from a neurocognitive perspective , finger counting provides multisensory input that conveys information on both cardinal and ordinal aspects of numbers . here , the number of fingers and their arrangement on both hands plays a fundamental role in first applications of externalized representations of numerical magnitude in initial counting and calculation . the importance of such embodied finger - based representations of number magnitude is further illustrated by findings suggesting that blind children ( crollen et al . , 2011 ) and even children with amputated hands and forearms ( poeck , 1964 ) use their ( phantom ) hands and fingers as external quantifiers . in line with such findings , recent neurocognitive data indicate that finger gnosis is associated with children s numerical / arithmetical competencies , including computational skills ( e.g. , nol , 2005 ; penner - wilger et al . , 2007a , b ) . even in adults , recent evidence suggests that the link between finger ( counting ) patterns and semantic cardinal number magnitudes is stronger for canonical ( e.g. , 7 represented by 5 and 2 fingers ) as compared to non - canonical finger patterns ( e.g. , 7 represented by 4 and 3 fingers ; di luca and pesenti , 2008 ; di luca et al . , 2010 ) . this suggests a close link between finger counting and the representation of abstract number magnitude in healthy adults ( di luca et al . however , the exact origin of this link is still debated . on the one hand , 2008 ) showed that the association of numbers and fingers is modulated by palm orientation . this suggests a more general association of relatively smaller and larger numbers with  left  and  right , respectively , independent of specific fingers ( e.g. , see also ishihara et al . , ( 2006 , 2010 ; see also di luca and pesenti , 2008 , 2010 ) have presented converging evidence indicating a direct association of specific numbers embodied as specific fingers / finger patterns . finally , recent neuroimaging data suggest that the neural correlates of finger and number representations are located in neighboring or even overlapping cortex areas ( see e.g. , kaufmann et al . , 2008 ) . from a neurocognitive view , this link seems to be functional and not exclusively correlational . for instance , gracia - bafalluy and nol ( 2008 ) observed that systematic training of finger gnosis led to an improvement of numerical performance . moreover , badets and pesenti ( 2011 ) showed that learning to associate certain finger movements with meaningless syllables automatically associated certain magnitudes with the same syllables ( see also andres et al . , 2008b for a review on finger - based numerical associations ) . yet , recent research not only investigated associations between finger - based representations and numerical / arithmetical competencies but also specified the importance of mode and structure in the interrelation of finger and number representations , with space and base being particularly relevant in this context . in terms of space , recent studies indicate a reliable influence of finger - based representations on the spatial representation of number magnitude . for individuals ( fischer , 2008 ) and cultures ( lindemann et al . , 2011 ) that start counting predominantly with their left hand  that is , associating small numbers with the left a successful and functioning spatial representation of numbers in children is associated with more elaborate calculation skills ( e.g. , bachot et al . , 2005 ; booth and siegler , 2008 ) . thus , an indirect influence of embodied finger - based representations of numbers on general numerical / arithmetical competencies can be expected ( gracia - bafalluy and nol , 2008 , see also fischer and brugger , 2011 for a review ) . second , with respect to the representation of base the german and many other , but not all , finger counting systems are so - called sub - base 5 systems . in sub - base 5 systems , numbers such as 7 are always coded as 5  +  2 ( i.e. , one whole hand and two more fingers ) , but never as , for instance , 4  +  3 ( e.g. , brissiaud , 1992 ) . moreover , the finger symbol for 2 within the finger symbol for 7 is identical to the finger symbol for the number 2 . interestingly , this structural ambiguity seems to influence the processing of symbolic arabic digits : specific sub - base 5 effects have been observed for deaf signers ( iversen et al . , 2004 , 2006 ; domahs et al . , 2010 ) , children ( domahs et al . , 2008 ) , and even healthy adult participants ( domahs et al . , 2010 ) . in the latter study , sub - base 5 effects in a magnitude comparison task were found to be more pronounced for german as compared to chinese adults . importantly , the german , not the chinese , finger counting system involves a sub - base of 5 . in the chinese system numbers between 6 and 10 are coded symbolically using only one rather than two hands . finally , in the study by domahs et al . ( 2008 ) , primary school children were required to solve simple ( sum  <  10 ) or complex ( sum  >  10 ) addition problems . importantly , the probability that numbers differing from the correct result by 5 , and thus by one hand , were produced as erroneous responses was reliably higher than expected on the basis of their distance to the correct result . this increased probability of wrong - by-5 errors is interpreted as a direct influence of the structure of finger - based representations on mental arithmetic involving symbolic arabic digits ( see also klein et al . , 2011 for sub - base 5 effects in addition in adults ) . in summary , the above - reviewed evidence indicates that there is a functional link between both the spatial layout as well as the base structure of finger counting systems and numerical / arithmetical competencies . therefore , neurocognitive researchers conclude that successful finger counting and finger - based arithmetic serve as building blocks for later numerical / arithmetical development and thus should be taught early in kindergarten and primary school . ( 2008 ) is also relevant to the direction of embodied finger - based influences . generally proficient finger gnosis and finger counting / calculation abilities are regarded as beneficial for numerical / arithmetic development in the neurocognitive literature ( see above ) . however , the data of domahs et al . ( 2008 ) suggest that this may only be part of the story : driven by the sub - base 5 structure of the finger counting system , the probability of specific split-5 errors is increased and finger - based representations are seen as the reasons for these errors . thus , in this specific case finger - based representations are not beneficial but detrimental instead  a view held prominently in the literature of mathematics education research presented in the following section . research in mathematics education assumes that young children begin performing calculations by counting , mostly using their fingers ( schipper , 2005 ) . nevertheless , a problem may arise when first elements of numeracy anchored on finger counting are perpetuated to the point of hindering the necessary passage to an understanding of numerosity , of operations , and of computational strategies . krauthausen and scherer ( 2001 , see also padberg and benz , 2011 ) report several findings that reveal the tension between the relevance of counting on the one hand and the problems this may cause on the other : counting is a fundamental competency . however , persistent use of this strategy alone may lead to severe problems with computational tasks . weaker children have trouble generating computational strategies from finger counting . in the long run , children who only use counting strategies tend to obtain fewer correct results than those who also use other computational strategies . mathematics education research recommends fostering mental numerical representations so as to induce children to abandon finger counting at the end of first or beginning of second grade , at the latest . more precisely , this entails shifting from finger counting to performing computations with the help of concrete structured representations and finally , to base computations on abstract mental representations of numbers ( e.g. , floer , 1995 ; kaufmann and wesselowski , 2006 ) . at the end of these phases numbers should no longer be represented as sequences of single units ( e.g. , fingers ) , but as decomposable into larger entities . in fact , the failure to abandon finger - based representations is seen as one possible reason for children s computing errors in second grade . the goal in current mathematics education is to ensure that children understand their computations and acquire computational flexibility . one priority is that early counting should implicitly convey mathematical features , such as the associative and commutative properties of addition and multiplication . for example : these examples illustrate that adequate decompositions and compositions of numbers become a basis for flexible calculation , where , for example , 8 can be seen as 3  +  5 , 7  +  1 , or as 5  +  3 . making conscious use of these decompositions this automatism should not be acquired through memorization but rather emerge through children s handling of representations both enactively and mentally ( wessolowski , 2010 ) . most frameworks for early arithmetic in mathematics education are characterized by a debate between the positions of different representation methodologies ( maier , 1990 ; butterworth et al . , 2011 ) . accordingly , it is questioned whether such decompositions can be developed through mere use of one s fingers ? some mathematics educators do propose using fingers not only for sequential counting but also for representing numbers ( lorenz and radatz , 1993 ) . schipper ( 2005 , 2009 ) describes exercises with fingers for subitizing and quasi - subitizing numbers up to 10 ( see also eckstein , 2011 for finger - based strategies for numbers up to 1,000 ) . however , comparable to neurocognitive research it has been observed that the use of fingers for counting and performing numerical operations stresses a cognitive anchoring on 5 and 10 , that is , on sequential strategies that do not foster representations of numbers as cardinal entities ( moser opitz , 2009 ; gaidoschik , 2010 ) . in the end , flexibility with regard to representational changes is considered a core component of performing successful arithmetical operations . yet , in the first phase of this debate methodologies strictly based on counting had a strong impetus ( eckstein , 2011 ) and were practiced for more than four decades . the birth of the  new math  era in the 1960s led educators to promote a methodology whereby counting with fingers was to be exclusively used at an early stage . the new math era stressed the importance of developing a feeling for both cardinality and ordinality rather than for just ordinality . an interest in finger counting re - emerged in the 1990s , especially in connection with dyscalculia . recent studies on early arithmetic , in particular those that bridge mathematics education with cognitive psychology and neuroscience , indicate that the brain contains a special device for making sense of numbers ( e.g. , dehaene , 1997 ; butterworth , 1999 ) . children begin to enumerate objects at an early age , just as they begin to differentiate between colors ( e.g. , wynn , 1998 ; feigenson et al . , 2002 ) . nevertheless , instruction in numerical operations is indispensible for acquiring the basic competency called numeracy ( e.g. , floer , 1995 ) . successful primary school children acquire flexibility for juggling between different types of representations when counting and operating with numbers ; this is not true of dyscalculic children . to foster elementary numeracy in dyscalculic children , educators propose working mainly with one central representational framework , namely , number imaging , both enactively ( i.e. , using their hands to operate with structured materials , such as blocks and cubes ) and iconically ( i.e. , looking at pictorial representations , like dots or icons ) . a typical symptom of dyscalculic children is firmly consolidated sequential counting , often anchored on finger counting . adequate treatment with structured , enactive materials that foster number images , number decompositions and the cardinal aspect of numbers can enhance the understanding of numbers , so that decompositions like 8  =  7  +  1  =  5  +  3  =  3  +  5 are handled more easily . furthermore , the participation of parents is crucial in fostering their children s early computational abilities and in motivating them to successfully implement the aforementioned representational changes ( e.g. , mehlhuish et al . , 2008 ) . it is known , for instance , that boys tend to abandon finger counting earlier than girls ( pawelec and kurz - milcke , 2009 ) , apparently because parents are somewhat stricter with boys . the feminist literature tends to view this unequal treatment and corresponding effects as disadvantageous for girls ( carr and jessup , 1997 ; martignon , 2010 ) . there is obviously some discrepancy between neurocognitive and mathematics education communities concerning the beneficial and/or detrimental influences of finger - based counting / calculation strategies on numerical development . on the one hand , much of the neurocognitive literature indicates a functional and beneficial interrelation between finger - based numerical representations and numerical / arithmetical development in terms of an embodied numerosity representation ( e.g. , domahs et al . , 2010 ) . on the other hand , mathematics education research sees finger - based strategies in counting and calculation as a starting point that should be overcome in favor of more elaborate and abstract representations upon which numerical cognition is assumed to operate . basically , the rationale behind the neurocognitive argument is based on correlational associations between different cognitive measures or different brain activations of numerical / arithmetical competencies and indicators of finger - based representations . to our knowledge , in the neuorocognitive literature , there is only one intervention study that has trained finger gnosis and showed transfer effects ( gracia - bafalluy and nol , 2008 ) . in contrast , the view held by mathematics education researchers is often falsely drawn on the observation that children using finger - based back - up strategies show poor numerical / arithmetical performance . both views are problematic : in first case causal conclusions may be drawn on correlational data , while in the latter case it is unknown whether children exhibit poor numerical / arithmetical competencies because they still use their fingers or whether they use their fingers because this is their only available cognitive strategy . despite such methodological constraints , the current state of the art in both neurocognitive as well as mathematics education research suggests that the question whether or not finger - based counting and/or calculation strategies are beneficial may be too broad to be answered definitively at the moment . moreover , differences may arise with the differing presuppositions employed by neurocognitive and mathematics education researchers . mathematics education research suggests that reliance on finger - based representations should be overcome and replaced by more abstract numerical representations by the end of first grade to prevent detrimental influences . neurocognitive research , however , predicts that finger - based representations influence number processing and arithmetic even in numerate adults as an additional , and sometimes helpful , embodied representation ( i.e. , without excluding the role of other representations , such as place - value representation ) in arithmetic development . moreover , in contrast to mathematics education research , the neurocognitive perspective does not consider fingers as just another external material for learning to count and/or to calculate ( like blocks or marbles , for example ) . instead , based on the concept of embodied cognition , finger - based representations are considered to be a natural numerical representation , which is firmly grounded on sensory - bodily experience , and prevails even when more abstract or conceptual representations are built up . finger - based representations may be more beneficial for some subgroups of children , for instance less skilled children , as a multisensory experience that helps build abstract mental representations . furthermore , finger - based representations may be particularly useful for specific tasks but not for others . for example , finger - based representations may be more beneficial in operations involving addition than multiplication , as even most single - digit multiplications exceed the number range possible to code easily by two hands . however , considering that multidigit numbers are processed decomposed into the single digits of units , tens , hundreds , etc . , 2011 for a review ) , which can be represented by fingers , and considering that the same representations of these digits are recruited in single- as well as in multidigit number processing ( e.g. , verguts and de moor , 2005 ) , it is conceivable that influences of finger - based representations may not only be limited to numbers up to 10 . even though neurocognitive and mathematics education research agrees that children make use of finger - based numerical representations , they disagree on the consequences of reliance on such numerical representations . on the one hand , the neurocognitive literature suggests that embodied numerical representations , including finger - based ones , are important in numerical cognition in general ( even present in educated adults , see domahs et al . , 2010 ) . on the other hand , mathematics education research recommends the reliance on external representations , including finger - based ones , only as an aid in the transition to mental representations of numbers . these are then assumed to underlie adult numerical cognition ( see also rips et al . , 2008 , for the development of number concepts ) . in sum , the different views clearly show that there is a lack of systematic communication between the two disciplines . further , the theoretical postulates and assumptions arising from the two different fields need to be addressed . to remedy this situation , interdisciplinary discourse between neurocognitive science and mathematics education is urgently needed . the authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest .
children typically learn basic numerical and arithmetic principles using finger - based representations . 
 however , whether or not reliance on finger - based representations is beneficial or detrimental is the subject of an ongoing debate between researchers in neurocognition and mathematics education . from the neurocognitive perspective 
 , finger counting provides multisensory input , which conveys both cardinal and ordinal aspects of numbers . 
 recent data indicate that children with good finger - based numerical representations show better arithmetic skills and that training finger gnosis , or  finger sense ,  enhances mathematical skills . 
 therefore neurocognitive researchers conclude that elaborate finger - based numerical representations are beneficial for later numerical development . 
 however , research in mathematics education recommends fostering mentally based numerical representations so as to induce children to abandon finger counting . 
 more precisely , mathematics education recommends first using finger counting , then concrete structured representations and , finally , mental representations of numbers to perform numerical operations . taken together 
 , these results reveal an important debate between neurocognitive and mathematics education research concerning the benefits and detriments of finger - based strategies for numerical development . in the present review 
 , the rationale of both lines of evidence will be discussed .