Staircases and Cuisenaire
We've come to staircases, which are covered in exercises 10-12 of chapter 2 in Gattegno's Mathematics Textbook 1. When I used to think of Cuisenaire Rods, this is what came to mind. If you do a google search for Cuisenaire images, you will find lots and lots of staircases. So what's the deal with the staircase?
Something I wish Gattegno had done differently is tell everyone to slow down. Don't rush through these exercises, even though they look simple. Additionally, he should have denoted certain activities to inform you "Hey this is important, don't skip this." Staircases are one of the exercises you should repeat and take slowly. Gattegno doesn't tell you why you are doing an exercise either, which is also frustrating. Here is what I see happening with the staircases at this stage:
- Students are discovering relationships between quantity.
- There is a fixed quantity between each rod.
- Each successive rod is one quantity larger than the one before.
- Rods have a regular order of size.
- The distance between rods can be measured, and it's consistent.
- Before/after depend on context. Orange is before blue going down the staircase. Orange is after blue going up the staircase.
- Beginning work with number bonds.
Building Our First Staircases
We begin by asking the student to select one rod of each color. This seems like a simple task, and it should be if your blocks come in a case. If your blocks are in a tub or a ziplock bag, this can be a challenge for the student, especially if the student is 5 or younger.You can ask the student to sort the blocks according to color or length first, which is what we did, and then have him select one of each color. Now the student must arrange the rods in order from smallest to largest so that it looks like a staircase. For younger students, this can be a sticking point as well. You will have to fight the temptation to intervene. Unless the student asks for help, let them figure it out themselves.
Types of questions we'll ask:
- Which rod is biggest?
- Which rod is smallest?
- Which rod comes before the biggest rod?
- Which brick comes after the smallest brick?
- Which block comes before red?
- Which block comes after dark green?
- Which rod is bigger a yellow or a black?
We will say the colors of the rods going forward and then backward. When we can do this exercise fairly quickly, we will close our eyes, then repeat the process from memory.
After we have worked with the rods forward and backward, Gattegno suggests that we say to the student, “Now you know the colors of the rods in a sequence." He intends for you to introduce the word sequence. Gattegno is adamant that we use correct mathematical terminology from the beginning. I agree. The only place where I have changed this is adding Jerry Mortensen's emphasis on equal being the same. We interchange same and equal at will.
In mathematics, a sequence is a list of objects, usually numbers, but not always, that are in order. The order depends on context. A sequence can be every odd number, every 3rd odd number, even numbers under 20 listed backward, the alphabet or the letters in your name. A sequence can be finite as in the letters of your name, or infinite, as in every odd number.
We don't need to explain all of this to the student, but you should know what a sequence is. When I introduced the term, I stated that now we know the colors of the rods in sequence. Then asked, "Can you say that for me - sequence?" When we returned to this activity, my terminology changed. "Please say the sequence of blocks going down, starting with the orange brick."
Now that we've made our staircase, we will ask the students to find the rods that will make every step of the staircase even with the orange rod. In the next section, Gattegno will call these compliments. The student can start at the blue rod and work backward; this is the easiest place to start. Gattegno asks that the student removes the blocks before asking questions. We didn't do this as my students would have to take their rods and measure to get the correct answer. Instead, we answered questions with the compliments visible to the student. The following are the types of questions I would ask:
- Which rod does the red one need to be equal/even to the orange one?
- Which rod does the brown one need to make same with the orange one?
- Will the brown one always need a red to be equal with the orange or could you use a different brick?
- What does the yellow one need to be even?
- What about a dark green?
Once the student has done this a few times with the blocks visible, remove the compliments. Now ask the student to name which block goes with a green to make same with an orange block, etc.
Gattegno doesn't do this, but we did and sometimes still do this activity. Instead of removing the compliments I slide them over so that they became a mirror image of the staircase only going down. Ask the student to recite the colors starting on the left with the smallest rod going up the staircase and then down the other side. This activity helps the student understand that the compliments are just the staircase in reverse.
Why are staircases important for later work and why do we need to build a lot of them?
We'll go into this more as we get into chapter 3 and 4 of Gattegno's Textbook 1. Right now our students are just beginning to manipulate numbers, by way of the rods, and be flexible in their thinking. One of our goals is to develop the skills of rapid mental calculation in our children, all of them; my child and the children I tutor. Why? If a child can break apart and rearrange numbers so that the 4 basic operations can be done quickly and accurately, the worst part of math education is over. I hate drill and kill as much as anyone, but the reality is that children must know their basic math facts and how to add, subtract, multiply and divide. That doesn't mean I think timed tests are the way to get that done. Rather, I am with Gattegno in that the students who can manipulate numbers, because they recognize the relationships found in those numbers, will be much quicker and more accurate than the students who were drilled only.
Which is easier to calculate mentally? 303 - 98 or 305 - 100. The 305 - 100. Which one carries the lowest mistake risk? 305 - 100. You can borrow, subtract and do all that jazz if you want, but Gattegno's way is much faster and more accurate. If you get my Math Monday's newsletter, you can see the many applications of this kind of flexibility.
Right now our students are internalizing the idea of quantity, and how one quantity is related to another. So, build staircases. Count them forward, count them backward. Skip count forward and backward on them. Recite the colors with eyes open and with eyes closed. Wear a math mask to make it fun. But don't just breeze over this.