By now you can tell about my fascination with the Montessori Math materials, when you’re reading a second post on Math within two days.
Up until now, almost all of my posts are triggered by my experiences in the classroom. If you have attended a traditional school as a child, just like me. I’m sure, you learned to exchange (by the method that is called ‘carrying’.)
Have you ever wondered… If someone asks you what ‘carrying’ means, you would not know how to explain it to them, because you were just following the algorithm. It was just another process for you, that was being followed by mere memorization.
The golden beads introduce the child to the place value of quantities and the decimal system. We start with the presentation tray. Beginning from the right to left, we place a golden bead ( a unit bead), a bead bar (a bar consisting of ten beads), one hundred square (a square representing 10 bead bars together) and a cube of one thousand ( a cube representing 10 one hundred squares together). This lesson is given in the primary classroom, usually to the 4 year olds.
Moving on to the lessons following in your math ‘scope and sequence,’ as your students have mastered, ‘static addition,’ you can introduce the exchange game for addition to them.
Exchange Game for Addition ( building up the one thousand cube.)
This is a very interesting concept. Beginning with the counting of the units, you will tell the children about the magic number. ‘The magic number is 10. As we count the unit beads, we will stop as we count 10 beads. Now do you see anything on the tray that is similar to these 10 beads?’ The 10 bead bar is similar to these 10 beads. Since we agreed that they look the same, can we give the bank these 10 beads and exchange it for a 10 bead bar? Yes we can!
Moving on to the 10 bead bars. ‘The magic number is still 10. As we count the 10 bead bars, we will stop as we count 10 bars. Now do we see anything on the tray that is similar to these 10 bead bars?’ The 100 square is similar to these 10 bead bars. Since we agreed that they look the same, can we give the bank these 10 bead bars and exchange them for a 100 square? Yes we can!
Moving on to the 100 squares. ‘The magic number is still 10. As we count the 100 squares, we will stop as we have counted 100 squares. Now do we see anything on the rug that is similar to these 100 squares?’ The 1000 cube is similar to these 100 squares. Since we agreed that they look the same, can we give the bank these 100 squares and exchange them for a 1000 cube? Yes we can! Did you observe, how we started our counting with one unit bead and were able to build a 1000 cube with the help of exchanging. This is the exchange game for addition.
Recently, I was introducing a group of children (6-7 year olds) to the concept of subtraction with zeros. Now let me tell you this, these children have already mastered static and dynamic addition and have also been working on multiplication. They have mastered their static subtraction, and have been introduced to dynamic subtraction as well in the recent past and have been working on it for sometime.
For me it was surprising to still see them, being thrown off, as they saw the ‘zeros’ in their subtraction problems. The zeros in the unit column, made them move to the tens column… well there is zero there as well. This was just the best time to introduce them to the exchange game for subtraction.
Exchange game for Subtraction (going down to one unit bead).
I start this game with a story.
Once upon a time, there was a rich king, who made 1000 cubes of gold. One day, one of his knights came to him and requested the king to give him just one tiny piece of gold (one unit bead) from one of his one thousand cubes. Now, the only way to do that is to take this thousand cube apart. To do so, we will have to exchange.
So, lets give this one thousand cube to the bank and the bank will give us, 10 one hundred squares. Do you think now we can give the knight a piece of gold (one unit bead?) Its still not possible, so lets continue to exchange.
We will give the bank a one hundred square and exchange it for 10 bead bars. Do you think now we can give the knight a piece of gold (one unit bead?) Its still not possible, so lets continue to exchange.
We will give the bank a 10 bead bar and exchange it for 10 unit beads. Do you think now we have enough unit beads to take one out for the knight? Yes we do! This is the exchange game for subtraction.
These two games with the ‘Golden Beads,’ are my favorite and some of my students who sometimes struggle with exchange, recall these themselves and often are able to work on the concept. Please remember practice and repetition play a big role in mastering concepts.