Number 4 and Number 5 Study
This post continues our study of Gattegno's Mathematics Textbook 1, chapter 4, starting with exercise 5 where we begin the study of the number 4. If you've just stumbled upon my blog, you can find all the posts in this series by clicking on Gattegno Textbook 1 under the categories tab.
Gattegno introduces the student to the study of the number 4 and the number 5 in the same way he introduces the numbers 2 and 3: we make partitions of the rods. Here the student is expected to write, using figures, the patterns for each rod; dictation is also allowed if the student is an emerging writer. The patterns for the purple rod are as follows:
After the student completes the partitions for the purple rod, Gattegno follows with a few observational questions regarding how many rods are in each line. This is an interesting line of questioning as Gattegno doesn't throw in questions just for busy work. If he asks a question, he intends to force an awareness in the student even if he offers no explanation to the teacher. What purpose could be served by asking the following questions?
- How many red rods are there in the red line?
- How many rods are in the white line?
- How many rods are in the white and red line?
What possible observations, based on the number of rods in each line, could be made with rods other than the purple rod? I'm not offering an answer here, just noting it for your speculation.
Subtraction within the Number 4
Exercise 6 covers a review of difference using written problems. Here Gattengo wants the student to look at the written statement and find the solution. There is one problem that involves brackets. If you didn't spend enough time on brackets in the previous chapter, you might want to revisit them using letters then, when the student is ready, apply that knowledge to problems using numbers.
There isn't a section dedicated to subtraction for the number 5, addition and subtraction are tied together in exercise 10 just as they are in exercise 7. In both of these exercises, the student is expected to complete the patterns in writing AND from memory. Those tasks are nearly impossible for a young student unless sufficient time has been spent in chapters 2 and 3 working on their memory skills and rod fluency. Also, if a student is an emerging writer, expecting them to both write and work from memory is taxing on the short term memory. It might be better to spread these sections out over a couple days or write some and dictate the others. Gattegno does allow for using the rods if it is too difficult for the student. I wouldn't default to that option. It's good for the student to struggle and think.
In exercise 9, Gattegno asks the student to read written addition and subtraction statements using the words one, two, three, etc. as well as plus, minus, and equivalent to or equals. Reading expressions is important and should not be skipped. We want students who can read mathematical statements, take those statements and translate them into the rods, and read rods and translate those rods into written symbols. Eventually, we will dispense with the rods altogether.
Another Option for Studying The Numbers
When we begin a study of a new number, the first day I ask P. to partition the rod and come up with as many statements as he can for that number. With the numbers 2 and 3, a young student can pretty much exhaust the patterns. With the number 4, it's possible but more difficult for a 6-year old, and the number 5 starts to get very tricky. I'm not looking for exhaustion, but I want P. generating as much of his own math as possible. He needs to know that he can approach a mathematical situation and come up with just as much and sometimes more than the book would give him. It is the difference between filling-in-the-blank and writing a composition.
Goutard is of the same mind here and she encourages exploration of the numbers. The textbook is a guide and we should cover what is in there, but there is a lot of flexibility in how the books are applied. Gattegno never meant for the textbooks to be covered in a linear fashion. In the next post, we'll be covering multiplication and fractions inside 2-5, but that doesn't mean you should wait to cover those topics until you get to the exercises that contain them. If the student has spent a lot of time in chapter 3, then bringing the knowledge gained from those chapters to this one should be easy.
Again, just because a student covers the basic operations and fractions in compositions, doesn't mean you should skip those exercises when you get to them because Gattegno makes sure that he covers any areas of possible confusion. Right answers do not equate to thorough understanding.
If you want to connect with other parents and tutors who are using Cuisenaire Rods and other base-ten blocks, you can visit us in the Arithmophobia No More Facebook Group; we'd be glad to see you there.