…to teach mathematical operations.

Hello and welcome to the 54th edition of our fortnightly newsletter, Things in Education.

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Long ago we had written something about number sense of students in foundational years and how it develops into adulthood. We had also written about the need to marry this number sense to other mathematical and quantitative skills like numbers and mathematical operations, spatial reasoning and geometry, or measurement and data interpretation. So today, we write about what it means to marry one of these skill sets to number sense, why it is important and what it means in a foundational years classroom.

Before we begin, let’s do a quick recap on what number sense is. Stare at the plus sign in the image below for 30 seconds and only then scroll down to the next figure.

Look at the plus sign in the second figure and estimate which circle - left or right has more number of dots.

It is likely that you selected the right circle. Actually, both circles have the same number of dots. But your mind was trained on the earlier image and it quickly compared the 10-dot circle to the 40-dot circle on the right and “knew” that it was “more”, while it compared the 100-dot circle to the 40-dot circle on the left and “knew” that it was “less”. This quick estimation that our brain does is one of the aspects of number sense (more on number sense).

Relationship of number sense and mathematical operations in the brain

One day at our office, our academic team was trying to figure out how helpful generative AI could be for some tasks. They were using specific examples and getting generative AI to respond. When the group reached an impasse, they asked another team member (not previously involved in the conversation) to check out an AI-generated response.

The new entrant in the conversation looked at the response and immediately said, “That obviously won’t work, as the answer is wrong.” This person did not really calculate the answer. They did not know what the correct answer was. But they “knew” that if it takes four workers three hours to complete a task, it should take six workers less time than that. This is where their number sense kicked in. Let’s see if the same happens with you. Look at the equation below:

13 + 26 = 99

Were you able to tell that this equation was wrong almost instantly? Mostly, yes. However, when to actually calculate, it takes time for us to figure out if the right answer is 39 or 49.

The part that says that 13 + 26 is definitely not 99 is your number sense, while the part that gives you the accurate answer to 13 + 26 is the part of the brain that does the mathematical operations.

These processes seem to happen at different locations in the brain. Recent studies have shown that the parts of the brain that are responsible for number sense are very different from the part of the brain that is responsible for mathematical operations. However, it is important that both these regions are activated when we do mathematical calculations. So, the connections between the number sense regions and the mathematical operations regions need to be well established for students to be good at addition, subtraction and other operations.

What does this mean for a teacher in the classroom?

The innate number sense part of the brain is fairly well developed in all children, while the education system should help with the development of the mathematical operations regions and the connections between the mathematical operations region and the innate number sense region. Making connections between two parts of the brain is its way of quick recall in the future for any situation that needs number sense and mathematical operations. The deeper this connection in the brain, the quicker the recall (more details here). So how do teachers do this?

One controversial example of this is learning to count on one’s fingers. There is a school of thought that frowns upon this practice and wants students to wean away from finger counting as soon as possible. We argue that counting on fingers is probably one of the best techniques to make some of the intuitive ideas of number sense concrete. We work with a decimal number system. There are ten fingers. This is not a coincidence. So why not leverage this? Students should use their fingers to count – this will help them understand why numbers are cyclical at intervals of ten. This type of counting also helps make abstract numbers concrete, which helps strengthen the connections between the number sense and mathematical operations part of the brain.

It is important to use finger counting or even day-to-day events to show the utility of numbers and mathematical operations. Digging a little deeper into this strategy of making numbers and mathematical operations concrete, it is also important to choose the right example for students to understand abstract mathematical concepts.

For example, if a teacher wants to introduce the concept of negative numbers, there are multiple ways to do it. However, it would be unintuitive for a student to understand that there were three apples in the basket and you take away six, which leaves you with minus three apples. There is no such thing as minus three apples. A better way to explain negative numbers would be to use the temperature scale. Once we explain that zero degrees celsius is the temperature at which water freezes, it becomes pretty straightforward to explain what minus three would mean. It is a temperature that is three degrees colder than the temperature at which water freezes. This example intuitively conveys what minus three can be, but it also gives a concrete example of where such a concept may be useful.

So what can teachers do to leverage number sense to teach mathematical operations in class:

Connect concrete to abstract (fingers to counting)

Use authentic concrete examples (temperature for learning about negative numbers)

Repeated but diverse concrete examples for abstract concepts (find the middle number between 11 and 19; can use groups of marbles, can use number line, can count on fingers, etc.)

Number sense is innate, and preschool and school mathematics should focus on making connections between the innate number sense and the other mathematical skills. This edition focussed on mathematical operations. In the future we will write about spatial reasoning, geometry, measurement and data analysis and interpretation.

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Edition: 3.2