viaggio

Giulia Vigna Lab Meeting January 30th, 2024

From Nothing to Something

A brief journey into Nothing

Welcome on board!

Creating a general overview

(from multiple points of view!)

What is known

My ideas / questions

Overview

What is known

Nieder, 2016

• 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.

Zero in Math

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 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!)

Zero in Math Education

• 1222 Fibonacci

• Muslims

• 600 a.C., India: Zero as a number

• 300 b.C. - 900 b.C. Maya

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

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.

Kaplan, 1999; Seife, 2013; Neider, 2016

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

Zero: cultural progression

L'essere è e non può non essere

Zero in Philosophy

Zero in Philosophy

Single neurons responding to empty sets

Absence as a behaviorally relevant category

Absence as a number

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

Using symbols or words to refer to absence

Empty sets along a numerical continuum

Zero-like concepts in animals

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

Absence

Negation

See Szabó (2021) for a Review

How is it possible? Maybe focus on a context?

Precursors of Zero in infants and toddlers

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

• 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

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

• Empty post-it to indicate the null quantity

• Linguistic problems?

Children's understanding of Zero as a number

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

Children's understanding of Zero as a number

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.

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

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.

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

• 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

Item response theory

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.

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

Children's understanding of Zero as a number

• 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

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

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

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.

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

Adults' understanding of Zero

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.

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

Zero and machine learning

Creating an overview!

Nieder's Summary

Zero as a multiform concept

Relevance

Creating an overview

Relevance

Creating an overview

Zero as a multiform concept

Creating an overview

Semantics

Syntatic

Ordinal

Cardinal

Non-Symbolic representations

Symbols

Words

Representations

Nieder's Summary

Creating an overview

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

Nieder's Summary

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

Thank you!!!

A brief journey into Nothing

Frequently reported MISCONCEPTIONS and DIFFICULTIES about Zero

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

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

• 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.

Frequently reported MISCONCEPTIONS and DIFFICULTIES about Zero

End effect

Distant effect

The numerial continuum

Symmetry point

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

Empty set(focus on the container)

Absence(focus on what is missing )

Nothing No stimultion

LITTLE ATTENTION dedicatedto Zero in Numerical Cognition Research