Accounting for individual variability in inversion shortcut use

This study investigated whether children's inversion shortcut use (i.e., reasoning that no calculations are required for the problem 4 × 8 ÷ 8, as the answer is the first number) is related to their analogical reasoning ability, short-term memory capacity, and working memory capacity. Children from Grades 6 and 8 solved multiplication and division inversion problems and classical analogy word problems and completed memory tasks. Analogical reasoning ability and working memory functioning both accounted for individual variance in inversion shortcut use. These findings suggest that the ability to understand relationships and executive functioning may enable children to internally represent and manipulate mathematical problems, facilitating the application of conceptual mathematical knowledge to generate the inversion shortcut.     

Inversion in mathematical thinking and learning

Inversion is a fundamental relational building block both within mathematics as the study of structures and within people’s physical and social experience, linked to many other key elements such as equilibrium, invariance, reversal, compensation, symmetry, and balance. Within purely formal arithmetic, the inverse relationships between addition and subtraction, and multiplication and division, have important implications in relation to flexible and efficient computation, and for the assessment of students’ conceptual understanding. It is suggested that the extensive research on arithmetic should be extended to take account of numerical domains beyond the natural numbers and of the difficulties students have in extending the meanings of operations to those of more general domains. When the range of situations modelled by the arithmetical operations is considered, the complexity of inverse relationships between operations, and the variability in the forms that these relationships take, become much greater. Finally, some comments are offered on the divergent goals and preoccupations of cognitive psychologists and mathematics educators as illuminated by research in this area.     

A catalog of models for multiplication and division of whole numbers

Summary The above catalog contains fifteen headings, each of which indicates a collection of families of models for multiplication and division of whole numbers. The catalog refers to somewhat more than sixteen families of models which are easily distinguished one from the other.Not included in the catalog thus far developed are several interpretations of multiplication and division that are also of interest. Among these are models based on the equivalency of denominations of money and various units of measurement. Other interpretations which are of historical interest are those of McLellan and Dewey [15] and Thorndike [24]. The relation between models of operations on whole numbers and models of operations defined on larger universal sets is also of interest. One aspect of this area of interest is the process of constructing models of multiplication and division of whole numbers from such models by altering the rules of the model or delimiting its universal set. For example, one can begin with one of Diénès' models of multiplication of integers [8, pp. 57–58] and make approapriate adjustments and result in a model of multiplication of whole numbers. Other interpretations developed by Diénès are of interest because they involve concretizations of whole numbers which are operators as opposed to states [8, pp. 12, 30; 9, p, 36].These are a great many strategies available for the use of models in teaching the operations on whole numbers. In one such strategy, an educator can define either multiplication or division on some basis (most likely in terms of a model) and then the other can be defined as its inverse.Another strategy is to define each operation in terms of a different model. For example, one might define multiplication in terms of the repeated addition model and division in terms of the repeated subtraction model.Still another type of procedure involves a multiple embodiment strategy in which several interpretations are taught as representing each operation.The choice of a particular strategy would depend upon a great many factors. Some of the factors would be the type of culture and students for which the program is written, the psychological assumptions adopted by the writer, and the writer's knowledge of the domain of models for the operations as well as their relation to the abstract mathematical domain which they represent. This article has contributed to a basis for intelligent decisions in this area by presenting a characterization of the domain of models for multiplication and division of whole numbers and their relation to the abstract operations.     

Dutch sixth graders’ use of shortcut strategies in solving multidigit arithmetic problems

Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth graders (12-year-olds) use shortcut strategies in solving multidigit addition, subtraction, multiplication, and division problems, to what extent student factors and task instructions affected this frequency of shortcut strategy use, and to what extent the strategies differed in performance. A sample of 648 sixth graders from 23 Dutch primary schools completed a paper-and-pencil task of 12 multidigit arithmetic problems, designed to elicit specific shortcut strategies such as compensation. Based on the students’ written work, strategies were classified into whether a shortcut strategy was used or not. Results showed that the frequency of shortcut strategies ranged between 6 and 21% across problem types, and that boys and high mathematics achievers were more inclined to use shortcut strategies. An explicit instruction to look for a shortcut strategy increased the frequency of these strategies in the addition and multiplication problems, but not in the subtraction and division problems. Finally, the use of shortcut strategies did not yield higher performance than using standard strategies. All in all, spontaneous as well as stimulated use of shortcut strategies by Dutch sixth graders was not very common.     

Acquisition and use of shortcut strategies by traditionally schooled children

This study aimed at analysing traditionally taught children’s acquisition and use of shortcut strategies in the number domain 20–100. One-hundred-ninety-five second, third, and fourth graders of different mathematical achievement levels participated in the study. They were administered two tasks, both consisting of a series of two-digit additions and subtractions that maximally elicit the use of the compensation and indirect addition strategy (, so the answer is 2 + 1 or 3). In the first task, children were instructed to solve all items as accurately and as fast as possible with their preferred strategy. The second task was to generate at least two different strategies for each item. Results demonstrated that children of all grades and all achievement levels hardly applied the compensation and indirect addition strategy in the first task. Children’s strategy reports in the second task revealed that younger and lower achieving children did not apply these strategies because they did not (yet) discover these strategies. By contrast, older and higher achieving children appeared to have acquired these strategies by themselves. Results are interpreted in relation to cognitive psychological and socio-cultural perspectives on children’s mathematics learning.

How reform curricula in the USA and Korea present multiplication and division of fractions

In order to give insights into cross-national differences in schooling, this study analyzed the development of multiplication and division of fractions in two curricula: Everyday Mathematics (EM) from the USA and the 7th Korean mathematics curriculum (KM). Analyses of both the content and problems in the textbooks indicate that multiplication of fractions is developed in KM one semester earlier than in EM. However, the number of lessons devoted to the topic is similar in the two curricula. In contrast, division of fractions is developed at about the same time in both curricula, but due to different beliefs about the importance of the topic, KM contains five times as many lessons and about eight times as many problems about division of fractions as EM. Both curricula provide opportunities to develop conceptual understanding and procedural fluency. However, in EM, conceptual understanding is developed first followed by procedural fluency, whereas in KM, they are developed simultaneously. The majority of fraction multiplication and division problems in both curricula requires only procedural knowledge. However, multistep computational problems are more common in KM than in EM, and the response types are also more varied in KM.     

Young children's knowledge of fair sharing as an informal basis for understanding division: A latent profile analysis

This study examined patterns of individual differences in young children's early understanding of division. Two hundred and thirty-seven 5- and 6-year-old children completed division tasks that assessed their understanding of the direct and inverse relations in division in two different situations – partitive and quotitive. Two main results emerged from our latent profile analyses. First, all children who had good performance in the inverse-relation problems also performed well in the direct-relation problems, but the converse was not true. Second, all children who performed well in the inverse-relation problems in quotitive situations also performed well in the inverse-relation problems in partitive situations, but not vice versa. The findings highlight the importance of situations in determining children's success in recognizing the inverse relation in division. Several theoretical implications for the development of children's division concepts and educational implications for assessment and teaching are discussed.     

Elementary School Students’ Mental Computation Proficiencies

Mental computation helps children understand how numbers work, how to make decisions about procedures, and how to create different strategies to solve math problems. Although researchers agree on the importance of mental computation skills, they debate how to help students develop these skills. The present study explored the existing literature in order to identify key points that are related to students’ use of different mental calculation strategies in a variety of settings and their conceptual understanding of those strategies.     

Random quadralinear forms and schur product on tensors

In this work, we made progress on the problem that lr(○×)lp(○×)lq is a Banach algebra under schur product. Our results extend Tonge's results. We also obtained estimates for the norm of the random quadralinear form A:lMr×lNp×lKq×lHs→ C, defined by: A(ei, ej, ek, es)=aijks, where the (aijks)'s are uniformly bounded, independent, mean zero random variables. We proved that under some conditions lr(○×)lp(○×)lq (○×)ls is not a Banach algebra under schur product.     

Teaching children how to include the inversion principle in their reasoning about quantitative relations

The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research on improving children’s understanding and use of the principle of inversion through interventions has only been applied to the solving of a + b − b = ? sums. The main aim of the intervention described in this article was to study the effects of teaching children about the explicit use of inversion as part of the relational calculus needed to solve inverse addition and subtraction problems using a calculator. The study showed that children taught about relational calculus differed significantly from those who were taught numerical procedures, and also that effects of the intervention were stronger when children were taught about relational calculus with mixtures of indirect and direct word problems than when these two types of problem were given to them in separate blocks.     

Historical Objections Against the Number Line

Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students’ difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians such as Arnauld, Leibniz, Wallis, Euler and d’Alembert. Not only does division by negative numbers pose problems for the number line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics, we argue for the introduction of negative numbers in education within the context of symbolic operations.     

Study of factors influencing research productivity of agriculture faculty members in Iran

The purpose of this research is to analyze the relationship between individual, institutional and demographic characteristics on one hand and the research productivity of agriculture faculty members on the other. The statistical population of the research comprises 280 academic staff in agricultural faculties all over Tehran Province. The data regarding research productivity and demographic characteristics were extracted from the faculty members’ profiles. Questionnaires were utilized to collect information concerning individual and institutional variables. The reliably of the questionnaire was calculated to be between 0.74 and 0.97 using the Cronbach’s Alpha. The regression analysis revealed that from among demographic characteristics two variables, namely, academic rank and age ( \textR_\textAD ² {\text{R}}_{\text{AD}}^{ 2}  = 0.265), among individual characteristics, three variables, namely, working habits, creativity as well as autonomy and commitment ( \textR_\textAD ² {\text{R}}_{\text{AD}}^{ 2}  = 0.097), and among institutional characteristics four variables namely, network of communication with colleagues, resources of facilities, corporate management and clear research objectives ( \textR_\textAD ² {\text{R}}_{\text{AD}}^{ 2}  = 0.151) were significant predictors for agricultural faculty members’ research productivity.     

Computational skills, working memory, and conceptual knowledge in older children with mathematics learning disabilities

Knowledge and skill in multiplication were investigated for late elementary-grade students with mathematics learning disabilities (MLD), typically achieving age-matched peers, low-achieving age-matched peers, and ability-matched peers by examining multiple measures of computational skill, working memory, and conceptual knowledge. Poor multiplication fact mastery and calculation fluency and general working memory discriminated children with MLD from typically achieving age-matched peers. Furthermore, children with MLD were slower in executing backup procedures than typically achieving age-matched peers. The performance of children with MLD on multiple measures of multiplication skill and knowledge was most similar to that of ability-matched younger children. MLD may be due to difficulties in computational skills and working memory. Implications for the diagnosis and remediation of MLD are discussed.     

Equivalent Structures on Sets: Equivalence Classes, Partitions and Fiber Structures of Functions

This study reports on how students can be led to make meaningful connections between such structures on a set as a partition, the set of equivalence classes determined by an equivalence relation and the fiber structure of a function on that set (i.e., the set of preimages of all sets {b} for b in the range of the function). In this paper, I first present an initial genetic decomposition, in the sense of APOS theory, for the concepts of equivalence relation and function in the context of the structures that they determine on a set. This genetic decomposition is primarily based on my own mathematical knowledge as well as on my observations of students’ learning processes. Based on this analysis, I then suggest instructional procedures that motivate the mental activities described in the genetic decomposition. I finally present empirical data from informal interviews with students at different stages of learning. My goal was to guide students to become aware of the close conceptual correspondence and connections among the aforementioned structures. One theorem that captures such connections is the following: a relation R on a set A is an equivalence relation if and only if there exists a function f defined on A such that elements related via R (and only those) have the same image under f.     

Number lines,but not area models,support children’s accuracy and conceptual models of fraction division

The Common Core State Standards in Mathematics recommends that children should use visual models to represent fraction operations, such as fraction division. However, there is little experimental research on which visual models are the most effective for helping children to accurately solve and conceptualize these operations. In the current study, 123 fifth and sixth grade students solved fraction division problems in one of four visual model conditions: number lines, circular area models, rectangular area models, or no visual model at all. Children who solved the problems accompanied by a number line were more accurate and showed evidence of consistently producing sound conceptual models across the majority of problems than did children who completed problems with either area model or no visual model at all. These findings are particularly striking given that children have experienced partitioning area models into equal shares as early as first grade, thus circles and rectangles were likely familiar to children. The number line advantage may stem from the fact that they afford the ability to represent both operand magnitudes in relation to one another and relative to a common endpoint. Future work should investigate the optimal order that instructors should introduce various visual models to promote children’s representational fluency across number lines and area models.     

Semantic structures of one-step word problems involving multiplication or division

From classifications of word problems in international discussion of elementary mathematics instruction as well as from conceptual elaborations of didactical analyses in Germany, a classification of semantic structures of one-step word problems involving multiplication or division is proposed, comprehending four main classes: Forming the n-th multiple of measurers, combinatorial multiplication, composition of operators, and multiplication by formula. This classification is more comprehensive and differentiated than the classifications of Vergnaud (1983), Nesher (1988), and Bellet al. (1989) — aiming at a better assignment between diverse contextual circumstances and conceptual demands of mathematics and at compatibility with the well-known semantic structures of addition and subtraction word problems.     

The role of number words in preschoolers’ addition concepts and problem‐solving procedures

Preschoolers’ conceptual understanding and procedural skills were examined so as to explore the role of number‐words and concept–procedure interactions in their additional knowledge. Eighteen three‐ to four‐year‐olds and 24 four‐ to five‐year‐olds judged commutativity and associativity principles and solved two‐term problems involving number words and unknown numbers. The older preschoolers outperformed younger preschoolers in judging concepts involving unknown numbers and children made more accurate commutativity than associativity judgements. Children with conceptual profiles indicating a strong understanding of concepts applied to unknown numbers were more accurate at solving number‐word problems than those with a poor conceptual understanding. The findings suggest that an important mathematical development during the preschool years may be learning to appreciate addition concepts as general principles that apply when exact numbers are unknown.     

Procedural and conceptual changes in young children’s problem solving

This study analysed the different types of arithmetic knowledge that young children utilise when solving a multiple-step addition task. The focus of the research was on the procedural and conceptual changes that occur as children develop their overall problem solving approach. Combining qualitative case study with a micro-genetic approach, clinical interviews were conducted with ten 5–6-year-old children. The aim was to document how children combine knowledge of addition facts, calculation procedures and arithmetic concepts when solving a multiple-step task and how children’s application of different types of knowledge and overall solving approach changes and develops when children engage with solving the task in a series of problem solving sessions. The study documents children’s pathways towards developing a more effective and systematic approach to multiple-step tasks through different phases of their problem solving behaviour. The analysis of changes in children’s overt behaviour reveals a dynamic interplay between children’s developing representation of the task, their improved procedures and gradually their more explicit grasp of the conceptual aspects of their strategy. The findings provide new evidence that supports aspects of the “iterative model” hypothesis of the interaction between procedural and conceptual knowledge and highlight the need for educational approaches and tasks that encourage and trigger the interplay of different types of knowledge in young children’s arithmetic problem solving.     

图的整数值控制函数

For an arbitrary subset P of the reals, a function f ： V →P is defined to be a P-dominating function of a graph G = （V, E） if the sum of its function values over any closed neighbourhood is at least 1. That is, for every v ∈ V, f（N[v]） ≥ 1. The definition of total P-dominating function is obtained by simply changing ‘closed＇ neighborhood N[v] in the definition of P-dominating function to ‘open＇ neighborhood N（v）. The （total） P-domination number of a graph G is defined to be the infimum of weight w（f） = ∑v ∈ V f（v） taken over all （total） P-dominating function f. Similarly, the P-edge and P-star dominating functions can be defined. In this paper we survey some recent progress on the topic of dominating functions in graph theory. Especially, we are interested in P-, P-edge and P-star dominating functions of graphs with integer values.     

Early learning and awareness of division: A phenomenographic approach

Interviews with 72 pupils in grade 2–6 were used to investigate awareness of the relation between situation and computation in simple quotitive and partitive division problems as informally and formally experienced. The research approach was phenomenographic. Most second graders counted or made drawings, and related these methods to the situation described in the problems. Several of the older children, on the contrary, experienced a conflict between computation and situation in partitive division. Most second graders, but also some third, fourth and sixth graders, could still not carry out repeated addition, the precursor of multiplication. The data are finally viewed from two theoretical perspectives other than phenomenography. It was concluded that formal division, understood as related to everyday situations, only develops in interplay with informal knowledge.This revised version was published online in September 2005 with corrections to the Cover Date.