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Math Instruction

Math Instruction


William McCallum on Focus & Coherence
William McCallum on Coherence
Phil Daro on Rigor
William McCallum on
  Focus & Coherence
William McCallum on
 Phil Daro on Rigor

The Common Core State Standards for Mathematics call for shifts that can be categorized into three major areas:

  1. A greater focus on fewer topics:
    1. Grades K-2: Concepts, skills & problem solving related to addition and subtraction
    2. Grades 3-5: Concepts, skills & problem solving related to multiplication and division of whole numbers and fractions
    3. Grade 6: Ratios & proportional relationships, and early algebraic expressions and equations
    4. Grade 7: Ratios & Proportional relationships, and arithmetic of rational numbers
    5. Grade 8: Linear algebra & linear functions
  2. Coherence: Linking topics and thinking across grades
    1. Shift away from teaching a list of disconnected topics, tricks, or mnemonics
    2. Teaching within a context of the coherent progression of standards from grade to grade
  3. Rigor: Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity
    1. Conceptual Understanding: Help students see math as more than a set of mnemonics or discrete procedures
    2. Procedural Skills & Fluency: Develop core functions in order to have access to complex concepts and procedures
    3. Application: Students learn to correctly apply mathematical knowledge in appropriate situations


Common Core initiatives also require a shift from expecting students to passively absorb information from teachers to actively becoming involved in the pedagogical process, thereby becoming practitioners of mathematics. Developing students into becoming practitioners of mathematics is enhanced by incorporating the eight math practices into instructional methods on a regular basis as a means to engage students in their learning experience.

Examples (videos and activities) from Inside Mathematics (University of Texas at Austin):

  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.


Principles to Action: Ensuring Mathematical Success for All outlines instructional recommendations that teachers, coaches, interventionists, and all who help form the knowledge base of student learning of mathematics; these reflect a range of instructional approaches necessary to promote high levels of learning.

  1. Establish mathematics goals to focus learning.
  2. Implement tasks that promote reasoning and problem solving.
  3. Use and connect mathematical representations
  4. Facilitate meaningful mathematical discourse.
  5. Pose purposeful questions
  6. Build procedural fluency from conceptual understanding.
  7. Support productive struggle in learning mathematics.
  8. Elicit and use evidence of student thinking.

 Discussion Tool posted by System Implementation and Monitoring (under general resources), K-12


...term that describes the relationships and “interactions between teachers, students, and the learning environment and the learning tasks" (Murphy, 2009. p 35)

Teacher-centered, Learner-centered, Learning-centered

Resource  Source  Description
Five Practices  NCTM Publication (Mathematics Teacher) One structure that helps sequence student work in a group environment
Math Practice Standards Math Frameworks Summary of the eight math practice standards applicable to all grade levels
Math Teaching Standards NCTM Summary of the eight teaching practices that help foster student learning

Other Thoughts

Dan Finkel, Ph.D. in Algebraic Geometry University of Washington Five Principles of Extraordinary Math Teaching (TEDx Talks - 2017, Seattle, Washington): "...approach learning and teaching math with courage, curiosity, and a sense of play..."
Dan Meyer, Ph.D. Stanford, Secondary math teacher, speaker, consultant, writer The Learning Exchange Thoughts about inquiry, 2014