Diagnostic assessment of children’s proportional reasoning

A proportional reasoning item bank was created from the relevant literature and tested in various forms. Rasch analyses of 303 pupils’ test results were used to calibrate the bank, and data from 84 pupils’ interviews was used to confirm our diagnostic interpretations. A number of sub-tests were scaled, including parallel ‘without models’ and ‘with models’ forms. We provide details of the 13-item ‘without models’ test which was formed from the ‘richest’ diagnostic items and verified on a further test sample (N=212, ages 10-13). Two scales were constructed for this test, one that measures children’s ‘ratio attainment’ and one that measures their ‘tendency for additive strategy.’ Other significant errors — ‘incorrect build-up,’ ‘magical doubling/halving,’ ‘constant sum’ and ‘incomplete reasoning’ — were identified. Finally, an empirical hierarchy of pupils’ attainment of proportional reasoning was formed, incorporating the significant errors and the additive scale.     

Using the Relational Paradigm: effects on pupils’ reasoning in solving additive word problems

Pupils’ difficulties in solving word problems continue to attract attention: while researchers highlight the importance of relational reasoning and modelling, school curricula typically use short word problems to develop pupils’ knowledge of arithmetic operations and calculation strategies. The Relational Paradigm attributes the leading role in mathematics learning to the development of relational thinking. Using this perspective, we implemented a new approach to teaching additive word problem-solving in primary school, encouraging relational thinking and modelling. We compared the overall results of additive word problems solved by Grade 2 elementary pupils in the experimental group (N?=?216) and in the control group (N?=?196). Our data show: (a) on average, the experimental group performed significantly better in problem-solving than the control group; and (b) in the control group, there was a considerable lack of success in solving problems that require relational thinking—there was no such effect in the experimental group.     

Numerical scaling of ratio scale utilities in multi-criteria decision analysis with geometric model

The ratio scale measurement of utilities in multi-criteria decision analysis has been criticised due to theoretical and practical interpretation of the required assessments. Compared to the ratio scale models, for example, the ‘0–1-scale’ models based on interval scale have been more generally accepted. On the other hand, under some conditions it has been shown that the results of multi-criteria decision support based on ratio scale comparisons are actually independent of the numerical magnitude of the ratio scale assessments. This paper reviews and interprets these results and proposes new solutions for the remaining problems, for which the numerical magnitude of the ratio scale utilities is required. Moreover, we interpret the scale-independent ratio scale models with respect to some other multi-criteria decision support models. The theoretical analysis is carried out by statistical models and illustrated by numerical examples.     

A SCALE FOR ASSESSING PROBABILISTIC THINKING AND THE REPRESENTATIVENESS TENDENCY

We report a study of children's probability conceptions and misconceptions using a diagnostic instrument developed from the literature on the representativeness heuristic. Rasch measurement methodology was used to develop the 13-item open response instrument with a sample (n=116) of 12-15 year olds. The result is that a hierarchy of responses at two levels is confirmed for this sample, and a third level is hypothesised. Each level is characterised by the ability to overcome typical 'representativeness' effects, namely 'recency', 'random-similarity' (at level 1), 'base-rate frequency' and 'sample size' (at level 2-3). The validity of our interpretations was tested and some anomalies were identified through clinical interviews with children making the errors (n=8). Another Rasch ability measure, which we named the 'representativeness tendency', was constructed from 11 multiple-choice errors.     

METAPHORS AND OTHER LINGUISTIC POINTERS TO CHILDREN'S MENTAL REPRESENTATIONS

The mental representations that 6- and 7-year-old pupils form as a result of their interactions with their teacher's verbal, written, pictorial and concrete material representations has to be inferred from the language they use. In this study many pupils seem to have mental representations which capture surface characteristics of a particular teachers’ representation and use metaphoric language associated with that representation when describing their calculations. Pupils’ use of ‘you’ is characteristic of those who adopt a representation-specific procedure, whilst ‘if’ and ‘like’ are linguistic pointers to their use of generic examples to describe a procedure. Individual pupils show a preference for the same style of mental representation when describing images and procedures in mathematical and non-mathematical contexts.     

Pupils’ attitudes towards mathematics: a comparative study of Norwegian and English secondary students

Comparing English and Norwegian pupils’ attitude towards mathematics, in this article I develop a deeper understanding of the factors that may shape and influence ‘pupil attitude towards mathematics’, and argue for it as a socio-cultural construct embedded in and shaped by students’ environment and context in which they learn mathematics. The theoretical framework leans on work by Zan and Di Martino (The Montana Mathematics Enthusiast, Monograph 3, pp. 157–168, 2007) to elicit Norwegian and English pupils’ attitude of mathematics as they experience it in their respective environments. Whilst there were differences which could be seen to be accounted for by differently ‘figured’ environments, there are also many similarities. It was interesting to see that, albeit based on a small statistical sample, in both countries students had a positive attitude towards mathematics in year 7/8, which dropped in year 9, and increased again in years 10/11. This result could be explained and compared with other larger scale studies (e.g. Hodgen et al. in Proceedings of the British Society for Research into Learning Mathematics. 29(3), 2009). The analysis of pupils’ qualitative comments (and classroom observations) suggested seven factors that appeared to influence pupil attitude most, and these had ‘superficial’ commonalities, but the perceptions that appeared to underpin these mentions were different, and could be linked to the environments of learning mathematics in their respective classrooms. In summary, it is claimed that it is not enough to identify the factors that may shape and influence pupil attitude, but more importantly, to study how these are ‘lived’ by pupils, what meanings are made in classrooms and in different contexts, and how the factors interrelate and can be understood.     

Deterministic macroeconomic dynamic equations from a stochastic microeconomic description: The role of time scales

In the present paper, we develop the ‘stochastic’ microeconomic approach to obtained a deterministic macrodynamic equation. Three time scales have to be introduced. To be valid, the dynamic equation has to be written on the meso-scopic time scale. ‘Discrete’ and ‘continuous’ time versions are given. On the other hand, the comparison between the ‘stochastic’ and the ‘average’ (all the agents have the same characteristics, the ‘average’ ones) approaches is presented. Both methods give the same form for the dynamic equation, nevertheless the ‘average’ approach gives conditions of validity which are not strict enough. An advantage of the ‘stochastic’ approach is to inforce this point and to propose a better framework for writing down deterministic macrodynamic equations and for making further developments if necessary.     

Happy and sad thoughts: An exploration of children's integer reasoning

The purpose of this study was to investigate elementary children's conceptions that might serve as foundations for integer reasoning. Working from an abstract algebraic perspective and using an opposite-magnitudes context that is relevant to children, we analyzed the reasoning of 33 children in grades K-5. We focus our report on three prominent ways of reasoning. We do this by describing and analyzing the responses of three particular children (in Grades 1, 3, and 5) who exemplify these ways of reasoning. We view each of the three ways of reasoning as rich and interesting, and we see relationships of each to formal integer reasoning. At the same time, we view these ways of reasoning in terms of increasing levels of sophistication, potentially belonging to a single learning trajectory. Thus, we see the roots of more sophisticated integer reasoning in children's early intuitions about opposite magnitudes.     

Optimal Conditional Estimation: Average Case Setting

We consider an estimation problem which appears in the identification of systems by means of restricted complexity models: find the optimal approximation to an element of a linear normed space (a system) based on noisy information, subject to the restriction that approximations (models) can be selected from a prescribed subspace M of the problem element space. In contrast to the worst-case optimization criterion, which may be pessimistic, in this paper the quality of an identification algorithm is measured by its local average performance. Two types of local average errors are considered: for a given information (measurement) y and for a given unknown element x, the latter in two versions. For a wide spectrum of norms in the measurement space, we define an optimal algorithm and give expressions for its average errors which show the dependence on information, information errors, unmodelled dynamics, and norm in the measurement space.     

Comparative statics of a signaling game: An experimental study

In this paper a simple and basic signaling game is studied in an experimental environment. First, we check whether we can replicate some of the findings in the literature concerning equilibrium selection and the use and impact of costly signals. Second, and foremost, the comparative statics implications of the game are studied. The experimental results are related to the predictions of two competing behavioral models: a game model, in which subjects are assumed to behave in line with (refined) sequential equilibrium theory, and a decision model, in which subjects are assumed to behave as non-strategic decision makers. The experimental outcomes replicate the finding in the literature that costly messages are sent more frequently by ‘higher’ sender types (whose information is such that persuasion is also profitable to the responder), and that such messages have an impact on the behavior of the responder. These results are consistent with (versions of) both the game model and the decision model. The comparative statics results, however, clearly point in the direction of the decision model. Play is most strongly affected by ‘own’ payoff parameters, as predicted by the decision model, and less so by opponent's payoff parameters, as predicted by the mixed strategies of the refined sequential equilibrium. Particularly, a decision model in which players are assumed to adapt beliefs about opponents' choice probabilities in response to experience in previous play, appears to succeed best in organizing the data.     

Bayesian prediction of crack growth based on a hierarchical diffusion model

A general Bayesian approach for stochastic versions of deterministic growth models is presented to provide predictions for crack propagation in an early stage of the growth process. To improve the prediction, the information of other crack growth processes is used in a hierarchical (mixed‐effects) model. Two stochastic versions of a deterministic growth model are compared. One is a nonlinear regression setup where the trajectory is assumed to be the solution of an ordinary differential equation with additive errors. The other is a diffusion model defined by a stochastic differential equation where increments have additive errors. While Bayesian prediction is known for hierarchical models based on nonlinear regression, we propose a new Bayesian prediction method for hierarchical diffusion models. Six growth models for each of the two approaches are compared with respect to their ability to predict the crack propagation in a large data example. Surprisingly, the stochastic differential equation approach has no advantage concerning the prediction compared with the nonlinear regression setup, although the diffusion model seems more appropriate for crack growth. Copyright © 2016 John Wiley & Sons, Ltd.     

Unique finite conjoint measurement

This paper studies one of the most important types of measurement that has arisen from the social sciences, the additive conjoint measurement introduced by Debreu (1960) and Luce and Tukey (1964). It also considers the variant we call additive conjoint extensive measurement. Both types of measurement are based on qualitative comparisons between multiattribute alternatives in Cartesian products of sets. This paper initiates a study of their uniqueness for the case in which all sets are finite. It considers uniqueness up to similar positive affine transformations for additive conjoint measurement, and uniqueness up to similar proportionality transformations for additive conjoint extensive measurement. Both types of uniqueness are related to sets of ‘indifference’ comparisons that correspond to sets of linearly independent equations for the measurement representation. After we explicate necessary and sufficient conditions for uniqueness, we explore specific aspects of sets of unique solutions for two-factor (two-set) additive conjoint measurement and two-factor additive conjoint extensive measurement.     

The relationship between DEA efficiency and the type of production function, the degree of homogeneity, and error variability

In this paper, we use simulations to investigate the relationship between data envelopment analysis (DEA) efficiency and major production functions: Cobb-Douglas, the constant elasticity of substitution, and the transcendental logarithmic. Two DEA models were used: a constant return to scale (CCR model), and a variable return to scale (BCC model). Each of the models was investigated in two versions: with bounded and unbounded weights. Two cases were simulated: with and without errors in the production functions estimation. Various degrees of homogeneity (of the production function) were tested, reflecting a constant increasing and decreasing return to scale. With respect to the case with errors, three distribution functions were utilized: uniform, normal, and double exponential. For each distribution, 16 levels of the coefficient of variance (CV) were used. In all the tested cases, two measures were analysed: the percentage of efficient units (from the total number of units), and the average efficiency score. We applied a regression analysis to test the relationship between these two efficiency measures and the above parameters. Overall, we found that the degree of homogeneity has the largest effect on efficiency. Efficiency declines as the errors grow (as reflected by larger CV and of the expansion of the probability distribution function away from the centre). The bounds on the weights tend to smooth the effect, and bring the various DEA versions closer to one other. The type of efficiency measure has similar regression tendencies. Finally, the relationship between the efficiency measures and the explanatory variables is quadratic.     

A diagrammatic view of the equals sign: arithmetical equivalence as a means,not an end

It is recommended in the mathematics education literature that pupils be presented with equality statements that can be assessed for numerical balance by attending to notational structure rather than computation. I describe an alternative, diagrammatic approach in which pupils do not assess statements but instead use them to make substitutions of notation. I report on two trials of a computer-based task conducted with pairs of pupils and highlight two findings. First, the pupils found it useful to articulate the distinct substitutive effects of commutative (‘swap’, ‘switch’) and partitional (‘split’, ‘separate’) statements when working on the task. Secondly, the pupils did not notice that some of the statements presented were in fact false, which suggests their substituting activities were independent of numerical equivalence conceptions. This demonstrates that making substitutions offers task designers a mathematical utility for equality statements that is distinct from, but complementary to, assessing numerical balance.     

Coding strategic behaviour in mathematical problem solving

There has been some debate about the extent to which contextual'stories' and graphical images can help pupils to think their way through numerical problems. The degree to which contexts stimulate useful'models to think with' may vary considerably. Some contexts seem to encourage children to reason more effectively, but others have little impact either on pupils'performance, or on the way in which they tackle the computations required. In this study three sets of four questions involving subtraction and division were trialled with a total sample of 1795 Year 6 to 9 pupils. In each comparison, about a quarter of the pupils could answer the question only when it was presented in context, or only out of context, but not both. Pupils' methods of working were analysed to reveal differences in their approach to the different questions.     

Heteroscedastic replicated measurement error models under asymmetric heavy-tailed distributions

We propose a heteroscedastic replicated measurement error model based on the class of scale mixtures of skew-normal distributions, which allows the variances of measurement errors to vary across subjects. We develop EM algorithms to calculate maximum likelihood estimates for the model with or without equation error. An empirical Bayes approach is applied to estimate the true covariate and predict the response. Simulation studies show that the proposed models can provide reliable results and the inference is not unduly affected by outliers and distribution misspecification. The method has also been used to analyze a real data of plant root decomposition.     

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The additive model is a more flexible nonparametric statistical model which allows a data-analytic transform of the covariates.When the number of covariates is big and grows exponentially with the sample size the urgent issue is to reduce dimensionality from high to a moderate scale. In this paper, we propose and investigate marginal empirical likelihood screening methods in ultra-high dimensional additive models. The proposed nonparametric screening method selects variables by ranking a measure of the marginal empirical likelihood ratio evaluated at zero to differentiate contributions of each covariate given to a response variable. We show that, under some mild technical conditions, the proposed marginal empirical likelihood screening methods have a sure screening property and the extent to which the dimensionality can be reduced is also explicitly quantified. We also propose a data-driven thresholding and an iterative marginal empirical likelihood methods to enhance the finite sample performance for fitting sparse additive models. Simulation results and real data analysis demonstrate the proposed methods work competitively and performs better than competitive methods in error of a heteroscedastic case.