The question of how the number zero is represented in the brain has only recently been studied in the field of neuroscience. Last year, researchers examined the ability of human brains to represent the concept of zero. In the study, they found neurons in the brain specifically responded to the concept of zero both in symbolic (the digit 0) and nonsymbolic (the empty set) forms, suggesting that the brain interprets zero as part of the number system. When zero was presented to participants symbolically, neurons processed zero just like other numbers. On the other hand, when zero was presented as an empty set, the neurons had a slower response, suggesting that the brain requires more time to process abstract representations of zero.
This new study offers math educators with valuable insight into how the human brain interprets abstract representations of zero and why children take longer to grasp the concept of zero compared to other numbers.
“It took a long stretch of human history for zero to be recognized and appreciated. Children show a delayed understanding of the concept of zero numerosity, long after they comprehend positive integers,” researchers Florian Mormann and Andreas Nieder, explain to PsyPost about their study. “For a brain that has evolved to process sensory stimuli (‘something’), conceiving of empty sets (‘nothing’) as a meaningful category demands high-level abstraction. It requires the ability to represent a concept independently of experience and beyond what is perceived. The brain needs to interpret ‘nothing’ as ‘something,’ as a mathematical object.”
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The History of Zero
Although zero can be taken for granted, with the number appearing in everyday calculations and conversations, the concept of zero was not always present in the field of mathematics. The historical origins of zero are contested: the earliest known representation of the number zero is approximately 300 BC, when it was used as a placeholder by the Sumerian civilization. On the other hand, ancient Greeks and early Christians rejected the concept of zero as a number. The idea of nothing was antithetical to early Christian beliefs — the existence of God was incompatible with the notion of the void, and as a result, the number zero had no place in the cosmology of the church.
The invention of zero as a number, rather than a placeholder, is credited to the Indian mathematician Brahmagupta, whose work in the year 628 AD paved the way for modern mathematics. In his book Brāhma-sphuṭa-siddhānta, he developed a system of rules for calculations involving zero and introduced the idea of negative numbers. His rules for dealing with zero remain fundamental to the field of mathematics today: 1 + 0 = 1, 1 – 0 = 1, and 1 × 0 = 0.
Zero represents the absence of quantity, yet it is the cornerstone of every technological advancement we know today. The emergence of zero unlocked new fields for mathematicians and created the foundation required for all modern technology. Today, the number zero continues to puzzle both young learners and expert mathematicians alike.
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Zero in the Classroom
“Awareness of the meaning behind the number zero develops in phases,” says Colleen Uscianowski Maas, a researcher with Development and Research in Early Mathematics Education (DREME) based at Stanford University. “Around age 4 is when children begin to understand the meaning of an empty set as the lack of something that would typically be there. Children initially learn to recognize the zero symbol (0) without knowing what it means.”
The team of researchers at DREME explores how children learn the concept of zero and develop an understanding of the empty set. Even after formally learning the meaning of zero (“none, nothing, nil, zilch”), children struggle to understand its function as a number with a place value and position on the number line.
Maas suggests that children may struggle with such tasks because the mathematical term zero feels distinct from the concept of nothing or negation. For instance, in one study by DREME, children were asked to not give balls to the bird in two different ways: “give zero balls to the bird” and “do not give any balls to the bird”. The children understood the directions when the word “zero” was not used.
Video: What is Zero?
A quick look at how zero came about, from its origins in ancient civilizations to being the cornerstone of calculus.
Helping children understand zero
Creating Conversation
Discussing what zero is and means can lead to the sharing of interesting ideas. These conversation starters can help educators gauge students’ understanding of the concept:
- How can you represent the concept of nothing?
- What do you think the world would look like without zero?
- Can you find examples of zero in everyday life (e.g., zero sugar, zero waste, zero emissions)? What do these examples have in common, and why is zero important in these contexts?
- What does it mean to have zero dollars?
- What’s the difference between having “nothing” and having zero? Are they always the same? Why or why not?
- Think about a timer counting down, a score in game or money in a bank account — what would happen if we stopped using zero in these everyday scenarios?
Model the Language
Teachers can establish connections between the language of math and everyday language in the classroom. This will help students associate the term zero with the concept of nothing when describing empty sets.
Example: When describing empty sets: “There are zero apples on the tree. No more apples. They are all gone. No apples are left!”.
Use Games
Children can also engage in games to help understand the magnitude of zero compared to other positive numbers.
Example: Children can fill labelled containers with different amounts of items and compare the quantities. For example, the container with the label 0 should have zero items at the end of the game, the container with the label 1 should have one item, and so on. Ask students questions to help solidify their understanding of zero: “Which container has the most items? Which container has the least items? Which container is empty? Which container has zero items?”.
Begin Counting at Zero
In early years, children should be encouraged to count numbers from zero to ten (0-10) rather than one to ten (1-10), so they can begin to acknowledge the role of zero in the number system.
Example: Ask students to represent each number with their fingers while counting. When they count the number five, five fingers should be up. When they count the number zero, all fingers should be down.
Explore Arithmetic Operations
As students get older, they can explore different arithmetic operations with the number zero, such as addition, subtraction, multiplication, and division. Such inquiries around the number zero will help students foster a more nuanced understanding of its role in mathematics.
Example: Ask students the following inquiry questions regarding addition and subtraction with the number zero: “What happens when you add zero to a number? What happens when you subtract zero from a number?”. Once students are ready, you can introduce inquiry questions on multiplication and division: “What does it mean to multiply a number by zero? What happens when you multiply a number by zero? Can you divide a number by zero?”. Encourage students to refer to the definitions of multiplication and division.
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Author
Bisman Kaur
Student Teacher, Master of Teaching Program (OISE/U of T)
