Explaining Place Value in Numbers to a Mixed Age Group

© Grace Chapman

The greatness of teachers is measured not by how much they know but by how much they share.
Rev. Jesse Jackson

I recall bedtime with our first born. Life was different then. We'd spend an hour in bed together, singing songs and nursery rhymes we knew, making up songs, telling stories, reading stories, playing games and eventually drifting into sleep. When our second child was born I still did all that - but not quite as often. I had a left and a right side so that each child could still snuggle in close to Mum. I didn't always lead the singing - sometimes I was tired enough to just let the eldest child lead the singing and often she sang us to sleep.

The children were writing out an invoice for some work they'd done. They had to find 15 lots of $3.50. They couldn't do it. I showed them how I would do it on paper. Nick showed them his way. The children followed our steps but I could see that they still hadn't understood the procedure. (That is, they wouldn't be able to do it on their own.) That's no worry because I don't expect mastery straight away- 12 year old has been shown examples of multiplication before this.  The children were happy enough to have an answer so that they could get on with other things.

I slept on this and woke with an idea. I would show the children, physically, what I understand about place value- eventually arriving at two digit multiplications like the given example. (They understand that multiplication is addition and can do simple multiplications like 7x 89 in their heads.)

I invited the children to hear me tell a story about numbers. I asked them to gather large and small rocks, shells and coral from their vast collection.

I drew a picture of a house and called it the House of Ones. I asked the children  to put the pebbles  into groups of 10. I then went off on a tangent, showing how these could be recorded using tally marks-they were excited by this and so we played with  Roman numerals for a while. That was a major distraction of excitement. We saw how the X is made from two Vs .. We saw many things. Time for a glass of water and a promise to return to Roman numerals another day.

Back to our clusters of ten pebbles, I went on with the 'story'. ( I really can't repeat the story-I just bungled along, took lots of slow breaths, trusting that the words would come  - being careful not to judge my performance but rather just to enjoy it.) When the ones become a cluster of ten, they have to move into a new house, the House of Tens. (This makes me think of the three little pigs who had to move on, it makes me think of our own children, who once they have amassed enough knowledge, experience etc, move on from familiar houses.) We swapped each cluster of ten for a scallop shell. The scallop shells were gathered into groups of ten and moved on to the new house, the House of Hundreds. They became pieces of coral, then onto the House of Thousands where they became big rocks.

At this stage, everyone was still actively involved and enjoying the pattern. As much as possible, I made sure the younger ones were actively involved in moving the pieces around.

We then played a game of placing pieces in the houses and reading the number. I said things like, "I'm thinking of a number that has 8 ones, 3 tens, no hundreds and 7 thousands." I then took them one step further- "Given the number 7 038, add on 2 tens. Now add on 7 ones" etc. At this stage, we were still manipulating the pieces. I could see that the two eldest were calculating the answers in their heads but were still enjoying manipulating the pieces and racing each other. Seven year old was losing interest but I didn't call for her attention, aware that she was still absorbing what we were doing. She was playing a quiet game with her own collection of pieces.

We then went on to removing pieces-subtraction. (By now 7 year old was saying, "I can't do this". My response was to smile and add that she couldn't do this 'yet' and reminded her  that her mind was still absorbing what we were doing. I wanted her to know that I had no expectation of her achievements.)  I went one step further and showed how I would record these actions. The children could see very easily what is really happening when they are subtracting or adding and carrying and borrowing from houses. We'd  had enough. Time for a drink of water.

Twelve year old hung around. She  wanted to see how this could work with money. We went on together for another quarter hour or so-working around the original question of 15 x $3.50.

As I'm writing this account, 24 hours later, what remains with me is the pleasure we had together. We took time to laugh at jokes ( there are lots of them when 9 year old son is around). We went off on tangents, we 'went with the flow'. We placed no judgement on anyone's performance. I know there will be many more such times.

I'm sure this sort of thing happens in lots of homes. If my account has served to help someone appreciate what they are doing, or if it has inspired someone to share what they do, then my mission is accomplished. Can we ever really  grasp the full effect of our actions?