Testimony	and	children's	acquisition	of	number	concepts Helen	De	Cruz This is the final, non-copyedited draft of a paper now published in S. Bangu (Ed). Naturalizing Logico-Mathematical Knowledge. Approaches from Philosophy, Psychology	and	Cognitive	Science.	London:	Routledge. Abstract An enduring puzzle in philosophy and developmental psychology is how young children	acquire	number	concepts,	in	particular	the	concept	of	natural	number.	Most solutions	to	this	problem	conceptualize	young	learners	as	lone	mathematicians	who individually	reconstruct	the	successor	function	and	other	sophisticated	mathematical ideas.	In	this	paper,	I	argue	for	a	crucial	role	of	testimony	in	children's	acquisition	of number concepts, both in the transfer of propositional knowledge (e.g., the cardinality	concept),	and	in	knowledge-how	(e.g.,	the	counting	routine). 1.	The	puzzle	of	number	acquisition At around three-and-a-half to four years of age, children in western and other numerate cultures	experience	a	profound shift in their	understanding	of	numbers: they come to understand how counting works. They can use number words to denote the cardinality of collections of items in a precise fashion, by placing each item	to	be	counted into	a	one-to-one	correspondence	with	elements	of	a	counting list,	and	using	the	last	item	to	denote	the	cardinality	of	the	set	(see	e.g.,	Sarnecka	in press, Le Corre 2014). Children's acquisition of number concepts is often conceptualized in terms of individual discovery and personal reconstruction. For example, Carey (2009, 302) writes that children learn to individuate three items "before	figuring	out	how	the	numeral	list	represents	natural	number".	Davidson	et	al. (2012, 163) put it this way, "sometime between the ages of 3-and-a-half and 4, children	discover that	counting	can	be	used	to	generate	sets	of	the	correct	size	for any	word	in	their	count	list"	(emphasis	added	in	both). While young children's understanding of natural numbers is remarkable, it is misleading to suggest that each child figures this out individually. Young children grow up in an environment with numerate adults and older children, who use counting systems that are transmitted over many generations. Across cultures, humans use a variety of ways to count, including tallying, body-part counting, counting	rods,	and	abacuses.	Children	are	born in	these	rich	cultural	environments and adopt the counting systems of their parents, further extending them and building on them.	Our counting systems are in line	with other cultural inventions, where successive generations build on what earlier generations have already achieved.	Tomasello	et	al.	(2005,	688)	describe	this	process	of	learning	as	the	ratchet effect:	children	are	born	in	an	environment	of	collective	artifacts	(e.g.,	abacuses)	and social	practices	(e.g.,	body-part	counting),	which	structure	their	cognitive	ontogenies, and	allow	them	to	build	on	the	cultural	achievements	of	previous	generations. It is not clear when the earliest counting systems arose, but archaeological evidence 2 suggests that numeracy arose substantially earlier than writing. Bone, antlers, or ochre sticks	with regular incisions appear in the archaeological record from about 77,000	years	ago,	but it is	unclear	whether these	notches	are	purely	decorative	or might	have	numerical	meaning	(Cain	2006).	From	about	25,000	years	ago,	artifacts appear that	have groupings	of	markings, indicating that	quantities	may	have	been meaningful.	Two	bones	from	Ishango,	Congo	(25,000	years	BP)	have	notches	that	are grouped in interesting	ways.	For instance,	one	of these	bones	has two	of its three sides	covered	with	notches	that	add	up	to	60	(Pletser	and	Huylebrouck	1990). In this paper, I	will argue that testimonial transmission	plays a crucial role in how children learn about numbers. This involves both transmission of propositional knowledge (knowledge-that), and the transmission of skills (knowledge-how). The paper is structured as follows: In section 2, I review children's acquisition of the cardinal principle (CP). Section 3 describes how this cognitive change is usually framed	in	terms	of	individual	discovery.	This	view-which	I	term	the	child	as	a	lone mathematician-fits in a broader picture of children as individual learners and discoverers. Section 4 presents an alternative account of children's learning, the child as a social learner who acquires knowledge through testimony. While traditionally	(particularly	in	philosophy),	testimonial	transmission	has	been	framed	in terms of propositional knowledge, I argue for a broader view where testimonial transmission also involves demonstration and transmission of knowledge-how. In section	5, I	outline	how	this	approach is fruitful in	explaining	how	children	acquire number	concepts. 2.	Children's	acquisition	of	the	cardinal	principle Children	typically	learn	to	count	through	a	stable	developmental	sequence.	Toddlers between	18	months	and	two	years	can	recite	counting	lists	(e.g.,	"one,	two,	three") as	a	meaningless	string	of	words;	they	cannot	use	the	number	words	to	determine the cardinality of collections. At around two years of age, they become subsetknowers (Wynn	1990). The first stage is to be a one-knower: a child can correctly give one item	when requested, e.g., "Can you give	me one toy", but performs at chance level for higher quantities (e.g., the child gives three toys when two are requested).	The	next stage is to	become	a two-knower:	children	can	correctly	give one	or two	toys	but	perform	at	chance level for three	or	more; they	then	become three-knowers,	and	occasionally,	four-knowers-always	in	that	order. One	would	expect that the	next step is	becoming	a fouror five-knower,	but	after three or four, a crucial cognitive change takes place. Children become cardinal principle (CP) knowers: they understand that counting can be used to denote the correct	cardinality	of	any	given	collection	of	items.	The	cardinal	principle	states	that if n is followed by	m in the counting sequence, a set with cardinality n will have cardinality	m	if	an	item	is	added. While	this	seems	straightforward	(indeed,	we	use this	principle	without	conscious	reflection	when	we	count),	no	other	animal	seems to	understand it, in spite	of extensive	efforts	by	experimenters to train animals in numeracy.	For	example,	Ai,	a	female	chimpanzee,	learned	to	use	Arabic	digits	1	to	9 to	denote	quantities,	but	had	to learn	each	separate	number	by	brute	association. She	did	not	make	the	shift	to	CP	at	the	numbers	4	or	5	(Biro	and	Matsuzawa	2001). 3 CP	builds on	phylogenetically older numerical abilities,	which	we share	with	many other	animals.	Even	insects,	such	as	honeybees,	can	recognize	small	quantities	up	to 3 or 4 precisely, a capacity that is termed 'subitizing' (e.g., Dacke and Srinivasan 2008).	Humans	also subitize: they	are	more confident, less error-prone, and	much quicker	when	handling	small	sets	up	to	four	items,	compared	to	larger	sets	(Revkin et	al.	2008).	Human	newborns	can	distinguish	between	sets	up	to	3	or	4	items	(Antell and Keating 1983). Numerical cognition for larger quantities is approximate: for instance,	we	can	see	that	putting	16	+	16 together is smaller than	40,	but	without counting,	we	don't	know	how	much	smaller	(Pica	et	al.	2004).	Young	chicks	(Rugani et al. 2013) and guppies (Bisazza et al. 2010) can estimate and compare different quantities,	such	as	four	versus	eight	without	any	prior	training.	However,	animals	are limited in their capacity to distinguish between numerosities larger than 3 or 4 exactly.	A	wide	range	of	studies	indicates	that	animals	can	distinguish	between	2	and 3,	but	fail	to	tell	the	difference	between	5	and	6	(e.g.,	Petrazzini	et	al.	2015).	Animal numerical cognition above 3 or 4 is approximate, and becomes increasingly less precise	as	cardinalities increase	(see	e.g.,	Scarf	et	al.	2011,	for	a	comparative	study with	pigeons	and	primates). Since only humans are able to represent cardinalities above 3 or 4 precisely, cognitive	scientists	and	philosophers	have	speculated	how	children	acquire	number concepts.	Experimental	studies	on	children's	numerical	cognition	suggest	that	it	is	a mixture of knowing-that (propositional knowledge) and knowing-how (procedural knowledge).	Examples	of	propositional	knowledge	are	knowing	that	numbers	can	be extended	indefinitely,	or	that	number	words	denote	precise	cardinalities	rather	than approximate	ones.	Already	before	they	are	able	to	count,	young	children	realize	that if you	add items to	a collection	of "five", the resulting	cardinality cannot	be	"five" anymore,	and	that	"six	plus	more"	is	no	longer	"six",	but	that	"a	lot	plus	more"	is	still "a lot" (Sarnecka and Gelman 2004). As children grow older and have more experience	with formal	mathematics, they learn additional properties of numbers, such	as that	2 is	prime,	or that	3 is	both	prime	and	odd.	Knowledge-how involves elements	of the	counting	routine,	such	as	being	able	to	recite	number	words from memory in the correct order. It also involves the ability to keep track of what is already counted	when counting a collection of items. In	many cultures, numerical competence includes being able to use correct procedures, for instance, to	make calculations on an abacus or counting board, being able to carry numerals when multiplying, or using the correct procedure for a long division. Such skills require extensive	practice,	often	in	a	formal	school	context. 3.	The	child	as	a	lone	mathematician How do we explain young children's acquisition of knowledge about numbers, in particular the acquisition of CP? Standard accounts tend to conceptualize young children as lone mathematicians, who come up with the requisite concepts independently.	This is in line	with	the	popular	view	of	the	child	as	a little	scientist, introduced	by	the	developmental	psychologist	Piaget	(1929),	but	going	back	as	early as Rousseau (1762/1999). Piaget was impressed by how quickly children pick up information	about	their	environment,	and	he	was	intrigued	by	the	errors	they	made in the process. These errors suggested to him that children are not passive 4 receptacles of knowledge, but active learners. Children learn by actively engaging with	the	world,	and	changing	their	ideas	about	it	as	a	result	of	their	experience,	a	bit like scientists	who	conduct	experiments	and formulate	hypotheses	on the	basis	of them. This view has contemporary defenders, for example, Gopnik, Meltzoff and Kuhl (2001), who draw an explicit comparison between babies and scientists engaged in individual experimentation. Carey (2009, 20) developes the analogy between scientists and	children further, conceptualizing	developmental changes in individual	children's	minds in	terms	of	mini-scientific	revolutions,	with	processes in individual	minds being similar to "those described in the literature on history and philosophy	of	science"	(Carey	2009,	20). Carey	(2004,	2009)	regards	children's	acquisition	of	the	CP	as	a	cognitive	revolution. She uses a Quinean bootstrapping account to explicate it. Bootstrapping is the process	whereby	a	child	acquires	a	new	concept	by	first	using	a	system	of	external symbols (counting words), initially without knowing what they mean. Through analogical	and	inductive	inference,	children	come	to	realize	what	the	symbols	stand for. First, children learn the counting list as	meaningless lexical items. During the next	phase,	they	realize	that	"one"	corresponds	to	one	item	in	a	collection	(e.g.,	one biscuit), and they do the same	with "two" and "three". Children probably rely on their	subitizing	capacity to	associate	the	number	words	"one",	"two",	"three" (and sometimes	"four")	with	the	corresponding	quantities.	However,	subitizing	ends	at	3 or	4,	so	for	larger	quantities	this	system	does	not	work	anymore,	which	may	explain why children	become	one-knowers, two-knowers, three-knowers, and	occasionally four-knowers,	but	never	five-knowers.	According	to	the	bootstrapping	account,	it	is at this point that a crucial induction takes place: the child notices an analogy between	the	next	in	the	numeral	list	and	next	in	series	of	objects	to	be	counted.	She realizes	(for	small	numerosities	within	the	subitizing	range)	if	"x"	is	followed	by	"y"	in the	counting	sequence,	adding	an	individual	to	a	set	with	cardinal	value	x	results	in	a set with cardinal value y. She generalizes this for quantities > 4. In this way, the successor function is	established.	The	successor function	plays	a	crucial role in the Dedekind–Peano axioms of arithmetic, a 19th century formalization of arithmetic. The successor function is a primitive function, which takes that if a given n is a natural	number,	so	is its	successor, i.e.,	S(1)	=	2,	S(2)	=	3,	and	so	on.	The	successor function obeys several of the Dedekind-Peano axioms of arithmetic, for instance, that	the	successor	of	a	natural	number	is	also	a	natural	number,	and	that	there	is	no natural	number	whose	successor is	0. It is	remarkable,	given	that	this	function	was spelled	out	only in the	19th century, that threeand four-year-old children	around the	world	routinely	hit	upon	this	principle	through	induction. The	bootstrapping	account	is	not	without	controversy.	Rips	et	al.	(2006),	for	example, have	argued	that it is insufficient	to	explain	how	children	actually learn	the	natural numbers.	They	present	a	rather	contrived	case	of	three-year-old	twins	(Fran	and	Jan) to argue their case. Fran learns the standard natural numbers, while Jan learns a peculiar, modular system in which the count sequence cycles back around from ''nine''	to	"none",	and	then	goes	back	to	"one",	"two",	etc.,	again.	Clearly,	Jan	ends up	with	a	badly	underdetermined	system,	since	"two"	can	refer	to	collections	of	2, 12, 22, ... items.	Although	both children	have	used induction,	only Fran learns the natural	numbers.	Thus,	according to	Rips	et	al. (2006), induction	by itself	does	not 5 suffice	to	establish	natural	numbers.	Rips	et	al.	(2008)	instead	propose	that	children use abstract mathematical schemas, such as commutativity (a + b = b + a), to understand natural numbers. How do children learn such abstract schemas? According	to	Rips	et	al., they	do	so	through	physical	experiences	and	manipulating the	environment: Suppose that children initially	notice that two similar sets	of	objects	– for	example,	two	sets	of	three	toy	cars	–	can	be	matched	one-to-one.	At a	later	stage,	they	may	extend	this	matching	to	successively	less	similar objects	–	three	toy	cars	matched	to	three	toy	drivers	–	and	eventually	to one-to-one	matching for any two sets of three items. This could yield the	general	concept	of	sets	that	can	be	matched	one-to-one	to	a	target set of three objects – a possible representation for three itself (Rips 2008,	637). Again,	children	are	conceptualized	as	individual	discoverers	who	actively	experiment and	acquire	sophisticated	mathematical	knowledge	through	this. The picture of preschoolers who learn about numbers, including sophisticated concepts as the successor function and commutativity is that of the lone mathematician-akin to Srinivasa Ramanujan (1887-1920) who, seemingly on his own,	made	substantial	contributions	to	several	areas	of	mathematics.	The	tradition of the child as lone scientist/mathematician is not only descriptive, but also normative:	children	ought	to	learn	in	this	way.	Educators,	such	as	Maria	Montessori, have	devised	education	systems	that	encourage	children	to	learn	through	individual discovery.	The	idea	behind	this	is	that	individual	discovery	is	a	form	of	learning	that is	superior	to	social learning,	which	is	regarded	as	mere	parroting,	that	precludes	a genuine	understanding	of	what	is	being	learned. 4.	Learning	through	testimony An	alternative	developmental	picture	sees	young	children	primarily	as	social	learners, who	benefit from	the	vast	knowledge	and	experience	of	adults	and	older	children. This tradition	goes	back to	Thomas	Reid (1764)	who	regarded testimony	as	one	of the	most important channels through	which	we learn; like	perception	or	memory, testimony is a basic source of knowledge. Although we may occasionally be led astray by liars or people who are genuinely mistaken, testimony is an important source of information for facts outside of our immediate experience. Indeed, without testimony,	we	would	not	even	know	the	place	and	date	of	our	own	birth. Reid (1764) formulated	several	arguments for	why testimony is	a	crucial	and	basic source	of	knowledge:	humans	have	a	natural inclination	to	believe	what	they	learn from	others,	and	they	only	become	sceptical	when	they	have	reasons	to	be	doubtful (e.g., the testifier is a known liar). Young children are more gullible than adults, showing	that	it	is	an	innate	tendency.	As	he	wrote: It is evident, that in the matter of testimony, the balance of human judgment is by nature inclined to the side of belief; and turns to that side of itself when there is nothing put in the opposite scale. ... if credulity	were	the	effect	of	reasoning	and	experience, it	must	grow	up 6 and	gather	strength, in the	same	proportion	as reason	and	experience do.	But if it is the gift of nature, it	will be strongest in childhood, and limited	and restrained	by	experience;	and the	most superficial view	of human	life	shows,	that	the	last	is	really	the	case,	and	not	the	first	(Reid 1764,	478–479). Reid's	account	of	testimony	is	both	descriptive	and	normative:	humans	acquire	vast stores	of	knowledge	through	testimony,	and	it is	proper	to	do	so.	For	testimony	to work, two elements need to be in place: trustfulness and truthfulness, or as Reid calls them, the principle of credulity and the principle of veracity, "two principles that tally with each other" (Reid 1764, 474). Children need to be willing to learn counterintuitive,	strange ideas,	such	as that the	Earth is round	even	though it is in their own experience flat, if they want to learn the correct scientific view of the Earth. Parents and teachers must be "benevolent communicators" (Csibra and Gergely 2009, 152), who are willing to teach and not to deceive. The strange scenario	Rips	et	al.	(2006)	offer	of	one	of	the	twins	being	taught	a	modular	counting method	(by	a	"diabolic	parent",	as	they	put	it)	is	unlikely	to	occur	in	real	life. This Reidian picture of children as trustful social learners is gaining increasing popularity	in	cognitive	science	(see	e.g.,	Harris	2012,	for	a	comprehensive	overview). Csibra and Gergely (2009) have argued that humans are natural pedagogues, and that communication has evolved to facilitate the transmission of knowledge. Applying	this	model	of	the	child	as	social	learner	to	numerical	cognition,	we	come	to the view of children as socially-embedded mathematicians. Likening children to grown-up scientists or mathematicians is inherently problematic; for one thing, children do not really conduct experiments. However, even if one were to insist upon	such	a	close	analogy,	it	is	important	to	point	out	that	mathematicians	rely	on testimony and other forms of socially transmitted information. Thurston (2006) observes Mathematical knowledge can be transmitted amazingly fast	within a subfield. When a significant theorem is proved, it often (but not always)	happens	that	the	solution	can	be	communicated	in	a	matter	of minutes from one person to another within the subfield. The same proof	would be communicated and generally understood in an hour talk to members of the subfield ... When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are usually not very	good	in	checking	formal	correctness	of	proofs,	but	they	are	quite good	at	detecting	potential	weaknesses	or flaws in	proofs (Thurston, 2006,	43,	46). Geist	et	al.	(2010)	argue	that	mathematicians	frequently	rely	on	testimony	to	decide matters	such	as	the	soundness	of	a	mathematical	proof.	For instance,	many	proofs rely on theorems that have been proved before, and individual mathematicians using these theorems do not meticulously check for themselves whether these earlier	proofs	are	correct.	Moreover,	reviewers	for	papers	for	mathematical	journals 7 do	not	check	all	proofs	in	detail	for	themselves,	but	rely	on	the	author's	own	scrutiny, only	checking	results	in	detail	when	it	seems	warranted	to	do	so. Given that children's early knowledge of numerical concepts is largely procedural (correctly using the counting routine to denote cardinalities of sets), one	may ask whether testimony is	a	good	model for the	acquisition	of	number	concepts.	When we	think	of	testimony,	we	tend	to	think	of	propositional	knowledge.	Indeed,	Poston (2016)	argues	that	knowing-how	is	not	transmitted	as	easily	as	knowing-that.	When	I tell my daughter that Jimi Hendrix was a famous guitarist, she thereby comes to know	that	Hendrix	was	a	famous	guitarist.	However,	when	someone	demonstrates	a guitar	riff	from	Hendrix'	repertoire	to	her,	she	thereby	does	not	come	to	know	that riff. It	would	likely	take	her	many	weeks	to	learn	the	riff,	and	if	her	guitar	skills	are not up to it, she may never learn how to perform it correctly. We can through testimony learn fairly complex pieces of propositional knowledge,	which does not seem	to	be	the	case	for	knowledge-how.	This	disanalogy	between	the	transmission of	knowing-that	(which	you	can	do	by	simply	stating	that	p)	and	knowing-how	(which you cannot do by simply demonstrating φ) leads Poston (2016) to conclude that knowing-how	is	not	reducible	to	knowing-that. While	Poston	(2016)	is	correct	that	learning	a	skill	requires	additional	resources	from the learner (e.g., practice), there	are	nonetheless striking	parallels	between	verbal testimony and demonstration of knowing-how. Buckwalter and Turri (2014) argue that demonstration has a close analogue to the knowledge norm of assertion. According	to	the	knowledge	norm	of	assertion,	when	you	assert	that	p you	should know	that	p	(Williamson	2000).	There	are	several	motivations	for	this,	e.g.,	it	would be	paradoxical	to	say:	p,	but	I	don't	know	that	p,	and	people	will	often	challenge	an assertor by asking, "How do you know that?" Similarly, in showing a skill (demonstration), you need to know how to perform that skill. It would be paradoxical	to	say,	"I	will	show	you	how	to	φ,	but	I	don't	know	how	to	φ".	You	can also challenge a	demonstrator, e.g., one can ask a	bungling ski instructor "Do you actually	know	how	to	ski?" If we look at testimony in the broader context of social learning, it is clear that demonstrating	skills	preceded	verbal	testimony,	perhaps	by	over	one	million	years. Evidence for explicit teaching and demonstration appears as early as 2.5 million years	ago.	Early	human	stone	tool	technology,	such	as	Oldowan	(ca.	2.5	million	years ago)	and	Acheulean	(ca.	1.5	million	years	ago),	require	some	form	of	social	learning that	almost	certainly involved	demonstration,	and	very likely	also	explicit teaching, e.g.,	how	to	correctly	strike	the	core	(demonstrating	the	angle	with	which	you	have to	hit	the	core	so	as	to	obtain	a	razor-sharp	flake	that	can	be	used	to	cut	meat),	or which	pebbles to flake from.	Demonstration	and trusting	demonstration	may	be	a uniquely	human	capacity.	While	chimpanzees	are	sophisticated	social	learners,	they do not faithfully follow demonstrators. For example, Horner and Whiten (2005) showed	children	and juvenile	chimpanzees	a	complex	way to	open	a	puzzle	box in order	to	retrieve	a	candy.	Both	chimps	and	children	opened	the	box	in	the	way	that was	demonstrated.	However,	in	a	second	try,	a	transparent	box	was	used	that	made it obvious that	part of the actions	were superfluous. The children continued	using the way that was demonstrated to them (this tendency of sticking to a learned 8 suboptimal	routine	is	called	overimitation).	The	chimpanzees	went	for	a	more	direct way	to	obtain	the	candy	that	deviated	from	the	demonstration.	The	human	default position is one of trust. Children show a high degree of trust when an adult demonstrates	them	the	"proper	way"	to	do	something:	they	assume,	by	default,	that demonstration	presupposes	knowing-how.	Overimitation	has	not	only	been	found	in western	children (who	might	be influenced	by	schooling),	but	also in	children	who live in groups	with little formal education, such as in South-African and	Australian hunter-gatherer	societies	(Nielsen	et	al.	2014). Humans	are	the	only	species	that	have	cumulative	cultural	transmission	in	a	variety of domains: they build on the accomplishments of earlier generations and create more	complex	cultural	elements,	including	artifacts	(e.g.,	the	atl	atl,	a	small	artifact that	attaches	to	a	spear	to	make	it	go	much	further	and	faster	when	thrown),	ways of preparing food (to make it less toxic, tastier, or easier to digest), and stories (including very long and complex epic tales). Cumulative culture is not altogether absent in nonhuman animals. For example, New Caledonian crows (Corvus moneduloides)	which	use	tools	made	of	the	edges	of	Pandanus	leaves	to	help	them catch invertebrates, use design innovations by conspecifics to improve their own tools (Hunt and Gray 2003). However, their cumulative culture is limited to the domain of tools. By contrast, human learning takes place in an environment populated with potential teachers and artifacts, and concerns a wide range of domains.	According to	Sterelny's (2012)	apprenticeship	model,	humans	grow	up in environments	that	are	"seeded"	with	artifacts	that	help	and	support	learning	novel skills	and	information.	Children	learn	by	combining	information	from	the	social	world with	information	from	the	physical	world.	For	example,	in	a	workshop,	tools	tend	to be laid	out in	an	order that facilitates	completing	specific tasks, such	as	making	or repairing	furniture.	Through	observing	accomplished	woodworkers	or	through	active teaching,	or	likely	a	combination	of	both,	young	apprentices	learn	both	the	requisite propositional knowledge (e.g., distinguishing between the top rail, splat, and back rail	of	a	chair)	and	knowledge-how	(e.g.,	how	to	use	a	hole-saw	properly).	In	this	way, the testimonial transmission of	many complex domains of culture requires both a transmission of propositional knowledge, and a transmission of skills through demonstration. 5.	The	testimonial	transmission	of	number	concepts We	are	now in	a	position to	examine	how	the testimonial transmission	of	number concepts	might	work.	When	children	learn	to	count	at	about	3.5-4	years	of	age,	they do	not	yet	have	a	fully-fledged	concept	of	natural	numbers.	For	example,	children	of this	age	tend	to	fail	the	Piagetian	conservation	task	(Piaget	1941):	when	one	lays	out a row of items, they tend to think that its number changes if the spatial configuration of the row changes (e.g., by	widening the gaps between the items). Although	Piaget's	theoretical	rationale	for	why	young	children	fail	this	task,	and	only succeed in it about age 5-7 is no longer generally accepted, failure of toddlers to pass the conservation task is robust and has often been replicated. Young CPknowers	do	not	realize	that	number	words	further	in	the	counting	list	are	larger	than numbers earlier in the counting list (the later-greater principle), even though it 9 seems like this could be easily inferred from the CP and its applications (Le Corre 2014). Moving to	more sophisticated concepts, the	development	of children's concept of zero	follows	a	different	trajectory	compared	to	other	natural	numbers. It	seems	to proceed in three phases (Wellman and Miller 1986). In the earliest phase, preschoolers	recognize	the	Arabic	numeral	0	and	the	noun	"zero",	but	they	do	not know what it means. In the second phase, children understand zero to refer to nothing, but they	do	not	understand its relationship to	other small numbers (e.g., they are equally likely to say that 0 is smaller than 3 or vice versa,	whereas they understand	that	1	is	smaller	than	3).	The	third	phase	is	when	children	realize	zero	is the smallest natural number, and they can make accurate comparisons between zero	and	other	small	numbers. Children's understanding of infinity also appears several years after they learn to count. Many counting systems, be they number words or other ways to denote numbers (e.g., pointing to body	parts,	where each	body	part symbolizes a natural number)	are	quite limited,	often	not	exceeding	20	or	30. In these	cultures,	people can	come	up	with	higher	numbers	on	an	ad	hoc	basis	(e.g.,	counting	fingers	and	toes of	several	people	present).	Children's	understanding	of	infinity	only	arises	at	about	8 years	of	age,	many	years	after	their	ability	to	count	collections	has	developed.	Eightyear-olds	have	some	understanding	of	potential	and	actual	infinity,	but	they	still	do not appreciate that an infinite set is immeasurably bigger than a finite set (for example,	they	mistakenly	think	that	a	very	large	set,	such	as	the	number	of	grains	of sand on a beach, is almost infinite)-indeed, even many numerate adults fail to grasp	this	unbridgeable	gap	between	very	large	sets	and	infinite	sets	(Falk	2010).	It	is worth	pointing	out	that	the	concepts	of	zero	and	infinity	are	lacking	in	many	cultures that use natural numbers (e.g., Roman numerals do not have a symbol for zero). Thus,	knowledge	of	numbers	is	not	all-or-nothing.	It	has	many	components,	such	as understanding how counting works, what cardinalities are, how to perform calculations, as	well as	more complex, culturally restricted concepts, such as zero and	infinity. How	do	children learn	about	natural	numbers in	a	social	context?	Children	tend	to grow	up in environments that are seeded	with artifacts and ideas that help them acquire number concepts. These include not only special learning tools, such as abacuses for children, or foam numbers for the bathtub, but also the symbolic representations of numbers that adults use in their day-to-day lives. In this environment, children acquire knowledge	of numbers through their parents, older siblings,	and	more	formal	learning	situations	(e.g.,	preschool	and	elementary	school). Learning the	counting list	by rote	memory is	a	crucial step towards learning	about natural	numbers.	Counting is	ostensive	and	deliberate:	across the	world, there	are counting	songs	and	routines	to	help	children	memorize	the	first	few	counting	words. In	English,	this	is	usually	the	first	five	(Five	little	monkeys,	Five	little	ducks)	or	the	first ten counting words (Ten in a bed, This old man). During these early encounters, children do not yet grasp the semantic content of these number words, but the songs do help them familiarize themselves with counting routines. Interestingly, 10 such counting routines and songs are absent in languages without exact number words,	and	in	homesigners	(deaf	children	and	adults	who	live	with	hearing	parents and	carers,	without	using	a	fully-developed	sign	language)	(Spaepen	et	al.	2011). Levine	et	al.	(2010)	recorded	7.5	hours	of	natural	conversations	(in	several	sessions) between parents and toddlers aged 14 to 30 months to understand how talking about numbers ("number talk") influences numerical cognition. Parents were informed	that	this	was	a	study	on	language	development,	but	they	did	not	learn	that it was specifically about number	words, so as to not influence them to use	more number	words	than	they	would	do	under	ordinary	circumstances.	The	authors	found that parents used an average	of 90.8	number	words	over the	7.5	hours recorded. Remarkably, only an average of 6.3 instances of this number talk consisted of prompts	where children had to give cardinal values, e.g., "how	many cars do you see?"	Most	of	the	time,	number	talk	was	testimonial.	About	50%	of	number	talk	by parents consisted of stating cardinal values ("Look, three fish!"); 32% consisted of counting songs, rhymes and routines.	When the toddlers engaged in number talk, about 62% consisted of counting routines, and 28% of cardinal values. These utterances demonstrate the importance of transmitting knowing-how to count in children's	early	acquisition	of	number	concepts.	Both	the	toddlers	and	their	parents included counting routines in their number talk. Levine et al. (2010) found that children whose parents engaged more in number talk performed better at a cardinality test at 46	months. Given that high socioeconomic status (SES) parents tend	to	talk	more	to	their	children,	they	controlled	for	the	amount	of	talking	and	for SES.	They found that	parents	who talked	more	about	number, taking into	account SES,	had	children	with	more	knowledge	of	the	cardinal	meanings	of	number	words	at 46	months. Number talk not only helps children to	master the counting routine, it also helps them	to	become	aware	of	the	fact	that	number	words	and	symbols	refer	to	discrete quantities.	This	has	been	demonstrated in the	blocks	and	water task (Slusser	et	al. 2013), where young children are presented with two bowls, one containing countable items (blocks	of different colors) and	another containing colored	water. The	experimenter	asks	the	toddlers	either	a	number	question	"Which	one	has	five?" or	a	question	that	does	not	involve	numbers	("Which	one	has	orange?")	N-knowers, who	can	only	enumerate	collections	up	to	three	or	four,	do	well	within	their	range but perform less consistently at higher numbers (five, six). However, when experimenters start with small numbers and work their way up, these younger children tend to choose the bowl with the blocks when asked "Which one has five/six", indicating they	have some	understanding that these cardinalities refer to discrete	entities,	not	to	uncountable	quantities.	Number	talk,	where	parents,	other carers, and older siblings refer to discrete sets, may help young children to understand	this	property	of	number	words. Next to linguistic input, artifacts can help support children's developing numerical cognition.	Board	games,	such	as	the	Game	of	Goose,	which	often	involve	numerical properties	(e.g.,	throwing	a	die	that	shows	n	requires	the	child	to	move	her	piece	n spaces forward), have a positive impact on children's later mathematical achievements. As Siegler and Booth (2004, 441) point out, this may be because 11 "Board games provide children with strongly correlated spatial, temporal, kinaesthetic, and verbal or auditory cues to numerical magnitude." Siegler and Ramani (2009) found that especially designed games that	were linear, rather than having numbers in a spiraling or circular pattern, were particularly effective in helping	young	children from low	SES	backgrounds to close the	numeracy	gap	with their high SES peers. Under the apprenticeship model of human learning, the environment is seeded with artifacts that help to support the development of numeracy. However, the lack of knowledge of natural numbers in homesigners indicates	that	an	environment	seeded	with	cues	for	numbers	is	not	sufficient.	There needs to be active testimonial transmission as well, which is lacking in these transmission situations where there is no adequate sign language to transmit number	concepts. If the acquisition of number concepts is the result of testimony to skills and propositional knowledge, we can understand why not all human cultures have natural	numbers.	Some	cultures,	such	as	the	Pirahã	(e.g.,	Frank	et	al.	2008)	and	the Mundurucú (e.g., Pica et al. 2004), lack words to denote exact cardinalities. In a video1	recorded	by	anthropologist	Pierre	Pica,	an	old	man	and	the	medicine	man	of	a Mundurucú	village	attempt	to	count	10	and	13	seeds,	with	limited	success.	As	can	be seen, the	men	are	not	used to count, e.g., they	do	not separate	what	has	already been counted and what still has to be counted, something young children in numerate cultures learn early on. Mundurucú do not have counting routines, although, intriguingly, their language does have a count/mass noun distinction, indicating	they	see	the	difference	between	discrete	and	continuous	quantities	(Pica and	Lecomte	2008).	While	this	distinction	is	vital	for	children	to	understand	natural numbers,	it	is	clearly	not	enough	to	help	them	count. The	testimonial	model	of	knowledge	transmission	can	also	explain	why	non-human animals fail to understand CP. Learning CP, as we have seen, depends on an understanding of the counting routine. Children learn the counting routine when they	are young toddlers, far too	early to	grasp its	meaning.	Humans	are	willing to copy	actions	even	if	they	do	not	understand	them,	which	helps	them	to	acquire	the correct	sequence	of	counting	words.	This	is	a	big	difference	with	the	chimpanzee	Ai who	was	taught	to	use	Arabic	digits in	Biro	and	Matsuzawa's	(2001)	studies.	Ai	did not learn the counting routine, instead relying on estimation to enumerate collections of items and assigning them to the correct numeral. As Biro and Matsuzawa (2001)	observe,	while	Ai	may	have	been similar to children in the fact that she learned symbols for numbers, this is just one aspect of learning how to count. She	was not raised in a supportive, rich environment filled	with numerical cues. 6.	Conclusions The	question	of	how	children learn	number	concepts is	unresolved. In	this	paper, I have	argued	that testimony	plays	an important	role in the	transmission	of	number concepts',	both	to	knowing-how	(the	skills	involved	in	counting)	and	to	knowing-that 1	https://www.youtube.com/watch?v=9iXh8wte3gM 12 (propositional knowledge	about	number	words, such	as that they refer to	discrete quantities).	Children	grow	up in	a	world	seeded	with	artifacts,	counting	songs,	and other	cultural	features	that	help	them	to	count.	I	have	pointed	out	the	shortcomings of the model of children as lone mathematicians, who discover sophisticated mathematical	principles	by	themselves.	In	order	to	understand	numerical	cognition, we	should	not	only	pay	attention	to	what	may	go	on	in	the	minds	of	young	children, but	also	to	what	goes	on	in	their	broader	learning	environment. Acknowledgments Many thanks to Sorin Bangu, Max Jones, Dirk Schlimm, Johan De Smedt, and audiences in Bergen,	Norway and	Bristol, for their helpful comments to an earlier version	of	this	paper. References Antell,	S.E.,	&	Keating,	D.P.	(1983).	Perception	of	numerical invariance	in	neonates. Child	Development,	54,	695–701. Biro,	D.,	&	Matsuzawa,	T.	(2001).	Chimpanzee	numerical	competence:	Cardinal	and ordinal	skills.	In	T.	Matsuzawa	(Ed.),	Primate	origins	of	human	cognition	and	behavior (pp.	199–225).	Tokyo:	Springer. Bisazza,	A.,	Piffer,	L.,	Serena,	G.,	&	Agrillo,	C.	(2010).	Ontogeny	of	numerical	abilities in	fish.	PLoS	ONE,	5,	e15516. Buckwalter,	W.,	&	Turri,	J.	(2014).	Telling,	showing	and	knowing:	A	unified	theory	of pedagogical	norms.	Analysis,	74,	16–20. Cain,	C.	R. (2006). Implications	of the	marked	artifacts	of the	Middle	Stone	Age	of Africa.	Current	Anthropology,	47,	675–681. Carey,	S.	(2004).	Bootstrapping	and	the	origin	of	concepts.	Daedalus,	133,	59–68. Carey,	S.	(2009).	The	origin	of	concepts.	Oxford:	Oxford	University	Press. Csibra,	G.,	&	Gergely,	G.	(2009).	Natural	pedagogy.	Trends	in	Cognitive	Sciences,	13, 148–153. Dacke, M., & Srinivasan, M.V. (2008). Evidence for counting in insects. Animal Cognition,	11,	683–689. Davidson,	K.,	Eng,	K.,	&	Barner,	D.	(2012).	Does	learning	to	count	involve	a	semantic induction?	Cognition,	123,	162–173. Falk, R. (2010). The infinite challenge: Levels of conceiving the endlessness of numbers.	Cognition	and	Instruction,	28,	1–38. Frank,	M.	C.,	Everett,	D.	L.,	Fedorenko,	E.,	&	Gibson,	E.	(2008).	Number	as	a	cognitive technology:	Evidence	from	Pirahã	language	and	cognition.	Cognition,	108,	819–824. 13 Geist, C., Löwe, B., & Van Kerkhove, B. (2010). Peer review and knowledge by testimony	in	mathematics.	In	B.	Löwe,	&	T.	Müller	(Eds.),	PhiMSAMP.	Philosophy	of mathematics: Sociological aspects and mathematical practice (pp. 155–178). London:	College	Publications. Gopnik, A.,	Meltzoff, A.	N.,	& Kuhl, P. (2001).	The scientist in the crib.	What early learning	tells	us	about	the	mind.	New	York:	Perennial. Harris, P.L. (2012). Trusting what you're told. How children learn from others. Cambridge,	MA:	Harvard	University	Press. Horner, V., & Whiten, A. (2005). Causal knowledge and imitation/emulation switching in chimpanzees (Pan troglodytes) and children (Homo sapiens). Animal Cognition,	8,	164–181. Hunt,	G. R.,	&	Gray, R. D. (2003). Diversification and cumulative evolution in	New Caledonian crow tool	manufacture.	Proceedings of the Royal Society of London B: Biological	Sciences,	270,	867–874. Le Corre,	M. (2014). Children acquire the later-greater principle after the cardinal principle.	British	Journal	of	Developmental	Psychology,	32,	163–177. Levine, S.C., Suriyakham, L.W., Rowe, M.L., Huttenlocher, J., & Gunderson, E.A. (2010).	What counts in the development of young children's number knowledge? Developmental	Psychology,	46,	1309–1319. Nielsen,	M.,	Mushin,	I.,	Tomaselli,	K.,	&	Whiten,	A.	(2014).	Where	culture	takes	hold: "Overimitation" and its flexible deployment in	Western, Aboriginal, and Bushmen children.	Child	Development,	85,	2169–2184. Petrazzini,	M.	E.	M.,	Lucon-Xiccato,	T.,	Agrillo,	C.,	&	Bisazza,	A.	(2015).	Use	of	ordinal information	by	fish.	Scientific	Reports,	5,	15497. Piaget,	J.	(1929).	The	child's	conception	of	the	world	(J.	Tomlinson	and	A.	Tomlinson, trans).	London:	Routledge	and	Kegan	Paul. Piaget,	J.	(1941/1952).	The	child's	conception	of	number	(trans.	C.	Gattegno	and	F.M. Hodgson).	London	and	New	York:	Routledge. Pica,	P.,	&	Lecomte,	A.	(2008).	Theoretical	implications	of	the	study	of	numbers	and numerals	in	Mundurucu.	Philosophical	Psychology,	21,	507–522. Pica,	P.,	Lemer,	C.,	Izard,	V.,	&	Dehaene,	S.	(2004).	Exact	and	approximate	arithmetic in	an	Amazonian	indigene	group.	Science,	306,	499–503. 14 Pletser,	V.,	&	Huylebrouck,	D.	(1999).	The	Ishango	artefact:	The	missing	base	12	link. Forma,	14,	339–346. Poston,	T.	(2016).	Know	how	to	transmit	knowledge?	Noûs,	50,	865–878. Reid,	T.	(1764).	An	Inquiry	into	the	Human	Mind,	on	the	Principles	of	Common	Sense. Edinburgh:	Millar,	Kincaid	&	Bell. Revkin, S.K., Piazza,	M., Izard, V., Cohen, L.,	&	Dehaene, S. (2008).	Does subitizing reflect	numerical	estimation?	Psychological	Science,	19,	607–614. Rips,	L.J.,	Asmuth,	J.,	&	Bloomfield,	A.	(2006).	Giving	the	boot	to	the	bootstrap:	How not	to	learn	the	natural	numbers.	Cognition,	101,	B51–B60. Rips,	L.J.,	Bloomfield,	A.,	&	Asmuth,	J.	(2008).	From	numerical	concepts	to	concepts of	number.	Behavioral	and	Brain	Sciences,	31,	623–642. Rousseau,	J-J.	(1762/1999).	Emile.	In	Oeuvres	Complètes,	Volume	4.	Paris:	Pléiade. Rugani, R., Vallortigara, G., & Regolin, L. (2013). Numerical abstraction in young domestic	chicks	(Gallus	gallus).	PloS	One,	8,	e65262. Sarnecka,	B.W.,	&	Gelman,	S.A.	(2004).	Six	does	not	just	mean	a	lot:	Preschoolers	see number	words	as	specific.	Cognition,	92,	329–352. Sarnecka,	B.W.	(in	press).	Learning	to	represent	exact	numbers.	Synthese. Scarf, D., Hayne, H., & Colombo, M. (2011). Pigeons on par with primates in numerical	competence.	Science,	334,	1664-1664. Siegler, R.S., & Booth, J.L. (2004). Development of numerical estimation in young children.	Child	Development,	75,	428–444. Siegler,	R.S.,	&	Ramani,	G.	B. (2009).	Playing linear	number	board	games-but	not circular	ones-improves	low-income	preschoolers'	numerical	understanding.	Journal of	Educational	Psychology,	101,	545–560. Slusser, E., Ditta, A., & Sarnecka, B. (2013). Connecting numbers to discrete quantification:	A	step	in	the	child's	construction	of	integer	concepts.	Cognition,	129, 31–41. Spaepen, E., Coppola,	M., Spelke, E. S., Carey, S. E., & Goldin-Meadow, S. (2011). Number	without	a	language	model.	Proceedings	of	the	National	Academy	of	Sciences, 108,	3163–3168. Sterelny, K. (2012). The evolved apprentice. How evolution made humans unique. Cambridge,	MA:	MIT	Presss. Thurston,	W. (2006). On proof and progress in	mathematics. In R. Hersh (Ed.),	18 unconventional	essays	on	the	nature	of	mathematics	(pp.	37–55).	New	York:	Springer. 15 Tomasello,	M., Carpenter,	M., Call, J., Behne, T.,	&	Moll,	H. (2005).	Understanding and sharing intentions: The origins of cultural cognition. Behavioral and Brain Sciences,	28,	675–691. Williamson,	T.	(2000).	Knowledge	and	its	limits.	Oxford:	Oxford	University	Press. Wellman, H.M., & Miller, K.F. (1986). Thinking about nothing: Development of concepts	of	zero.	British	Journal	of	Developmental	Psychology,	4,	31–42. Wynn,	K.	(1990).	Children's	understanding	of	counting.	Cognition,	36,	155–193.