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(BeishuizenMMental strategies and materials or models for addition and subtraction up to 100 in Dutch Second GradesJournal for Research in Mathematics Education19932429432310.2307/749464)
BeishuizenMMental strategies and materials or models for addition and subtraction up to 100 in Dutch Second GradesJournal for Research in Mathematics Education19932429432310.2307/749464BeishuizenMMental strategies and materials or models for addition and subtraction up to 100 in Dutch Second GradesJournal for Research in Mathematics Education19932429432310.2307/749464, BeishuizenMMental strategies and materials or models for addition and subtraction up to 100 in Dutch Second GradesJournal for Research in Mathematics Education19932429432310.2307/749464
Background: The objective of this study was to scrutinize number line estimation behaviors displayed by children in mathematics classrooms during the first three years of schooling. We extend existing research by not only mapping potential logarithmic-linear shifts but also provide a new perspective by studying in detail the estimation strategies of individual target digits within a number range familiar to children. Methods: Typically developing children (n = 67) from Years 1-3 completed a number-to-position numerical estimation task (0-20 number line). Estimation behaviors were first analyzed via logarithmic and linear regression modeling. Subsequently, using an analysis of variance we compared the estimation accuracy of each digit, thus identifying target digits that were estimated with the assistance of arithmetic strategy. Results: Our results further confirm a developmental logarithmic-linear shift when utilizing regression modeling; however, uniquely we have identified that children employ variable strategies when completing numerical estimation, with levels of strategy advancing with development. Conclusion: In terms of the existing cognitive research, this strategy factor highlights the limitations of any regression modeling approach, or alternatively, it could underpin the developmental time course of the logarithmic-linear shift. Future studies need to systematically investigate this relationship and also consider the implications for educational practice. Background Waheed [6] propose the tendency of skilful estimators Estimation is a required skill for everyday life. Numeri- to have a better conceptual understanding of mathe- cal estimation skills are an example of what Piaget [1] matics, as well as better counting and arithmetic skills. described as logico-mathematical knowledge. While Pia- Here we provide an investigation of numerical estima- get did not carry out numerical estimation tasks specifi- tion skills at the beginning of primary school. We used cally he considered logic-mathematical knowledge to be a number range familiar to the children and analyzed the mental relationships between and among objects/ dependent variables for each target digit in depth. This representations. Understanding the development of approach goes beyond studying a potential logarithmic- numerical estimation is particularly important to psy- linear representational shift in estimation and allows chologists and educators, as several studies indicate the further insight into the development of children’sesti- mation strategies. benefits of advanced estimation skills. For example, many studies (e.g. [2-5]) have determined a strong, posi- Several studies (e.g. [2,5,7-11]) have investigated devel- tive correlation between the accuracy of numerical esti- opmental changes in numerical estimation in school- mation and standardized tests of mathematics aged children. Estimation requires the translation achievement. Furthermore, LeFevre, Greenham and between alternative quantitative representations. For example presenting a child with a number and asking * Correspondence: sl.white@qut.edu.au; ds377@cam.ac.uk them to position it on a number line can be described University of Cambridge, Department of Experimental Psychology, Centre as a translation from a numerical to spatial representa- for Neuroscience in Education, Downing Site, CB2 3EB, UK tion [5]. Full list of author information is available at the end of the article © 2012 White and Szűcs; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 2 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 Much of the research into numerical estimation (here- count to 100. While this might question the validity of after: estimation) has focused on how magnitudes might the logarithmic-linear claim, Berteletti et al. [7] utilized be mentally represented and how this representation 1-10, 1-20 and 0-100 number lines in an estimation task changes with maturity. It is assumed that estimation is and found evidence for the logarithmic to linear repre- based on internal models of magnitudes. Two models sentational shift. Of the preschool aged children (3.5-6.5 attempt to describe the internal representation of num- years) who participated in the 0-100 task all groups dis- played a logarithmic dominant representation. For the ber, namely the accumulator (linear) model [12] and the 1-10 task the youngest group (approximately 4 years) logarithmic model [13]. The accumulator model sug- was best fit by both models, the middle and older gests that magnitudes are represented linearly and that the accuracy of this mental representation decreases groups (approximately 5 and 6 years respectively) were with increasing magnitude [8]. The variability of estima- now demonstrating a linear preference for this reduced tions in relation to the magnitudes estimated remains in number scale. In the 1-20 estimation task the youngest a constant ratio; this is termed ‘scalar variability’ [14]. group was best fit by a logarithmic model, whereas the Dehaene [13] argued that quantities are represented in a middle and oldest groups were equally well represented logarithmic fashion. This mental representation results by logarithmic and linear models. These findings in an exaggeration of the distance between small num- [2,5,7,10] reinforce the belief that child estimation beha- ber magnitudes in comparison to distances between viors demonstrates a logarithmic phase prior to linearity large number magnitudes. In relation to the core sys- and that this transition is evident first with familiar and tems of number; namely the small number system for then unfamiliar number contexts. small number enumeration and the approximate num- Berteletti et al. [7] acknowledges that the exact path ber system (ANS) for larger numerosities [15], it is the that leads from logarithmic to linear representations is approximate number system that would encode the still unclear. Siegler, Thompson and Opfer [18] argue it numerosities in an estimation task. Specifically, Halberda to be a process important to education. Thompson and andFeigenson [16]foundthatANS acuity wasstill Siegler [11] interpret the flexibility/variability of beha- developing in children aged 3-6 years and speculated viors within Siegler’s [19] overlapping waves theory; that sharpening of the ANS was not complete until late whereby representations and strategies are used selec- in adolescence. Furthermore, Berteletti et al. [7] argues tively when most effective and that individual choice of for an approximate number system that is a logarithmic learned (external) mechanisms contribute to numerical representations. In various adult studies of numerical representation first, with numerate children and adults processing (e.g. [20-23]), it has been determined that acquiring greater precision, and thus a linear representa- tion. This shift to a linear representation is evident first many different strategies can be used to solve a single with familiar number contexts and then subsequently problem, whether that be estimation, multiplication or with less familiar number ranges [17]. equation solving. Dowker [20] in her study of expert Many of the developmental studies have used pure mathematicians found that individual strategy selection numerical estimation with large number scales (e.g. 0- for the same problem can vary between trials. Smith 100 and 0-1000: [2,5,10]). On a 0-100 number line, this [24],inastudywith schoolagedchildrenand adoles- research ([2,5,10]) purports that both representations cents, investigated reasoning with rational numbers are evident and pinpoints a logarithmic-linear shift at (expressedasfractions)andfoundthathigherlevel around Grade 2 (7-8 years). Booth and Siegler [2] competence required rich and diverse knowledge, declare a linear best fit for 74% of Grade 2 children in numerically specific and invented strategies, as well as their study; with the remainder of participant behaviors those general strategies learned through educational being best represented by a logarithmic model or in a instruction. Huntley-Fenner [25] attributes the variability minority of cases, an exponential model. With younger children demonstrate in estimation tasks as being the participants the logarithmic-linear distinction is less result of reduced knowledge of estimation strategies. clear; for example in Siegler and Booth [5], 5% of kin- While the influence of individual strategies was men- dergarteners produced a series of estimates better char- tioned in the numerical estimation child studies of Sieg- acterized by the linear than the logarithmic model and ler and Booth [5] and Siegler and Opfer [10], Barth and 45% were best modeled by a logarithmic representation. Paladino [26] and Ebersbach et al. [8] discussed the use It could also be argued that these results, particularly of estimation strategies and proposed the need for an alternative modeling approach in order to capture child those of children in kindergarten, could be influenced behaviors. The alternative models were found to factor by the unfamiliarity of the number ranges. Verifying this in the use of a half-way reference point [26] and num- proposition Ebersbach et al. [8], in a similar 0-100 task, found that only 17% of kindergarteners and 38% of ber familiarity [8]. Ebersbach et al. [8] posited a model Grade 1 children who participated in the study could composed of two linear segments, with the change point White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 3 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 an indicator of number familiarity. Thompson and This study further examines the putative logarithmic- Opfer [17] found that the segmented model change linear shift of mental representations for the familiar point varied depending on number scale and questioned number range (0-20) with children in Years 1, 2 and 3; the claims of Ebersbach et al. [8]. Importantly, Opfer, with a focus on determining if external factors (i.e. strat- Siegler and Young [27] still maintain the validity of the egy) may be at play during the estimation of specific tar- transition from logarithmic to linear representations, get digits. Using the most established logarithmic and linear regression analysis of Berteletti et al. [7], Booth and caution that the fit of power models, as used by and Siegler [2]; Siegler and Booth [5] and Siegler and Barth and Paladino [26], could be influenced by the Opfer [10] as a foundation, this research extends the noise created during averaging procedures. Using an eye-tracking methodology, along with gaze pattern and analysis systematically to investigate estimation beha- fixation analysis Schneider et al. [28] found that children viors of individual target numbers. First, a logarithmic in the first years of school do spontaneously use orienta- or linear best fit model will be determined for individual tion points (external markers) to support spatial-numer- data and used as a reference point to the Berteletti et al. ical processes such as numerical estimation. In light of [7] study that also utilized a number line with a maxi- these studies ([17,27,28]) we chose to focus on the mum of 20. Second, the residuals to each target digit established logarithmic and linear (not segmented) mod- from both the logarithmic and linear models will be els and have only included the power model at the coef- investigated to identify how well the two models repre- ficient of determination analysis, with subsequent sent the estimates of specific numeric values. Third and analyses focusing on individual target digits which were finally, the accuracy of estimation for individual target likely to have been positioned with the aid of an exter- digits will be scrutinized, without any regression model- nal factor or strategy. ing. It is proposed that employing a familiar number A common strategy would be to utilize the number range will increase the likelihood of strategy application line itself, providing two external anchor points that and the novel analysis will indicate individual digits could improve estimation accuracy of numbers located which might be the target of selective strategy use as within close proximity to the end numbers (e.g. 97 is 3 suggested by Ebersbach et al. [8] and Thompson and units back from 100). The youngest participants may Siegler [11]. It is hypothesized that the developmental be limited to a counting strategy and may not factor in shift from logarithmic to linear mental representation, the spatial representation, as is their level of concep- after approximately two years of school ([2,5,10]), will also be the transition period where strategy use becomes tual understanding. A more advanced possibility, as evident. Based on existing speculation (e.g. [8,26]), this posited by both Barth and Paladino [26] and Ebersbach et al. [8], could be the application of mental anchor is likely to be located in close proximity to external points, however, such a strategy would require under- anchor points or, in an advanced circumstance, mental standing of the part-whole (proportion) relation. For anchor points which require the division of the number example on a scale of 0-100, 50 could be a mental line into equal portions. For this reason, the present anchor point that arises from the participant dividing study is interested in main effects, but also the interac- the number scale into two equal parts and matching tion between the variables, with a particular focus on this spatial division with the knowledge that 50 is half separate year group behaviors. For example, it may be of 100. As with external anchor points, estimations for particular target digits that the linear and logarith- that occur near these mental partitions are likely to mic models demonstrate the greatest disparity, and that have increased accuracy in comparison to those num- this varies developmentally. It is intended that this infor- bers located a greater distance away from a reference mation will inform subsequent investigations that will point; as suggested earlier, further individual variation seek to determine the various components (e.g. cogni- is likely to be associated with number knowledge. Stu- tive mechanisms, learned strategies)thatcontributeto dies ([29,30]) have given corrective feedback for target the developmental path from logarithmic to linear digits near a particular landmark (150 on a 0-1000 representations and/or to the development of strategy number line) and found that estimates were less accu- use in numerical estimation, and any relationship rate when the target numbers were more distant from between mental representations and strategy use. the landmark. This evidence from feedback studies (e. g. [29,30]) indicates that children can utilize both Methods external and mental anchor points. In the Barth and Participants Paladino [26] study the midpoint was highlighted to Participants were 67 British children from Years 1, 2 participants prior to commencing the task, but this and 3 (Year 1: n = 20, mean age 6.4 ± 0.24 years, 11 prompt was not an inclusion within the present females. Year 2: n = 24, mean age 7.3 ± 0.33 years, 13 investigation. females. Year 3: n = 23, mean age 8.5 ± 0.36 years, 10 White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 4 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 females). All participants performed within one and a be estimated more accurately due to the contribution of half standard deviations of the mean on a brief version external factors, such as learned strategy. rd of the Wechsler Intelligence Scale for Children, 3 edi- Coefficient of Determination tion (WISC-III: Vocabulary, Blocks and Digit Span); This analysis began with fitting linear and logarithmic with no significant difference between the year groups models to the target number estimates, for each partici- for the WISC-III triad (F (2, 64) = 1.592, p = 0.211). pant. Then for individual models (linear and logarith- Written informed consent was obtained from the par- mic) a coefficient of determination (R ) was calculated. ents/guardians of the children in this study. The study Comparing the linear coefficient (R ) and logarithmic Lin obtained ethical approval from the Cambridge Psychol- coefficient (R ), for each child, it could be determined Ln ogy Research Ethics Committee. which model best represented each child’s mental repre- sentation. The coefficient of determination values were Task and procedures entered in a 3 × 2 ANOVA. Factors were: Year (Year 1, The assessment instrument was a number-to-position 2 or 3) × Model (Linear or Logarithmic). (NP) pencil and paper task, where participants were In addition to this analysis, we explored the propor- given a number and had to mark its position on a 160 tion judgment power model adopted by Barth and Pala- mm number line. The number line had the range of 0- dino [26]. On an individual basis, for both 1-cycle and 20 and the target digits were evenly distributed across 2-cycle models, values of R were determined, along the scale with 2, 4, 7, 8, 11, 13, 16 and 17. Two exam- with the parameter b (the exponent determining the ples (3 and 9) on a 0-10 scale were completed together shape of the power function relating psychological to by the participant and researcher prior to commence- physical magnitude). We selected the b with the highest 2 2 ment; this was discussion based to ensure understanding R and subsequently compared the best R for the 1- of the task and did not propose any strategies. During cycle and 2 cycle models (Figure 1)[31]. These findings the testing trials (0-20) no corrective feedback was pro- were interpreted in relation to the logarithmic-linear vided, just encouragement to continue onto the next shift, with b = 1 corresponding to a linear model and item. then the further the value from 1, the closer to a loga- rithmic model. This additional model was then entered Analysis into a separate 3 × 3 ANOVA. Factors were: Year (Year The NP task was assessed in the following way: The dis- 1, 2 or 3) × Model (Linear, Logarithmic or Power). tance from the left end point (zero on the number Model Residuals scale), to the participant marking was measured in milli- Following the approach of Siegler and Opfer [10] the meters. The distances to the line markings were con- target number estimates were tabulated and analyzed at verted to numerical estimates for each target number. a group level. The median score for each year group Mirroring the method employed by Siegler and collea- was used to generate a graph of estimates versus actual gues [2,5,10], the following equation was used to deter- target numbers. The median was selected because it is mine the target number estimates: less affected by outliers, which could occur in this type of task. Each year group graph was then used to calcu- Distance from left end point to marking (mm) × scale of number line late both a linear and logarithmic regression model. The Total length of line (mm) Siegler and Opfer [10] method calculated residuals to the group level median values and entered this into a An example of this calculation if a mark was placed 25 paired-samples t-test. This would only determine if mm from the left end point of a 0-20 number line there significant difference between the residuals of the would be 25 ÷ 160 × 20; which means the target num- two models overall, omitting the variation of model resi- ber estimate equates to 3.125. duals that could occur for individual target numbers. The target number estimates were used in two main Extending the approach of Siegler and Opfer [10], the analyses; regression modeling and estimation accuracy present study calculated residuals to individual target for individual target digits. The regression modeling number estimates of each participant. This allowed the built on the method initially utilized by Siegler and residuals to be entered into a more powerful 3 × 8 × 2 Opfer [10] and two derived measures were used: the repeated measures ANOVA. Factors were: Year group Coefficient of Determination values for individual parti- (Year 1, 2 or 3) × Target Number (2, 4, 7, 8, 11, 13, 16 cipants, as well as Model Residuals for group level mod- or 17) × Model (Linear or Logarithmic). This analysis els. The unique component of the analysis was that of allowed for a more detailed examination of how the Estimation Accuracy. Taking the foundation from Sieg- model residuals varied for each target number and ler and Booth [5], this research gained a more detailed whether this interacted with year group. perspective into the numerical values that are likely to White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 5 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 Figure 1 Prediction proportion judgment cyclic power models as used in Hollands and Dyre [31]and Barth and Paladino [26]. a) 1-cycle model, with no central reference point; b) 2-cycle model, with one central reference point. In both a) and b) b is the exponent in the power function describing the relationship of psychological to physical magnitude. Legend: Green b = 0.1, Aqua b = 0.3, Blue b = 0.5, Red b = 0.7, Black b = 1.0. NB. When b = 1, the function is linear. To add further descriptive detail we followed the conducted on a year group basis in conjunction with the approach of Geary, Hoard, Byrd-Craven, Nugent and percent absolute error; lower standard deviations could Numtee [32] and used the absolute residuals (from the pinpoint the location of an external or mental anchor group level models) to classify all trials as linear or loga- point that was consistently applied by members of a rithmic based on whether the child’s estimate was closer year group. to the predicted value of the linear of logarithmic model (i.e. which model produced the smaller residual). When Results the residuals of the linear and logarithmic models had a Coefficient of Determination difference less than ± 0.4 units these trials were classi- 50% of Year 1, 75% of Year 2 and 74% of Year 3 chil- fied as ambiguous; this value was determined based on dren had higher coefficients of determination for the the distribution of the individual residuals. linear rather than the logarithmic models. The analysis Estimation Accuracy of the coefficient of determination values revealed that The absolute percent error for each target number was the main effect of Model was significant (F (1, 64) = calculated according to the equation: 9.96, p = 0.002, h = 0.129). That is, the linear model (R = 0.87 ± 0.20) explained a greater degree of var- Lin Estimate − Target Number × 100 iance than did the logarithmic model (R =0.84 ± Ln Scale of number line 0.18). There was a main effect of Year (F (2, 64) = If the estimate was determined to be 3.125 and the 10.63, p < 0.001, h = 0.249) because the amount of var- iance explained by either model was significantly lower target number 2, the equation would be |(3.125-2) ÷ 20| for Year 1, than Year 2 (p < 0.001) and Year 3 (p < × 100 to obtain the error of 5.625%. In an attempt to 0.001); there was no significant difference between Years reveal information about estimation accuracy of indivi- 2 and 3. There was no Model × Year interaction. dual target numbers, and thus the potential application As the power model is sensitive to noise [27], some of external and mental anchor point strategies this study participants were discarded from the analysis leaving 51 extends the method to reveal accuracy details about in total. In terms of 1- or 2-cycle models majority of the individual target numbers. This more detailed informa- participants were best represented by a 1-cycle model, tion was maintained and the percent absolute error with b values often very close to 1 (Table 1). Table 1 values were entered into a 3 × 8 ANOVA, with factors: Year group (Year 1, 2 or 3) × Target Number (2, 4, 7, 8, presents the individual power model results in compari- 11, 13, 16 or 17). The Greenhouse-Geisser epsilon (ε) son to the coefficient of determination values (R ) of the correction for sphericity was used in all ANOVAs when- best fit logarithmic and linear models reported in the ever necessary. Reporting indicates the original degrees previous paragraph. The separate 3 × 3 ANOVA of freedom, the epsilon value, followed by the corrected returned similar results. There was a main effect of (more conservative) significance level. All post hoc ana- model and a main effect of year (Model: F (2, 96) = lyses were Tukey HSD tests. In the results section values 11.92, ε = 0.83, p < 0.001, h = 0.196; Year: F (2, 96) = represent mean ± standard deviation, unless otherwise 5.19, p = 0.009, h = 0.177). Overall, the linear model stated. (R = 0.90 ± 0.13) explained more variance than both Lin To further investigate the potential for external and the power (R = 0.86 ± 0.15, p < 0.001) and logarith- Pwr mental anchor points, we sought to explore the standard mic models (R = 0.88 ± 0.11, p = 0.007). The main Ln deviation of estimates as a function of target number, effect of year also indicated that the variance explained used by Cohen and Blanc-Goldhammer [33]. This was by any of the three models was significantly lower for White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 6 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 Table 1 Fits of power, logarithmic and linear models for individual children in Years 1-3 Year 1 Year 2 Year 3 n=13 n=19 n=19 Power Log Linear Power Log Linear Power Log Linear 2 2 2 2 2 2 2 2 2 1-cycle or 2-cycle b R R R 1-cycle or 2-cycle b R R R 1-cycle or 2-cycle b R R R 1 0.78 0.70 0.85 0.78 2 0.53 0.69 0.85 0.94 11 0.88 0.98 0.97 1 0.7 0.62 0.71 0.77 1 0.73 0.73 0.85 0.79 1 0.82 0.88 0.89 0.88 1 0.84 0.91 0.93 0.95 11 0.98 0.90 0.98 11 0.98 0.94 0.99 1 0.88 0.71 0.74 0.73 2 0.81 0.96 0.89 0.98 1 0.88 0.63 0.68 0.79 11 0.85 0.75 0.86 1 0.97 0.93 0.96 0.93 11 0.99 0.95 0.99 1 0.92 0.95 0.89 0.97 1 0.89 0.84 0.87 0.93 11 0.92 0.92 0.98 2 0.75 0.89 0.92 0.92 2 0.75 0.89 0.88 0.97 2 0.8 0.91 0.93 0.98 1 0.47 0.24 0.28 0.22 1 0.74 0.96 0.92 0.99 1 0.67 0.43 0.76 0.62 11 0.75 0.77 0.87 2 0.93 0.98 0.91 0.99 1 0.93 0.79 0.89 0.86 1 0.73 0.74 0.79 0.73 1 0.69 0.80 0.89 0.79 1 0.98 0.97 0.96 0.97 11 0.74 0.77 0.77 11 0.94 0.89 0.98 1 0.83 0.81 0.92 0.98 2 0.69 0.96 0.94 0.97 11 0.91 0.88 0.94 1 0.76 0.80 0.85 0.84 11 0.93 0.96 0.95 11 0.98 0.91 0.99 11 0.99 0.94 0.99 2 0.53 0.87 0.91 0.95 2 0.65 0.96 0.86 0.96 1 0.95 0.81 0.92 0.99 11 0.97 0.96 0.98 2 0.96 0.97 0.94 0.99 1 0.84 0.87 0.97 0.99 11 0.95 0.94 0.98 11 0.93 0.94 0.95 11 0.97 0.87 0.90 1 0.96 0.99 0.94 0.99 1 0.93 0.88 0.95 0.90 11 0.96 0.92 0.97 For the power model it indicates whether a 1- or 2-cycle model was better supported by the data and the corresponding coefficient of determination (R )and shape of the function (b). For logarithmic and linear models best supported by individual data, the R values are reported. Year 1, than Year 2 (p = 0.01) and Year 3 (p = 0.02); require the Year 1 participants to have the highest var- there was no significant difference between Years 2 and iance explained for a logarithmic model; what these 3. There was no Model × Year interaction. findings argue for the dominance of a linear representa- Using individual data and individual regression mod- tion in Years 2 and 3 only. els, these results indicate that in Years 2 and 3 the majority of participants had higher R values derived Model Residuals from a linear model, indicating a linear best fit model. This approach utilized individual residuals for each tar- This wasnot thecasewithYear1as thevariance get number, calculated from group level median models explained, by any model, was significantly lower. This (Figure 2). The linear model had a significantly smaller information supports our hypothesis that Year 2 indi- mean residual than did the logarithmic model (F (1, 51) cates the onset of the dominance of the linear mental = 36.71, p < 0.001, h = 0.388. Linear vs. Logarithmic: representation. Importantly, these findings do not clearly 1.72 ± 1.96 and 2.01 ± 1.78 units, respectively). The argue for a logarithmic to linear shift, as that would Model × Year interaction was significant (F (2, 51) = Figure 2 Median estimates for each target number, with linear and logarithmic regression equations for Years 1-3 White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 7 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 3.43, p = 0.040, h = 0.073). Linear model residuals were produced the lowest absolute errors. Error for target smaller than logarithmic model residuals for Years 2 and number 2 was significantly lower than for numbers 7, 8 3, but not Year 1. For Year 1 participants, the mean lin- and 13 (p < 0.01), and errors for target number 4 were ear residual (2.45 ± 2.33 units) was found to be not sig- significantly lower than for digits 7 and 13 (ps < 0.05). nificantly smaller than the mean logarithmic residual The mean estimation error rates decreased as years of (2.56 ± 2.18 units). The main effect of Year was mar- education increased (Year 1: 12.40 ± 7.51%, Year 2: ginal (F (2, 51) = 3.00, p = 0.058, h = 0.105). 8.39 ± 6.55%, Year 3: 6.47 ± 3.91%), but the main effect Overall the interaction of Target Number × Model of Year did not reach significance (F (2, 51) = 2.66, p = 2 2 was significant (F (7, 357) = 8.77, ε = 0.55, p < 0.001, h 0.079, h = 0.095). = 0.143). Post hoc comparisons indicated that linear Figure 3 shows the marginally significant Target Num- residuals were significantly smaller than logarithmic for ber × Year interaction (F (14, 357) = 1.56, p = 0.087, h target numbers 2 (p < 0.001) with a residual difference = 0.051). To increase the confidence of this marginal of 1.1 units, and 17 (p < 0.001) with a 0.7 unit difference finding and protect against a potential type 2 error, uni- in residuals. This difference was not significant for any variate analyses were completed and indicated that there other target numbers. These numbers were the focus of were significant differences between the year groups for separate year group analyses (Table 2), with significant target digits 11 and 13 (F (2, 53) = 3.38, p = 0.042, h = effects in Years 2 and 3 only. 0.117 and F (2, 53) = 6.10, p = 0.004, h = 0.193, respec- Using the approach of Geary at al. [32] we further tively). Closer scrutiny demonstrated that for target digit explored how model residuals would be distributed if 11 the Year 1 mean error was higher and marginally sig- we included an ambiguous category in addition to linear nificant in comparison to Year 2 (p = 0.09) and signifi- and logarithmic. As described in the methods, an cantly higher when compared to Year 3 (p = 0.01). A ambiguous trial would occur when the residuals of the similar pattern was evident with target digit 13; Year 1 linear and logarithmic models had a difference less than children had a mean error which was significantly ± 0.4 units. The overall percentages of trials classified as higher than both Years 2 and 3 (p < 0.01). Years 2 and linear, logarithmic or ambiguous are provided in Table 3 produced no significant differences for target digits 11 3, including a breakdown by individual target digits in and 13. Table 4. These classifications support the residual Separate Year group ANOVAs indicated a main effect Model × Year analysis, with Years 2 and 3 demonstrat- of target number for Year 1 children (F (7, 91) = 4.33, p ing a higher percentage of trials classified as linear in = 0.006, h = 0.250). Post hoc analysis of Year 1 data contrast to logarithmic, and the linear dominance less again pointed towards target digits 11 and 13 and this is clear in Year 1. Overall, this analysis of model residuals evident in Figure 3. The accuracy of estimating 13 was has further explored the linear and logarithmic mental significantly poorer than the estimation accuracy of representations and the transition at Year 2, but this numbers 2, 4 and 17 (ps < 0.05). This significant accu- time highlighting specific target digits in relation to a racy difference was also evident with positioning target group level model. numbers 11 and 2 (p < 0.001). In contrast to the Year 1 children, there was no effect of target number in Years Estimation Accuracy 2 and 3. Overall, this analysis has examined number line There was a main effect of Target Number (F (7, 357) = estimation, without regression modeling, in order to 6.61, ε = 0.46, p < 0.001, h = 0.109). Tukey post hoc pinpoint individual digits that might be the target of comparisons indicated that the significant differences selective strategy use. The year group comparisons indi- were in reference to the target numbers 2 and 4, which cated that digits 11 and 13 were poorly estimated by the Table 2 Statistical results for separate year group analyses for the Target Number × Model interaction (including post hoc analyses for target numbers 2 and 17) Residuals from group level models (mean ± SD) Group Interaction Target Number × Model Model Tukey HSD Target number 2 Target number 17 Year 1 F (7, 91) = 2.02, ε = 0.54, p = 0.06, h = 0.134 Linear 0.9 ± 0.7 ns 2.1 ± 2.2 ns Logarithmic 1.5 ± 1.1 2.6 ± 1.8 Year 2 F (7, 133) = 5.30, ε = 0.45, p < 0.001, h = 0.218 Linear 0.5 ± 0.7 p < 0.001 1.8 ± 2.4 p = 0.09 Logarithmic 2.1 ± 1.7 2.5 ± 1.9 Year 3 F (7, 133) = 3.59, ε = 0.43, p = 0.001, h = 0.159 Linear 0.7 ± 0.4 p < 0.001 1.0 ± 0.9 p = 0.001 Logarithmic 1.8 ± 0.7 1.9 ± 0.6 White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 8 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 Table 3 Analysis of model residuals, by year group Group Residual fit (trial-by-trial) Residual from group level models Linear Logarithmic Ambiguous Linear Logarithmic % % % M SD M SD Year 1 46 37 17 2.5 2.3 2.6 2.2 Year 2 56 25 19 1.5 1.8 2.0 1.6 Year 3 54 21 25 1.3 1.6 1.7 1.4 Percentage of trials with best residual fit for linear and logarithmic models, or ambiguous (both linear and logarithmic model residuals within ± 0.4 units of one another). The overall mean ± standard deviation of the model residuals are summarized by year group. Year 1 participants, thus separating them from Years 2 and 3. This was further confirmed through the compari- Figure 3 The absolute percent error for each target number, son of the standard deviation of estimates for each tar- by year group. The error bars represent ± 95% confidence interval get digit, with 11 and 13 with the highest standard from the mean. deviation in Year 1 (Figure 4). categorization of residuals into linear, logarithmic and Discussion ambiguous (Table 4). For selected target digits, there The aim of this investigation was to explore the devel- were a high percentage of Year 1 participants who opmental transitions of the mental representations asso- demonstrated linear-like behaviors however this was not ciated with numerical estimation (logarithmic-linear always the case (e.g. target digits 7 and 8, Table 4). In shift). This was achieved by focusing on the estimation contrast, Years 2 and 3 demonstrated a consistently behaviors of individual target digits within a familiar high percentage of participants with trials that were best number range (0-20), adapting and extending the meth- represented by a linear model. This matches the trend ods of Siegler and colleagues [2,5,10] and building on observed across the three groups of preschool children the ideas of Barth and Paladino [26], Berteletti et al. [7], in the Berteletti et al. study [7]. In the familiar number Ebersbach et al. [8] and Thompson and Siegler [11]. scale tasks used in that study (1-10 and 1-20) the The statistical analysis of individual numbers inferred youngest group (mean age 4 years) did not demonstrate that there is merit in future in depth analyses of strate- a bias towards either representation, meanwhile the gies application in conjunction with regression middle and oldest groups (4.5-6.5 years) indicated a sig- modeling. nificantly lower linear residual. Furthermore, the present On a developmental front, Year 1 children did not findings are in line with the 0-100 number line develop- demonstrate a dominance of any representation in either mental findings of Siegler and colleagues [2,5,10] except the individual regression models or group level resi- in Year 1 with no significant bias. duals. In fact, looking at the individual regression mod- Extending the analysis to include 1- and 2-cycle power els alone, the variance explained by any model (linear, functions [26], in this case, did not create any further logarithmic or power), was significantly lower in Year 1 clarity in terms of R values. Perhaps it was the reduced than in Years 2 and 3. When examining the group level number range (0-20) and minor differences in task model, children in Years 2 and 3 demonstrated the low- instructions that limited the potential for power models est residuals from a linear model, in comparison to a to represent the data, as Barth and Paladino [26] logarithmic model. The complexity of this transition to focused on a 0-100 number line and indicated 50 as a linear mental representation is indicated by the Table 4 Percentage of trials with best residual fit for linear and logarithmic models, or ambiguous, by year group and target number Group 2478 11 13 16 17 Lin Ln Amb Lin Ln Amb Lin Ln Amb Lin Ln Amb Lin Ln Amb Lin Ln Amb Lin Ln Amb Lin Ln Amb Year 1 55 25 20 68 32 0 25 69 5 33 61 6 56 44 0 0 0 100 67 27 6 61 39 0 Year 2 96 0 4 63 29 8 57 39 4 60 36 4 50 37 13 0 0 100 63 29 8 65 26 9 Year 3 70 0 30 74 26 0 43 39 17 57 29 14 52 22 26 0 0 100 50 36 14 83 17 0 The ambiguous category was used when both the linear and logarithmic residuals were within ± 0.4 units of one another. Please note, that for target number 13, all are ambiguous because the residuals are identical as this is the location where the linear and logarithmic models intersect. White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 9 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 Figure 4 The standard deviation of estimates for individual target numbers, by year group being ‘halfway’ at the beginning of the experiment. The Parallels can be drawn between the observations of the individual data in the present study was typically best fit present study and the developmental progression seen bya1-cycle model(Table1), whichalignswiththe fact with arithmetic strategies in the classroom. Existing that participants were not directed towards the ‘halfway’ research (e.g. [34-36]) purports the importance of point when task instructions were given. Further explora- sequences and counting in the early stages of develop- tion into the use of proportional power models is required, ment, but also identifies that the application of the particularly in relation to the appropriateness of using base-ten structure in constructing novel relationships such an approach, as highlighted by Opfer et al. [27]. among numbers up until approximately 9 years of age. Extending the work of Siegler and Opfer [10] the As a further example, Dutch mathematics education results from the individual target digits are a unique programs teach mental arithmetic strategies that contribution to the body of literature, as this begins to employs decomposition as the basis of instruction. From explore the possibility that the development of mental Grade 2, Dutch children are encouraged to use mental representations could be marked by the use of the exter- jumps and decomposition, often beginning on a number line, in order to encourage flexible mental strategies nal ‘anchor points’ as described by Ebersbach et al. [8]. [37]. Given this information and linking back to the pre- In the case of this research application of external sent data, it is proposed that the Year 1 children did not anchor points should be observed for target digits 2 and 17. According to existing research [2,5,7,8,10], a low lin- demonstrate any clear anchor point application because ear residual would indicate a more accurate mental they were limited to counting strategies and were unable representation. The number scale 0-20 is familiar to stu- to link the numerical value to the spatial cues provided dents in Years 1-3, however, the strategies involved in by the number line. Subsequently, in Years 2 and 3 we positioning numbers accurately on an externally repre- do see evidence of the continued development of more sented number line may not be fully established. For flexible strategies, and use of anchor points, that utilize Year 1 the linear and logarithmic residuals for both of decomposition and part-whole relations. these numbers were similar (Table 2). Year 2, showed a A question that comes out of this discussion is, given significantly lower linear residual for target number 2, the flexibility of strategy application, is it in fact mean- and perhaps as a function of the magnitude effect or ingful to try and model the mental representation of incomplete transition, only a marginal preference for a numbers using a fixed linear/logarithmic model? What linear model for target number 17 (Table 2). Finally the previous paragraph has posited is that specific num- then, for Year 3, the difference between linear and loga- bers could exhibit unique behaviors as a function of the rithmic residuals for target numbers 2 and 17 was sig- familiarity with the number range, proximity to either nificant with a linear model providing the lowest external or mental anchor points, as well as knowledge residual (Table 2). This indicates a developmental transi- of arithmetic strategy. While Ebersbach et al. [8] focused tion, but also highlights that the greatest disparity on the role of external anchor points, the mental anchor between the logarithmic and linear models is likely to points in particular would relate to more advanced strat- occur in close proximity to external anchor points. It is egy application, such as knowledge of proportions (e.g. half, quarter etc.) and ability to mentally partition the this external anchor point reasoning that we use to external number. This potential for individual difference speculate that both target digits 2 and 17 have lower lin- ear residuals in comparison the logarithmic model represents a limitation of the linear/logarithmic model- residuals. ing approach. In the followings we discuss what a more White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 10 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 detailed approach could add to the current knowledge furthernumberestimationstrategy could be to apply base. mental anchor points and divide/partition the external Siegler and Booth [5], in their first experiment with 0- number line into segments which would increase the 100 scale, had error rates of 27% for Kindergartners, accuracy of positioning. The Year 2 and 3 children of 18% for Grade 1 and 15% for Grade 2. The later Booth this study were probably exhibiting these behaviors. A and Siegler [2] study demonstrated a more obvious pla- potential mental partition would be that of halfway, and number 11 was the closest target digit to the mental teau with Kindergartners: 24%, Grade 1: 12%, Grade 2: anchor point of 10. It is proposed that, in Year 1, as 10%, and Grade 3: 9%. This could be a function of the both a result of both mental representation (no clear 0-100 number scale being in the unfamiliar range. In these two studies [2,5], the youngest groups were always logarithmic-linear preference) and educational experi- significantly different to the subsequent year levels; how- ence, children lacked the requisite representations/skills ever, this was not the case in the present research, as to apply mental anchor point strategies and accurately average percent absolute error was lower and there was estimate these central numbers (i.e. 11 and 13). The no significant group difference. core learning concepts for Year 1, as prescribed by the Overall, the results from the percent absolute error National Framework in England focus on counting and data indicated that the most accurately estimated num- skills related to addition and subtraction, not division or bers on the 0-20 number line were digits 2 and 4. This the part-whole relation. The early understanding of divi- could be for two reasons; first, referencing the lower sion principles often stems from the relationships values (2 and 4) to the external lower anchor point of between doubling and halving, which is promoted in zero or their knowledge of one. Second, it could be that Year 2. This knowledge of ‘half’ is required for the appli- number magnitudes up to 4 have a stronger representa- cation of mental anchor point strategies, which as a tion, because these quantities are understood from function of educational experience the Year 1 children infancy [38-40] in what Feigenson et al. [15] describes do not typically possess. The fact that the formal teach- as the small number system. This could also be linked ing of this concept in Year 2, coincides with increased to how frequently these numbers are encountered in a accuracy of estimation for central digits (Figure 3), lends young child’s everyday language (e.g. [13,25]). This also further support to this argument. Further detailed ana- fits with theories of enumeration and subitizing abilities lyses would be required to strengthen these proposals being present prior to verbal counting (e.g. [41-43]). and will be the focus of subsequent studies. Examining percent absolute error for specific target The fact that these enumeration and subitizing abilities numbers allowed the discussion to go beyond the limita- are limited to numerosities of four (e.g. [44,45]), and that they are present from infancy, could explain the tions of structured modeling to further explore the strength of the representation of digits 2 and 4, and potential of strategy application. The merits of both thus producing more accurate estimations. This is external and mental anchor points, as estimation strate- further evidenced in the categorization of trials (Table gies were supported. In line with the initial hypothesis, 4), where all year groups demonstrated a high percen- Year 2 appeared to represent a transitional phase, with tage of linear classifications of digits 2 and 4. Then for the apparent onset of part-whole strategies to aid the the subsequent numbers 7 and 8, Year 1 stood out with creation of mental anchor points. It was hypothesized a high percentage of logarithmic classifications, which that the developmental shift from logarithmic to linear were not evident in Years 2 and 3 (Table 4). It is our mental representation would likely coincide with evi- interpretation that for Year 1 digits 7 and 8 are likely to dence of strategy use; the present study supports this. be less frequently encountered and could contribute to The development of the anchor point strategy applica- the variation. tion could be described as follows; the first would be to On the developmental front, the Year 1 children again utilize the external anchor points to assist positioning of showed separation from the older year groups in the the numbers. Children in the first year of school may estimation accuracy of individual target digits. For the only use the left most point, rather than having the stra- positioning of numbers 11 and 13, Year 1 children pro- tegic knowledge to employ both extremes. The develop- duced estimates that were significantly less accurate mental progression, along with the logarithmic-linear than Years 2 and 3 (Figure 3). This gained further sup- shift, extends to include the use of both anchor points port when investigating the standard deviations for each after Year 1. This is followed by some level of mental partitioning, which again advances in complexity and is target digit, with the greatest variation evident for digits the result of educational experience, which became evi- 11 and 13 for Year 1 participants (Figure 4). The idea of dent in Year 2 (Figure 3). It would be an interesting applying external anchor points to aid in the estimation of numbers has been purported, although only briefly in investigation to more closely map the development of existing studies [5,8]. However, it could be argued that a these individual strategies and educational experience White and Szűcs Behavioral and Brain Functions 2012, 8:1 Page 11 of 12 http://www.behavioralandbrainfunctions.com/content/8/1/1 part of the PhD project completed by Sonia White at the University of with the logarithmic and linear modeling scenarios. This Cambridge, who was supported by the Cambridge Commonwealth Trust. extension would ideally include saturation of all target This research was also partly funded by the Medical Research Council, Ref. digits within the prescribed number range and a direct G90951. means of determining the strategy used to solve the pro- Author details blem, whether requesting verbal reports from partici- University of Cambridge, Department of Experimental Psychology, Centre pants or applying the use of eye-tracking technology as for Neuroscience in Education, Downing Site, CB2 3EB, UK. Queensland University of Technology, School of Early Childhood, Faculty of Education, introduced by Schneider et al. [28]. Furthermore, it Victoria Park Road, Kelvin Grove, 4059, Australia. would be worthwhile to investigate whether the type and flexibility of strategies used during these first years Authors’ contributions Both SW and DS took part in planning and designing the experiment. SW of school could predict later mathematical achievement. completed the data collection and analyses and drafted the manuscript. DS It is the combination of these proposals that could facili- assisted in preparing the manuscript. All authors read and approved the final tate the most meaningful insights for informing manuscript. education. Competing interests The authors declare that they have no competing interests. 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Published: Jan 4, 2012
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