Symposium 2013 > Abstracts
Foundations for symbolic mathematics: development and evolution of our primitive number sense
I will present a body of data that demonstrates that there are strong developmental and evolutionary precursors to adult mathematical cognition that can be uncovered by studying human infants and nonhuman primates. I will take the position that the approximate number system serves as a developmental foundation for the uniquely human numerical faculty. Implications for education will be explored by describing a) a longitudinal study exploring the relationship between infants’ number sense and later developing mathematical cognition in childhood and b) a set of training studies exploring the link between primitive number sense and symbolic mathematics.
Nonnumerical properties of the mental number line and the relativity of left and right
After 20 years of hunting spatialnumerical associations of response codes (SNARCs) we are still unable to precisely delineate the object of the hunt. In fact, the continuous systematic (but relatively uncoordinated) approach of hundreds of hunters has only scared up more game. In addition to SNARC (lefthand reaction time advantage for small compared to large numbers), there are SNARC’s siblings: SMARC (spatialmusicalassociation of response codes; lefthand advantage for low tones compared to high tones), SPARC (SpoliticalARC; lefthand advantage for leftwing over rightwing parties) and STEARC (StemporalARC; lefthand advantage for brief compared to long intervals), among many others. I discuss the discovery of some of these effects and their significance for theories of the spatial properties of abstract concepts.
Horizontal spatialconceptual associations are modulated by culturally dependent directional habits. Famously, the SNARC effect reverses for persons with a reading/writing direction opposite to ours’ (Westerners’). Also, the “arrow of time” expands from left to right for people with a lefttoright oriented reading/writing habit, but points to the opposite direction for people with an opposite directional habit. These crosscultural observations in healthy research subjects, among other evidence collected in the course of hunting the SNARC and its siblings, seem to have ruled out a significant role of the two cerebral hemispheres for the formation of spatialconceptual associations. Yet, studies with patients with hemispatial neglect after unilateral hemispheric lesions have taught us otherwise: those, who display attentional deficits towards the left side of space also show similar deficits along the mental number line or along other lines of symbolic horizontal extension.
I briefly review the literature on the neglect syndrome to show how abstract the meanings of “left” and “right” may become and introduce ongoing experimentation on the spatialization of verbal memory: in healthy participants, there is not only a lefthand recognition advantage for events early as compared to late in their autobiography, but also for events acquired just a few seconds earlier than others (or a few seconds later than others in the case of Hebrew participants). In verbal list learning, the serial position curve is influenced by leftsided hemispatial neglect; patients with the disorder showed worse access to words from the first half relative to the second half of a list. These results indicate that unilateral lesions can compromise access to items on the “left” side of highly abstract, culturally shaped spaces and illustrate the interplay between brain and culture in the organization of thought.
The Acquisition of Mathematical Concepts: Beyond Logical Construction
On one dominant view of conceptual development within the cognitive sciences, it is logically impossible for learning to increase the expressive power of the innate conceptual system. This is so IF learning consists of creating new concepts by logical combination of those one already has, and IF expressive power is a function of the stock of innate primitives and those constructable from them given the combinatorial resources of language and logic. Call this the logical construction position. Piantadosi et al. recently proposed a logical construction model of how the child learns the meanings of verbal numerals, given by their place in a count list, in which they explicitly commit themselves to the position that new concepts arise only by conceptual combination. I will argue that contrary to the logical construction position, it is possible to learn new conceptual primitives through processes that do not involve making innate primitives manifest (or logical construction from conceptual primitives, of course), and thus increases in expressive power are common in conceptual development. I will illustrate these points in terms of the construction of meanings of verbal numerals, and will present new data from animals (an African grey parrot and rhesus macaques) that show that the learning mechanisms needed for such construction predate human language and cognition.
Whorf in sheep’s clothing: The role of language in creating large exact number
Nearly a decade ago, a landmark study by Peter Gordon suggested that the Pirahã, who lack words for exact numbers in their language, may as a consequence lack the capacity to mentally represent exact numerosities greater than three. The author interpreted his data as supporting “the strongest version of Benjamin Lee Whorf’s hypothesis that language can determine the nature and content of thought.” Critics suggested that the data had been overinterpreted. Several years later, Michael Frank and colleagues returned to the Amazon and replicated Gordon’s findings, clarifying vagaries of the original data set. Surprisingly, however, the authors interpreted their data as supporting a very weak Whorfian position, according to which number words merely allow for “efficient coding of information about quantity.” In this talk, I will suggest that these data have been underinterpreted. Frank et al.’s data indicate that the Pirahã’s numerical language and thought are vastly different from our own. In combination with data on exact enumeration in home signers (from Spaepen and colleagues), these data provide the best evidence to date for strong linguistic determinism. Beyond helping us to use number representations efficiently, number words appear to play a crucial role in creating more powerful representational resources — resources that transform the mind, and which have enabled us to transform the world.
Dissociations between mathematical and language syntax
“Words and language, whether written or spoken, do not seem to play any part in my thought processes.” According to Albert Einstein’s introspection, mathematical reasoning is entirely dissociable from linguistic processing. In my talk, I will examine this hypothesis in the light of contemporary cognitive neuroscience. A basic number sense is certainly present in preverbal infants and many animal species in the absence of any linguistic competence – but what role does language play in the construction of more advanced numerical and mathematical concepts? Research in the Munduruku, an Amazon tribe with reduced mathematical language and little access to education, suggests that even advanced numerical and geometrical intuitions are present prior to any linguistic formulation. Although that they can be improved and refined by education, particularly the introduction of a counting routine, it is still unclear whether language itself is the key factor in this development. In Western adults, new results indicate that even the syntax of written numerical expressions does not seem to make use of the same resources as the syntax of language. Using a novel technique which allows continuous tracking of the finger trajectory during a numbertoline test, I will show that multidigit numerals are processed very quickly and in parallel, and that their syntactic processing can be preserved even when linguistic processing is impeded. Using fMRI, we find that even complex parenthesized expressions such as ((1+4)(32)) are analyzed solely by the visual and parietal systems, without recourse to linguistic networks housed in the left temporal and inferior frontal areas. Thus, evidence suggests that mathematical processing, at least in its most elementary forms, is largely dissociable from language syntax.
Why integers require set representations
Humans have a system for representing approximate cardinalities and a system for precisely representing individual objects. Neither of these systems is capable of representing integers like “exactly seventeen.” Instead, integer representations are typically constructed over a protracted period in early childhood. Here, I suggest that in order for children to build integer concepts they must first have the concept of a set of individuals. Set representations allow thinkers to represent individual entities (like individual objects), then combine these individuals into hierarchically organized units whose cardinality can count as a feature. In this talk I offer evidence that children spontaneously represent such sets starting in infancy, and that young children can use a variety of cues in order to bind individuals into sets. Further, I present evidence that, before having mastered integer representations, toddlers can not only remember the number of entities contained within a set, but also can bind verbal labels to sets containing different numbers of items. These abilities may serve as key ingredients for the eventual creation of integer concepts.
A Hierarchical View of Grounded, Embodied, and Situated Numerical Cognition
There is much recent interest in the idea that we represent our knowledge together with the sensory and motor features that were activated during its acquisition. This paper reviews the evidence for such “embodiment” in the domain of numerical cognition, a traditional stronghold of abstract theories of knowledge representation. The focus is on spatialnumerical associations, such as the SNARC effect (small numbers are associated with left space, larger numbers with right space). Using empirical evidence from behavioral research, I first describe sensory and motor biases induced by SNARC, thus identifying numbers as embodied concepts. Next, I propose a hierarchical relationship between grounded, embodied, and situated aspects of number knowledge. This hierarchical conceptualization helps to understand the variety of SNARCrelated findings and yields testable predictions about numerical cognition. I report several such tests, ranging from crosscultural comparisons of horizontal and vertical SNARC effects (Shaki & Fischer, 2012) to motor cortical activation studies in adults with left and righthand counting preferences (Tschentscher et al., 2012). It is concluded that the diagnostic features for each level of the proposed hierarchical knowledge representation, together with the spatial associations of numbers, make the domain of numerical knowledge an ideal testing ground for embodied cognition research.
References:
 Shaki, S., & Fischer, M. H. (2012). Multiple spatial mappings in numerical cognition. Journal of Experimental Psychology: Human Perception and Performance, 38(3), 804809.
 Tschentscher, N., Hauk, O., Fischer, M. H., & Pulvermüller, F. (2012). You can count on the motor cortex: fMRI reveals embodied number processing. NeuroImage, 59, 31393148.
Numbers as tools for thinking
What is the relationship between language and thought? Traditional approaches to this question have staked out extreme positions: either that language determines the shape of the thoughts you can entertain, or else that natural language is only an thin overlay on top of a more basic "language of thought." Work in the domain of numerical cognition supports a middle view: that language is a tool that can help with complex cognitive tasks by supplementing core nonlinguistic numerical abilities. But if number systems are tools, then the way these tools are structured should make a big difference to how they are used and what they are good for. In this talk, I'll describe crosslinguistic and crosscultural evidence from Brazil, India, and Papua New Guinea, showing some of the incredible variation in number representations across the world and how these representations affect the cognition of their users. I'll end by presenting some new data on whether alternative numerical representations can be valuable in math education.
Mental Magnitudes and the Problem of Reference
The representations of space, time, number, rate and probability are the foundational abstractions, the primitives in the brain’s representation of the experienced world. They are prominent among the developmentalists’ core domains, and they are present even in insects. The representations in these inherently quantitative domains must be founded on a common system for the representation and arithmetic processing of magnitude, that is the computable numbers, those real numbers that can be physically represented. Objective numerical magnitude (numerosity) is unusual in being discrete. Objective temporal, spatial, rate and probability magnitudes are continuous. However, the subjective (mental) magnitudes that represent numerosity must feed into the same arithmetic machinery as the subjective magnitudes that represent the objectively continuous magnitudes. This follows from the fact that the derived magnitudes like rate, probability and density arise from the arithmetic processing of, in the first case, a symbol for a discrete magnitude (numerosity) divided by a symbol for a continuous magnitude (duration, distance or area), and, in the second case (probability), a discrete magnitude divided by a discrete magnitude. The discovery of this mental magnitude system and the appreciation of its primitive and foundational character are notable achievements of modern cognitive science.
It seems plausible to assume that the abstract representations of space, time, number, rate and probability built on this magnituderepresenting system are the foundations of many abstract word meanings. That is, a spatial word or a temporal word or even grammatical tenses may gain their meaning from the constructs within this system to which they refer. Under this assumption words gain their seeming objective referents indirectly. They refer directly only to mental constructs, which constructs refer in complex ways to the represented world. This assumption may go some way toward explaining referential opacity.
Communicating about quantity without a language model: The ways in which number devices are, and are not, like words in homesing
Homesigns are communication systems created by deaf individuals without access to conventional linguistic input. Homesigners use their gestures to communicate not only about objects, actions, or attributes, but also about number. In the first part of my talk, I will show that adult homesigners in Nicaragua produce gestures that enumerate sets (cardinal number marking), as well as gestures that signal one vs. more than one (noncardinal number marking). Both types of gestures are fully integrated into each homesigner’s gesture system and, in this sense, behave like words. But the gestures homesigners use to represent number do not behave like words in all senses. In the second part of the talk, I will show that these number gestures do not function as single units in shortterm memory––more specifically, they do not function as summaries of the cardinal values of sets (four), but rather as indexes of items within a set (oneoneoneone). Thus, when developed without linguistic input, gestures for number can function like number words in that they are fully integrated into sentences, but do not function like number words in terms of what they represent––they stand for a collection of individuated items rather than a set.
Quantities and Quantifiers: Weber's Law, Monotonicity and Modularity
I will describe a series of experiments that aimed to study the relation between language and numerical cognition. We manipulated the properties of instruction probes for a numerical comparison task, which enabled us to test of the relation between behavioral reflexes of linguistic and numerical processes, with unprecedented resolution.
Participants performed speeded numerical comparisons between visually displayed discrete quantities, guided by auditory instruction sentences with complex quantifiers that contrasted in Polarity (e.g., more and lessthanhalf), and with analogous nonverbal instructions with arithmetical inequality symbols (<, >). Our results demonstrate the distinctness (modularity) of linguistic and numerical processes: core temporal parameters of numerical comparison (abiding by Weber’s Law) remain constant across a range of verbal and nonverbal instruction probes, as well as across various levels of perceptual difficulty of a visual numerosity task. However, performance parameters are affected by the manipulation of Polarity, a logical property of verbal instructions.
I will also report related results we obtained in fMRI and from tests of patients with aphasia. They all support a cognitive architecture in which language and numerical cognition constitute separate processing modules in the brain. Finally, I will discuss the linguistic significance of these results, and try to present a remaining puzzle.
Neural Basis for Quantifiers and Number Knowledge
We hypothesize that quantifier processing depends in part on a largescale neural network that involves number processing in the parietal lobe, executive resources in dorsolateral prefrontal cortex that helps manipulate this knowledge, and a simple form of logic that may also contribute to grasping certain pragmatics of communication. We find that patients with parietal disease due to corticobasal syndrome are impaired with numbers and quantity. Moreover, these nonaphasic patients appear to have difficulty understanding firstorder quantifiers like “at least three X.” This is regardless of the verbal or visual modality of presentation, and this is equally evident in multiple domains of knowledge (e.g. time, distance). In another group of nonaphasic patients with dorsolateral prefrontal disease, there seems to be an impairment understanding higherorder quantifiers like “at least half.” Finally, patients with disease in anterior prefrontal regions appear to have a relative deficit appreciating logical quantifiers such as “some.” “Some” may be used as a logically correct description of a situation, even though this may not provide as informative a description of the situation as a quantifier like “all.” In addition to patient studies, I will discuss converging evidence from fMRI studies of healthy adults. These findings are consistent with the hypothesis that a largescale neural network involving number knowledge appears to contribute to the comprehension of quantifiers.
Possible Number Systems
Number systems—such as the natural numbers or the reals—play a foundational role in mathematics, but these systems present barriers for students. The first part of this talk looks at the idea that children in kindergarten and the early grades have a concept of the integers 1100 that is dominated by a mental number line (or approximate number system). According to this theory, children’s understanding of the integers is systematically distorted because their number line is nonlinearly (e.g., logarithmically) related to the numbers’ true magnitude. An alternative interpretation, however, is that children have a grasp of the underlying structure of the integers but are unsure about the meaning of the larger numerals in this range. I’ll present results favoring the second alternative. If the right math structures are in place, though, why do students have trouble learning new systems? One possibility is that they have incorrect preconceptions about what a number system can be, based on earlier knowledge. The second part of the talk presents evidence that even college students, who have a generally correct view of number systems, may retain some incorrect ideas about their properties.
Eventive enumeration
Counted at the gate, three million passengers, flew National Airlines, (1). Counting their frequent flyer numbers, they turned out to be one million, (2):

(1) Three million passengers crowded National Airlines’ routes in 1980.
F Three million frequent flyers crowded National Airlines’ routes in 1980.(after Gupta 1980: 23, Moore 1994)

(2) The three million passengers who crowded National Airlines’ routes in 1980 were the one million frequent flyers loyal to it.
Yet, this cannot be, if the number words are the simple arithmetic predicates ‘3,000,000(X)’ and ‘1,000,000(Y)’, as what is three million is not what is one million:
(3) (3,000,000(X) & 1,000,000(Y)) → X ≠ Y
Rather, number words join other measure phrases, (Aaron— 26 inches in 1988; 71 inches in 2013), in denoting the result of a dated event of measurement, so that three million translates as “now counted to three million,” ‘count(e,X, 3000000)’. The same things once counted as passengers to number three million may then be recounted under a different protocol as frequent flyers to number one million.
Only if number words denote events can they in turn be recruited as temporal adverbs to explain the contrast between (4) and (5):

(4) The three million passengers were frequent flyers.

(5) #The three million passengers are still frequent flyers.

(6) The one million frequent flyers are still frequent flyers.
In the present tense, (5) cannot count more passengers than persons who fly, despite their having remained frequent flyers to this day (cf. (6)). But, if sentences turn noun phrases into adverbs (adverbialization), with counting now part of it, sentences (4) and (5) become (7) and (8) respectively:

(7) The three million passengers when counted to three million were frequent flyers.

(8) #Three million passengers when counted to three million are still frequent flyers.
Counting the passengers at the gate, they were then frequent flyers, (4)/(7). There is no counting to three million that counts them as they now are, which, according to (8), is what defeats (5).
It is safe to refer to naïve, familiar objects and to the identities it implies such as (2); but only if number words denote events of measurement and occur as adverbs.
Why is a raven like writing desk? Parallels and divergences between number words and scalar quantifiers
Number words and scalar quantifiers have parallel patterns of interpretation. They both are often interpreted as having an upper bound (12). But, in some contexts, this upper bound seems to disappear, giving rise to a lowerbounded reading (34).
1. I know six of the deadly sins. (but not 7)
2. I ate some of the cookies. (but not all)
3. If you know six of the deadly sins, you will pass the quiz. (if you know 7, you pass as well)
4. If you ate some of cookies, you can't have cake. (if you ate them all, you can't have cake)
This pattern has led many linguists to propose the same mechanism accounts for the two interpretations. Specifically, both kinds of phrases are argued to have a lowerbounded semantic interpretation, which is can be supplemented by upperbounding implicature in supportive contexts. We have explored this hypothesis by studying: 1) the momenttomoment processing of scalar and numerical quantifiers in adults, and 2) how the interpretation of number and scalars changes over development. Our findings support the conventional analysis of scalar quantifiers. Adults rapidly access the lowerbounded interpretation of "some" but are slow to calculate the scalar implicature. Children typically fail to do so. However, our findings suggest that the interpretation of numerical quantifiers follows a different path: adults immediately access the upper bound of a number and children reliably infer that numericallyquantified phrases are exact by three years of age. We suggest that the similarities between numerical and scalar quantifiers (14) reflect an abstract similarity in their logical properties, rather than a shared cognitive process.