## 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.

When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”

Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.

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.

The basic idea of addition is that we are combining similar things. Once again, we meet the counting models from lesson 1.1: sets, measurement, and the numberline. As homeschooling parents, we need to keep our eyes open for a chance to use all of these models — to point them out in the “real world” or to weave them into oral story problems — so our children gain a well-rounded understanding of math.

Addition arises in the set model when we combine two sets, and in the measurement model when we combine objects and measure their total length, weight, etc.

One can also model addition as “steps on the number line”. In this number line model the two summands play different roles: the first specifies our starting point and the second specifies how many steps to take.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

## 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 education.

Our decimal system of recording numbers is ingenious. Once learned, it is a simple, versatile, and efficient way of writing numbers. … But the system is not obvious nor easily learned. The use of place value is subtle, and mastering it is the single most challenging aspect of elementary school mathematics.

Ironically, these challenges are largely invisible to untrained parents and teachers — place value is so ingrained in adults’ minds that it is difficult to appreciate how important it is and how hard it is to learn.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

In other words, we take place value for granted. I know this was true of me when I started teaching my kids. Every year, their textbooks would start with the obligatory chapters on place value, which seemed to me just busywork. I began to appreciate the vital importance of place value when I read Liping Ma’s book and saw how the American teachers were unable to properly explain subtraction or multi-digit multiplication.

Place value is the heart of our number system, the foundation on which all the rest of arithmetic must be built. Because of place value, “The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.”

## 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.

Many things in mathematics need to be understood relationally — that is, in relationship to other concepts. But some things just need to be memorized. How do you know which is which? A homeschooling friend pointed out that one thing children definitely need to memorize is the counting sequence from 1-100 and beyond. While there are some patterns that make counting easier, one does just have to memorize which “nonsense sounds” we have attached to each number.

Another sort-of counting that young students should master is subitizing — recognizing at a glance how many items are in a small group. Children do this instinctively, but we can help them develop the skill by playing subitizing games.

[Aside: In writing this blog post, I ran into some nostalgia. Back when we first did these PUFM lessons, my daughter Kitten was only a toddler. I wrote, “I’ve tried to do lots of counting with my youngest, who hasn’t quite gotten beyond, ‘…eleven, twelve, firteen, firteen, nineteen, seven,…’ The numbers tend to start appearing randomly after she gets past 10.” Ah, memories.]

## 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 understanding exhibited by several of the Chinese teachers she interviewed.

You gain PUFM the hard way: by teaching. The Chinese teachers with PUFM were the ones who had taught for years, taught multiple levels, and studied intensively the materials they taught. I doubt there’s any other way to do it. Home schooling is great for developing PUFM because you teach for years and teach multiple levels. The problem is, by the time you really understand the stuff, the kids are grown. Here are a few hints to help speed up the process a little bit:

• Learn as much as you can, wherever you can, even when the topic doesn’t seem to relate to what your kids are studying now. Ask questions.
• Pick up library books on math (510-519 on the Dewey Decimal shelves), some of which you’ll find helpful and some will bore you to distraction. Read the helpful ones and return the others — but try to get through at least 10 pages of a math book before giving up. You’ll learn a lot that way.
• Always look for connections between topics. Think about how addition and subtraction are related, or addition and multiplication, or fractions and division. Think about how the different levels of understanding a topic are related. (Hint: Start by reading the lesson titles as well as the lessons themselves. Lay out a few years’ worth of math books and just read lesson titles, to see how it all goes together.)
• Work on picking up the math vocabulary (distributive property, factors, sum, numerator, etc.) yourself and using it as you teach. Having the right words will help you hold ideas in your mind.

## PUFM 1.0 Introduction

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 understanding exhibited by several of the Chinese teachers she interviewed.

The Chinese teachers with PUFM didn’t get it automatically. It grew over many years of teaching several levels of elementary math and of studying their textbooks and teaching materials. They met weekly in teaching research groups to learn from each other’s experience, to find multiple ways to solve problems, and to broaden their mathematical understanding.

More than eight years ago, a group of homeschooling friends started a Yahoo “teaching research group” to discuss math in hope of deepening our own understanding and learning to better help our students. We had a good time, but the busy-ness of everyday life eventually won out. The group has mostly disbanded, though the archives remain. Now I’d like to bring that study to my blog, bit by bit, updated with things I’ve learned in the years since.

## New Edition of Must-Read Math Book

I thought I knew math fairly well.

I thought arithmetic was boring.

I thought the reason other nations beat America in international math tests was that their students worked harder than ours.

I thought all sorts of silly things before I read Liping Ma’s Knowing and Teaching Elementary Mathematics. Now this must-read book is coming out in a new edition, due in bookstores next week.

I can hardly wait!

In American elementary mathematics education, arithmetic is viewed as negligible, sometimes even with pity and disdain—like Cinderella in her stepmother’s house. Many people seem to believe that arithmetic is only composed of a multitude of “math facts” and a handful of algorithms. . . Who would expect that the intellectual demand for learning such a subject actually is challenging and exciting?

## It’s Elementary (School), My Dear Watson

[Rescued from my old blog.]

From Time magazine, June 18, 1956:

“[M]athematics has the dubious honor of being the least popular subject in the curriculum… Future teachers pass through the elementary schools learning to detest mathematics… They return to the elementary school to teach a new generation to detest it.”

Quoted by George Polya in How to Solve It. I finally got my very own copy of this excellent book, so I can quit pestering the librarian to let me order it from library loan again…

Blogger Rudbeckia Hirta teaches math to pre-service teachers, and it seems that not much has changed since 1956. Hirta says the test answers shown were representative of her class — for instance, 25% of her students missed the juice problem. Too bad these students never read Polya’s book, in which he discusses a four-step method for solving problems. Step four is to look back and ask yourself whether the answer makes sense. Good advice!