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PUFM 1.0 Preface
Profound Understanding of Fundamental Mathematics (PUFM) is a phrase coined by Liping Ma in her landmark book, Knowing and Teaching Elementary Mathematics, to describe the deep, broad, and thorough...
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PUFM 1.1 Counting
Photo by Iain Watson via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education....
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PUFM 1.2 Place Value
Photo by Chrissy Johnson1 via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home...
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PUFM 1.3 Addition
Photo by Luis Argerich via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education....
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PUFM 1.4 Subtraction
Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education....
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PUFM 1.5 Multiplication, Part 1
Photo by Song_sing via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education. My...
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PUFM 1.5 Multiplication, Part 2
Poster by Maria Droujkova of NaturalMath.com. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home...
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Understanding Math: A Cultural Problem
All parents and teachers have one thing in common: we want our children to understand and be able to use math. Counting, multiplication, fractions, geometry — these topics are older than the pyramids....
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Understanding Math: What Is Your Worldview?
Click here to read Part 1: Understanding Math: A Cultural Problem. Educational psychologist Richard Skemp popularized the terms instrumental understanding and relational understanding to describe these...
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Understanding Math: Is There Really a Difference?
Click to read the earlier posts: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview? From the outside, it’s impossible to tell how a person is thinking....
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Understanding Math: Area of a Rectangle
Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; and Understanding Math, Part 3: Is There Really a...
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Understanding Math: Multiplying Fractions
Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; Understanding Math, Part 3: Is There Really a...
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Understanding Math: Algebraic Multiplication
Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; Understanding Math, Part 3: Is There Really a...
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Understanding Math: Conclusion
Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; Understanding Math, Part 3: Is There Really a...
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Making Sense of Arithmetic
Homeschoolers have an advantage in teaching math: As our students grow, our own understanding of math grows with them because we see how the ideas build on each other. This is especially true for those...
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