Mathematics for Teaching

Which concept is easier for students to understand and perform operations on, numbers in fraction form or negative numbers? I think fractions may be harder to work with, but people understand what it is; at least, as an expression to describe a quantity that is a part of a whole. Like the counting numbers, fractions came into being because we needed to describe a quantity that is part of a whole or a part of a set. The fraction notation later became powerful also in denoting comparison between quantities (ratio) and even as an operator. See What are fractions and what does it mean to understand them?  And negative numbers? Do we also use them as frequently like we would fractions? I think not. People would rather say ‘I’m 100 bucks short’ than ‘I have -100 bucks’.

How did negative numbers come into being? As early as 200 BCE the Chinese number rod system represented positive numbers in Red and Negative numbers in black. There was no notion of negative numbers as numbers, yet. The Chinese just use them to denote opposites. There was no record of calculation involving negative numbers.  Although the first set of rules for dealing with negative numbers was stated in the 7th century by the Indian mathematician Brahmagupta, it was only in the middle of the 19th century, when mathematicians began to work on the ‘logic’of arithmetic and algebra that a clearer definition of negative numbers and the nature of the operations on them began to emerge (you may want to read the brief history of negative numbers). It was not easy for many mathematicians before that time to accept negative numbers as ‘legitimate’ numbers. Why did it take that long? In her article Negative numbers: obstacles in their evolution from intuitive to intellectual constructs, Lisa Hefendehl-Hebeker (1991) identified the hurdles in the acceptance of negative numbers:

1. There was no notion of a uniform number line.The English mathematician, John Wallis (1616 – 1703) is yet to invent the number line which helps give meaning to the negative numbers. Note that it did not make learning operations easy.The preferred model was that of two distinct oppositely oriented half lines. This reinforced the stubborn insistence on the qualitative difference between positive and negative numbers. In other words, these numbers were not viewed as “relative numbers.”  You may want to read Historical objections against the number line.
2. A related and long-lasting view was that of zero as absolute zero with nothing “below” it. The transition to zero as origin selected arbitrarily on an oriented axis was yet to come. There was attachment to a concrete viewpoint, that is, attempts were made to assign to numbers and to operations on them a “concrete sense.”
3. In particular, one felt the need to introduce a single model that would give a satisfactory explanation of all rules of computation with negative numbers. The well-known credit-debit model can “play an explanatory but not a self-explanatory role”.  [Until now, no such model exists. More and more math education researchers are saying that you need several models to teach operations on integers]
4. But the key problem was the elimination of the Aristotelian notion of number that subordinated the notion of number to that of magnitude.

Lisa Hefendehl-Hebeker #4 statement is very important for teachers to understand. If you keep on teaching the concept of negative numbers like you did with the whole numbers and fractions which naturally describes magnitude, the longer and harder it would take the students to understand and perform operations on negative numbers. The notion of negative numbers as representing a real-life situation say, a debt, becomes a cognitive obstacle when they now do operations on these numbers. I am not of course saying you should not introduce negative numbers this way. You just don’t over emphasize it to the point that students won’t be able to think of negative numbers as an abstract object. I would even suggest that when you teach the operation on negative numbers, make sure the introduction of it as representation of a real-life situation has been done a year earlier. Here’s one way of doing it – Introducing negative numbers.

Here’s Brahmagupta (598 – 670) rules for calculating negative and positive numbers. See how confusing the rules of operations are if  students think of negative numbers as representing magnitude.

 Image from Nrich.