** 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.

## Preview of Coming Attractions

We’re almost ready to dig into our lessons in how to understand and teach elementary mathematics. If you want to join us, these PUFM lessons will follow the textbook Elementary Mathematics for Teachers, by Thomas H. Parker and Scott J. Baldridge.

Do you read the front matter of your textbooks, or do you just dive into Chapter 1? I’m compulsive: If it’s in print, I have to read it. In the *Preface *of our textbook, the authors make several points that are worth mentioning.

In order to teach elementary mathematics, we need to understand much more than simply how to solve math problems. We need to know:

- How to present the ideas clearly.
- How each concept fits into the overall plan of mathematics.
- And where our students are likely to struggle, so we can ward off trouble in advance (or at least recognize it when it happens).

And what is the overall plan of elementary mathematics? In Grades K-4, our students lay a solid foundation for the four basic arithmetic operations (+/-, ×/÷) and begin working with fractions and decimals. In grades 5-7, our goal is to develop deeper insights into all these topics and to encourage more abstract thinking, taking initial steps toward a proof-based approach to algebra and geometry.

[Measurement, geometry, probability, and statistics are covered in a separate textbook.]

Like a stool which needs three legs to be stable, mathematics education needs three components: good problems, with many of them being multi-step ones, a lot of technical skill, and then a broader view which contains the abstract nature of mathematics and proofs. One does not get all of these at once, but a good mathematics program has them as goals and makes incremental steps toward them at all levels.

— Richard Askey

quoted in Elementary Mathematics for Teachers

## How is Math Different?

Math is different from many other topics that our students study, in that it builds on itself — topics naturally grow from each other in logical steps, each one building a stronger foundation for the next. While some of the “big ideas” of math can be studied at a surprisingly young age, it is still necessary to help our students build the foundations of math firmly, one stone at a time.

Also, math is logical. Math is consistent — it makes sense. In everything we teach, our goal must be that our students “Know HOW, and also know WHY.” That is, they must know the steps to use in solving a math problem, and they also must know why those steps make sense. A student (or teacher) who only knows HOW does not understand mathematics.

I like to ask my students, “How did you figure that out?” When students explain the reasoning behind their answer to a math problem, they are giving a mini-proof. And if they get used to doing this in elementary school, they will find it much easier to think through more abstract proofs later on.

Finally, math requires practice to make the basics automatic:

Repetition progressively frees the mind from attention to details, makes facile the total act, shortens the time, and reduces the extent to which consciousness must concern itself with the processes.

— E. B. Huey

quoted in Elementary Mathematics for Teachers

## Think like a Teacher

In addition to our textbook, the PUFM course will refer to several books from the Singapore Primary Mathematics series. At times, we will work homework problems from the books, imagining how we would help our students understand. Other assignments will involve “studying the textbook” — that is, reading it carefully in order to see how the mathematics is presented.

As you read through the Singapore math books, think like a teacher. Notice how the ideas are presented in small increments and how the illustrations add to the student’s understanding. What are the key steps in solving each problem? How could you explain these clearly and simply? Remember, the longer the explanation, the more likely it will confuse the student.

Pay special attention to the “concrete → pictorial → abstract” approach. Topics are introduced through hands-on experiences and real-world items, so that children experience the ideas directly. Then the book moves to a pictorial/symbolic approach, using diagrams and models. The book does not rush to a purely-numbers approach to arithmetic, but gives the student plenty of time to build up to such abstraction.

Notice also the many word problems in these books. Never underestimate the importance of word problems. They serve as a sort of mental manipulative, helping students wrap their minds around ideas that would be much more difficult to understand as pure abstraction.

For example, coins, nuts and buttons are clearly distinct and countable and for this reason are convenient to represent relations between whole numbers. The youngest children need some real, tangible tokens, while older ones can imagine them, which is a further step of intellectual development. That is why coin problems are so appropriate in elementary school.

Pumps and other mechanical appliances are easy to imagine working at a constant rate. Problems involving rate and speed should be (and in Russia are) common already in middle school. Trains, cars and ships are so widely used in textbooks not because all students are expected to go into the transportation business, but for another, much more sound reason: these objects are easy to imagine moving at constant speeds and because of this are appropriate as reifications of the idea of uniform movement, which, in its turn, can serve as a reification of linear function.

Thus, we can move children further and further on the way of de-reification, that is development of abstract thinking.

— Andrei Toom

Word problems: Applications vs. Mental Manipulatives

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

Will we need all of the required Singapore math books to begin…or will we start with 3a and progress? I own 3a and am wondering if I need the others right away or if I can buy them as we go? Thanks, Shalynn

The first chapter’s homework refers to 3A, 4A, and 5A, but 3A is definitely the most frequent. At one lesson per week, it will take me almost two months to get through that chapter. The second chapter uses all three textbooks and the 5A workbook, too. The 6A textbook isn’t used until chapter 4.

Okay thanks Denise. I will order them all now just to make sure I’m covered.