Wednesday, September 2, 2020

How I Teach Place Value

Our first main lesson block this year is on Place Value and Computation. In Waldorf, Second Grade is when students begin to do Column Algorithms (math problems are presented horizontally in first grade). In Montessori, students begin to work with place value right away in Kindergarten.

Having been trained in both, I prefer the Montessori hands on materials because I believe the color coding is so valuable. You can, of course, use this color coding and these materials in a Waldorf school or when homeschooling.

Montessori Color Coding

Units place is green
Tens place is blue
Hundreds place is red

This is introduced at the very beginning of Montessori math.

Wooden Numeral Cards

Students use these numeral cards when building numbers with the Golden Bead Material (which is only gold, and focuses on having children be able to SEE the difference between 1, 10, 100, 1000). Visually, you want them to focus on that one aspect of the material. Thus the color is the same.

Golden Bead Material

When children solve problems with the GBM, they write the answer using the correct colored pencil for each digit. When you write the problems for them, you write them using the correct colored pencil for each digit.

The numeral cards can also be slid apart for expanded notation or stacked together for standard notation. So convenient!

Montessori math materials are most famous for slowly moving children from concrete materials to abstract. One way you can do this with a child is to write their math problems in regular pencil instead of colored. If they mess up their place value, you should go back to using the colors. It is also a handy visual cue to help them line up their digits when writing their answer. Some children also benefit from having you write U above the units place, T above the tens place, H above the hundreds place, and Th above the thousands place.

As children move on to the Stamp Game -- the next step in abstraction because the stamps are all the same size (the ten tile is not ten times larger than the units tile) but their value is written on them -- the same color coding helps them know which tiles to use when building numbers. And it helps them keep track of which piece is which when doing the actual computation (addition, subtraction, multiplication, division).

Stamp Game

Here, you build the number by putting down the appropriate number of stamps and then manipulate them to solve. You can easily make exchanges to solve dynamic problems (borrowing and carrying). The color coding is maintained for this as well as more complex Montessori materials such as the Checker Board (for long multiplication to the hundred millions place).

Here's how you use the wooden tiles to solve math problems:

Dynamic Addition (5:16)

Dynamic Subtraction (5:26)

Double Digit Multiplication (5:39)

Stamp Game Division (8:54)

In Montessori, all table and floor work is always done on a mat. This shows respect for the material and helps to delineate your work space from a neighbor's. A placemat is fine.

Reading Large Numbers

Lots of adults find it confusing at first that the thousands place is green, just as the units place is. However, if there was a different color for every place value from units to hundred millions... THAT would be a lot of colors to remember. This simple green, blue, red is infinitely scalable. In fact, it is used even in the advanced multi-base bead frame for working with numbers from binary to hexadecimal.

Thousands is green because is is the units place of the thousands family. Think about where the comma would go.

We present the numbers as being in families. The simple family has "no last name." The thousands family is next to it to the left. When you read a large number, you read the number in betwen the commas as you "usually would" and then when you get to the comma you "say its last name." I'll give some examples below. You can go as high as you want when kids get interested in it. I have had children who wanted to read numbers to the quintillions place! I help children read large numbers by putting my hands over everything except the bit that's in between the commas.

So for 156 , 725 I would cover 725 with my hand. I would have them just read "one hundred fifty-six" aloud and then I would point to the comma and we would say the last name of the family: thousand. Then I would cover 156 with my hand and have them read "seven hundred twenty-five." You don't say anything at the end of the simple family because it has no last name.

For 134 , 237 , 984 I would write M for millions under 134 and I would write Th for thousands under 237 and I would write S for simple family under 984. Then I would cover everything except 134. The child would read aloud "one hundred thirty-four" then I would point to the comma and we would say million (the last name of that family). Then I would use one hand to cover 134 and the other hand to cover 984 and the child would read "two hundred thirty-seven" and I would point to the comma and we would say thousand. Then I would cover everything except 984 and they would read "nine hundred eighty-four." Easy!

The patterns are even easier to see if you write the digits in color.

134 , 237 , 984

And, as I say, you can go as high as you like. And there's a whole Montessori lesson called Infinity Street which covers this.

The Dice Game

This is my absolute favorite game for practicing place value and it is great for younger kids AND older kids, especially ones who like to go to a large number of digits. The more digits there are in the number, the harder it is to strategize. You can use a six-sided die or a ten-sided die. I bought ten-sided dice for my students but they didn't arrive in time to go in the first set of tote bags. But families can begin with a six-sided and then I can give them a ten-sided die in the second tote bag!

The idea is simple. You decide how many digits you want to go up to. Every person who is playing (there is no limit to how many people can play but you need at least two) has a piece of paper which is hidden from view of everyone else. A clipboard works nicely. Draw a line for each digit. Let's say you are playing to the ten millions place.

___ ___ , ___ ___ ___ , ___ ___ ___

The youngest person, who is the one this lesson is for, rolls the die and announces what number is showing. Everyone has to write that number in one of the spaces on their paper. Careful! Once you put it down you cannot erase it. Let's say it is a five. I have to decide if I want to put that to the farthest left, or if I think there's a chance a six may be rolled. Hmmmm.

___ _5_ , ___ ___ ___ , ___ ___ ___

Of course, other people may be making a different decision and you can't see their number.

You continue to roll the die and decide where you want to put each digit. When the number is full, each person reads his/her number aloud. Then the youngest person has to figure out who won. Whoever's number is the LARGEST is the winner.

If needed, the child can make the numbers with the math material to compare them and see visually who won, like you would in a Crocodile Game problem.

This game is absolutely fantastic for place value.

In order to win, you have to understand that the more valuable digits are to the left. You have to work a little with probability when coming up with your strategy. You have to practice reading big numbers when you read your number aloud, and you have to work with place value when deciding greater than or less than, and figuring out the winner. A small number will often result in a tie, as multiple players have the same strategy. When you do a really large number, you can get very interesting results!

Using Glass Gems

This year all of my students have to have their own personal set of math materials. Because of COVID, we can't used a shared Stamp Game box. Usually, the Montessori approach advocates having only one or two of each material in a classroom so that there's a built-in social negotiation of either working together or finding something else that's productive to do while you're waiting your turn. However, this is a different time.

In order to have enough materials for everyone and have it not completely break the bank, I chose to make up little bags of colored glass gems. I'm using the Montessori colors, of course. The gems don't have numbers written on them, so families can either write them on the flattened glass bottom side with a silver sharpie, or just leave them unlabeled. Since they don't come with values written on them, the pieces can represent ten thousand, hundred thousand, million, or even higher!

There are decimal colors in the Decimal Stamp Game as well. These are a light red, a light blue, and a light green. I got gems in those colors too.

Decimal Stamp Game

I'm really pleased with this glass gem idea because each child can have a bag of stamp game gems for work at home and also bring them to school. They can be washed with water and bleach if needed. And they can be used in craft projects when all this is over! So the money isn't wasted.

our white burlap tote bags for distance learning materials

our little burlap drawstring bags for Stamp Game gems

red - hundreds place

dark blue - tens place

dark green - units place
light green - thousandths place

light blue - tenths place

pink - hundredths place

If you are a Waldorf person who is doing the Math Gnomes block in grade 1, I suggest using green gems for your math work in all of grade 1, and then switching to the color coding for place value when you introduce Column Algorithms in grade 2. I have some notes about how to introduce this on the Column Algorithms page on my website. Let me know if you have questions!

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