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, tenbars, hundredsquares, and thousandcubes.)
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, fivebar, tenbar, 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 coincounting 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 skipcounting 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 tenbar 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 fourdigit number. I had to help her realize that all she had to do was think of a number between 09 for her units, then do the same for each column. (We used fourcolumn 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 twoperson 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 skipcounted 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.

Loved the way you changed the games up. I think it is important to find innovative ways to make numbers real for the student. Whenever we encounter numbers in our everyday life I try to make it a demonstration of how numbers and mathematics operations are useful in understanding the relationships between objects we come into contact with in the world (a la your girl scout cookies–unfortunately, I live with Yuen, so we cannot even use those to teach object permanence).
Unsurprisingly, when multiplication was the best tool for the problem at hand, I would introduce it, but Trevor did not understand and we had to revert to adding multiple times. I tried several times to make the link, but did not seem to be getting anywhere. Do not get me wrong, I was not worried, just noted that it was not working.
Then one day Yuen and I were discussing how much Trevor has been into football this year and I realized that on those occasions when we let him watch part of a game (and he has started to notice that he is only seeing part of them, damn) touchdowns were the perfect opportunity to teach multiplication.
There turned out to be more challenges than I realized. Unfortunately, there were few touchdowns in the next game I let him watch. Then I had to teach him how we generally assume that a touchdown is 7, even though it is 6 +1, but we can assume 7 because pros always make the extra point, unless they decide not to, then it is 6+2, but we will assume that it is just 7 or 6+1, okay (The first time I realized I had to do this I was overcome by fear that I was about to make it more opaque, rather than use it to make multiplication more clear). Fortunately, Trevor is a sports nut, so he was eager to learn all the possible point combinations. Come up with the most complex scenario you can think of, he has proposed it and relished in the answer.
At any rate, once he understood why a touchdown is generally counted as 7, he actually took very quickly to memorizing how many points 2, or 3, or 4, or 5, or 6 touchdowns would be. As he started getting consistent, I would then say, “So, 6 touchdowns is how many points?” Answer, “42.” “Ok, so what is 6 times 7?” “Um,” was the answer the first time, but some helpful questioning led him to link the two concepts. He does not have his 7 table down solid yet, but he is getting the concept…and he is having fun.
The moral of my story is that the integration that they do so need, is often assisted by relating the abstract concept to a concrete they value….now we just have to keep him interested in actually playing ice hockey because it is less violent than football!..I mean so he can work on his multiples of 1.

Hi Amy,
Could you please email me your contact details? I have a new phone and would like to send you some info from MSO
Angela
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