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Educator’s Handbook to Quality Math Instruction

by Cali Kulavic

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THE EDUCATOR'S
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HANDBOOK
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TO QUALITY MATH INSTRUCTION
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Using the concrete-representational-abstract framework for learning
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Written by: Cali Kulavic
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Mathematical Mindset . . . . . . . . . . . . . . . . . . .

Equitable Strategies . . . . . . . . . . . . . . . . . . . . .

Concrete Learning . . . . . . . . . . . . . . . . . . . . . . .

Representational Learning. . . . . . . . . . . . . . .

Abstract Learning . . . . . . . . . . . . . . . . . . . . . . .

Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Common Questions . . . . . . . . . . . . . . . . . . . . .

Additional Resources . . . . . . . . . . . . . . . . . . . .

About the Author . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
Mathematics instruction in the last decade has lost its sense of purpose in teaching students to fully understand numbers and the values they represent. Often times I observe math “tips and tricks” that devalue the foundational learning of numbers and hinder students from understanding more complex ideas in mathematics as they progress through the grades.
When students are taught tricks to solve math problems, they are unable to use a variety of thinking strategies to deepen their understanding. Across districts, schools, and even classrooms within the same school, we can see stark differences in the way children are taught mathematics - and that begins with the way they are taught to approach mathematics.
Differentiated instruction has become a crucial concept in education as we address the needs of students with diverse backgrounds, needs, and content knowledge.
This idea of differentiated instruction is at the core of almost all trainings and professional development that educators attend, yet we fail our students every day that we do not provide them the opportunities to practice math in meaningful ways, using a variety of techniques in the classroom.
“... we can see stark differences in the way children are taught mathematics - and that begins with the way they are taught to approach mathematics."
To fully understand how we as educators can provide enhanced learning opportunities to our students in math,
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we must first address how we approach the subject, first with ourselves and then with our students.

It is my belief that math curriculum needs to be reformed to allow students ample time to practice concepts using a three-way, non-linear approach to thinking and learning. In this book, we will discuss the concepts surrounding mathematic curricula reform through discussion of mathematical mindset and concrete, representational, and abstract thinking.

In addition, we will address crucial components of education such as ethics, diversity, cultural relevance, academic standards, rigor, and student performance.

The concrete-representational-abstract thinking model is a non-linear model of thinking and learning. The purpose of this model allows students to build on skills and strategies they already know and feel comfortable maneuvering, before attempting additional strategies to deepen understanding and effectiveness.
● Concrete thinking
● Representational thinking
● Abstract thinking
Non-linear


Ellipse;
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Mathematical Mindset
Attempting to tackle a new framework of teaching and learning is not an easy feat, nor can it be accomplished without restructuring the way we think and feel about mathematics - not just for ourselves, but in the way we approach math with our students as well.

Following the framework and concepts presented by education author and professor Jo Boaler, we can begin to understand how the brain functions and what that means for learning in mathematics.
When you engage in authentic, deep learning, synaptic activity in our brains creates lasting connections and forms structural pathways (Boaler, 2016). This type of learning not only happens in traditional classroom activities, but through peer-to-peer conversations, building, manipulating, and game play. Through research, we have discovered that the brain continues to grow and show incredible capacity even within a relatively short period of time (Boaler, 2016). Imagine if we harnessed that power and utilized our time effectively with our students - imagine the growth we would see.

Additional scientific evidence suggests that the “difference between those who succeed and those who don’t is not the brains they were born with, but their approach to life, the messages they receive about their potential, and the opportunities they have to learn.” (Boaler, 2016 p. 6)
Comic Panel 1
Figure 1.1 Brain activity in individuals with a fixed and growth mindset
Source: Moser et al., 2011.
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