Visual messages: Making sense with young children

Whether we know it or not, the environment teaches children how to read. The symbols, pictures, and print in our classrooms are a foundation for later reading. Children need useful and practical information communicated through visual messages.Sandra L. Gordon and Beth C. Anderson are Associate Professors in the Education Department at Moorhead State University in Minnesota.     

Making sense of student mathematical thinking: the role of teacher mathematical thinking

Educational Studies in Mathematics - In mathematical whole-class discussions, teachers can build on various student ideas and develop these ideas toward mathematical goals. This requires teachers...     

Reading fluency: implications for the assessment of children with reading disabilities

The current investigation explored the diagnostic utility of reading fluency measures in the identification of children with reading disabilities. Participants were 50 children referred to a university-based clinic because of suspected reading problems and/or a prior diagnosis of dyslexia, where children completed a battery of standardized intellectual, reading achievement, and processing measures. Within this clinical sample, a group of children were identified that exhibited specific deficits in their reading fluency skills with concurrent deficits in rapid naming speed and reading comprehension. This group of children would not have been identified as having a reading disability according to assessment of single word reading skills alone, suggesting that it is essential to assess reading fluency in addition to word reading because failure to do so may result in the under-identification of children with reading disabilities.     

Subtraction by addition in children with mathematical learning disabilities

In the last decades, strategy variability and flexibility have become major aims in mathematics education. For children with mathematical learning disabilities (MLD) it is unclear whether the same goals can and should be set. Some researchers and policy makers advise to teach MLD children only one solution strategy, others advocate stimulating the flexible use of various strategies, as for typically developing children. To contribute to this debate, we compared the use of the subtraction by addition strategy to mentally solve two-digit subtractions in children with and without MLD. We used non-verbal research methods to infer strategy use patterns, and found that both groups of children switch between the traditionally taught direct subtraction strategy and subtraction by addition, based on the relative size of the subtrahend. These findings challenge typical special education classroom practices, which only focus on the routine mastery of the direct subtraction strategy.     

Building number sense with Number Worlds: a mathematics program for young children

Number sense is easy to recognize but difficult to define and hence, to teach. In this article, number sense is defined in terms of the knowledge known to underlie it, as identified in cognitive developmental theory and research. A preK-2 mathematics program, called Number Worlds, that was specifically developed to teach this knowledge is described and five instructional principles that lie at the heart of the program, and that are grounded in recent theories of learning and cognitive development, are used to illustrate several of its features. Evidence of the program’s effectiveness in teaching number sense and in enhancing the mathematics learning and achievement of at-risk children is briefly presented.     

Feeling number: grounding number sense in a sense of quantity

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.     

Mathematical thinking intervention programmes for preschool children with normal and low number sense

This study investigated the possibility of enhancing the level of preschoolers’ number sense by introducing two intervention programmes, Let’s think! and Young children with special educational needs count, too! .Forty‐five preschoolers, mean age 66.4 months, were randomly assigned to the experimental and control groups. The experimental group received instruction twice a week, for half an hour, and for a period of nine months. Pretest–postest comparison revealed that the experimental group showed enhanced number‐sense performance immediately after the instruction ended, but the between‐group difference faded after six months. There were no statistically significant differences between the groups in general mathematical thinking abilities (transfer tasks) after the intervention.     

Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities

The study sought out to extend our knowledge regarding the origin of mathematical learning disabilities (MLD) in children by testing different hypotheses in the same samples of children. Different aspects of cognitive functions and number processing were assessed in fifth- and sixth-graders (11–13 years old) with MLD and compared to controls. The MLD group displayed weaknesses with most aspects of number processing (e.g., subitizing, symbolic number comparison, number-line estimation) and two cognitive functions (e.g., visual–spatial working memory). These findings favor the defective approximate number system (ANS) hypothesis, but do not fit well with the access deficit hypothesis. Support is also provided for the defective object-tracking system (OTS) hypothesis, the domain general cognitive deficit hypothesis and to some extent the defective numerosity-coding hypothesis. The study suggests that MLD might be caused by multiple deficits and not a single core deficit.     

A model for nurturing and assessing multidigit number sense among first grade children

Based on a synthesis of the literature and on the results of a two-year teaching program working with young primary children, a framework was developed, refined and validated for nurturing and assessing multidigit number sense. The major constructs incorporated in this framework were counting, partitioning, grouping, and number relationships. For each of these constructs, four different levels of thinking were established which, in essence, reflected a learning apprenticeship for multidigit number sense. At each level, and across all four constructs, learning indicators were developed and matched to distinctive problem tasks that went beyond the four basic operations.The framework was validated through data obtained from six case studies of grade 1 children. The thinking of these children was assessed and analyzed on the problem tasks for the four constructs and four levels. While the students were at different levels, all but one showed striking consistencies across the four constructs. Moreover, no student was able to solve a problem at a higher level when they had not solved a lower-level problem in the same category. The present framework for multidigit number sense covers only the lower primary grades, but research and instruction would benefit from an extended framework across the elementary grades.     

Assessing preschool number sense: skills demonstrated by children prior to school entry

Components of early number sense, as identified in two Delphi studies and in the number sense literature related to mathematics difficulties, were assessed for 176 children in preschools and childcare centres across one local government area in Sydney, Australia, using tasks or modifications of tasks reported in the number sense literature. In addition, the children’s receptive vocabulary was measured using The Peabody Picture Vocabulary Test (third edition) and math reasoning was measured using Woodcock‐Johnson III Tests of Achievement. Although the children demonstrated a broad range of skills, there were no significant differences between children attending childcare and preschools for any of the measures. However, boys performed significantly better than girls in quantitative concepts and girls performed better than boys in subitising. In discussing the data, a comparison is made of the skills demonstrated by children and skills that were highlighted in the two Delphi studies and in the early number sense literature as being essential components of number sense prior to school entry. Implications for kindergarten mathematics curricula and approaches to the teaching of early number skills are discussed.     

Making sense of separation

Judith Leipzig is an Infant/Parent, Development Specialist in West Nyack, New York.     

Making sense in education: Deleuze on thinking against common sense

According to a widespread view, one of the most important roles of education is the nurturing of common sense. In this article I turn to Gilles Deleuze’s concept of sense to develop a contrary view of education—one that views education as a radical challenge to common sense. The discussion will centre on the relation of sense and common sense to thinking. Although adherents of common sense refer to it as the basis of all thought and appeal to critical thinking as instrumental in eliminating its occasional errors, I shall argue, following Deleuze, that common sense education in fact thwarts thinking, while only education which revolves around making sense may provoke thinking that goes beyond the self-evident. I demonstrate how making sense can become an educational encounter that breaks hierarchies and generates thinking independently of the thinker’s knowledge and place in the sociopolitical order. The present article attempts, therefore, to put some sense into Deleuzian education for thinking, and thereby shed new light on its radical-political, counter-commonsensical power.     

Making sense of iconic symbols: a study of preschool children conducting a refuse-sorting task

This article is part of a larger project focusing upon explanatory illustrations that children encounter in pre- and primary school education. The research questions concerned (a) how preschool children make sense of iconic symbols when placing items of refuse on illustrations of refuse bins in a sorting task and (b) what stumbling blocks they encounter when interpreting these symbols. Video data were collected with 30 children between four and five?years of age. From the children’s verbal and non-verbal interactions, four different categories of sense-making were constructed: by material, by object type, by appearance and by function. Three stumbling blocks were identified. The first had to do with giving the symbols a different logical meaning to the intended one; the second related to what materials the different refuse items were made of; the third was being able to stick to one correct way of interpreting each symbol.     

Making sense of graphs: does metacognitive instruction make a difference on students' mathematical conceptions and alternative conceptions?

The present study investigates the differential effects of cooperative learning with or without metacognitive instruction on making sense of graphs. Participants were 196 eighth-graders who studied in six classrooms. Data were analyzed by using quantitative and qualitative methods. Results indicated that students who were exposed to the metacognitive instruction within cooperative learning (COOP + META) significantly outperformed their counterparts who were exposed to cooperative learning with no metacognitive instruction (COOP). The positive effects of COOP + META were observed on both graph interpretation and graph construction (transfer task) with regard to alternative conceptions. Furthermore, observations indicated differential characteristics of discourse behaviors during small group interaction under these methods. The practical implications of the study are discussed.     

Social acknowledegments for children with disabilities: effects of service dogs

While service dogs are known to perform important tasks for people using wheelchairs, such as retrieving dropped items or pulling a wheelchair, they may also serve as an antidote for social ostracism. Adults in wheelchairs have been found to receive many more social acknowledgements when a service dog is present than when not. This study examined whether disabled children in wheelchairs with service dogs receive more frequent social acknowledgment than when no dog is present. Behaviors of passersby in response to children in wheelchairs were recorded in shopping malls and on school playgrounds. In both settings, social acknowledegments (e.g., friendly glances, smiles, and conversations) were substantially more frequent when a service dog was present. Social effects of the dog were more pronounced in shopping malls, typical of unfamiliar settings where the child would be likely to experience being ignored or overlooked. Service dogs may assist in normalizing the social interactions for children with disabilities producing social isolation.