How to Teach Factorials with Cuisenaire Rods
"Mom, what's a factorial?" Um, let me get out my colorful box of Cuisenaire rods.
In case you thought you found a mathematical whizzy mom, I have to confess I didn't know the answer.
Well, I didn't know that I already knew the answer to be exact.
That's the nature of Cuisenaire rods. You are playing with factorials long before you know their name.
So how long have we been building factorials?
Since our free permutation exercise, of course. There are so many shades of wonder in that activity.
But even before that, we built trains and mats at the start of Gattegno's Textbook 1.
Here in lies the mathematical beauty of Cuisenaire rods. They do not break down but carry the student far into the fascinating world of numbers. You can not do that with counting bears.
How to Teach Factorials with Cuisenaire Rods
So how did I present factorials to my children?
The factorial problem that was presented to them was something like how many ways could three monsters be ordered for a game of tug-a-war. It came out of a comic style math book. Immediately, this factorial problem was familiar to me but not so much for the children.
I pulled out 3 different color rods, a white, a red and a light green. Then I named each rod after one of the monsters. I asked them to build as many different 3 rod trains as they could using one rod of each of the 3 colors.
They looked puzzle because they were all too familiar with the activity. How was this related to factorials?
Factorials with Cuisenaire Rods Develops Organized Thought
But they began to build....and organize. Here I noticed a huge improvement in their organization since the permutation exercise. They began to sort the rods with much more thought to ensure they had found all the rods.
First, my children did all trains that started with the white rod. Then, the red rod. Finally, the light green. Last time, as seen in the picture above, it was a hot mess. It was haphazardly done. If they hadn't figured out a pattern from Pascal's triangle, they may never had been able to answer the most important question.
What is that ALL important question? >>How do I know I found all the trains?<<
But they did wonderful sort it out this time. That is the beauty of mat building. It develops organized thought.
We continued to explore factorials. We tackled the next question. What if we added another monster? That is, what if we added another color rod, the purple rod? How many different trains could we build now?
The two mats were finished. It was now time to hone in on their habit of observation. For the first mat, they noticed the trains could only start 3 different ways, white, red or light green. Next, they observed the second part of the train only had two options which finally left one option. Yes, 3-2-1.
The notation they saw for factorials in a comic style math book now had meaning. They connected the 3 x 2 x 1 with the 6 trains it took to create all the possible ways. It became even more solid when they saw the pattern extend itself in the next mat. 4 x 3 x 2 x 1 = 24 trains
That left us off to wonder. How many different trains for a mat with 5 different color rods? And this exploration went on all the way to 10. Did we build them all these trains? Obviously not.
The rods gave us enough understanding that we didn't have to do all the tedious work of building all those trains. No, we capitalize on that laziness and move to the realm of notation.
5! = 1 x 2 x 3 x 4 x 5 = 120 different trains (or ways)
6! = 1 x 2 x 3 x 4 x 5 x 6 = 720 different trains (or ways)
Factorials with Cuisenaire Rods and Storytelling
Lastly, we moved into storytelling. What other situations would we find factorials? Here they created their own word problems.
How many different ways can 7 friends line up for the merry-go round? How many ways can a person eat 6 scoops of ice cream? How many ways can you order a deck of 52 cards? (Check out this video for more on that).
Factorials, Cuisenaire Rods and Symbol Sense
Gattegno gave my youngest a love for notation. He felt quite fancy creating this notation after learning about factorials. I wasn't 100% sure he understood factorials, so I asked him to extend his equation. He understood much more than I thought. He wrote out 64 x 63 x 62 x 61 x 60 x 59 x 58 x 57 x 56...
This proves another distinct difference in Gattegno's method. Gattegno develops a person's conceptual understanding so that they read meaning into notation.
There is an excellent paper by Dr. Mary Shephard on the ability to read meaning into notation as a high indicator of success in mathematics. You can check out this podcast on that paper HERE.
My Takeaways from Teaching Factorials with Cuisenaire Rods
The basic structure of trains and mat building with Cuisenaire rods builds an intuition for math. When you play with rods, you understand what they can and can not do. Because they are the most mathematically accurate manipulative, they also give you a sense for what numbers can and can not do.
At some point, you have to put the manipulative away. It is just too tedious and cumbersome to do really large problems with rods. But too often, kids are holding on to counting objects in order to do math.
When we use a manipulative for the purpose of calculating, we cripple the student. A manipulative's only purpose is teach how numbers behave. Anything more and you might as well just give them a calculator.