viaggio

A brief journey into Nothing

From Nothing to Something

Giulia Vigna Lab Meeting January 30th, 2024

Welcome on board!

Overview

Creating a general overview

What is known

(from multiple points of view!)

My ideas / questions

What is known

Zero in Math

• Zero is a number to describe ‘no quantity’ or ‘null quantity’.
• Zero is the only natural number (by most definitions) that is not positive. 
• Zero is neither positive nor negative.
• Zero is the integer immediately preceding 1. 
• Zero is an even number because it is divisible by 2.
• Zero is the only number that can be divided by every other number.
• Zero is the only number that can divide no other number.
• Zero is a prerequisite to understanding negative numbers.
• Zero is a placeholder sign (or digit) in other numbers (e.g., 30, 103).

• The empty set is not nothing, because a set containing no element still is a set.

Nieder, 2016

Zero in Math Education

• Zero is the same as nothing 1
• Zero is not a number 2
• Zero is even or odd? 3
• How to divide by Zero? 4
• How to use zero in exponential? 5
• Where to place Zero on a number line?
• How to transcode numbers with Zeros? 6
• When using a ruler, should we start measuring from 0 or from 1?7
• On a number line, should we count spaces or notches? Negative numbers in the transition from arithmetic to algebra or in the transition from natural numbers to integers 8
• Teachers exhibited confusion about whether zero is a number, and showed stable patterns of calculation errors using zero.
• No big differences between children with and wihtout learning disabilities in processing Zero (both of them make mistakes!)

1. Pogliani et al., 19982. Wheeler & Feghali, 1983; Blake et. Al., 1985, Catteral, 2006; Russell & Chernoff, 2011; Sen & Agarwal, 2015; Seidelmann, 20043. Levenson, et al., 2007 4. Henry, 1969; Reys & Grouws, 1975; Knifong & Burton, 1980; Wheeler & Feghali, 1983; Blake & Verhille, 1985; Tirosh & Graeber, 1989; Duncan, 1971; Ball, 1990; Sundar, 1990; Watson, 1991; Simon, 1993; Even & Tirosh, 1995; Tsamir & Sheffer, 2000; Crespo & Nicol, 2006; Kim, 2007; Quinn, Lamberg, & Perrin, 2008; Cankoy, 2010; Chernoff, 2011; Thanheiser et al., 2014; Karakus, 2018. 5. Levenson, 2012. 6. Smith, Solomon & Carey, 2005 7. Clements & Stephan, 2004 8. Gallardo & Hernández, 2005; Gallardo & Hernández, 2006

Zero: cultural progression

The realization that zero has its own quantitative (null) value marks a cognitive turning point because it requires the insight that even if a set is empty, it still is a quantitative set.

• 500 b.C., China: Zero in positional system is rapresented as empty space

• 400 b.C., Babylonian: Zero as placeholder (never alone)

• 600 a.C., India: Zero as a number

• 300 b.C. - 900 b.C. Maya

• Muslims

• 1222 Fibonacci

Kaplan, 1999; Seife, 2013; Neider, 2016

Zero in Philosophy

L'essere è e non può non essere

Zero in Philosophy

Zero-like concepts in animals

Absence as a number

Using symbols or words to refer to absence

Empty sets along a numerical continuum

Single neurons responding to empty sets

Absence as a behaviorally relevant category

Pepperberg, 1988; Pepperberg and Gordon, 2005; Pepperberg, 2006; Biro and Matsuzawa, 2001; Merritt et al., 2009; Okuyama et al., 2015; Ramirez-Cardenas et al., 2016; Howard et al., 2018; Ramirez-Cardenas and Nieder, 2019; Cammaerts et al., 2020; Kirschhock et al., 2021, Szabó et al. 2022

Precursors of Zero in infants

How is it possible? Maybe focus on a context?

Although 8-month olds responded to the "magical" disappearance of an object as an unexpected event, they did not so respond to a magical appearance.

Wynn, 1998

Precursors of Zero in infants and toddlers

Negation

How is it possible? Maybe focus on a context?

Absence

See Szabó (2021) for a Review

Children's understanding of Zero as a number

Wellman & Miller, 1986 Thinking about nothing: Development of concepts of zero

• Zero is harder to understand compared to other numbers
• Role of Zero in understanding basic algebra --> zero holds a special status in fostering their reasoning about such rules

Stadial model of acquisition of Zero:

1. Familiarity with the name and symbol for zero.
2. Conceptual understanding that zero refers to a unique numerical quantity, none or nothing.
3. Realization that zero is the smallest number in the series of (non-negative) integers.

Children's understanding of Zero as a number

Bialystok & Codd, 2000Representing quantity beyond whole numbers: Some, none, and part

• Empty post-it to indicate the null quantity

• Linguistic problems?

Children's understanding of Zero as a number

Merritt & Brannon, 2013 Nothing to it: Precursors to a zero concept in preschoolers

Do 4-year old children understand the numerical value of empty sets prior to developing a concept of symbolic zero? Childrens’ performance with empty sets was variable; children exhibiting poor numerosity performance were at chance on empty-set comparisons and showed no distance effect. However, children who were more proficient with comparisons of countable numerosities also showed higher accuracy levels and distance effects with empty sets. Are empty sets represented as mental magnitudes? Yes! Both children and adults showed distance effects for empty sets.

Children's understanding of Zero as a number

Krajcsi et al., 2021 Development of Preschoolers’ Understanding of Zero

Preschoolers may understand numbers as the properties of items or objects in a set --> zero is not considered as a number because missing items cannot have any properties.

• Can children use empty sets in numerical tasks (i.e. comparison, addition, and subtraction tasks)? Yes!

• Can children use empty sets soon after they understand the cardinality principle? Yes!

• Do children know what the word “zero” refers to? Some of them.

• Can children categorize zero as a number (in smallest-number and is-it-anumber tasks)? Some of them.

Children's understanding of Zero as a number

Pixner et al., 2018 Differential Development of Children’s Understanding of the Cardinality of Small Numbers and Zero

Do the acquisition of cardinality knowledge on small numbers and the concept of zero share cross-domain and domain-specific numerical predictors? Domain specific numerical precursor skills seem to be more important for children’s development of an understanding of the cardinality of small numbers as well as of the concept of zero than language and visuospatial abilities.

Children's understanding of Zero as a number

Hartmann et al. 2022 Zero – an Uncommon Number: Preschoolers' Conceptual Understanding of Zero

When does a conceptual understanding of “zero” as number word for an empty set emerge?

• 107 preschoolers
• Compared to the natural numbers, zero was substantially more difficult.
• Children are able to list zero in a number word sequence (0, 1, 2, 3 .... or 3, 2, 1, 0), but were unable to describe a set as having zero numbers.

Item response theory

Adults' understanding of Zero

Benavides-Varela et al., 2016Zero in the brain: A voxel-based lesion–symptom mapping study in right hemisphere damaged patients

• What is the role of right hemisphere in numerical processing?
• Transcoding numerals containing zero is more problematic than transcoding numbers formed by non-zero digits.

• Most common zero-errors maximally associated to the right insula which was previously related to sensorimotor integration, attention, and response selection.
• Damage to the right hemisphere impairs the mechanism of parsing, and the ability to set-up empty-slot structures required for processing zeros in complex numbers

Adults' understanding of Zero

• Brysbaert (1995): reading times for to 99 depends on magnitude, frequency, and syllable length of the number name. Reading times for 0 are slower than expected --> 0 not in the mental number line?

• Fischer and Rottmann (2005): Is the integer (-9 to 9) smaller or larger than 0? --> distance effect only when positive numbers were compared to 0 --> mental number line starts with 0 but does not extend to the left of it

Adults' understanding of Zero

• Pinhas & Tzelgov (2012): end effect in automatic processing of numbers (Size congruity effect, SCE) suggests that 0, or 1 in the absence of 0, is perceived as the smallest entity on the mental number line

• Zaks-Ohayon et al. (2021): magnitude comparison task. Distance effects for comparisons to empty sets only when responding to target location, regardless of the response mode, indicating that spatial features should be primed in order to perceive an empty set as a numerical entity.

• Zagury et al. (2022): comparisons to empty sets produced an inverted SCE in the physical comparison task, whereas no SCE was found for comparisons to empty sets in the numerical comparison task.--> Empty sets are not perceived as “zero,” but rather as “nothing,” when processed automatically.

• Zaks‑Ohayon et al. (2022): perceiving an empty set, but not 0, as a numerical entity is determined by notation homogeneity and task requirements.

Adults' understanding of Zero

Csink et al., 2022Remembering nothing: Encoding and memory processes involved in representing empty locations

How unoccupied locations are encoded and maintained in memory? Adults encode exact coordinates of items in memory. Empty locations are represented as a property of the global configuration of items and empty space, and not as independent units of information.

Zero and machine learning

Nasr and Nieder, 2021Spontaneous representation of numerosity zero in a deep neural network for visual object recognition

Zero network units show maximum activity to empty sets and a gradual decrease in activity with increasing countable numerosities. --> network spontaneously ordered numerosity zero as the smallest numerical value along the number line. --> processing visual information is sufficient for a visual number sense that includes zero to emerge and explains why cognitively advanced animals with whom we share a nonverbal number system exhibit rudiments of numerosity zero.

Creating an overview!

Relevance

Creating an overview

Zero as a multiform concept

Nieder's Summary

Creating an overview

Relevance

Creating an overview

Zero as a multiform concept

Semantics

Cardinal

Ordinal

Syntatic

Representations

Non-Symbolic representations

Words

Symbols

Creating an overview

Nieder's Summary

Nieder's Summary

How do we transform "Nothing" into "Something" (i.e., an abstract concept)?

The concept of zero shows how the brain, originally evolved to represent stimuli (‘something’), detaches from empirical properties to achieve ultimate abstract thinking.

My ideas/questions

A brief journey into Nothing

Thank you!!!

Frequently reported MISCONCEPTIONS and DIFFICULTIES about Zero

Zero consists of MULTIPLE CONCEPTIONS which are involved in a wide range of mathematical fields

• Central concept in mathematics and science.
• If you don't understand zero, limited opportunities for learning (and the other way around)
• Misrepresentative metaphors --> Misconceptions ?
• Polysemous concepts and symbols in mathematics are more difficult to learn because of the constant shift the learner has to make between multiple everyday and mathematical meanings.

Chorney, 2023; Ruttenberg-Rozen, 2018; Byers, 2007; Mamolo, 2010

Frequently reported MISCONCEPTIONS and DIFFICULTIES about Zero

Symmetry point

The numerial continuum

Distant effect

End effect

Nothing No stimultion

Mathematical value: property of collections instantiated by empty collections (Barton, 2019)

Absence(focus on what is missing )

Empty set(focus on the container)

LITTLE ATTENTION dedicatedto Zero in Numerical Cognition Research