Influences of basic numerical abilities on graph reading performance.

Understanding graphically presented information is an important aspect of modern mathematical and science literacy. In our study, we investigated the influence of basic numerical abilities on students' ability answer mathematical tasks with information presented in graphs. We analyzed data of 750 German students (grades 9-11) and evaluated the determinants of graph reading performance with multiple regression analysis using predictors of basic numerical abilities (such as number line estimation, basic arithmetic operations, etc.), considering also the influences of general cognitive ability, age, and gender. We found that number line estimation, subtraction, and conceptual knowledge were significant predictors of graph reading performance beyond the influences of general cognitive ability. This indicates that basic numerical abilities are still relevant for real-life problem solving in secondary school. We discuss possible mechanisms which directly (through respective arithmetic procedures) as well as indirectly (through mathematization of the problem) effectuate that basic numerical abilities graph reading performance.

Training the equidistant principle of number line spacing.

The characteristics of effective numerical trainings are still under scientific debate. Given the importance of number line estimation due to the strong relation between task performance and arithmetic abilities, the current study aimed at training one important number line characteristic: the equidistant spacing of adjacent numbers. Following an embodied training approach, second-graders were trained using a randomized crossover design to divide a presented line into different numbers of equal segments by walking the line with equally spaced steps. Performance was recorded, and feedback as to the correct equidistant spacing was provided using the Kinect sensor system. Training effects were compared to a control training with no involvement of task-specific whole-body movements. Results indicated more pronounced specific training effects after the embodied training. Moreover, transfer effects to number line estimation and arithmetic performance were partially observed. In particular, differential training effects for bounded versus unbounded number line estimation corroborate the assumption that not only bodily experiences but also the need for a flexible adaption of the perspective on the training material might influence training success. Hence, more pronounced training effects of the embodied training might stem from different cognitive processes involved.

The relevance of basic numerical skills for fraction processing: Evidence from cross-sectional data.

Recent research indicated that fraction understanding is an important predictor of later mathematical achievement. In the current study we investigated associations between basic numerical skills and students' fraction processing. We analyzed data of 939 German secondary school students (age range = 11.92 to 18.00 years) and evaluated the determinants of fraction processing considering basic numerical skills as predictors (i.e., number line estimation, basic arithmetic operations, non-symbolic magnitude comparison, etc.). Additionally, we controlled for general cognitive ability, grade level, and sex. We found that multiplication, subtraction, conceptual knowledge, number line estimation, and basic geometry were significantly associated with fraction processing beyond significant associations of general cognitive ability and sex. Moreover, relative weight analysis revealed that addition and approximate arithmetic should also be considered as relevant predictors for fraction processing. The current results provide food for thought that further research should focus on investigating whether recapitulating basic numerical content in secondary school mathematics education can be beneficial for acquiring more complex mathematical concepts such as fractions.

The Use of Local and Global Ordering Strategies in Number Line Estimation in Early Childhood.

A lot of research has been devoted to number line estimation in primary school. However, less is known about the early onset of number line estimation before children enter formal education. We propose that ordering strategies are building blocks of number line estimation in early childhood. In a longitudinal study, children completed a non-symbolic number line estimation task at age 3.5 and 5 years. Two ordering strategies were identified based on the children's estimation patterns: local and global ordering. Local ordering refers to the correct ordering of successive quantities, whereas global ordering refers to the correct ordering of all quantities across the number line. Results indicated a developmental trend for both strategies. The percentage of children applying local and global ordering strategies increased steeply from 3.5 to 5 years of age. Moreover, children used more advanced local and global ordering strategies at 5 years of age. Importantly, level of strategy use was related to more traditional number line estimation outcome measures, such as estimation accuracy and regression fit scores. These results provide evidence that children use dynamic ordering strategies when solving the number line estimation task in early stages of numerical development.

Place-value understanding in number line estimation predicts future arithmetic performance.

In multi-digit numbers, the value of each digit is determined by its position within the digit string. Children's understanding of this place-value structure constitutes a building block for later arithmetic skills. We investigated whether a number line estimation task can provide an assessment of place-value understanding in first grade. We hypothesized that estimating the position of two-digit numbers requires place-value understanding. Therefore, we fitted a linear function to children's estimates of two-digit numbers and considered the resulting slope as a measure of children's place-value understanding. We observed a significant correlation between this slope and children's performance in a transcoding task known to require place-value understanding. Additionally, the slope for two-digit numbers assessed at the beginning of grade 1 predicted children's arithmetic performance at the end of grade 1. These results indicate that the number line estimation task may indeed constitute a valid measure for first-graders' place-value understanding. Moreover, these findings are hard to reconcile with the view that number line estimation directly assesses a spatial representation of numbers. Instead, our results suggest that numerical processes involved in performing the task (such as place-value understanding) may drive the association between number line estimation and arithmetic performance.

An integration of competing accounts on children's number line estimation.

Children's estimation patterns in bounded number line estimation (NLE) reveal marked developmental changes. Three different theoretical accounts were proposed to explain these changes: a log-to-linear shift account, a proportion-judgment account and a two-linear account considering familiarity with numbers or the understanding of the place-value structure of the Arabic number system. However, only the first two accounts are considered prominently in the ongoing scientific debate. Therefore, we first present a reanalysis of NLE data of Austrian first-graders contrasting all three accounts. Results indicate that the two-linear account is a reliable alternative to the log-to-linear shift as well as the proportion-judgment account. However, we do not claim the two-liner account to provide an exhaustive explanation for the observed developmental changes. We rather introduce the idea that aspects of all three accounts may complement - instead of exclude - each other. Jointly considering conceptual (i.e., familiarity, place-value) and procedural (i.e., proportion-judgments) aspects will allow for a more comprehensive understanding of children's development in NLE.