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Topic: Treatment of Negative Numbers
Replies: 12   Last Post: Dec 5, 2012 2:13 AM

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Robert Hansen

Posts: 7,573
From: Florida
Registered: 6/22/09
Treatment of Negative Numbers
Posted: Dec 1, 2012 10:59 PM
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The following are the chapters on negative numbers from 3 different textbooks dating from 1952 through 1973. The first book is the predecessor to the Dolciani series. Two of its 3 authors (Freilich & Berman) are two of the three authors of the first batch of Dolciani S&M books in the 1960's. The other author or the S&M books is of course Dolciani.

https://www.dropbox.com/sh/hz6q14yeafltfmn/Nd3JCDzftM/Freilich-1952.pdf

https://www.dropbox.com/sh/hz6q14yeafltfmn/_Y8FmVMapj/Dolciani-1965.pdf

https://www.dropbox.com/sh/hz6q14yeafltfmn/zGUCZGvmEs/Dolciani-1973.pdf

Interestingly, the formula for the Dolciani books, with the oral and written exercises, followed by word problems, along with the extra for experts sections, was already established in the much earlier Freilich books. In fact, several of the "History of Mathematics" inserts in the later Dolciani books are from the Freilich books. All of the books were published by Houghton Mifflin. These are all Algebra 1 books. The Algebra 2 books had only a review section on real numbers (including signed numbers).

In my most modern Algebra 1 book, a 2012 Holt version on the iPad, negative numbers are not covered much at all. They are listed in the introduction to Algebra section along with their arithmetic rules, that's it. Likewise with the Discovering Nothing book. In fact the Discovering Nothing book makes more of a deal about the difference between the "minus" key and the "sign" key on the calculator than it does about negative numbers.

I don't have any pre-algebra texts, and even though I know that negative numbers are introduced quite early (by 3rd grade) in order to close subtraction, you can't really treat them properly till algebra. So add another deficiency to the list of deficiencies in modern curriculums. Also, I note that Joe mentioned the case of Not Not, which is a good point. Unfortunately, truth tables are a scarcity in modern curriculums as well.

Back to the classics...

All three of the texts spend 30 to 40 pages on the subject. The two earlier books (1952, 1963) deal with negative numbers separately while the later book (1973) deals with real numbers altogether. I think the later approach is better.

The later (1973) book is great on the axioms but short on providing exemplary applications of negative numbers in the text. The 1963 and 1952 books spend a lot more time with examples of applying negative numbers. The Freilich (1952) book doesn't use the axioms like the Dolciani (1965) book does. The influence of SMSG and New Math no doubt.

As a mix of "key points" (for Joe's satisfaction) I like the 1965 book the best, but I would would rework it (hindsight is 20/20, right?). The following elements would still be my focus...

1. The Number Line

2. Adding and Subtracting Signed Numbers

I would use many examples of this, math examples and application examples. The students need to learn how to state something negatively. The first two books did a good job of challenging the students to keep up with the signage.

3. Ample Use of Parenthesis

This drives home the fact that the "minus sign" is part of the number, not a subtraction operation. Also, it helps the student traverse situations like (-2) - (-3).

4. The Axioms

By this time the student has a solid feeling for most of the axioms, and they are not only equally valid here but helpful as well. And this is the perfect time to state them more formally, although I would drop the "For every x in ...".

5. Exemplary Applications Throughout

Money, Temperature, etc. (see item 8)

6. Multiplication as Repeated Addition (Repeated Subtraction in the Case of Negative Numbers).

3 * (-2) = (-2) + (-2) + (-2) <- Everything is easier with parenthesis.

7. Division in the Context of Multiplication

If (-2) * (7) = (-14) then (-14) / (-2) must equal (7) <- This derivation dates back to 3rd and 4th grade.

8. Word Problems and Examples

Write an expression for the fuel left in a tank that is leaking 100 gallons an hour. If the tank has 1000 gallons now, how much did it have 3 hours ago.? I saw problems like this in the Freilich book. The student must learn how to think negatively in either direction and to keep that all straight in the algebra.

9. Close it all up with a review of the math and application of negative numbers and leave the students realizing that all we did was expand the set of numbers, not create two different sets.

And this all should come earlier than later. The Freilich book is odd in that it hits formulas, exponents and graphs all in the first two chapters. I would talk about numbers before that.

PS: I used one of these wand scanners to scan these chapters...

http://www.vupointsolutions.com/browse.php?cat=scanner&page=PDS-ST415-VPS

It is quite productive at scanning pages from books (10 minutes to scan a chapter) and it can literally hold 1000's of pages before you need to copy them to your computer.

Bob Hansen



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