Date: Oct 7, 2012 11:24 PM
Author: Robert Hansen
Subject: Topic 5 - The Beginning of the End
So here is the beginning of the end for many students.
In enVision (Grade 4 Florida Version), topic 5 is 16 pages about multiplication and here is what is covered.
1. Expanded algorithm using partial products.
2. Standard algorithm (like we do it).
3. two, three and four digit numbers by one digit.
4. Algebra (Compare 2 * 90 to 89 + 89)
5. Solving compound problems (algebraish).
6. Estimating and checking.
7. Extra or missing information.
8. Using a calculator.
This year I have been backing off a bit and letting my son take more responsibility for his work and this weekend I asked him what they did in class Friday. He said "We started topic 5, it was multiplication but he was having a problem with some of it." I asked him to explain to me what the problem was and he said that the teacher was showing them how to multiply 3 times 15 and he didn't understand. It was different than how we had done it."
I know what you are thinking, something like grid-multiplying, but it wasn't that, enVision is pretty traditional in method. It was just multiplication using partial products, what they call the "expanded algorithm", but it was being taught with pictures and the pictures are just not that helpful. I am of the opinion that if you add a visual to a lesson it should help the student to see the original point, not add yet a second point to figure out. I also don't think he was paying attention as well as he should have.
So today we opened up topic 5 for a little help. The topic begins with the idea of multiplying a one digit number by a two digit number using partial products and rather than stick with a clear description of the steps, it uses pictures of unit cubes (grouped in tens and ones) that do not do a very good job at all in representing a sequence of STEPS. I have attached the 2 pages that start this topic and at the top there is a brief description of what is happening.
I've complained before that this book avoids descriptive prose. It is written like a comic book that jumps from one situation to another. But my point in this post isn't the lack of prose, it is how many situations (as listed above) it has managed to stuff into 16 pages. This is just one topic out of 14. These kids will have witnessed many things in this class. They will master none of them, except of course, unless they have someone making sure that they do.
The problem obviously has to do with using topic lists to define a math curriculum. Too much focus on the trees and not enough on the forest. But the problem also has to do with injecting later topics into the same space where you are supposed to be developing earlier topics. In the third page I attached, they compare the expanded algorithm to the standard algorithm. Where do they teach the standard algorithm? Right there! Do you not see the comic book panel containing the standard algorithm at the top of the page? Now, I know that no 4th grade book can replace a teacher explaining these algorithms at the board but that doesn't mean it should avoid explanations altogether.
When I first started reading this section and it said "Compare the Expanded Algorithm to the Standard Algorithm", I asked my son "What's the standard algorithm?" and he said "I guess they mean the way we do it." I replied, "But where does it say that?" There is nothing but that brief mention on the following page (page 95). Can you imagine what a typical parent goes through? I have been studying curriculums for how long?
All of us (I assume) have read an advanced math text at some time.There are two ways to read an advanced math text. The first way is to breeze through it and when you finish you will know "of" the topics in the text. The second way is to work every problem offered by the text and when you finish you will "know" the topics in the text. This book is written with the first way in mind. Unfortunately, the topics are fundamentally important and practical. After following this series for 3+ years, I am convinced that much was lost when we abandoned the systematic approach to teaching arithmetic. I am also convinced that this plays into the very poor results in algebra.