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