negative numbers Archives - Mathematics for Teaching

Which is easier to teach and understand – fractions or negative numbers?

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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:

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.
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.”
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]
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.

Are negative numbers less than zero?

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I found this interesting article about negative numbers. It’s a quote from the paper titled The extension of the natural number domain to the integers in the transition from arithmetic to algebra by Aurora Gallardo. The quote was transcribed from the article Negative by D’Alembert (1717-1738) for Diderot Encyclopedia.

In order to be able to determine the whole notion, we must see, first, that those so called negative quantities, and mistakenly assumed as below Zero, are quite often represented by true quantities, as in Geometry where the negative lines are no different from the positive ones, if not by their position relative to some other line or common point. See CURVE. Therefore, we may readily infer that the negative quantities found in calculation are, indeed, true quantities, but they are true in a different sense than previously assumed. For instance, assume we are trying to determine the value of a number x which, added to 100, gives 50, Algebra tells us that: x + 100 = 50, and that: x = –50, showing that the quantity x is equal to 50, and that instead of being added to 100, it must be subtracted from that number. Consequently, the problem should have been stated in the following way: Find the quantity x which, subtracted from 100, gives 50. Thus, we would have: 100 – x = 50, and x = 50. The negative form for x would then no longer exist. Thus, the negative quantities really show the calculation of positive quantities assumed in a wrong position. The minus sign found in front of a quantity is meant to rectify and correct a mistake in the hypothesis, as clearly shown by the above example. (quoted in Glaeser, 1981, 323–324)

Interesting, isn’t it? Numbers are abstract ideas. They get their meanings from the context we apply them to. Of course from the school mathematics point of view we cannot start with this idea.

Here are the different meanings of the negative number that students should know before they leave sixth grade: 1) it is the result of subtraction when a bigger number is taken away from a smaller number; 2) it is the opposite of a counting/ natural number; 3) that when added to its opposite counting number results to zero; and 4) it represents the position of a point to the left of zero.

Likewise for the minus sign which indicates subtraction. Subtraction has three meanings: take away, find the difference, and inverse operation of addition. For further explanation read the post What exactly are we doing when we subtract?

Number Sense

Introducing negative numbers

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One of the ways to help students to make connections among concepts is to give them problem solving tasks that have many correct solutions or answers. Another way is to make sure that the solutions to the problems involve many previously learned concepts. This is what makes a piece of knowledge powerful. Most important of all, the tasks must give the groundwork for future and more complex concepts and problems the students will be learning. These kinds of task need not be difficult. And may I add before I give an example that equally important to the kind of learning tasks are the ways the teacher facilitates or processes various students’ solutions during the discussion.

I would like to share the problem solving task I made to get the students have a feel of the existence negative numbers.

We tried these tasks to a public school class of 50 Grade 6 pupils of average ability and it was perfect in the sense that I achieved my goals and the pupils enjoyed the lessons. This lesson was given after the lesson on representing situations with numbers using the sorting task which I describe in my post on introducing positive and negative numbers.

Sorting is a simple skill when you already know the basis for sorting which is not case in the task presented here.

Just like all the tasks I share in this blog, it can have many correct answer. The aim of the task is to make the students notice similarities and differences and describe them, analyse the relationship among the numbers involved, be conscious of the structure of the number expressions, and to get them to think about the number expression as an entity or an object in itself and not as a process, that is speaking of 5+3 as a sum and not the process of three added to five. The last two are very important in algebra. Many students in algebra have difficulty applying what they learned in another algebraic expression or equation for failing to recognize similarity in structure.

Here are some of the ways the pupils sorted the numbers:

1. According to operation: + and –

2. According to the number of digits: expressions involving one digit only vs those involving more than 2 digits

3. According to how the first number compared with the second number: first number > second number vs first number < second number.

4. According to whether or not the operation can be performed: “can be” vs “cannot be”.

5. This did not come out but the pupils can also group them according to whether the first/second term is odd or not, prime or not. It is not that difficult to get the students to group them according to this criteria.

Solution #4 is the key to the lesson:

During the processing of the lesson I asked the class to give examples that would belong to each group and how they could easily determine if a number expression involving plus and minus operation belongs to “can be” or “cannot be” group. From this they were able to make the following generalizations: (1) Addition of two numbers can always be done. (2) Subtraction of two numbers can be done if the second number is smaller than the first number otherwise you can’t. You can imagine their delight when they discovered the following day that taking away a bigger number from a smaller number is possible.

One pupil proposed a solution using the result of the operations but calculated for example 3-10 as 10-3. This drew protests from the class. They maintained that 3-10 and similar expressions does not yield a result. Note that class have yet to learn operations on integers. And obviously they could not yet make the connection between the negative numbers they used to represent situation from the lesson they learned the day before to the result of subtracting a bigger number from a smaller. To scaffold this understanding I ask them to arrange the number expressions from the smallest to the biggest value. This turned out to be a challenging task for many of the students. Only a number of them can arrange the expressions for smallest to the biggest value. My next post will show how the task I gave to enable the class to make the connection between the negative number and the subtraction expressions.

Elementary School Math, High school mathematics, Number Sense

Teaching positive and negative numbers

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A popular approach for teaching numbers is to use it to describe a property of an object or a set of object. For example, numbers are used to describe the amount or quantity of fruits in a basket.

In introducing integers, teachers and textbooks presents integers as a set of numbers that can be used to describe both the quantity and quality of an object or idea. Contexts involving opposites are very popular situations to show the uses and importance of positive and negative numbers and the meaning of its symbols. For example, a teacher can tell the class that +5 represents going 5 floors up and -5 represents going five floors down from an initial position.

Mathematics is a language and a way of thinking and should therefore be experienced by students as such. As a language, math is presented as having its own set of symbols and “grammar” much like our spoken and written languages that we use to describe a thing, an experience or an idea.But apart from being a language, mathematics is also a way of thinking. The only way for students to learn how to think is for them to engage them in it! Here’s my proposed activity for teaching positive and negative numbers that engages students in higher-level thinking as well.

Sort the following situations according to some categories

3^o below zero
52 m below sea level
$1000 net gain
$5000 withdrawal from ATM machine
$1000 deposit in savings account
3 kg weight loss
2 kg weight gain
80 m above sea level
37^o above zero
$2000 net loss

The task may seem like an ordinary sorting task but notice that the categories are not given. Students have to make their own way of grouping the situations. They can only do this after analyzing each situation, noting commonalities and differences.

Possible solutions:

1. Distance vs money (some students may consider the reading the thermometer under distance since its about the “length” of mercury from the “base”)

2. Based on type of quantities: amount of money, temperature, mass, length

3. Based on contrasting sense: weight gain vs weight loss, above zero vs below zero, etc.

The last solution is what you want. With very little help you can guide students to come-up with the solution below.

Of course, one may wonder why make the students go through all these. Why not just tell them? Why not give the categories? Well, mathematics is not in the curriculum because we want students to just learn mathematics. More importantly, we want our students to think critically and creatively hence we need to give them learning experiences that develops good thinking habits. Mathematics is a very good context for learning these.

Here are my other posts about integers:

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