I am still unsure about how much understanding a Montessori Primary student should have about numbers and the basic math operations. I know what exercises they are capable of, but what I’ve learned is that children are able to do these exercises with very little real grasp of what they are doing with quantities. The whole point of doing the same exercises over and over is to get them to where they **do** have that grasp, which is great. But just because your child comes out of Montessori able to work the multiplication bead board doesn’t mean he or she has mastered multiplication, or even really understands that 3 times 4 is the same as 4+4+4.

Sam came out of Montessori without understanding any of the operations – not even that addition means putting together. I freaked out a little bit, but I’ve been able to observe her for many months now, and there is no real problem with her or with the Montessori method. Sam is on the slow side for math, for sure. And I’m peeved at her teacher for not being able to observe what I have about her lack of understanding. But we’ve been plugging away at all the Montessori math work, and Sam is beginning to get it. If she had continued for another year in Montessori Primary as her teacher recommended, she might have gotten up to speed. But maybe not, because I found a better way to get through to her: combining and integrating the math exercises.

First of all, here are the things that she has done for the past five months:

Exchange game with a die: we play a game together where we take turns rolling a die – 1 through 6. Using single “unit” beads, you take the amount shown on the die on put it in your pile. When you reach ten, you trade in, or exchange, ten unit beads for a “ten bar.” Whomever gets to 100 first wins. (And there is a “one hundred square”.)

Exchange game with a deck of cards – aces through tens, four each. Same thing as above but with cards, and the person with the most at the end of the deck wins, instead of playing to 100.

Exchange game on paper – Same as above but write down the number on paper and sum it up each turn.

War – the standard card game. The concept to learn here is “higher or lower” which Sam was terrible at, coming out of Montessori.

The Bring Me game – I write down a four digit number and Sam has to bring it to me on a tray in the form of beads. (Unit beads, ten-bars, hundred-squares, and thousand-cubes.)

The Bring Me game in reverse – Sam writes the numbers and I bring them to her.

The Stamp Game – we use an iPad app for this. A stamp is just a rectangle with a number on it: 1, 10, 100, or 1000. There are boxes full of each at the top of four columns. Then you get an addition or subtraction problem. Say 1,111 plus 2,222. You start with units. You bring down one “1″ stamp, and then you bring down two “1″ stamps. You do the same with tens, hundreds, thousands. You “exchange” for higher quantities when you reach ten of anything. You have an answer and write it down.

Coin counting – pour out a random pile of coins. Group them by type. Count each pile. How much is each worth? Put the beads for each value next to the pile of each type: unit bead, five-bar, ten-bar, and I have beads tied together to represent a quarter. Mommy writes it down, but multiply to get the total value like this:

- pennies: 6 X 1 = 6
- nickels: 4 X 5 = 20
- dimes: 14 X 10 = 140
- quarters: 3 X 25 = 75

Mommy helps as necessary with the multiplication (many ways to help without telling or using a calculator and that is part of the value here). Then put the coins in a coin-counting machine to check your work. Fun!

Multiplication bead board – times tables using beads in columns and rows to count up the totals.

Girl Scout cookies: use the beads to understand what “four dollars per box” means and how to do the math.

Finger charts for addition and subtraction – these are just charts where you use your fingers to find an answer to a simple math problem. There are control charts to check your work. This is the beginning of memorization, not a tool for conceptual understanding, except that you can see patterns on the chart and that helps.

A skip-counting iPad game – not very useful for us.

Adding and subtracting on our fingers – we do this every day, all the time.

So a lot of that is pure Montessori. But Sam can do any of this by rote, and she gets it down so quickly. I thought that changing things up would help engage her mind more. That was how I decided to do the exchange game with cards after doing it with the die. I felt like she was in a rut with the die and getting to 100. So we used the cards, and she had to integrate that we were doing the same thing. And playing until we ran out of cards forced us to count up and compare our totals at the end (although we would do that most every turn anyway because Sam really loves to WIN right now.) Also, just the idea of turning up a ten card and getting to pull a ten-bar directly – that was different. At first, she couldn’t understand because she was used to taking units – just one through six.

Once I saw the success of this, I decided to do other things to change things up. Instead of Bring Me with Sam always bringing the beads, I had her make up and write the numbers for me to bring. I could see in her responses that this added a new level of understanding to quantities for her. It’s hard to just think of a random four-digit number. I had to help her realize that all she had to do was think of a number between 0-9 for her units, then do the same for each column. (We used four-column paper with headings of “units” “tens” etc. for this). Then, she still has trouble with the idea of not writing down a zero on the higher orders…so if her number is 456, she will want to write “0456.” She also had a lot of trouble reading the numbers. This is something I don’t see Montessori addressing: how we say the numbers is different. “Fifty four” is a weird thing to say rather than “four units and five tens.” What’s the best way to connect how we say the numbers with the decimal understanding of them? I don’t know, but this reverse game helped.

The most awesome thing I did was to make a two-person game out of the Stamp Game and the beads. We use the iPad app to generate some problems. Then one of us finds the answer with the Stamp Game while the other finds the answer with the beads. When we both come up with the same answer, Sam is thrilled. The Stamp Game is just one level more abstract than the beads, and one level less abstract than doing written problems with standard carrying. Doing them together links them.

Then I realized that this was how I would introduce carrying on paper. We played the Exchange Game. She used the beads and I used the beads **and** paper, summing my running total each turn. Whenever I had to carry, she would see me exchange ten unit beads for a ten bar, and she would see me write a tiny “1″ on top of my tens column. She got it immediately.

One day, we went crazy with the Girl Scout cookie game. One of us would sell the other a random number of boxes. Then we’d get out the “four” bead bars, since each box costs four dollars. We added all the fours. Then we skip-counted by four. Then we multiplied by four. We had huge piles of beads on the floor and did the problems every way we could think of. I could practically see the light bulb go off over Sammy’s head that day.

The moral of the story is that all the Montessori materials are great, but Sam needs more integration between them. What she needs now is more word problems like the cookies. I don’t have a source for those, except the random things that come up during the day. But usually we aren’t around our beads when those things come up. Finding a source for word problems, along with more memorization work, is the next order of business.

I am still unsure about how much understanding a Montessori Primary student should have about numbers and the basic math operations. I know what exercises they are capable of, but what I’ve learned is that children are able to do these exercises with very little real grasp of what they are doing with quantities. The whole point of doing the same exercises over and over is to get them to where they

dohave that grasp, which is great. But just because your child comes out of Montessori able to work the multiplication bead board doesn’t mean he or she has mastered multiplication, or even really understands that 3 times 4 is the same as 4+4+4.Sam came out of Montessori without understanding any of the operations – not even that addition means putting together. I freaked out a little bit, but I’ve been able to observe her for many months now, and there is no real problem with her or with the Montessori method. Sam is on the slow side for math, for sure. And I’m peeved at her teacher for not being able to observe what I have about her lack of understanding. But we’ve been plugging away at all the Montessori math work, and Sam is beginning to get it. If she had continued for another year in Montessori Primary as her teacher recommended, she might have gotten up to speed. But maybe not, because I found a better way to get through to her: combining and integrating the math exercises.

First of all, here are the things that she has done for the past five months:

Exchange game with a die: we play a game together where we take turns rolling a die – 1 through 6. Using single “unit” beads, you take the amount shown on the die on put it in your pile. When you reach ten, you trade in, or exchange, ten unit beads for a “ten bar.” Whomever gets to 100 first wins. (And there is a “one hundred square”.)

Exchange game with a deck of cards – aces through tens, four each. Same thing as above but with cards, and the person with the most at the end of the deck wins, instead of playing to 100.

Exchange game on paper – Same as above but write down the number on paper and sum it up each turn.

War – the standard card game. The concept to learn here is “higher or lower” which Sam was terrible at, coming out of Montessori.

The Bring Me game – I write down a four digit number and Sam has to bring it to me on a tray in the form of beads. (Unit beads, ten-bars, hundred-squares, and thousand-cubes.)

The Bring Me game in reverse – Sam writes the numbers and I bring them to her.

The Stamp Game – we use an iPad app for this. A stamp is just a rectangle with a number on it: 1, 10, 100, or 1000. There are boxes full of each at the top of four columns. Then you get an addition or subtraction problem. Say 1,111 plus 2,222. You start with units. You bring down one “1″ stamp, and then you bring down two “1″ stamps. You do the same with tens, hundreds, thousands. You “exchange” for higher quantities when you reach ten of anything. You have an answer and write it down.

Coin counting – pour out a random pile of coins. Group them by type. Count each pile. How much is each worth? Put the beads for each value next to the pile of each type: unit bead, five-bar, ten-bar, and I have beads tied together to represent a quarter. Mommy writes it down, but multiply to get the total value like this:

Mommy helps as necessary with the multiplication (many ways to help without telling or using a calculator and that is part of the value here). Then put the coins in a coin-counting machine to check your work. Fun!

Multiplication bead board – times tables using beads in columns and rows to count up the totals.

Girl Scout cookies: use the beads to understand what “four dollars per box” means and how to do the math.

Finger charts for addition and subtraction – these are just charts where you use your fingers to find an answer to a simple math problem. There are control charts to check your work. This is the beginning of memorization, not a tool for conceptual understanding, except that you can see patterns on the chart and that helps.

A skip-counting iPad game – not very useful for us.

Adding and subtracting on our fingers – we do this every day, all the time.

So a lot of that is pure Montessori. But Sam can do any of this by rote, and she gets it down so quickly. I thought that changing things up would help engage her mind more. That was how I decided to do the exchange game with cards after doing it with the die. I felt like she was in a rut with the die and getting to 100. So we used the cards, and she had to integrate that we were doing the same thing. And playing until we ran out of cards forced us to count up and compare our totals at the end (although we would do that most every turn anyway because Sam really loves to WIN right now.) Also, just the idea of turning up a ten card and getting to pull a ten-bar directly – that was different. At first, she couldn’t understand because she was used to taking units – just one through six.

Once I saw the success of this, I decided to do other things to change things up. Instead of Bring Me with Sam always bringing the beads, I had her make up and write the numbers for me to bring. I could see in her responses that this added a new level of understanding to quantities for her. It’s hard to just think of a random four-digit number. I had to help her realize that all she had to do was think of a number between 0-9 for her units, then do the same for each column. (We used four-column paper with headings of “units” “tens” etc. for this). Then, she still has trouble with the idea of not writing down a zero on the higher orders…so if her number is 456, she will want to write “0456.” She also had a lot of trouble reading the numbers. This is something I don’t see Montessori addressing: how we say the numbers is different. “Fifty four” is a weird thing to say rather than “four units and five tens.” What’s the best way to connect how we say the numbers with the decimal understanding of them? I don’t know, but this reverse game helped.

The most awesome thing I did was to make a two-person game out of the Stamp Game and the beads. We use the iPad app to generate some problems. Then one of us finds the answer with the Stamp Game while the other finds the answer with the beads. When we both come up with the same answer, Sam is thrilled. The Stamp Game is just one level more abstract than the beads, and one level less abstract than doing written problems with standard carrying. Doing them together links them.

Then I realized that this was how I would introduce carrying on paper. We played the Exchange Game. She used the beads and I used the beads

andpaper, summing my running total each turn. Whenever I had to carry, she would see me exchange ten unit beads for a ten bar, and she would see me write a tiny “1″ on top of my tens column. She got it immediately.One day, we went crazy with the Girl Scout cookie game. One of us would sell the other a random number of boxes. Then we’d get out the “four” bead bars, since each box costs four dollars. We added all the fours. Then we skip-counted by four. Then we multiplied by four. We had huge piles of beads on the floor and did the problems every way we could think of. I could practically see the light bulb go off over Sammy’s head that day.

The moral of the story is that all the Montessori materials are great, but Sam needs more integration between them. What she needs now is more word problems like the cookies. I don’t have a source for those, except the random things that come up during the day. But usually we aren’t around our beads when those things come up. Finding a source for word problems, along with more memorization work, is the next order of business.