Negative thinking for positive results

Sinny Mole

The weekly lesson plan review meeting of Math teachers was in progress. According to the plan the chapter on negative numbers was scheduled for the forthcoming week. We usually organize a meeting of math teachers to discuss innovative teaching techniques that can be applied in the classroom. The meeting helps pool different teaching ideas, standardizes the teaching methods and thus brings the teachers together on a common platform to challenge any learning difficulties that may arise in their classrooms. I was happy to note the active participation of the teachers in sharing their classroom experiences and also in seeking solutions to the difficulties encountered in previous occasions.

Before the meeting, I did some homework and listed out the probable learning issues that teachers may face. I could foresee some common difficulties that children would encounter in the mathematics operations of negative numbers. These issues were discussed at the meeting.

One of the major misconceptions that children have is that when you add numbers, the answer is always a bigger number. But this is not the case with negative numbers.

For example, when you add 6 and 4 you get 10, which is a bigger number. This is crystal clear to children. But this is true only if you’re working with whole numbers. Children have only been seeing and working with numbers that run from zero and up. So, a new lesson situation may confuse them making them think that math rules change all the time. This is why some children become frustrated and disconnected.

In many classes, children have a good procedural knowledge to perform math operations. But their misconceptions while working with positive and negative integers are due to the fact that they try to remember and apply rules that they don’t understand. Each child brings prior knowledge into a lesson and that knowledge can greatly influence what he or she gains or loses from the experience.

When dealing with negative numbers, children may not understand that -15 is smaller than -5 although 15 is bigger than 5. Teachers should think how to teach children and help them understand these concepts.

One way we can address the misconceptions is by highlighting the fact that rules change. For instance, just as the rules are different for cricket or hockey, so too the rules change for different number systems.

Numbers are abstract for some children, but when the numbers have meaning or relevance to their own lives, they become easier to understand.

We use physical or real objects as tools to make the learning of topics such as numbers and their operations interesting. But it becomes difficult to apply the same techniques to negative numbers. The problem is that the concept of a negative number cannot be easily demonstrated using apples and mangoes. Hence, teachers should explore innovative methods.

While learning negative numbers, the question that is always uppermost is: which number is big? The problem here is how children perceive “big”.

The number line is one of the traditional methods for teaching negative numbers. Teachers should emphasize when using the number line that signed values are the directed distances of the number line and not the points on a line. The length is the absolute value and the direction is the positive or negative sign of a number.

Once children recognize the meaning, it is easier to understand the concepts.

The classroom activity “Walk The Number Line” will be enjoyable for children. It makes them think about the different directions that indicate negative, positive, or zero at the centre.

You can show the number line on a vertical plane also. This will enable children to visualize the number line from a different perspective.

Another example teachers can use is travelling in an elevator up and down the various floors/basement of a shopping mall. The floor above ground level can be represented as positive and towards the basement as negative. Co-relate the case to the number line.

An interesting question – when you add two negative numbers you get a negative result. But when you multiply two negative numbers, it becomes positive! I threw the same question to the teachers. Most of them replied immediately.

“Ma’am, it’s a math rule – the product of two negative numbers is a positive.”

They are right. But it does not make any sense to the children. It only highlights the statement instead of conceptual understanding.

Children need answers to their questions left unresolved in their mind. Why?

Rules learned without understanding are the root cause of many of the misconceptions that children possess. To help overcome this problem, teachers have to get them thinking about the questions themselves.

I narrated a logical example of how to address the “why” part of the question.

Imagine that a boy offers food to his puppy.

If he says “Eat!” he is encouraging the puppy to eat (positive). But if he says “Do not eat!” he is saying the opposite (negative).

Now if he says “Do NOT not eat!” It means he doesn’t want the puppy to starve, so it is the same as saying “Eat” (positive). Hence, two negatives make a positive.

Such examples encourage children to think up their own examples and this is a much better way of dealing with misunderstandings and misconceptions.

To help children master the concepts they need to be familiar with the vocabulary inverse/analogy to positive and negative. Example: up/down, increase/decrease, rise/fall, above ground level/below ground level, hot/cold, good/bad, etc.

Mathematics is often considered by children as a set of rigid rules. In order to build proficiency with negative numbers, children must know what specific concepts mean rather than just memorize rules. The operations of negative numbers conflicts with their pre-knowledge and thus leads to difficulties in understanding. Hence it becomes very essential for the teachers to re-educate their children to correct mathematical thinking to help overcome their misconceptions.

Many children have misconceptions concerning negative numbers and hence they need sufficient instructions and support in their learning process. There are many activities that will help children learn about negative numbers, but teachers should carefully choose the ones that will facilitate conceptual thinking in building the foundation, for thinking about negative numbers.

BOX 1: Walk on the Number Line

This activity will be enjoyable for visual and kinesthetic learners. Ask each one to locate a number on a number line placed on the floor.

This gets them thinking about the different directions that indicate negative, positive, or zero. On a horizontal number line, negative would be walking left, 0 would be in the centre and positive would be walking right. However, on a vertical number line, students look at the number line from a different perspective.

BOX 2

Divide the students into groups of five or six. Distribute 10 soda bottle caps that are identical to each group. These bottle caps can be placed downwards or upwards.

Attach the value +1 to the caps in the upward position and -1 to those in the downward position. The game is that each player throws these 10 caps in a single move after shaking them vigorously. After the caps land, the child removes all pairs like (+1 and -1). Then she/he counts the remaining caps. If 3 pairs are formed and 4 caps facing downwards are left, she/he gets +4 points. If 2 pairs are formed and 6 caps are left facing upwards, then the child gets -6.

BOX 3: Cold cubes and hot cubes!

Imagine that you’re making soup, but not on a stove. You control the temperature of the soup with magic cubes.

These cubes come in two types: hot cubes and cold cubes.

If you add a hot cube (add a positive number), the temperature goes up. If you add a cold cube (add a negative number), the temperature comes down. If you remove a hot cube (subtract a positive number), the temperature comes down. And if you remove a cold cube (subtract a negative number), the temperature goes UP! That is, subtracting a negative is the same as adding a positive.

Imagine you have a bowl of soup that has a current temperature of 40 degrees. We have ice cubes, each of which takes 1 degree C off the temperature; have fire cubes (!!!), each of which adds 1 degree to the temperature. Now, what happens if you add 3 ice cubes to your soup? Soup gets 3 degrees colder, and we have 40 + -3 =37. And if we take out 4 fire cubes, we have 37 – +4, equal 33 degrees. What will be 40 – -5 5 ice cubes removed, temperature goes up to 45. “The minus of a minus is a plus.”

The author is Assistant Coordinator, Primary-2, K D Ambani Vidyamandir, Reliance Greens, Jamnagar. She teaches math to the primary classes and conducts in-house training for teachers. She can be reached at sinnymolepm@yahoo.in.