** 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.

## Would You Like to Study With Me?

If you want to follow along, these PUFM lessons will use Elementary Mathematics for Teachers, by Thomas H. Parker and Scott J. Baldridge.

You will also need five books from the Singapore Primary Mathematics series: textbooks 3A, 4A, 5A, and 6A, and the 5A workbook. You may get any edition (I have a mixture of the old Primary Maths and the U.S. Edition) — the page numbers may not line up with our text, but just look for the matching topic headings.

We use the Singapore math books to “study teaching materials intensively,” as Liping Ma described. If you don’t have the Singapore texts, then look up the lesson topics in whatever math book your children are studying and analyze how your book presents each concept.

Our “Homeschooling Math with Profound Understanding (PUFM)” lesson posts will be titled with lesson number and topic, as in the headline above, and you will be able to find them all at the PUFM archive page. Work through the lessons at your own speed, and please post any questions or comments. When one person asks a question we all learn a little more because it makes us think more deeply.

As you study, remember the PUFM goal: “Know HOW, and also know WHY.” Most of us already know the “how” of these topics, but we are constantly learning new things about the “why” (and the “how to explain”). In this, homeschool teachers may have an advantage over our public school counterparts, because we teach our way through the entire curriculum. We can see how the concepts develop from year to year. Watch for that development, and for the deep connections between math concepts.

One thing to keep in mind is that mathematics is a story and that teachers are a story tellers; the teaching and curriculum sequences are there to help you with the structure of the story. If you can bring the story of mathematics to life then you will have a much better chance of reaching all your students.

What is easier: memorizing the story of the three little pigs, or learning to tell the three little pigs story on your own? Which is more satisfying?

— Scott Baldridge

Department of Mathematics, Louisiana State University

comment on the ProfoundUnderstanding Yahoo group

## What if You’re Not a Math Person?

Then you’re in good company. Most homeschool teachers are not “math people” — but we can still enjoy the adventure of learning!

Don’t let Liping Ma’s book discourage you. The main thing you want to get from that book is a vision for the goal of PUFM. You want to see what it really means to understand elementary-school math: how the teachers who have a deep understanding of math make connections between topics, and how they build their explanations on a strong foundation of basic concepts.

Then, once you have the goal in mind, you need to work toward it at whatever pace you can. The Parker textbook is an excellent resource for this, and I hope our discussions here will also help. As you learn and as you teach, you will see your ability grow. By this time next year, you will be able to answer your child’s questions from a much stronger conceptual base than you can now — and you’ll still be learning more every year.

This is the first post in the *Homeschooling with a Profound Understanding of Fundamental Mathematics* Series. [Go to the next post.]

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Denise, this is great! Thanks. I’ve just bought the textbook. (I hope I have the Singapore books at home.) I blogged last week about how much more of a mathematician I’ve become these past few years by attending math circles and blogging. I will enjoy this immensely, and I’ll think about how to modify it to work with other community college math teachers. Maybe I can get a group going online next fall.

I am really excited about this. I will be following along for sure! Off to order the textbook.

Thank you for the encouragement, Sue and Shalynn. I’m aiming for a weekly post, at least for awhile, but I know how things come up to redirect the best of intentions…

Parker & Baldridge is on my wish list. Currently reading Hung-Hsi Wu’s “Understanding Numbers in Elementary School Mathematics”, and Lore Rasmussen’s “First Grade Diary.” No room to fit in another elementary math book. How you have fun.

s/How you have fun./Hope you have fun./