Multiplication and Fractions Within The Number 5
We have reached the section on multiplication and fractions within the number 5 in Gattegno Mathematics Textbook 1. These topics are covered in exercises 11 - 39 of chapter 4, which is the largest section in this chapter. You can find all my posts in this series by clicking on Gattegno Textbook 1 under categories. If you are new to Gattegno Cuisenaire, Welcome! We're glad to have you. Teaching math to youngsters isn't as difficult as we've made it.
In the last post, I mentioned that if your student wants to use multiplication in compositions that you should allow it, even if you haven't covered it in the textbook. But just because your student is using multiplication in his work doesn't mean you should skip the multiplication exercises in chapter 4.
For those who are new to Gattegno, a math composition is very much like any other composition. It is math created by the student, usually without the help of the rods. It looks something like the following. My son's are much shorter, and I scribe for him as he is an emerging writer.
There is a lot packed into this section which means I'll be covering it over the course of a few blog posts. These 27 exercises contain nearly a 1/3 of the work in this chapter. Gattegno in laying a firm foundation in this section and we want to suck every morsel of wisdom he offers.
Reading Multiplication Statements
Gattegno begins this part of our study with a lesson in reading written multiplication statements in exercise 11. He starts with 2 x 2 and states that we will use this for two red rods and that we'll read that as "two times two", 2 x 1 for two of the white rods, and 3 x 1 for three of the white ones, which we read as "three times one". Elsewhere Gattegno states that it is important that the student realizes that 3 x 1 is different than 1 x 3. Some curricula state that 3 x 1 means three 1's and 1 x 3 means one three. In Ray's arithmetic and several others, including Mortensen, you will find the language three taken one time or one three.
In the big picture, it doesn't matter since multiplication is commutative. We settled with Gattegno's representation since it is more consistent with how we write the symbols in algebra. If I write 3r, I mean there are 3 of the r kind. Students need to know that the statement can be written either way, but for communication purposes, you and your students should agree upon what you mean by statements like 3 x 1 and 1 x 2.
The student will come across a couple of statements that look like they need brackets but no brackets are present. If your student spent some time working with brackets in the last chapter, they might express the need for them here. If they do, it is a excellent time to talk about how the solution to the statement will change depending upon the placement of the bracket. If not, don't worry about it. The purpose of this exercise is to get the student to read the statements not find the solution. Since we are only looking to read the statements, it's best to avoid setting rules for the student regarding the order of operations and brackets at this time. That will come after much exploration.
Adding and Multiplication
In exercises 12-14, the student is exploring what happens to numbers when we do repeated addition and when we multiply. These are investigation activities and time should be given for the student to make observations about statements like 2 x 2 + 1 and 1 + 2 + 2. Not every child will make the transition from 2r + 1 to 2 x 2 + 1 easily. These exercises are meant to address any possible confusion or glitches in their thinking. We want to be paying attention to the things that they say and what they understand about the math. We are not here to tell them anything about addition and subtraction or explain how it works. The student needs the time to think and work through the ideas on their own.
Writing Addition and Multiplication Statements
In exercise 13, the student is expected to write two statements for a rod scenario. Gattegno expects that the student to write this from a verbal statement or from the written word. It was tough for P. to write from an oral expression, so we had to give it multiple attempts over the course of a couple of days. I found this curious as he hasn't seemed to have trouble with understanding multiplication and addition. I suspect it had to do with not being able to see the blocks in front of him: he had to rely on auditory information alone. This is good practice for him as it trains his listening skills. This also proves that Gattegno is brilliant at deepening a student's understanding with what seems like very simple exercises.
In exercise 14, the student is rewriting statements using addition or multiplication depending on the original statement.
Exercise 15 has the student manipulating rods to determine which statement of three makes the longest rod. This work deepens a child's understanding of addition and multiplication and what happens to numbers when we add and multiply. Once the student is proficient at this (it might require more than one math session), lesson 16 has the student repeating this activity with two statements and no rods.
Gattegno defaults to using the rods if the student has trouble doing this mentally. I usually opt to spend more time in previous exercises instead of pulling out the rods. I think this is where taking your time pays off. There is no hurry to get through these early chapters of the textbook, and the student needs time to internalize all the things they are becoming aware of.
In the Facebook group, I posted about P's work on common difference. Several parents had glossed over common difference as they thought it was too easy. I mentioned that when we first started working with common difference, it dawned on me what was happening with the equal sign in algebra. On one side of the equal sign we have a quantity, and on the other side there is a quantity, and these quantities are the same just written differently. When I add or subtract the same number from each side of the = sign the quantity changes, but they remain equivalent. I knew that before. I could have told you that and even explained it because I understood the whole idea of using the balance. But obviously, something wasn't quite clear and playing with the rods cleared it up. I knew it, but it wasn't internalized - that takes time and the time it takes is different for every student (and every adult).