.
- Create plans and procedures to logically order events, steps, or ideas to solve problems.
- Decompose a complex problem into manageable parts.
- Relate previously learned concepts to new concepts.
- Look for similarities among problems.
- Connect solutions of problems to more complicated large-scale situations.
Teachers who encourage students to use patterns and structure to help understand and connect mathematical concepts:
- Help students recognize the patterns in the world around them and connect these patterns to mathematical concepts.
- Support students to develop generalizations based on the similarities found among problems.
- Provide opportunities for students to create plans and procedures to solve problems.
- Develop students’ ability to construct relationships between their current understanding and more sophisticated ways of thinking.
Surface Success Criteria: Look for obvious patterns and use if/then reasoning strategies for obvious patterns. Look for structure within mathematics to help solve problems efficiently.
Deep Success Criteria: Find and explain subtle patterns. Compose and decompose number situations and relationships through observed patterns in order to simplify solutions.
Transfer Success Criteria: Discover deep, underlying relationships. See complex and complicated mathematical expressions as component parts.
MTR 6 (MA.K12.MTR.6.1): Assess the reasonableness of solutions.
Mathematicians who assess the reasonableness of solutions:
- Estimate to discover possible solutions.
- Use benchmark quantities to determine if a solution makes sense.
- Check calculations when solving problems.
- Verify possible solutions by explaining the methods used.
- Evaluate results based on the given context.
Teachers who encourage students to assess the reasonableness of solutions:
- Have students estimate or predict solutions prior to solving.
- Prompt students to continually ask, “Does this solution make sense? How do you know?”
- Reinforce that students check their work as they progress within and after a task.
- Strengthen students’ ability to verify solutions through justifications.
Surface Success Criteria: Use estimation and predictions prior to solving.
Deep Success Criteria: Explain methods used and identify strengths and weaknesses of the solution.
Transfer Success Criteria: Compare and contrast various solution strategies in relation to verifying solutions.
MTR 7 (MA.K12.MTR.7.1): Apply mathematics to real-world contexts.
Mathematicians who apply mathematics to real-world contexts:
- Connect mathematical concepts to everyday experiences.
- Use models and methods to understand, represent and solve problems.
- Perform investigations to gather data or determine if a method is appropriate.
- Redesign models and methods to improve accuracy or efficiency.
Teachers who encourage students to apply mathematics to real-world contexts:
- Provide opportunities for students to create models, both concrete and abstract, and perform investigations.
- Challenge students to question the accuracy of their models and methods.
- Support students as they validate conclusions by comparing them to the given situation.
- Indicate how various concepts can be applied to other disciplines.
Surface Success Criteria: Explore through models and methods to connect concepts and skills to everyday experiences.
Deep Success Criteria: Explain how various skills and concepts can be applied within and outside of mathematics.
Transfer Success Criteria: Question and justify that models and methods used are accurate and efficient.
It is important to embed these MTR standards into lessons every day. Realistically, MTRs 1, 4, and 6 could be part of every lesson. Teachers should look for ways to integrate appropriate MTRs in authentic ways to deepen students’ understanding of the mathematics content standards. Ultimately, the goal should be to move from the surface level of each MTR to the transfer level.
These descriptions are part of a SOLO matrix developed by Big Ideas Math co-author, Dr. Laurie Boswell, in her webinar titled, “Surface-Deep-Transfer: Making Connections to the Mathematical Thinking and Reasoning Standards.” Included in the webinar are examples of ways problems can be worded to help move students’ behaviors from the surface level to the transfer level of the MTRs.