integers 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.

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.

High school mathematics, Lesson Study

Pedagogical Content Knowledge Map for Integers

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I’m working with a group of Year 7 mathematics teachers doing Lesson Study for the first time. The teachers chose to do a lesson study for what they believe to be the most difficult topic in this year level – integers. Part of my preparation as facilitator is to draw a map of what I know about teaching the topic. The map is more than a concept map because it includes not just related big ideas or concepts but also how these are taught and learned. Hence, I call this pedagogical content knowledge map (PCK map).

The PCK map I present here is a product of my own readings and my own experiences of teaching the topic. This means that it may not be the same as other teachers especially the ‘teaching part’ of the map, the ones in orange colors. For example, experience and research results back my claim that the number line is a very good way of representing the set of integers but not in teaching operations. Click here for my post about this. Notice that I gave emphasis on students knowing when a negative, a positive or a zero result rather than the rules for operation. I believe that without this, a conceptual understanding of the operation involving integers will be weak. Also, experience has taught me that although integers are numbers, the teaching of it must be algebraic. The instructions should be so designed so that students are learning algebraic thinking as well. I have noted this in the PCK map.

The map is not yet complete. I intend to include descriptions of effective activities and students’ learning trajectory of the concept after my research with the teachers. Please feel free to give your comments and share experiences for teaching integers that I could look into in my study.

pedagogical content knowledge — PCK Map for Integers

Please click the link to see my PCK map for Algebraic Expressions.

Tag: integers

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

Introducing negative numbers

Pedagogical Content Knowledge Map for Integers

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