Teachers’ construction of meanings of signed quantities and integer operation

Abstract

Understanding signed quantities and its arithmetic is one of the challenging topics of middle school mathematics. The specialized content knowledge (SCK) for teaching integers includes understanding of a variety of representations that may be used while teaching. In this study, we argue that meanings of integers and integer operations form the foundation for the construction of SCK about representations used to teach integers. We report that teachers’ concerns about teaching the topic of integers implicate issues of meaning, although this may not always be explicitly acknowledged by teachers. We develop a framework of integer meanings synthesizing previous research, and describe how the framework allowed teachers to investigate a wide range of representations including contexts and thereby construct SCK in a professional development setting. Teachers constructed SCK by connecting various meanings of integers with one another and with representations including contexts. Teachers made two important shifts, from exclusively using the state meaning of integers to including the application of change meaning to representations and from exclusive use of formal models to including contexts to teach integer addition and subtraction. An implication of the study is that frameworks of meaning for key mathematical topics could be an important component of pre- and in-service teacher education.

Appendices

Appendix 1: Worksheet 1

One of the difficulties that children face is in interpreting negative numbers. What does “−2” exactly mean? There are broad senses in which negative numbers or more generally integers (positive, negative numbers, and zero) are interpreted.

• 1. As a change Change includes increase or decrease, movement up or down (or forward and backward), or positive or negative growth (e.g., total annual sales of a company).

  Think of situations which involve change and can therefore be described using integers. The situations should be meaningful and interesting. Some suggested examples are given below. Think of more such examples.

  • ∞ Increase–decrease: Make a table of the weight gained by a baby every week (may be negative, what does it indicate?),

  • ∞ Movement forward/backward or up/down: Change in tennis ranking of a tennis player, change in run rate from over to over.

Make a presentation of such data in a way that would be interesting to students.

• 2. As a state We can specify the state of something we are interested in using integers but only when it is meaningful to talk about positive and negative states.

  Think of such situations where integers represent sate. Some suggested examples are given below. Think of more such examples.

  • ∞ Position of a lift in a building which also has basement floors

  • ∞ Temperature of water in a freezer

Again think of ways in which such situations can be presented in an interesting way to students.

• 3. As relation between numbers and quantities An important point here is that this is a directed relation. The relation makes sense if we distinguish the direction of the relationship and use positive and negative numbers to indicate it.

  Consider these two examples:

  • ∞ Me and my sister are standing in a queue to buy ice-cream. How far is my sister from me?

  • ∞ Me and my sister are on different floors of a tall building with several basement floor levels for parking. How far away is my sister from me?

Why is it meaningful to give the answers to these questions using integers? Is there any difference between the two examples? Think of more examples where relations can be represented using integers.

Appendix 2: Class VI: integers

Daily Lesson Plan

Day 1

The chapter will be introduced using the DREAM MALL figure.

The child learns that to move upward there is a “+” button, and to move downwards “−” button is to be used. Using this idea, he can number the floors accordingly.

Movement problems

Suppose Sapna and Kiran are in the ice-cream parlor. Sapna wants to go to the movie hall, and Kiran wants to go for shopping. How many steps each would move?

Their attention can be brought to the point that each would move same number of steps but in different directions, they can answer using appropriate signs.

Discussion regarding the importance of using the correct sign will be done at this moment.

In the figure, the boat is on the sea level. The aeroplane is flying 2000 km above the sea level (sic). The submarine is at 800 km below sea level (sic). Express their distances from the sea level. Have you seen numbers with “−” sign earlier?

Every day we see the weather report in a newspaper or a TV. Do you know there are places where the temperature is <0 °C?

(Refer Text Book page 154) for the list of temperatures of 5 places in India.

Integers: The first numbers to be discovered were natural numbers i.e., 1, 2, 3, 4,… If we include 0 in this collection, we get a new collection of numbers known as whole numbers. Now we find that there are numbers like −1, −2, −3, −4,… known as negative numbers. If we put the whole numbers and negative numbers together, the new collection will look 0, 1, 2, 3, 4, … −1, −2, 3, −4, … and this new collection is called integers.