*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.”

## Study Teaching Materials

[Experienced Chinese teachers] study how each unit of the textbook is organized, how the content was presented by the authors, and why. They study what examples are in a unit, why these examples were selected, and why the examples were presented in a certain order. They review the exercises in each section of a unit, the purpose for each exercise section, and so on.

— Liping Ma

Knowing and Teaching Elementary Mathematics

This lesson from our textbook covers topics taught in *Singapore Primary Math (Third Edition) 3A* pp. 6-17; *4A* pp. 6-11; *5A* pp. 6-9. If you have a different edition of Singapore math or a different textbook series, look for sections with titles like “Thousands, Hundreds, Tens and Ones” or “Number Patterns” or “Whole Numbers to 100,000” or “Place Values”.

The following video shows how challenging it can be to tell for sure whether our children understand what we teach. When you first hear the girl in class, she sounds like she knows place value, doesn’t she? Yet under closer questioning, we see how fragile is her knowledge. Our students’ understanding of place value needs to become a foundation strong enough to build on, because so much of arithmetic depends on it.

This is one reason we must frequently ask our children, “How did you figure it out?” or, “Why does that work?” Besides, it can be fun to hear what they say.

## Teaching Place Value

Craft sticks (popsicle sticks) and rubber bands make a cheap and helpful set of manipulatives for bundling tens and hundreds, as shown in our textbook on page 7. I’ve also heard of people using coins and dollar bills to teach place value. The popsicle sticks show what’s really happening, because the student can see ten sticks bound together to make a ten, and ten sets of ten to make a hundred. Coins are more abstract, and for many young children, coin values don’t make sense.

My own children disliked using manipulatives of any kind, because it takes so much longer to count out the numbers with blocks or sticks than just to think through the problem. They have strong imaginations, and therefore the *Primary Math* textbook pictures communicated the concepts well enough for them.

There are three steps to counting and working with numbers in a place value system:

- Form bundles to represent the numbers you want to count or add or subtract.
- Rebundle if necessary.
- Record the number of bundles in each appropriate position.

The second step, what the Chinese teachers call composing and decomposing tens, is what gives elementary students the most trouble. When talking to my children, I called this step making and breaking tens. It is not only tens, of course: In any place value column, 10 of that size will bundle together to make one of the position to its left. And in any place value column, you can take one of that size and break it apart into 10 of the position to its right.

## I Love Funny Numbers!

On page 8, our textbook authors talk about the value of counting-by puzzles to cement a student’s understanding of place value. The “Number Patterns” pages in *Singapore Math 3A* (pages 14-17 in my edition) are well worth studying.

But then our authors claim, “The number after 39 is not ‘thirty-ten’.” 😦

At our house, we use “thirty-ten” and “eighty-fourteen” and whatever other funny numbers come up. I want to separate and emphasize that rebundling step. (Or renaming, regrouping, composing and decomposing, carrying and borrowing, whatever terms you like.) We work through our math books orally, and we often give answers such as “27 + 35 = fifty-twelve” — then we make it a separate step to change “fifty-twelve” into the standard form of “62.”

You can see funny numbers in action and pick up a few other mental math tips in Mental Math: Addition.

## Expanded Form

If our students thoroughly understand the expanded form of numbers, as shown in the middle of page 9, they will be well prepared for mental math. I don’t think our book goes quite far enough on this point.

For instance, 405 can also be thought of as 40 tens and 5 ones. 3784 might be 37 hundreds and 84 ones. How you look at a number depends on the problem you are trying to solve. Students must be able to take numbers apart and put them back together so they can work with them in flexible ways.

Here is an expanded form download you can print on cardstock and cut out:

Also, the tens combinations (number bonds) on page 11 are very important, though I seem to be in the minority among homeschoolers because I don’t stress memorizing math facts. I believe most math facts are best learned through continual use in solving problems — and through number bond games like Tens Concentration.

I do use focused memory work as a mop-up activity, after most of the math facts have been internalized through repeated use. For my children, however, such drill work has only been needed with multiplication, and even as we practice those facts I try to build a pre-algebra perspective of deeper understanding. (See my Times Table Series.)

**Summary**

As teachers, we have several tools to help our students master place value:

- Bundling (craft sticks and rubber bands)
- Coins (or number chips) in columns
- Expanded form in words (orally)
- Expanded form in numbers (number cards)
- Counting by ones, tens, hundreds
- Putting numbers in order (small to big, or big to small)
- What comes before, what comes next, number patterns of all types

Students need to be able to move back and forth between all these representations. Can you find all these represented in the *Primary Math* books — or in whatever elementary textbooks you use? Which seem the easiest to you? Which require the most thought?

## Homework

This was a LONG homework set! I was getting finger cramp by the time I was half-way through. I will only comment on a couple of the questions:

#1c) This highlights the difference between our number system and those that came before: The order of the numbers is more important than the value of each individual digit.

#6) I especially liked this problem, with the idea of using pennies, nickels, and quarters to represent base five numbers. Too bad we don’t have a $1.25 bill for the “thousands” column! In general, to convert any number to any other base, we have to think about what the place values are in that base, and so we are forced to notice principles that normally we take for granted.

## Coda: Letter to a College Student

Dear John,

I am sending you $50, as you requested. By the way, please remember that 50 is written with one zero, not with two.

Love, Dad

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. Or start at the beginning.]

While I believe knowing place value is important, sometimes saying big numbers is very intimidating to kids and adults for that matter and it could cause math anxiety. I have spent the last 15 years teaching kids how to accomplish amazing things with numbers and improve their memory and what I have found works best is when I have taught children to square numbers like 487 in their head it is so much less intimidating for them to answer 2 3 7 1 6 9 than say two hundred thirty seven thousand one hundred sixty nine. If you watch any of the clips of kids who learn the Brainetics method you will see they more often than not answer math problems in that manner. Once they start believing in themselves and feel they can accomplish ANYTHING and begin to think math is really cool, then it is so much easier to teach them place value or anything else.

Mike Byster

In my experience, many children enjoy working with big numbers, in much the same way that they are fascinated with long dinosaur names. But you are right, we mustn’t be too fussy about saying the number “just right”, or we will create insecurity. Reading off the digits of a big number is an easy way for children to communicate what they mean.

I don’t agree. I understand there may be some math anxiety associated with big numbers but students don’t gain confidence by avoiding things they find difficult. They gain confidence by slowing building on previous knowledge. Understanding place value is very important. Today, not enough people, children and adults, understand there is a huge difference between large numbers such as “million”, “billion” or “trillion.” Not knowing this can have severe consequences for people as voters.

Perhaps the trouble comes from pedantic teachers. I have heard some teachers, for instance, insist that children must never say the word “and” in the middle of a number, such as, “One hundred and twelve.” The word “and”, in these teachers’ world, only and ever must be used to signal the position of the decimal point. I can see how such unnatural demands could lead to math anxiety among students.

On the other hand, it is also true that to young children, “a hundred” means an unimaginably large number. Slightly older children will use “a million” that way, and to most adults, that’s what “billion” and “trillion” mean. We have no experience with this sort of number, so how can it have meaning to us?

I really appreciate the people who try to put such things in perspective, such as the two pages I linked in this post: How Much Is a Trillion?

Laughing here, because in Australia (and in the United Kingdom too I suspect) we ALWAYS say “and” after the hundreds and before the tens and/or units. We use “point” to signify the decimal.

Great video you have included here.

I agree that place value is the hardest thing to teach in math and must be covered and reviewed all year long. I think children have to reach a certain maturity to understand it.

I like the vocabulary of “breaking” tens, hundreds, etc. When working with younger kids sometimes the vocabulary throws them off (deconstructing, decomposing). I will have to try this out with my class this year!

My Kindergartners caught on to place value well during the past school year. However, manipulatives were always needed in each activity. It is a hard concept for them and thinking about it in their head only was a big challenge. I like the tips for having kids read/understand large numbers. They were able to be successful while having their own tools and drawings to work their way through the problems!

Place value is a concept that needs to be present in more than just a math lesson. I teach kindergarten, and there are numerous opportunities to “sneak” it in all throughout the day. The bundling activity suggested is a variation of a part of our calendar routine. It shows these young students on a daily basis the concept of place value. It is it’s own little “mini-lesson” each day, so to speak. Then towards the end of the year when we focus on place value in math it isn’t so foreign!

Place value is hard for some students to understand. I really enjoyed this article. I would also like to know more about using coins and dollar bills to teach place value.

Thank you everyone for your comments!

I would NOT suggest using coins and dollar bills to teach place value — craft sticks that your students can bundle and unbundle (using rubber bands) work much better. Young children don’t know money well enough to use it as a tool for understanding an abstract concept.

When your students are older, however, money will be a great tool for early work with decimals…