Strategies You Can Implement for Deeper Comprehension
Middle school math teacher Deb Maruyama is a shining example of teaching students to “Make sense of problems and persevere in solving them.” She devotes class time and conversation to the notion of getting “stuck and unstuck,” honoring that this is a process all mathematicians experience and providing students with an array of strategies to use to “unstick” themselves. When her students struggle to solve problems, they talk about being stuck, they talk about why they are stuck, what they tried, and how that worked to get them unstuck. In this way, Deb has brought the vague notion of perseverance down to a practical level for her eighth graders.
This is an example of how one teacher brings one of the Common Core Standards for Mathematical Practice to life throughout her math workshops. We can utilize the structure of workshop to enhance students’ abilities to embrace all eight of these practices promoted in the Common Core document. Let us consider opportunities to integrate each of these practices at each stage of our minds-on math workshops, many of which will be explored in greater detail in this chapter and others.
Integrating Standards of Mathematical Practice into Workshop Model Math Instruction
|Mini-Lesson||Work Time||Sharing & Relfection|
|1. Make sense of problems and persevere in solving them.
|• Explicitly teach strategies for dissecting text, representing problems and solving them.
• Discuss and intentionally build students’ endurance.
|• Scaffold students’ independence.
• Offer ample time to explore a variety of solutions.
• Recognize and encourage stamina.
|• Welcome many approaches to problem solving.
• Reflect upon the efficacy of specific strategies.
• Celebrate perseverance.
|2. Reason abstractly and quantitatively.||• Teach math as reasoning and sense-making.
• Model logical reasoning.
• Teach students to support claims with evidence.
|• Invite learners to solve problems using methods that make sense to them.
• Create opportunities for discourse about methods.
|• Ask learners to explain their logic, how they arrived at solutions|
|3. Construct viable arguments and critique the reasoning of others.||• Teach learners how to express their ideas and thinking.
• Teach learners to respond respectfully to the thinking of peers.
|• Promote a respectful community of learners.
• Maintain a safe climate for sharing thinking.
|• Have learners present their solutions and the thinking behind them to peers.
• Encourage respectful discussion of learners’ ideas.
|4. Model with Mathematics||• Introduce a variety of mental models.
• Demonstrate how to transfer information from one representation to another.
|• Offer learners opportunities to make meaning of a variety of models.
• Encourage learners to create models that they themselves find meaningful.
|• Reflect on the value of a range of models given a particular purpose.|
|5. Use appropriate tools strategically||• Introduce a variety of tools.
• Teach learners appropriate use of tools.
• Teach learners to care for tools.
|• Make all tools accessible to students throughout work time.
• Offer tasks that invite use of a variety of tools.
|• Invite learners to consider how helpful specific tools were or were not for a given task.|
|6. Attend to precision||• Model precision.
• Demonstrate techniques that ensure precision.
• Value accuracy over speed.
|• Allow ample work time.
• Encourage peers to attend to precision throughout work time.
• Pause to invite self-monitoring.
|• Reflect on factors that promote or detract from precision.
• Strategize around common errors and the means to avoid those.
|7. Look for and make use of structure.||• Model how to find patterns.
• Discuss how identifying patterns and structures can assist mathematicians in understanding information.
|• Challenge learners to use structure and patterns to help them to solve problems.
• Practice decomposing numbers, equations and expressions into their composite parts.
|• Explore how students made sense of numbers and problems by considering structure and patterns.
• Discuss how structure and patterns helped learners solve problems.
|8. Look for and express regularity in repeated reasoning||• Explore how repeated reasoning can help mathematicians make sense of situations and solve problems.
• Demonstrate how equations, graphs and other mathematical representations summarize patterns.
|• Offer in-depth tasks that invite learners to monitor the reasonableness of solutions.
• Invite learners to generalize from patterns and to express those generalization mathematically.
• Use regularity to monitor for accuracy.
|• Share how learners as mathematicians expressed regularity.
• Discuss how monitoring for regularity helped us to check our solutions.
• Reflect on how repeated reasoning helps us to solve problems.
While it would be impractical to envision ourselves highlighting each of these practices at each stage of our workshop every day, this table can serve as a menu of options for us as we consider ways to make explicit our instruction around the Common Core Standards of Mathematical Practice.
Excerpted from Wendy Ward Hoffer’s Minds on Mathematics: Using Math Workshop to Develop Deep Understanding (Heinemann, 2012).
Hoffer, Wendy Ward. Developing Literate Mathematicians: A Guide for Integrating Language and Literacy Instruction into Secondary Mathematics. Reston: The National Council of Teachers of Mathematics, Inc., 2016. (15-16)
Vygotsky, L. “Interaction between Learning and Development.” In Mind and Society, translated by M. Cole, pp 79-91. Cambridge: Harvard University Press, 1978
Boaler, Jo, and David Foster. “Raising Expectations and Achievement: The Impact of Wide Scale Mathematics Reform Giving All Students Access to High Quality Mathematics.” 2014.
Learn More at an Upcoming Institute
Minds on Math Institute
In this institute, you will learn how to address all eight of the common core standards for mathematical practice within workshop model instruction. We offer this institute twice a year, we hope you can join us at one!
November 14-15, 2017
March 6-7, 2018
Explore how explicit thinking strategy instruction can promote sutdents mathematical understanding while preparing them with problem-solving skills they can apply on standardized assessments and real world situations that require them to think as mathematicians.