A visual description of early-number counting, addition and subtraction knowledge

Early-number is a rich fabric of interconnected ideas that is often misunderstood and thus taught in ways that do not lead to rich understanding. In this presentation, a visual language is used to describe the organisation of this domain of knowledge. This visual language is based upon Piaget’s notion of reflective abstraction (Dubinsky, 1991; Piaget, 1977/2001), and thus captures the epistemological associations that link the problems, concepts and representations of the domain. The constructs of this visual language are introduced and then applied to the early-number domain. The introduction to this visual language may prompt reflection upon its suitability and significance to the description of other domains of knowledge. Through such a process of analysis and description, the visual language may serve as a scaffold for enhancing pedagogical content knowledge and thus ultimately improve learning outcomes.

Virtual world process perspective visualization

Product Lifecycle Management has been developed as an approach to providing timely engineering information. However, the number of domain specializations within manufacturing makes such information communication disjointed, inefficient and error-prone. In this paper we propose an immersive 3D visualization of linked domain- specific information views for improving and accelerating communication processes in Product Lifecycle Management. With a common and yet understandable visualization of several domain views, interconnections and dependencies become obvious. The conceptual framework presented here links domain-specific information extracts from Product Lifecycle Management systems with each other and displays them via an integrated 3D representation scheme. We expect that this visualization framework should support holistic tactical decision making processes between domain-experts in operational and tactical manufacturing scenarios.

Towards a controlled growth of self-assembled nanostructures: shaping, ordering and localization in Ge/Si heteroepitaxy.

The strain-induced self-assembly of suitable semiconductor pairs is an attractive natural route to nanofabrication. To bring to fruition their full potential for actual applications, individual nanostructures need to be combined into ordered patterns in which the location of each single unit is coupled with others and the surrounding environment. Within the Ge/Si model system, we analyze a number of examples of bottom-up strategies in which the shape, positioning, and actual growth mode of epitaxial nanostructures are tailored by manipulating the intrinsic physical processes of heteroepitaxy. The possibility of controlling elastic interactions and, hence, the configuration of self-assembled quantum dots by modulating surface orientation with the miscut angle is discussed. We focus on the use of atomic steps and step bunching as natural templates for nanodot clustering. Then, we consider several different patterning techniques which allow one to harness the natural self-organization dynamics of the system, such as: scanning tunneling nanolithography, focused ion beam and nanoindentation patterning. By analyzing the evolution of the dot assembly by scanning probe microscopy, we follow the pathway which leads to lateral ordering, discussing the thermodynamic and kinetic effects involved in selective nucleation on patterned substrates.

Quantification of particle emission characteristics and development of an emission model for use in transport microenvironments affected by traffic emissions

Vehicle emitted particles are of significant concern based on their potential to influence local air quality and human health. Transport microenvironments usually contain higher vehicle emission concentrations compared to other environments, and people spend a substantial amount of time in these microenvironments when commuting. Currently there is limited scientific knowledge on particle concentration, passenger exposure and the distribution of vehicle emissions in transport microenvironments, partially due to the fact that the instrumentation required to conduct such measurements is not available in many research centres. Information on passenger waiting time and location in such microenvironments has also not been investigated, which makes it difficult to evaluate a passenger’s spatial-temporal exposure to vehicle emissions. Furthermore, current emission models are incapable of rapidly predicting emission distribution, given the complexity of variations in emission rates that result from changes in driving conditions, as well as the time spent in driving condition within the transport microenvironment. In order to address these scientific gaps in knowledge, this work conducted, for the first time, a comprehensive statistical analysis of experimental data, along with multi-parameter assessment, exposure evaluation and comparison, and emission model development and application, in relation to traffic interrupted transport microenvironments. The work aimed to quantify and characterise particle emissions and human exposure in the transport microenvironments, with bus stations and a pedestrian crossing identified as suitable research locations representing a typical transport microenvironment. Firstly, two bus stations in Brisbane, Australia, with different designs, were selected to conduct measurements of particle number size distributions, particle number and PM2.5 concentrations during two different seasons. Simultaneous traffic and meteorological parameters were also monitored, aiming to quantify particle characteristics and investigate the impact of bus flow rate, station design and meteorological conditions on particle characteristics at stations. The results showed higher concentrations of PN20-30 at the station situated in an open area (open station), which is likely to be attributed to the lower average daily temperature compared to the station with a canyon structure (canyon station). During precipitation events, it was found that particle number concentration in the size range 25-250 nm decreased greatly, and that the average daily reduction in PM2.5 concentration on rainy days compared to fine days was 44.2 % and 22.6 % at the open and canyon station, respectively. The effect of ambient wind speeds on particle number concentrations was also examined, and no relationship was found between particle number concentration and wind speed for the entire measurement period. In addition, 33 pairs of average half-hourly PN7-3000 concentrations were calculated and identified at the two stations, during the same time of a day, and with the same ambient wind speeds and precipitation conditions. The results of a paired t-test showed that the average half-hourly PN7-3000 concentrations at the two stations were not significantly different at the 5% confidence level (t = 0.06, p = 0.96), which indicates that the different station designs were not a crucial factor for influencing PN7-3000 concentrations. A further assessment of passenger exposure to bus emissions on a platform was evaluated at another bus station in Brisbane, Australia. The sampling was conducted over seven weekdays to investigate spatial-temporal variations in size-fractionated particle number and PM2.5 concentrations, as well as human exposure on the platform. For the whole day, the average PN13-800 concentration was 1.3 x 104 and 1.0 x 104 particle/cm3 at the centre and end of the platform, respectively, of which PN50-100 accounted for the largest proportion to the total count. Furthermore, the contribution of exposure at the bus station to the overall daily exposure was assessed using two assumed scenarios of a school student and an office worker. It was found that, although the daily time fraction (the percentage of time spend at a location in a whole day) at the station was only 0.8 %, the daily exposure fractions (the percentage of exposures at a location accounting for the daily exposure) at the station were 2.7% and 2.8 % for exposure to PN13-800 and 2.7% and 3.5% for exposure to PM2.5 for the school student and the office worker, respectively. A new parameter, “exposure intensity” (the ratio of daily exposure fraction and the daily time fraction) was also defined and calculated at the station, with values of 3.3 and 3.4 for exposure to PN13-880, and 3.3 and 4.2 for exposure to PM2.5, for the school student and the office worker, respectively. In order to quantify the enhanced emissions at critical locations and define the emission distribution in further dispersion models for traffic interrupted transport microenvironments, a composite line source emission (CLSE) model was developed to specifically quantify exposure levels and describe the spatial variability of vehicle emissions in traffic interrupted microenvironments. This model took into account the complexity of vehicle movements in the queue, as well as different emission rates relevant to various driving conditions (cruise, decelerate, idle and accelerate), and it utilised multi-representative segments to capture the accurate emission distribution for real vehicle flow. This model does not only helped to quantify the enhanced emissions at critical locations, but it also helped to define the emission source distribution of the disrupted steady flow for further dispersion modelling. The model then was applied to estimate particle number emissions at a bidirectional bus station used by diesel and compressed natural gas fuelled buses. It was found that the acceleration distance was of critical importance when estimating particle number emission, since the highest emissions occurred in sections where most of the buses were accelerating and no significant increases were observed at locations where they idled. It was also shown that emissions at the front end of the platform were 43 times greater than at the rear of the platform. The CLSE model was also applied at a signalled pedestrian crossing, in order to assess increased particle number emissions from motor vehicles when forced to stop and accelerate from rest. The CLSE model was used to calculate the total emissions produced by a specific number and mix of light petrol cars and diesel passenger buses including 1 car travelling in 1 direction (/1 direction), 14 cars / 1 direction, 1 bus / 1 direction, 28 cars / 2 directions, 24 cars and 2 buses / 2 directions, and 20 cars and 4 buses / 2 directions. It was found that the total emissions produced during stopping on a red signal were significantly higher than when the traffic moved at a steady speed. Overall, total emissions due to the interruption of the traffic increased by a factor of 13, 11, 45, 11, 41, and 43 for the above 6 cases, respectively. In summary, this PhD thesis presents the results of a comprehensive study on particle number and mass concentration, together with particle size distribution, in a bus station transport microenvironment, influenced by bus flow rates, meteorological conditions and station design. Passenger spatial-temporal exposure to bus emitted particles was also assessed according to waiting time and location along the platform, as well as the contribution of exposure at the bus station to overall daily exposure. Due to the complexity of the interrupted traffic flow within the transport microenvironments, a unique CLSE model was also developed, which is capable of quantifying emission levels at critical locations within the transport microenvironment, for the purpose of evaluating passenger exposure and conducting simulations of vehicle emission dispersion. The application of the CLSE model at a pedestrian crossing also proved its applicability and simplicity for use in a real-world transport microenvironment.

The acute adaptations of normal and pathological human Achilles tendons to eccentric and concentric exercise

Eccentric exercise is the conservative treatment of choice for mid-portion Achilles tendinopathy. While there is a growing body of evidence supporting the medium to long term efficacy of eccentric exercise in Achilles tendinopathy treatment, very few studies have investigated the short term response of the tendon to eccentric exercise. Moreover, the mechanisms through which tendinopathy symptom resolution occurs remain to be established. The primary purpose of this thesis was to investigate the acute adaptations of the Achilles tendon to, and the biomechanical characteristics of, the eccentric exercise protocol used for Achilles tendinopathy rehabilitation and a concentric equivalent. The research was conducted with an orientation towards exploring potential mechanisms through which eccentric exercise may bring about a resolution of tendinopathy symptoms. Specifically, the morphology of tendinopathic and normal Achilles tendons was monitored using high resolution sonography prior to and following eccentric and concentric exercise, to facilitate comparison between the treatment of choice and a similar alternative. To date, the only proposed mechanism through which eccentric exercise is thought to result in symptom resolution is the increased variability in motor output force observed during eccentric exercise. This thesis expanded upon prior work by investigating the variability in motor output force recorded during eccentric and concentric exercises, when performed at two different knee joint angles, by limbs with and without symptomatic tendinopathy. The methodological phase of the research focused on establishing the reliability of measures of tendon thickness, tendon echogenicity, electromyography (EMG) of the Triceps Surae and the standard deviation (SD) and power spectral density (PSD) of the vertical ground reaction force (VGRF). These analyses facilitated comparison between the error in the measurements and experimental differences identified as statistically significant, so that the importance and meaning of the experimental differences could be established. One potential limitation of monitoring the morphological response of the Achilles tendon to exercise loading is that the Achilles tendon is continually exposed to additional loading as participants complete the walking required to carry out their necessary daily tasks. The specific purpose of the last experiment in the methodological phase was to evaluate the effect of incidental walking activity on Achilles tendon morphology. The results of this study indicated that walking activity could decrease Achilles tendon thickness (negative diametral strain) and that the decrease in thickness was dependent on both the amount of walking completed and the proximity of walking activity to the sonographic examination. Thus, incidental walking activity was identified as a potentially confounding factor for future experiments which endeavoured to monitor changes in tendon thickness with exercise loading. In the experimental phase of this thesis the thickness of Achilles tendons was monitored prior to and following isolated eccentric and concentric exercise. The initial pilot study demonstrated that eccentric exercise resulted in a greater acute decrease in Achilles tendon thickness (greater diametral strain) compared to an equivalent concentric exercise, in participants with no history of Achilles tendon pain. This experiment was then expanded to incorporate participants with unilateral Achilles tendinopathy. The major finding of this experiment was that the acute decrease in Achilles tendon thickness observed following eccentric exercise was modified by the presence of tendinopathy, with a smaller decrease (less diametral strain) noted for tendinopathic compared to healthy control tendon. Based on in vitro evidence a decrease in tendon thickness is believed to reflect extrusion of fluid from the tendon with loading. This process would appear to be limited by the presence of pathology and is hypothesised to be a result of the changes in tendon structure associated with tendinopathy. Load induced fluid movement may be important to the maintenance of tendon homeostasis and structure as it has the potential to enhance molecular movement and stimulate tendon remodelling. On this basis eccentric exercise may be more beneficial to the tendon than concentric exercise. Finally, EMG and motor output force variability (SD and PSD of VGRF) were investigated while participants with and without tendinopathy performed the eccentric and concentric exercises. Although between condition differences were identified as statistically significant for a number of force variability parameters, the differences were not greater than the limits of agreement for repeated measures. Consequently the meaning and importance of these findings were questioned. Interestingly, the EMG amplitude of all three Triceps Surae muscles did not vary with knee joint angle during the performance of eccentric exercise. This raises questions pertaining to the functional importance of performing the eccentric exercise protocol at each of the two knee joint angles as it is currently prescribed. EMG amplitude was significantly greater during concentric compared to eccentric muscle actions. Differences in the muscle activation patterns may result in different stress distributions within the tendon and be related to the different diametral strain responses observed for eccentric and concentric muscle actions.

Choosing landmarks for risky planning

This work examines the effect of landmark placement on the efficiency and accuracy of risk-bounded searches over probabilistic costmaps for mobile robot path planning. In previous work, risk-bounded searches were shown to offer in excess of 70% efficiency increases over normal heuristic search methods. The technique relies on precomputing distance estimates to landmarks which are then used to produce probability distributions over exact heuristics for use in heuristic searches such as A* and D*. The location and number of these landmarks therefore influence greatly the efficiency of the search and the quality of the risk bounds. Here four new methods of selecting landmarks for risk based search are evaluated. Results are shown which demonstrate that landmark selection needs to take into account the centrality of the landmark, and that diminishing rewards are obtained from using large numbers of landmarks.

Using Bayesian methods for the estimation of uncertainty in complex statistical models

The research objectives of this thesis were to contribute to Bayesian statistical methodology by contributing to risk assessment statistical methodology, and to spatial and spatio-temporal methodology, by modelling error structures using complex hierarchical models. Specifically, I hoped to consider two applied areas, and use these applications as a springboard for developing new statistical methods as well as undertaking analyses which might give answers to particular applied questions. Thus, this thesis considers a series of models, firstly in the context of risk assessments for recycled water, and secondly in the context of water usage by crops. The research objective was to model error structures using hierarchical models in two problems, namely risk assessment analyses for wastewater, and secondly, in a four dimensional dataset, assessing differences between cropping systems over time and over three spatial dimensions. The aim was to use the simplicity and insight afforded by Bayesian networks to develop appropriate models for risk scenarios, and again to use Bayesian hierarchical models to explore the necessarily complex modelling of four dimensional agricultural data. The specific objectives of the research were to develop a method for the calculation of credible intervals for the point estimates of Bayesian networks; to develop a model structure to incorporate all the experimental uncertainty associated with various constants thereby allowing the calculation of more credible credible intervals for a risk assessment; to model a single day’s data from the agricultural dataset which satisfactorily captured the complexities of the data; to build a model for several days’ data, in order to consider how the full data might be modelled; and finally to build a model for the full four dimensional dataset and to consider the timevarying nature of the contrast of interest, having satisfactorily accounted for possible spatial and temporal autocorrelations. This work forms five papers, two of which have been published, with two submitted, and the final paper still in draft. The first two objectives were met by recasting the risk assessments as directed, acyclic graphs (DAGs). In the first case, we elicited uncertainty for the conditional probabilities needed by the Bayesian net, incorporated these into a corresponding DAG, and used Markov chain Monte Carlo (MCMC) to find credible intervals, for all the scenarios and outcomes of interest. In the second case, we incorporated the experimental data underlying the risk assessment constants into the DAG, and also treated some of that data as needing to be modelled as an ‘errors-invariables’ problem [Fuller, 1987]. This illustrated a simple method for the incorporation of experimental error into risk assessments. In considering one day of the three-dimensional agricultural data, it became clear that geostatistical models or conditional autoregressive (CAR) models over the three dimensions were not the best way to approach the data. Instead CAR models are used with neighbours only in the same depth layer. This gave flexibility to the model, allowing both the spatially structured and non-structured variances to differ at all depths. We call this model the CAR layered model. Given the experimental design, the fixed part of the model could have been modelled as a set of means by treatment and by depth, but doing so allows little insight into how the treatment effects vary with depth. Hence, a number of essentially non-parametric approaches were taken to see the effects of depth on treatment, with the model of choice incorporating an errors-in-variables approach for depth in addition to a non-parametric smooth. The statistical contribution here was the introduction of the CAR layered model, the applied contribution the analysis of moisture over depth and estimation of the contrast of interest together with its credible intervals. These models were fitted using WinBUGS [Lunn et al., 2000]. The work in the fifth paper deals with the fact that with large datasets, the use of WinBUGS becomes more problematic because of its highly correlated term by term updating. In this work, we introduce a Gibbs sampler with block updating for the CAR layered model. The Gibbs sampler was implemented by Chris Strickland using pyMCMC [Strickland, 2010]. This framework is then used to consider five days data, and we show that moisture in the soil for all the various treatments reaches levels particular to each treatment at a depth of 200 cm and thereafter stays constant, albeit with increasing variances with depth. In an analysis across three spatial dimensions and across time, there are many interactions of time and the spatial dimensions to be considered. Hence, we chose to use a daily model and to repeat the analysis at all time points, effectively creating an interaction model of time by the daily model. Such an approach allows great flexibility. However, this approach does not allow insight into the way in which the parameter of interest varies over time. Hence, a two-stage approach was also used, with estimates from the first-stage being analysed as a set of time series. We see this spatio-temporal interaction model as being a useful approach to data measured across three spatial dimensions and time, since it does not assume additivity of the random spatial or temporal effects.

The effect of simulated astigmatic refractive error on reading performance

Background: Few studies have specifically investigated the functional effects of uncorrected astigmatism on measures of reading fluency. This information is important to provide evidence for the development of clinical guidelines for the correction of astigmatism. Methods: Participants included 30 visually normal, young adults (mean age 21.7 ± 3.4 years). Distance and near visual acuity and reading fluency were assessed with optimal spectacle correction (baseline) and for two levels of astigmatism, 1.00DC and 2.00DC, at two axes (90° and 180°) to induce both against-the-rule (ATR) and with-the-rule (WTR) astigmatism. Reading and eye movement fluency were assessed using standardized clinical measures including the test of Discrete Reading Rate (DRR), the Developmental Eye Movement (DEM) test and by recording eye movement patterns with the Visagraph (III) during reading for comprehension. Results: Both distance and near acuity were significantly decreased compared to baseline for all of the astigmatic lens conditions (p < 0.001). Reading speed with the DRR for N16 print size was significantly reduced for the 2.00DC ATR condition (a reduction of 10%), while for smaller text sizes reading speed was reduced by up to 24% for the 1.00DC ATR and 2.00DC condition in both axis directions (p<0.05). For the DEM, sub-test completion speeds were significantly impaired, with the 2.00DC condition affecting both vertical and horizontal times and the 1.00DC ATR condition affecting only horizontal times (p<0.05). Visagraph reading eye movements were not significantly affected by the induced astigmatism. Conclusions: Induced astigmatism impaired performance on selected tests of reading fluency, with ATR astigmatism having significantly greater effects on performance than did WTR, even for relatively small amounts of astigmatic blur of 1.00DC. These findings have implications for the minimal prescribing criteria for astigmatic refractive errors.

Idyll number eleven

This is a short horror story formulated in the research process for the novel "That Blackfella Bloodsucka Dance!"

Prandtl number scaling of natural convection of an inclined flat plate due to uniform surface heat flux

A new scaling analysis has been performed for the unsteady natural convection boundary layer under a downward facing inclined plate with uniform heat flux. The development of the thermal or viscous boundary layers may be classified into three distinct stages including a start-up stage, a transitional stage and a steady stage, which can be clearly identified in the analytical as well as numerical results. Earlier scaling shows that the existing scaling laws of the boundary layer thickness, velocity and steady state time scale for the natural convection flow on a heated plate of uniform heat flux provide a very poor prediction of the Prandtl number dependency of the flow. However, those scalings performed very well with Rayleigh number and aspect ratio dependency. In this study, a new Prandtl number scaling has been developed using a triple-layer integral approach for Pr > 1. It is seen that in comparison to the direct numerical simulations, the new scaling performs considerably better than the previous scaling.

In search of the antecendents for risky driving at railway level crossings

This paper discusses human factors issues of low cost railway level crossings in Australia. Several issues are discussed in this paper including safety at passive level railway crossings, human factors considerations associated with unavailability of a warning device, and a conceptual model for how safety could be compromised at railway level crossings following prolonged or frequent unavailability. The research plans to quantify safety risk to motorists at level crossings using a Human Reliability Assessment (HRA) method, supported by data collected using an advanced driving simulator. This method aims to identify human error within tasks and task units identified as part of the task analysis process. It is anticipated that by modelling driver behaviour the current study will be able to quantify meaningful task variability including temporal parameters, between participants and within participants. The process of complex tasks such as driving through a level crossing is fundamentally context-bound. Therefore this study also aims to quantify those performance-shaping factors that contribute to vehicle train collisions by highlighting changes in the task units and driver physiology. Finally we will also consider a number of variables germane to ensuring external validity of our results. Without this inclusion, such an analysis could seriously underestimate the probabilistic risk assessment.

The knock-on from the knock-off : recent shifts within Australian mass market fashion design practice

Australia’s mass market fashion labels have traditionally benefitted from their peripheral location to the world’s fashion centres. Operating a season behind, Australian mass market designers and buyers were well-placed to watch trends play out overseas before testing them in the Australian marketplace. For this reason, often a designer’s role was to source and oversee the manufacture of ‘knock-offs’, or close copies of Northern hemisphere mass market garments. Both Weller (2007) and Walsh (2009) have commented on this practice. The knock-on effect from this continues to be a cautious, derivative fashion sensibility within Australian mass market fashion design, where any new trend or product is first tested and proved overseas months earlier. However, there is evidence that this is changing. The rapid online dissemination of global fashion trends, coupled with the Australian consumer’s willingness to shop online, has meant that the ‘knock-off’ is less viable. For this reason, a number of mass market companies are moving away from the practice of direct sourcing and are developing product in-house under a Northern hemisphere model. This shift is also witnessed in the trend for mass market companies to develop collections in partnership with independent Australian designers. This paper explores the current and potential effects of these shifts within Australian mass market design practice, and discusses how they may impact on designers, consumers and on the wider culture of Australian fashion.

A human factors investigation into the unavailability of active warnings at railway level crossings

This paper discusses human factors issues of low cost railway level crossings in Australia. Several issues are discussed in this paper including safety at passive level railway crossings, human factors considerations associated with unavailability of a warning device, and a conceptual model for how safety could be compromised at railway level crossings following prolonged or frequent unavailability. The research plans to quantify safety risk to motorists at level crossings using a Human Reliability Assessment (HRA) method, supported by data collected using an advanced driving simulator. This method aims to identify human error within tasks and task units identified as part of the task analysis process. It is anticipated that by modelling driver behaviour the current study will be able to quantify meaningful task variability including temporal parameters, between participants and within participants. The process of complex tasks such as driving through a level crossing is fundamentally context-bound. Therefore this study also aims to quantify those performance-shaping factors that contribute to vehicle train collisions by highlighting changes in the task units and driver physiology. Finally we will also consider a number of variables germane to ensuring external validity of our results. Without this inclusion, such an analysis could seriously underestimate risk.

Creative suburbia : rethinking urban cultural policy – the Australian case

This article considers the question of whether creative workers demonstrate a preference for inner cities or suburbs, drawing upon research findings from the ‘Creative Suburbia’ project undertaken by a team of Australian researchers over 2008–2010 in selected suburban areas of Brisbane and Melbourne. Locating this question in wider debates about the relationship of the suburbs to the city, as well as the development of new suburban forms such as master-planned communities, the article finds that the number of creative industries workers located in the suburbs is significant, and those creative workforce members living and working in suburban areas are generally happy with this experience, locating in the suburbs out of personal choice rather than economic necessity. This runs counter to the received wisdom on creative cities, which emphasize cultural amenity in inner city areas as a primary driver of location decisions for the ‘creative class’. The article draws out some implications of the findings for urban cultural policy, arguing that the focus on developing inner urban cultural amenity has been overplayed, and that more attention should be given to how to better enable distributed knowledge systems through high-speed broadband infrastructure.

Symbolic number : the integration of magnitude and spatial representations in children aged 6 to 8 years

The process of learning symbolic Arabic digits in early childhood requires that magnitude and spatial information integrates with the concept of symbolic digits. Previous research has separately investigated the development of automatic access to magnitude and spatial information from symbolic digits. However, developmental trajectories of symbolic number knowledge cannot be fully understood when considering components in isolation. In view of this, we have synthesized the existing lines of research and tested the use of both magnitude and spatial information with the same sample of British children in Years 1, 2 and 3 (6-8 years of age). The physical judgment task of the numerical Stroop paradigm (NSP) demonstrated that automatic access to magnitude was present from Year 1 and the distance effect signaled that a refined processing of numerical information had developed. Additionally, a parity judgment task showed that the onset of the Spatial-Numerical Association of Response Codes (SNARC) effect occurs in Year 2. These findings uncover the developmental timeline of how magnitude and spatial representations integrate with symbolic number knowledge during early learning of Arabic digits and resolve inconsistencies between previous developmental and experimental research lines.