High Impact Instruction

Making Math Count in WHPS

Fundamentals of Great Teaching in Math

This is a part of a series on fundamental components of great math instruction. We started by focusing on goals that establish what we want students to understand and know in math. These goals are the foundation for effective learning and teaching. Today we will continue the series by focusing on instructional moves and practices that support student learning in math.
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Plan for instruction that supports students to achieve math goals.

In Math in Practice: A Guide for Teachers, Susan O'Connell highlights "research-informed instructional strategies" that are vital in guiding students to develop understanding of mathematics. They are:


  1. Ask deep questions
  2. Build conceptual understanding before focusing on procedural fluency
  3. Make connections between math ideas
  4. Connect math concepts to real situations
  5. Teach problem solving and integrate problems into daily lessons
  6. Use multiple representations of mathematics
  7. Encourage students talk and writing


Let's zoom in and focus on the first four of these strategies today.

Ask Deep Questions

In the past teachers focused on asking for answers and describing procedures. In order to promote understanding of math concepts research tells us teachers must shift instructional talk from telling to asking and guiding. We should plan for questions that will help students develop mathematical understanding. The questions should challenge students to think about the math and make connections between concepts.


On page 23 in Math in Practice: A Guide for Teachers, Susan O'Connell identifies the types of questions we should be asking:


  • We ask why. We expect students to justify their answers, defend their choices, and construct arguments to prove their thinking.
  • We ask how. We expect students to explain how they solve problems or arrive at answers, knowing not just what steps they took, but why each one made sense.
  • We ask what. Rather than having students copy definitions of math concepts, we expect them to define and describe key math concepts in their own words.
  • We ask for insights. We expect students to be able to observe data, notice patterns or repetition, make conjectures, and explain their insights.


The goal is to shift students from doing math, to thinking about and understanding math.

Build Conceptual Understanding Before Focusing on Procedural Fluency

I recall being in elementary school and the teacher showing me the steps for multi-digit addition. I can still here her saying, "Always start on the right. Add your ones, if it is 10 or more, carry the one to the next column," and on the step-by-step directions for adding went. I was taught the procedure. Some students seemed to have so much difficulty remembering and following these steps. It made these students feel like they were not "math people". How difficult it must of been for students to try to remember all of these steps without any understanding of the math that made sense of the steps!


We have many tools available to help students build conceptual understanding:

Show multiple strategies. When students develop a toolbox of strategies, they develop a deeper understanding of the mathematics. It is powerful when students see the connections between strategies and the mathematics.

Explain and defend your answers. While a right answer is necessary, it is not sufficient to show conceptual understanding. When students explain their thinking and defend the answers, then they can show their true understanding of the math.

Be flexible and nix the tricks. Susan O'Connell says, "There is nothing wrong with shortcuts, rules, and algorithms...Telling [students] how it all works robs them of the opportunity to make sense of the math they are learning." (p.14) Let students use strategies and solve problems in a way that values their understanding of the mathematics rather than follows a prescribed procedure that has little connection to their own understanding.

Arouse discovery. Develop lessons that are grounded in understanding mathematical processes. Give students opportunities to look for rules and patterns in the work through models, so they can discover the math ideas and shortcuts or rules. Then connect their understanding and rules to procedures that lead to efficiency. Susan O'Connell says," We want students to be involved in finding the shortcuts, not just using them." (p. 14) Go to www.heinemann.com and register Math in Practice: A Guide for Teachers (See page 12 of your guide for the registration code) to see video examples of guiding students to make discoveries and develop understanding in math (Chapter 1).

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From Principles to Actions: Ensuring Mathematical Success for All (2014) NCTM p. 47
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Make Connections Between Math Ideas

Students need to see math as an integrated body of understanding where one idea builds on another, rather than as separate skills to be mastered. This starts with the goal of lesson. Each learning target should progress from and build upon previous understanding. Today's learning should connect to and be a foundation for tomorrow's learning. Students progress from understanding with models to connecting the models to more abstract representations and eventually to symbolic representations with numbers and equations. Facilitate students as they focus on the big ideas in math and how today's learning supports and contributes to a developing knowledge about a bigger math concept.

Connect Math Concepts to Real World Situations

Ground the mathematical ideas in context. When introducing a new concept, start with a story problem that highlights the mathematical strategy. This helps students see how math is connected to their lives. Susan O'Connell says, "Through problem contexts, the use of data, and scenarios from children's literature, our students begin to see math as making sense in the world."


Examining context helps students really understand the operations. Having students visualize the story or context and see the action of the story helps builds students' understanding in order to apply and connect the mathematical operations to their real experiences. Understanding operations is a cornerstone in developing mathematical understanding and context helps students lay this foundation.

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