Trains and Equivalence

Over the next few weeks, possibly months, we’re going to be walking through Gattegno’s Mathematics Textbook 1.  This book is all of 102 pages, but if you are starting in the pre-k to the 1st-grade range it has enough material to last 18 to 36 months or so. One of the things that Gattegno doesn’t do, and neither does Chambers, is tell you why you are doing a certain exercise and how it fits into the general scheme of mathematics. We start off with trains, but why?

We’re going to cover the exercises themselves, how to do them,  what variations you can use and why, and other information that seems appropriate. This is no substitution for a parent playing with the blocks. And this is not going to be the quick and dirty way to learn either. But, if your time is limited to the time you have with your student/child doing the math, this blog series should serve as a traveling companion. The subtitle of each blog post will contain the chapter of Gattegno’s Mathematics  Textbook 1 from which we are working.

Chapter 1 of Gattegno’s Mathematics – Sessions of Free Play

When you turn to that page you’ll find it blank. Nothing. Nada. He isn’t just being cute. You will find that both Mortensen Math and Gattegno/Cuisenaire place a strong emphasis on free play. We no longer begin our math sessions with free play because the blocks are readily accessible at any time and the kids play with the blocks outside of math time. This time should not be cut short. There is a ton of research on children and building blocks. It is not the same as playing blocks with an app. Kids need to get their hands on them and manipulate them.  

Free play serves multiple purposes:

1. Develops spatial reasoning
2. Develops fine motor skills
3. Encourages patience and focus
4. Encourages problem-solving and reasoning skills
5. Develops creative divergent thinking

In addition to the above, our specific goals are to:

1. Develop familiarity with the rods and skill in handling them.
2. Understand that the rods of the same color are equal in length; rods of the same length are the same color.
3. The rods have an order of size.
4. The rods are interchangeable; we can replace one rod with a combination of two or more different rods.

Students should not be told this. Most children will pick this up on their own by playing with the blocks. If not, the pre-arithmetic activities will help them discover these ideas.

How long should free play last? The first few introductions to the blocks should be entirely free play. If the students do not have the opportunity to play with the rods outside of a designated math time, it is advisable to start each math period with a session in free play of at least 10 minutes.

Chapter 2 of Gattegno’s Mathematics Textbook 1

As we go through these exercises, they should be included in your bag of tricks as you work with your students. These are certainly NOT the only activities you can do with your children. As your students grow out of an activity, you can drop it from your rotation. We made activity cards for many of these plus a few of our own. We remove cards as the student grows past them or as they become incredibly boring to the student. Activities do not need to be done in a particular order, but we do need to aware that we are building on skills. We will repeat some activities over and over. But each time we are looking to gather more information.

Equivalence by Color and Length

This activity will only need to be done once or twice. Dump a pile of rods onto the table and ask the student to sort the bricks by color. Gather all the rods together and mix them up. Now ask the student to sort the brick by size.  What we want the child to understand is that rods of the same color are the same length. Questions you can ask if the child doesn’t immediately make the observation: What color is the smallest rod? What color is the longest rod? What color is the biggest rod? Which block is bigger — a red brick or a yellow brick? Are all the red bricks smaller than the yellow bricks? How do you know? We will be covering comparisons in later activities, so don’t feel you have to cover that right now.

Trains

When we ask a child to make a train we mean he should place two or more rods end to end in a line. The smallest train is a train of two white rods. Making trains is something your kids will get used to. They serve as building blocks to larger mathematical ideas. We want to build trains and lots of them. We are looking for trains that are the same as an individual rod; trains that are the same size as other trains; trains made of the same blocks that are the same length as a train made of 2 or more different blocks.

When we talk about trains we want to use the terms equal and same. Make a train that is the same length as a yellow rod. Make a train that is equal to 5 green rods. At this point, we also want to ask the student to read the train. When the student gets good at making trains and comparing trains we will move on to patterns.

The blue train would be read “blue and red and black”.

The above trains would be read as “blue and red and black” and “green and green and green and green and green and green”. It is important that the student reads trains of the same color as we have noted. In a short time, they will transition to 6 x g, or 6g. Saying it the long way helps a child cement in his mind what it is we are doing when we multiply. We will introduce the language of multiplication when the child asks and has need of it.

What Are We Learning?

• We can combine blocks to make same with other blocks. We want them to eventually understand that numbers are made up of other numbers.
• Equal means same. Not in exactly the same but same in a certain attribute. In this case, we mean the same length or same quantity.
• Some trains can equal trains with rods of all the same colors and some cannot. This with help us understand multiplication, division, ratios and fractions later.
• There are rules for making what we want out of the blocks. Our students may not understand what the rules are right now, but they intuit that there is order in the blocks.
• Reading trains begins the process of reading, writing, and speaking the language of mathematics. As soon as a child is putting trains together s/he should be asked to read some of the trains they have made. Not all, please, or you will soon irritate your child.

Don’t rush to get through these exercises. The students need enough time to start to get the math “in their fingers” as my husband says. Don’t worry too much about asking questions at this stage. Pay close attention to what your students are doing and try to figure out how they are thinking. Try not to make judgments about right and wrong. Right now we are just getting to know their math mind.  Don’t forget to play blocks yourself.