Cogn Process DOI 10.1007/s10339-016-0756-7
RESEARCH REPORT
Does finger sense predict addition performance? Sharlene D. Newman1
Received: 25 November 2015 / Accepted: 1 March 2016 Ó Marta Olivetti Belardinelli and Springer-Verlag Berlin Heidelberg 2016
Abstract The impact of fingers on numerical and mathematical cognition has received a great deal of attention recently. However, the precise role that fingers play in numerical cognition is unknown. The current study explores the relationship between finger sense, arithmetic and general cognitive ability. Seventy-six children between the ages of 5 and 12 participated in the study. The results of stepwise multiple regression analyses demonstrated that while general cognitive ability including language processing was a predictor of addition performance, finger sense was not. The impact of age on the relationship between finger sense, and addition was further examined. The participants were separated into two groups based on age. The results showed that finger gnosia score impacted addition performance in the older group but not the younger group. These results appear to support the hypothesis that fingers provide a scaffold for calculation and that if that scaffold is not properly built, it has continued differential consequences to mathematical cognition. Keywords Finger gnosia Cognition Number Arithmetic
Handling editor: Martin H. Fischer, University of Potsdam, Germany. Reviewers: Marco Fabbri, Second University of Naples, Italy; Ilaria Berteletti, University of Illinois at Urbana-Champaign, USA. & Sharlene D. Newman
[email protected] 1
Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN 47405, USA
Introduction Mathematical competence, like all of cognition, begins early and has a neurological basis that is itself linked to the active experiences of children. It is a general and well accepted fact that the activities that we engage in have a direct impact on brain development and future cognitive processing (Greenough et al. 1987). This is particularly true of children due to the rapid neural development that takes place. Here, we focus on finger processing and its relationship to mathematical competence. Because finger use as well as finger sense has been shown to positively predict mathematical achievement in children (Fayol et al. 1998; Noel 2005; Chinello et al. 2013; Penner-Wilger et al. 2007, 2008), it is important to understand its precise role in mathematical cognition. This is especially important because several studies (Penner-Wilger et al. 2007, 2008; Fuson et al. 1982; Fuson 1988; Butterworth 1999a, b, 2005) suggest that finger processing may play a role in setting up the neural networks on which more advanced mathematical computations are built. The relationship between fingers and number has received a great deal of attention recently. It has been assumed that fingers play a significant role in the development of a mature counting system (Fuson et al. 1982; Fuson 1988; Butterworth 1999a, b, 2005). There are a number of hypotheses to account for the role of fingers in number processing: they are a memory aid during counting (Fuson et al. 1982); they aid in understanding cardinality (Fayol and Seron 2005); in the development of the one-toone correspondence principle (Alibali and DiRusso 1999), among others. Additionally, it has been suggested that finger counting habits may influence the way numbers are represented and processed (Pesenti et al. 2000; Zago et al. 2001; Fias and Fischer 2005; Di Luca et al. 2006; Fischer
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2006; Domahs et al. 2010; Newman and Soylu 2014; Sato et al. 2007). Furthermore, there has been some suggestion that finger sense may play a role in arithmetic as well as number processing. A number of studies have confirmed Gerstmann’s (1940) findings of an association between finger agnosia and arithmetic. For example, Reeves and Humberstone (2011) demonstrated that finger sense changed between the ages of 5 and 7 and that those changes were related to finger use in arithmetic computation, suggesting an important role for finger sense in arithmetic. Additionally, Fischer and Brugger (2011) hypothesize that fingers are important for setting up the space number associations which have been shown to be extremely important in mathematical cognition from magnitude processing to calculation. These studies along with others demonstrate the importance of fingers in mathematical cognition. However, the underlying mechanism that supports this relationship is unclear. There is a growing body of research that suggests the use of concrete materials aids classroom learning, particularly in math (Suh 2007; Thompson 1994; Fuson 1990; Fuson and Briars 1990). The use of manipulatives is thought to help students ‘‘think, reason, and solve problems’’ (Burns 1996, p. 48). They are an additional resource for helping students construct ideas, giving meaning to mathematical concepts and subsequently facilitating performance (Sternberg and Grigorenko 2004). Fingers are, in essence, a manipulative that is always present and that has a well-connected internal representation. Fingers are a part of the body, always present; they do the manipulating (with other manipulatives such as counting counters or pieces of fraction pies). They allow for physical interaction with number (e.g., they can be moved) which has been shown to enhance memory and understanding (Glenberg et al. 2004) and because of their constant availability, experiences with fingers in number contexts are likely to far exceed other concrete aids. For example, in a recent study using the iCub child-like robot, it was found that number knowledge was more efficiently learned when number words are learned with finger counting as opposed to without finger counting (De La Cruz et al. 2014). If a similar mechanism is at play for children, fingers may be a good tool to aid number learning. In order to examine the impact of finger processing on mathematical competency the current study examined the relationship between finger sense, arithmetic performance and general cognitive ability in a group of children between the ages of five and twelve. The first goal of the current study was to assess how cognitive ability interacts with addition performance and finger sense. The general cognitive abilities examined included phonological processing, short-term and working memory as well as verbal and nonverbal IQ. These measures were chosen because they have
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previously been shown to be correlated with mathematical performance (De Smedt et al. 2009; Jordan et al. 2010; Passolunghi and Lanfranchi 2012; Passolunghi and Siegel 2004; Passolunghi et al. 2007; Robinson et al. 2002; Imbo and Vandierendonck 2007). However, few studies have explored how all of these factors—age, finger sense, general cognitive ability and mathematical ability—interact. Based on previous work, it was expected that age would correlate with all factors, but that finger sense and cognitive ability would have independent relationships with arithmetic performance. The second aim explored here was to test the hypothesis that fingers provide a ‘‘natural scaffold for calculation’’ (Jordan et al. 2008). In other words, fingers may provide the support necessary to build calculation skills; it is foundational to mathematical competency. To explore this hypothesis, the relationship between age, finger sense and addition performance was further explored. As Reeves and Humberstone (2011) reported, finger sense is still developing during the early elementary school years; however, it may be expected to be somewhat stable at older ages. Along with the development of finger sense, addition skill is also being developed in younger children. A recent study by Berteletti et al. (2015) found a relationship between subtraction performance and finger related activation in somatosensory cortex. They argued that children with lower performance engaged finger processing areas more than children with higher performance. Because age was correlated with subtraction accuracy, this differential involvement of finger processing may be a function of development. Therefore, studying both a younger (5–8 years old) and older group (9–12 years old) will allow for the examination of the importance of the finger scaffold in both the early learning of addition as well as the later instantiated addition skill of older children. Based on previous findings, it was expected that younger children would show a stronger relationship between finger sense and arithmetic processing performance than older children who may be expected to have developed more mature retrieval strategies.
Methods Participants Seventy-six children (5–12 years of age, M = 8.67 ± 2, 36 males) participated in the study for pay. Participants all attended local schools and had no history of neurological or psychiatric disorders or diagnosed dyslexia or dyscalculia as reported by parents. Written informed consent was obtained from parents and assent from each participant, as approved by the Institutional Review Board of Indiana University, Bloomington.
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Measures Finger gnosia The finger gnosia test is a standard assessment that dates back to Benton (1955). During the test, participants sat with both hands palm down on the table in front of them. They were instructed to close their eyes and to keep them closed during the entire procedure (eyes were checked regularly). There were two phases of the test. During the first phase, the experimenter, with a pointer, touched a single finger of the left hand in a pre-determined order, touching each finger (5 trials). After each finger touch the subject was instructed to indicate by moving the corresponding finger of the other hand (1 point per trial). During the second phase, the experimenter touched a combination of two fingers in succession (5 trials) and the participant was instructed to indicate which two fingers were touched and the order that they were touched by moving the corresponding fingers of the other hand (2 points per trial; 1 point for the correct fingers and 1 point for the correct order). The score was the total number of points earned divided by the total possible points. There were 15 possible points.
they were required to read aloud letter combinations that are phonically consistent patterns in English but are nonwords or low frequency words. The score was the percent correct. Vocabulary The vocabulary subtest of the Wechsler Intelligence Scale for Children was administered as a test of verbal IQ. This test measures verbal fluency, concept formation, word knowledge and usage. It is an untimed test in which participants are read a word and are asked to define it. The score was the percent correct. Matrix reasoning The matrix reasoning subtest of the Wechsler Intelligence Scale for Children was administered as a test of non-verbal IQ. This test measures visual processing and abstraction and spatial perception. Children are shown colored matrices or visual patterns with something missing. The child is then asked to select the missing piece from a range of options. The score was the percent correct. Timed addition test
Handedness The Edinburgh Handedness Inventory (Oldfield 1971) was administered to each participant. Each question was read to the participant, and they demonstrated how they would perform the task. For example, for the question which hand do you use to throw a ball? The participant would be encouraged to simulate throwing. All subjects were righthanded.
Participants were presented with 40 single-digit addition problems and given 1 min to complete as many as they can. The problems were organized with easy problems presented first with problems becoming more difficult, with difficulty being defined by the size of the operands (all 9 or less). Therefore, the largest magnitude of the answers was 18. No tie problems were presented (e.g., 2 ? 2). The total percent correct (out of 40) and the number of problems attempted were examined.
Digit span The forward (FDS) and backward digit span (BDS) tasks were administered to assess short-term and working memory, respectively. For both, a series of digits were read to the participant at a constant pace starting with two digits and increasing by a single digit until failure to recall occurs twice. For the FDS, participants were told to repeat the digits in the order read. For the BDS, they were told to repeat the digits in the reverse order read. The score was the percent correct. Word attack The word attack (Woodcock et al. 2001) task was administered to assess phonological skills. The initial items require participants to produce the sounds for single letters. Afterward, difficulty increases. For the remaining items
Results No significant relationship was found between any of the factors and gender or handedness. As a result, neither was further considered. Multiple regression analysis A stepwise multiple regression analysis using the PHREG procedure in SAS was performed. The independent variables examined—age, word attack, finger gnosia, FDS, BDS, vocabulary and matrix reasoning—were entered into the analysis to determine which predicted performance on the timed addition task. The stepwise selection process resulted in a model with four explanatory variables—age, word attack, FDS and BDS. The model with these four
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variables explained 61 % of the variance [F(4,71) = 28.31, p \ 0.0001, MSE = 0.561, R2 = 0.6147]. Effect of age Because age is expected to be responsible for a large portion of the variance in addition performance, a second analysis designed to explore the impact of age on the relationship between finger gnosia and addition performance examined older and younger children separately. The subjects were divided into two groups based on age: older group (N = 42; M = 10.2 ± 1.01), younger group (N = 34; M = 6.7 ± 1.1). A stepwise multiple regression analysis was performed on each group separately. For the older group vocabulary, matrix reasoning and forward digit span significantly predicted addition performance [F(1,40) = 5.09, p \ 0.005; accounted for 29 % of the variance]. For the younger group, age and word attack predicted addition performance [F(1,32) = 17.29, p \ 0.001; accounted for 54 % of the variance]. Finally, because one of the primary aims of the study was to explore the relationship between finger sense and arithmetic, this relationship was further explored. First, a correlation analysis was performed with age partialled out (see Fig. 1). In the full dataset the correlation between finger gnosia and addition performance was significant before controlling for age (r = 0.36), after controlling for age the correlation was only trending (r = 0.2; Table 1). However, the two factors were correlated in the older group (r = 0.32, p = 0.04) but not the younger group (r = 0.17, p = 0.35). Second, to further explore the interaction between age and addition performance, a regression analysis was performed with only finger gnosia entered as a predictor. While this is an atypical analysis, it was Table 1 Correlation matrix controlling for age (the top number is the r value bottom number p value)
performed to simply explore the data further. Finger gnosia was found to significantly predict addition performance for the older group [F(1,40) = 4.41, p \ 0.05; accounted for 10 % of the variance] but not the younger group [F \ 1; accounted for 3 % of the variance].
Discussion The primary aim of the current study was to explore whether finger sense as measured by the finger gnosia test contributes to arithmetic computation performance in children. The impact of finger sense on addition performance is currently not well understood. Studies have shown that it predicts later mathematical competency in children; also imaging studies have found evidence of finger processing in adults and children during calculation and when viewing numbers (Berteletti et al. 2015; Tschentscher et al. 2012). Additionally, a number of studies have shown a relationship between arithmetic performance and general cognitive ability including working memory and phonological processing. The findings presented here provide further insight into the relationship between these factors and how they impact arithmetic performance in young children. First, addition performance was found to be predicted by general cognitive ability, particularly language processes and short-term memory. Second, the predictors of addition performance varied as a function of age group. Finally, the impact of finger sense varied as a function of age group with it having no predictive power in the younger group and a modest impact in the older group. Previous studies have attempted to link finger processing during arithmetic to finger counting strategies (Imbo
Pearson correlation coefficients, N = 76 Prob [ |r| under H0: q = 0
WA FDS BDS GNOSIA ADD VOCAB
WA
FDS
BDS
GNOSIA
ADD
VOCAB
MATRIX
1
0.389
0.4162
0.056
0.252
0.179
0.214
0.0006
0.0002
0.63
0.03
0.13
0.066
1
0.244
-0.018
-0.121
0.083
0.078
0.035
0.88
0.30
0.48
0.51
1
0.066
0.312
0.225
0.396
0.58
0.0064
0.053
0.0004
1
0.192
0.106
0.238
0.0992
0.37
0.039
1
0.244
0.23
0.035
0.041
1
0.295 0.01
MATRIX
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1
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and Vandierendonck 2008; Reeves and Humberstone 2011). Here the older children (ages 9–12) were not expected to use finger counting during simple, single-digit addition. It is actually thought that finger use during addition is an indication of mathematical deficits in older children while finger use in younger children is helpful. For example, Jordan et al. (1992, 1994) found that finger use was linked to higher accuracy on number combinations in kindergarten and first-grade students. Those students who rarely spontaneously used finger counting had poorer performance and typically came from low-income households. However, by second grade there was a shift in that better mathematical performance was associated with less finger use (Jordan et al. 2008) and a greater reliance on retrieval strategies. These findings may reflect a developmental trajectory in which finger counting sets the stage for more advanced skills, but once those skills are acquired finger counting is no longer needed, so that children who are still finger counting are the children who have not acquired those more advanced grade level skills. By analogy, in a kindergartener, invented spelling (writing ‘‘kitten’’ as KTTN) is a sign of precocity and readiness to learn to read; the same behavior in a 4th grader is a negative indicator of age-appropriate literacy (Treiman and Zukowski 1991). Again, finger counting during addition performance was not assessed here. However, it was found that finger sense better predicted performance in the older children than the younger children. This suggests that poorer finger gnosia scores in older children is a better indicator of mathematical computation deficits than in younger children. Because both addition skills and finger sense are still developing in the younger group, finger sense may not be a good predictor of addition performance at younger ages. However, both skills should be developed within the older group (9–12 years old), making the relationship between these two factors more evident. As mentioned in the introduction, there is a growing body of research focused on the association between finger sense and numerical and mathematical competency, perhaps through the discrimination of numerical quantities (Halberda and Feigenson 2008; Mazzocco et al. 2011). This previous research along with the previous studies that demonstrate that finger sense at younger grades predict math performance later (Fayol et al. 1998; Noel 2005; Penner-Wilger and Anderson 2013) support the hypothesis that poorer finger sense in older children is an indication of significant mathematical deficits. The relationship between finger sense and finger counting has not been well studied. Finger sense has been found to be correlated with number knowledge which in turn is essential to mathematical performance and finger counting appears to be an important and possibly necessary part of early mathematical calculation skill development.
Fig. 1 Correlation between finger gnosia score and addition performance
But how finger sense and finger counting relate to each other has not been clearly articulated. One possibility is that finger counting depends on finger sense. Reeves and Humberstone (2011) suggest that finger counting and finger sense, and their relations to numerical and mathematical processing, co-develop but give no information regarding the causal direction. Finger sense may positively impact the use of finger counting in young children via two mechanisms. The first possible route is via motoric processing. For example, better finger sense allows for better fine motor skills which may be necessary for both finger counting and for counting small entities (like rows of counters). Finger sense and/or these activities of counting fingers and things may also foster an increased ability to individuate the fingers which in turn leads to better finger counting. In any case it appears that if finger sense is not developed by a particular age, it can have detrimental effects on arithmetic performance later. Together this suggests that finger sense is an important factor in the development of arithmetic skills. However, the mechanism that links finger sense to arithmetic is still not understood. Another, none contradictory, explanation for the differential effect of finger sense on the two age groups is that the younger and older groups are using different strategies to solve the addition problems. The younger children are likely using costly counting procedures that heavily rely on working memory and phonological processing. The use of such strategies may be related to their under developed finger processing skills as discussed above. As such, for younger children general cognitive ability would be expected to play a larger role. Conversely, older children may rely more heavily on more automatic procedures whether they are retrieval from long-term memory or automated counting procedures (Barrouillet and Thevenot 2013). It should be noted that there is some controversy as to whether finger processing is involved in these automatic
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procedures. Barrouillet and Thevenot (2013) argued that counting strategies cannot be ruled out as a mechanism for processing small operand, single-digit arithmetic problems, even in adults. Additionally, studies have demonstrated automatic recruitment of finger representations when processing numerical information (Di Luca et al. 2006; Di Luca and Pesenti 2008; Badets and Pesenti 2010; Badets et al. 2010). Therefore, it may be that finger sense plays an even larger role in addition for older children than younger children due to the underdeveloped finger sense/use in younger children; and this is what the results presented here demonstrate. These differences in strategy use may be expected to have differing relationships with finger sense. Interestingly, language factors predicted addition performance here as it has been found to do in previous studies. Language has been found to be an important predictor of arithmetic performance, particularly phonology (De Smedt et al. 2010; Simmons and Singleton 2008), and meaning-related skills (Aunola et al. 2004). For example, Geary (1993) hypothesized that developmental dyscalculia is due to a difficulty in representing and retrieving phonological information. Also, Krajewski and Schneider (2009) suggest that phonological awareness allows for the differentiation of individual words in a number sequence which supports arithmetic problem-solving. Here, different aspects of language was found to predict addition performance in the younger and older groups with phonological processing predicting performance in the younger group and vocabulary in the older group. These differences may be related to differences in strategy use with phonological processing being more related to short-term memory processing and vocabulary to long-term memory processing. Limitations There are some limitations of the current study that should be noted. Even though the test is a standard assessment (Benton 1955), the finger gnosia task used may not be sensitive enough to adequately assess finger sense in young children. In order to correctly respond during the task not only does the participant have to ‘‘sense’’ the touch but she also must generate an internal representation of her hand(s) and fingers and then map this internal representation of the hand and fingers onto another representation, either their opposite hand or a picture of a hand. Thus, finger sense tasks like the one used here are measuring more than just the ability to discriminate the finger touched, they also measure the ability to create an internal representation and then map it onto another representation which requires a host of processes including working memory and spatial processes. In fact, there is a strong correlation between finger sense and matrix reasoning (a test of non-verbal reasoning) scores—children with high
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finger sense also have higher matrix reasoning scores. Because these mapping processes may not be developed fully in the young group, the test may not be appropriate for younger children. A second limitation is the age range used. Although there was an older and younger group of children, having a narrower age range and larger N’s would provide a clearer picture of the relationship between age, finger sense and addition.
Conclusions The current study demonstrates that finger sense does indeed contribute to arithmetic performance; however, its direct impact varies with age. Finger sense impacted the performance in the older group but not the younger group. It may be that their well-developed finger sense has laid the foundation for mathematical skills, possibly by facilitating mapping processes that now allow for fact retrieval to be a more efficient strategy than the laborious finger counting strategy. It may also be that variance in finger sense in the older group is more meaningful because it may differentiate individuals who have finger sense deficits as well as arithmetic processing deficits. One hypothesis that deserves greater consideration in future studies is that fingers provide a ‘‘natural scaffold for calculation’’ (Jordan et al. 2008) such that it may lay the foundation for future mathematical as well as spatial skills. Support for this hypothesis comes from studies showing that calculation skills may actually derive from finger sequencing and from neuroimaging studies that show that finger and calculation skills have overlapping neural bases (Ardila 1993; Berteletti et al. 2015). The results here also seem to support this hypothesis. Additionally, the results presented here demonstrate that low finger gnosia scores in older children may indicate significant arithmetic deficits, suggesting that if this ‘‘natural scaffold’’ is not properly developed early that it can have important consequences later. Further research exploring those older children with low finger sense as well as the younger children with high finger sense may be important to understanding the necessity of finger processing to mathematical cognition. Acknowledgments This research was funded by a Grant from Indiana University (FRSP). I would like to thank Roy Seo, Jessica Denton, Galen Hartman, Priyanka Ghosh and Taylor Hurst for the assistance with data collection. Compliance with ethical standards Conflict of interest of interest.
The authors declare that they have no conflict
Informed consent Informed consent was obtained from all individual participants included in the study.
Cogn Process Ethical approval All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.
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