The Measurement of Volume: Why Do Young Children Measure Inaccurately?

A total of 257 children in grades 2–5 were individually interviewed to find the grade level at which they demonstrated transitive reasoning and unit iteration in the measurement of volume. In the transitivity task, the children were asked if a larger, empty container could be used to compare the quantity of popcorn kernels (about 350 cc) in two containers that looked very different. The unit‐iteration task was similar except that children were asked if a small cup could be used to compare similar quantities of rice in two containers. It was found that a majority of children (51%) demonstrated transitive reasoning by third grade and that a majority (56%) demonstrated unit iteration by fourth grade. A conclusion reached is that the standard of the National Council of Teachers of Mathematics (2000) expecting children to understand units of measurement by grade 2 is unrealistic. Better principles of teaching are also suggested to encourage children to think logically.     

The Measurement of Time: Children's Construction of Transitivity,Unit Iteration,and Conservation of Speed

One hundred twenty children in kindergarten and grades 2, 4, and 6 were individually interviewed with five Piagetian tasks to determine the grade level at which most have constructed transitive reasoning, unit iteration, and the conservation of speed. The responses were categorized as “successful,”“unsuccessful,” or “transitional.” By combining the “successful” and “transitional” categories, it was found that the children reasoned transitively by second grade (70.0%) and demonstrated unit iteration and conservation of speed by sixth grade (70.0% and 83.3%, respectively). It was concluded that the construction of the logic necessary to make sense of the measurement of time is generally not complete before sixth grade.     

Fuzzy processing in transitivity development

We review the literature on the development of transitive reasoning, and note three historical stages. Stage 1 was dominated by the Piagetian idea that transitive inference is logical reasoning in which relationships between adjacent terms figure as premises. Stage 2 was dominated by the information-processing view that memory for relationships between adjacent terms is determinative in transitivity performance. Stage 3 has produced data that are inconsistent with both the logic and memery positions, leading to a new theory that is designed to account for such findings, fuzzy-trace theory. The basic assumption of fuzzytrace theory is that reasoners rely on global patterns, or gist. We describe the tenets of fuzzytrace theory, and explore its implications for different theoretical conceptions of logical competence, concluding that young children possess transitivity competence. We discuss the connection between transitivity competence (cognition) and intransitive preferences (metacognition).     

Using Prediction to Promote Mathematical Understanding and Reasoning

Research has shown that prediction has the potential to promote the teaching and learning of mathematics because it can be used to enhance students' thinking and reasoning at all grade levels in various topics. This article addresses the effectiveness of using prediction on students' understanding and reasoning of mathematical concepts in a middle school algebra context. In the treatment classroom, prediction questions were utilized at the launch of each algebra lesson, and in the control classroom such questions were not used. Both classrooms were taught by the same teacher and used the same curriculum. After completing each of the linear and exponential units, the two classrooms were compared in terms of their mathematical understanding and reasoning through unit assessments. Overall, the treatment classroom outperformed the control classroom on the unit assessments. This result supports that prediction is a valid construct with respect to enhanced conceptual understanding and mathematical reasoning.     

Comparison of Existence Results for Efficient Points

Existence results of maximal points with respect to general binary relations were stated by Hazen and Morin (Ref. 1) and by Gajek and Zagrodny (Ref. 2). In this paper, we point out that the natural framework for this problem is that of transitive and reflexive relations (preorders). The aim of this paper is to discuss existence results for maximal points with respect to general transitive relations in such a way that, when considering them for preorders defined by convex cones, we are able to recover most known existence results for efficient points; the quasi-totality of them, with their (short) proofs, is presented, too.     

Transitivity is neither hereditary nor finitely productive

We construct a transitive space that is the union of two subspaces homeomorphic to the (non-transitive) Kofner plane. Moreover, we show that the product of two transitive spaces need not be transitive. Finally, we observe that results of E.K. van Douwen establish that, under b = c, there exists a locally countable locally compact non-transitive zero-dimensional space. It follows that under b = c neither a locally transitive nor a compact space need be transitive.     

几种模糊传递关系的收敛性

本文对几种模糊传递关系进行比较 ,从一个侧面说明了用于构造偏好关系的模糊关系应是互逆关系 .文中证明了对任意的 n阶互逆传递模糊关系 R,必有 Rn=Rn+1 .     

Children’s unit concepts in measurement: a teaching experiment spanning grades 2 through 5

We examined ways of improving students’ unit concepts across spatial measurement situations. We report data from our teaching experiment during a six-semester longitudinal study from grade 2 through grade 5. Data include instructional task sequences designed to help children (a) integrate multiple representations of unit, (b) coordinate and group units into higher-order units, and (c) recognize the arbitrary nature of unit in comparison contexts and student’s responses to tasks. Our results suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This research extends prior work addressing the growth of unit concepts in the contexts of length, area, and volume by demonstrating the viability of level-specific instructional actions as a means for promoting an informal theory of measurement.     

Investigating written expressions of mathematical reasoning for students with learning disabilities

Assessment results from two open-construction response mathematical tasks involving fractions and decimals were used to investigate written expression of mathematical reasoning for students with learning disabilities. The solutions and written responses of 51 students with learning disabilities in fourth and fifth grade were analyzed on four primary dimensions: (a) accuracy, (b) five elements of mathematical reasoning, (c) five elements of mathematical writing, and (d) vocabulary use. Results indicate most students were not accurate in their problem solution and communicated minimal mathematical reasoning in their written expression. In addition, students tended to use general vocabulary rather than academic precise math vocabulary and students who provided a visual representation were more likely to answer accurately. To further clarify the students struggles with mathematical reasoning, error analysis indicated a variety of error patterns existed and tended to vary widely by problem type. Our findings call for more instruction and intervention focused on supporting students mathematical reasoning through written expression. Implications for research and practice are presented.     

Constructing highly arc transitive digraphs using a direct fibre product

We introduce a construction of highly arc transitive digraphs using a direct fibre product. This product generalizes some known classes of highly arc transitive digraphs but also allows us to construct new ones. We use the product to obtain counterexamples to a conjecture advanced by Cameron, Praeger and Wormald on the structure of certain highly arc transitive digraphs.     

Variation of student numerical and figural reasoning approaches by pattern generalization type,strategy use and grade level

This paper explored variation of student numerical and figural reasoning approaches across different pattern generalization types and across grade level. An instrument was designed for this purpose. The instrument was given to a sample of 1232 students from grades 4 to 11 from five schools in Lebanon. Analysis of data showed that the numerical reasoning approach seems to be more dominant than the figural reasoning approach for the near and far pattern generalization types but not for the immediate generalization type. The findings showed that for the recursive strategy, the numerical reasoning approach seems to be more dominant than the figural reasoning approach for each of the three pattern generalization types. However, the figural reasoning approach seems to be more dominant than the numerical reasoning approach for the functional strategy, for each generalization type. The findings also showed that the numerical reasoning was more dominant than the figural reasoning in lower grade levels (grades 4 and 5) for each generalization type. In contrast, the figural reasoning became more dominant than the numerical reasoning in the upper grade levels (grades 10 and 11).     

The stochastic economic lot sizing problem for non-stop multi-grade production with sequence-restricted setup changeovers

We study a variant of the stochastic economic lot scheduling problem (SELSP) encountered in process industries, in which a single production facility must produce several different grades of a family of products to meet random stationary demand for each grade from a common finished-goods (FG) inventory buffer that has limited storage capacity. When the facility is set up to produce a particular grade, the only allowable changeovers are from that grade to the next lower or higher grade. Raw material is always available, and the production facility produces continuously at a constant rate even during changeover transitions. All changeover times are constant and equal to each other, and demand that cannot be satisfied directly from inventory is lost. There is a changeover cost per changeover occasion, a spill-over cost per unit of product in excess whenever there is not enough space in the FG buffer to store the produced grade, and a lost-sales cost per unit short whenever there is not enough FG inventory to satisfy the demand. We model the SELSP as a discrete-time Markov decision process (MDP), where in each time period the decision is whether to initiate a changeover to a neighboring grade or keep the set up of the production facility unchanged, based on the current state of the system, which is defined by the current set up of the facility and the FG inventory levels of all the grades. The goal is to minimize the (long-run) expected average cost per period. For problems with more than three grades, we develop a heuristic solution procedure which is based on decomposing the original multi-grade problem into several 3-grade MDP sub-problems, numerically solving each sub-problem using value iteration, and constructing the final policy for the original problem by combining parts of the optimal policies of the sub-problems. We present numerical results for problem examples with 2–5 grades. For the 2- and 3-grade examples, we numerically solve the exact MDP problem using value iteration to obtain insights into the structure of the optimal changeover policy. For the 4- and 5-grade examples, we compare the performance of the decomposition-based heuristic (DBH) solution procedure against that obtained by numerically solving the exact problem. We also compare the performance of the DBH method against the performance of three simpler parameterized heuristics. Finally, we compare the performance of the DBH and the exact solution procedures for the case where the FG inventory storage consists of a number of separate general-purpose silos capable of storing any grade as long as it is not mixed with any other grade.     

Predetermined accommodations with a standardized testing protocol: Examining two accommodation supports for developing fraction thinking in students with mathematical difficulties

This study examined the effects of two pre-determined accommodations that were provided in a standardized testing. The two accommodations were meant to help students with difficulties in mathematics (SDMs) engage in unit thinking, reasoning, and coordination and consequently improve their ability to process fraction tasks. 23 middle school SDMs took the following tests and were asked to explain their solutions: a baseline fraction test without any accommodation; an annotated test with bolded information and additional simplified explanations; and a warming- up test that involved whole-number multiplicative reasoning tasks followed by the baseline test. Results show that while SDMs were able to construct and coordinate fraction units to solve fraction problems when appropriate accommodations were provided, standardized assessment with a predetermined “one-size-fits -all” accommodation could not meet the specific needs of all students with mathematics learning difficulties.     

Polynomial alternatives for the group of affine motions

It is known that every polycyclic-by-finite group – even if it admits no affine structure – allows a polynomial structure of bounded degree. A major obstacle to a further development of the theory of these polynomial structures is that the group of the polynomial diffeomorphisms of , in contrast to the group of affine motions, is no longer a finite dimensional Lie group. In this paper we construct a family of (finite dimensional) Lie groups, even linear algebraic groups, of polynomial diffeomorphisms, which we call weighted groups of polynomial diffeomorphisms. It turns out that every polycyclic-by-finite group admits a polynomial structure via these weighted groups; in the nilpotent (and other) case(s), we can sharpen, by specifying a nice set of weights, the existence results obtained in earlier work. We introduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existence of simply transitive actions of the corresponding Mal`cev completion. This, and other properties, provide a strong analogy with the situation of affine structures and simply transitive affine actions considered e.g. in the work of Fried, Goldman and Hirsch. Received November 30, 1998; in final form March 10, 1999     

Transitivity of the norm on Banach spaces¶ having a Jordan structure

We study transitivity conditions on the norm of JB ^*-triples, C ^*-algebras, JB-algebras, and their preduals. We show that, for the predual X of a JBW ^*-triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB ^*-triple with transitive norm is a Hilbert space. We extend to arbitrary C ^*-algebras previously known characterizations of transitivity [20] and convex transitivity [36] of the norm on commutative C ^*-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB-algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB-algebra with almost transitive norm is isomorphic to the field of real numbers. Received: 9 June 1999 / Revised version: 20 February 2000     

Preschool children's collective mathematical reasoning during free outdoor play

This paper illustrates how young children (age 1–5) use mathematical properties in collective reasoning during free outdoor play. The analysis of three episodes is presented. The results from the analysis of the argumentation show that the children used a variation of mathematical products and procedures, to challenge, support and drive the reasoning forward. When needed, they utilise concrete materials to illustrate and strengthen their arguments, and as an aid in order to reach conclusions. The children also use abstract social constructs, such as jokes, as part of their reasoning.     

Operations on binary relations

The effects of five basic operations (asymmetrization, complementation, dualization, symmetrization, transitive closure) on binary relations are examined. Identifies between compound operations are developed (e.g. the symmetric part of the transitive closure of the complement of the transitive closure equals the transitive closure of the symmetric complement of the transitive closure), ordering aspects of compound operations are noted, and it is shown that in addition to the empty and universal relations at most 110 different relations can be generated from a given binary relation by sequential applications of the five basic operations. Moreover, 110 is the least upper bound, and none of these 110 requires more than seven applications of the basic operations for its expression. One of the potentially irreducible compound operations of length seven is cstcatc, the complement of the symmetric part of the transitive closure of the complement of the asymmetric part of the transitive closure of the complement.     

基于传递性的效率评价及改进

现有交叉效率矩阵中往往会存在一些同行效率评价远低于自我效率评价的情形,即出现同行间的恶意评价。本文将第三方作为间接元素,考虑到这些间接元素在效率评价矩阵中具有传递性,引入间接判断结果对原有的交叉效率矩阵进行反复迭代,并证明迭代过程的最终稳定性。迭代的最终结果可以消除效率矩阵中的恶意评价元素,得到新的效率评价矩阵。最后,通过算例来说明模型的可行性和实用性。     

Evaluation of three interventions teaching area measurement as spatial structuring to young children

We evaluated the effects of three instructional interventions designed to support young children’s understanding of area measurement as a structuring process. Replicating microgenetic procedures we used in previous research with older children to ascertain whether we can build these competencies earlier, we also extended the previous focus on correctness to include analyses of children’s use of procedural and conceptual knowledge and examined individual differences in strategy shifts before and after transitions, enabling a more detailed examination of the hypothesized necessity of development through each level of a learning trajectory. The two experimental interventions focused on a dynamic conception of area measurement while also emphasizing unit concepts, such as unit identification, iteration, and composition. The findings confirm and extend earlier results that seeing a complete record of the structure of the 2D array—in the form of a drawing of organized rows and columns—supported children’s spatial structuring and performance.