Learning disabled students' interactions with commercial courseware: A naturalistic study

A naturalistic study investigated the interactions of learning disabled students and their teachers with instructional elements of commercial courseware. During two academic years, 62 students aged 9 to 18 and seven teachers from five subject areas were observed and interviewed with regard to their day-to-day uses of 26 packages. Findings reported here relate to students' interactions with the stimulus, response, and feedback/reinforcement dimensions inherent in those packages. Details of students' problems and strategies with verbal, numeric, graphics, and problem-solving courseware are explored to elucidate the role of courseware design in students' pervasive reading difficulties, overreliance on teacher assistance, introduction of competition to provide motivation, and limited strategies for using courseware independently. Implications are noted for improving course-ware design not only for LD students, but also for others who would profit from more carefully designed packages. The relationship of design to the overarching issues of student independence, motivation, and strategy development for computer-based instruction is discussed. The original research upon which this article is based was supported in part by a Presidential Fellowship awarded by the Graduate School of The Ohio State University. Additional work was supported in part by a grant from the Office of Graduate Studies and Research of The University of Maryland, College Park.     

Linguistic and pedagogical factors affecting children's understanding of arithmetic word problems: A comparative study

2493 children aged from 5 to 15 years in 114 classes in 25 primary schools were asked to do a similar set of 22 arithmetic word problems. 1195 of the children were in Grades K through 6 in Victorian schools (Australia) and 1298 were in Grades 4 through 6 in Papua New Guinea (PNG) schools. For both samples the questions were posed in English. This was the first language for most children in the Victorian sample but, for the PNG sample, English was usually the second, third, or even fourth language (even though it is the language of instruction in PNG schools). While the test instrument was based on widely accepted information processing models of how children solve arithmetic word problems, the data obtained were not so much in accord with these models as with psycholinguistic theories on children's acquisition of polarised comparative pairs (like more and less). The data also indicate that children from the two samples used similar strategies and made similar errors, with the order of relative difficulty being the same for both samples, the main factor determining difficulty being the semantic structure of the questions. Differences in performance between corresponding grades from the two samples can be attributed to differences in the degree of English language competence rather than to numerical facility.     

Physics students' understanding of relative speed: A phenomenographic study

It is important that students of physics develop both quantitative and qualitative understanding of physical concepts and principles. Although accuracy and reliability in solving quantitative problems is necessary, a qualitative understanding is required in applying concepts and principles to new problems and in real-life situations. If students are not able to understand what underlies quantitative problem-solving procedures nor interpret the solution in physical terms, it is questionable whether they have developed an adequate understanding of physics. The research reported here is part of a larger phenomenographic study that is concerned with the assessment of physics students' understanding of some basic concepts and principles in kinematics. In this article students' understanding of the concept of relative speed is described. A variety of ways of understanding relative speed and of viewing a problem that dealt with this concept were uncovered. The results are used to suggest ways for teachers to proceed in assisting students to enhance their understanding of this concept. The teaching principles outlined concern both teaching relative speed, in particular, and teaching scientific concepts and principles, more generally.     

简单两步计算的应用题的基本训练

目前学生普遍感到解答应用题难、解答复合应用题就更难。如何教好简单应用题是最关键的一环。在教学简单应用题时需要一定的训练 ,才能提高解题能力 ,我常使用的解题训练主要有以下一些方法。一、抓关键词、句的训练。应用题是数量关系和语言文字的综合体。因此解答应用题必须具备坚实的语言文字基础 ,如果学生连题都读不通 ,那就谈不上正确的解答。例如 :一年级一班有 13名“三好学生” ,二班和一班同样多 ,两班一共有多少名“三好学生” ?让学生读题后 ,问这道题说了什么事 ?是哪个年级的事 ?有几个班 ?如果学生答不出来 ,再读 ,并让学生…     

Building middle school students' understanding of proportional reasoning through mathematical investigation

This article describes and shares an innovative pedagogical practice that holds promise in contributing to the teaching and learning of proportions in middle school. The teaching and learning of mathematics with understanding framework was used as a vehicle to help 21 seventh grade students reason proportionally. The findings of this unit suggest that the classroom culture, which encouraged the students to make connections between their existing and new ideas and reflecting and communicating their thinking, contributed to their emerging understanding of proportions. The use of an authentic and non-routine task involving liquid measurements also heightened their interest, curiosity and enthusiasm, thereby contributing to their excitement about the mathematics they were learning.     

Realistic considerations in mathematical modeling of school arithmetic word problems

The aim of the present study was to collect in a systematic way empirical data about the (lack of) activation of real-world knowledge during elementary school pupils' understanding and solution of school arithmetic word problems. Ten pairs of word problems were collectively administered to 75 fifth-graders during a typical mathematics lesson. While the first item of each pair could be modeled and solved in a straightforward and unproblematic way by one or two simple arithmetic operations with the given numbers, the second problem could not be modeled and solved in such a way, at least if one seriously takes into account the realities of the context called up by the problem statement. An analysis of the pupils' reactions to these problematic word problems shows an alarmingly small number of realistic responses or additional comments based on realistic considerations.     

培养学生数学问题表征能力的教学探索

在解决数学问题的过程中,表征能力能够优化问题结构,降低解决难度,使学生更好的理解数学问题。本文对培养学生数学问题表征能力的教学进行相应的探讨。     

Conceptualizing teachers' understanding of students' mathematical learning by using assessment tasks*

The study was designed to support teachers on conceptualizing their understanding of students' learning by the use of assessment tasks. A school-based assessment team consisting of the researcher and four third-grade teachers teaching in the same school was set up as a learning context of supporting teachers in developing assessment tasks integral to instruction. The assessment tasks along with students' responses to the task, classroom observations, interviews, routine weekly meetings, teachers' weekly reflective journals, and students' responses to the assessment tasks were the data collected in the study. The teachers' views of using assessment tasks and the generation of assessment tasks were developed in the course of the study. In the process of generating assessment tasks, teachers improved their awareness of students' various solutions and learning difficulties to a specific problem, their awareness of the importance of developing students' critical thinking, and their awareness of where students need to make a remedial instruction. The research reported in the paper was supported by the National Science Council of Taiwan under Grant NSC 90-2521-S134-003-. The opinions expressed in this paper are those of the author and do not necessarily reflect the view of funding agency.     

Deaf and hard of hearing students' performance on arithmetic word problems

There has been limited research into the intersection of language and arithmetic performance of students who are deaf or hard of hearing, although previous research has shown that many of these students are delayed in both language acquisition and arithmetic performance. The researchers examined the performance on arithmetic word problems of deaf and hard of hearing students in the South-East Queensland region of Australia; they also examined these students' problem-solving strategies. It was found that performance on word problems was similar for deaf and hearing students, but that deaf students experienced delays in achieving successful performance on word problems relative to their hearing peers. The results confirm the findings of other studies showing that students who are deaf or hard of hearing experience delayed language acquisition, which affects their capacity to solve arithmetic word problems. The study conclusions stress the need for greater use of direct teaching of analytic and strategic approaches to arithmetic word problems.     

Cognitive skills used to solve mathematical word problems and numerical operations: a study of 6- to 7-year-old children

This study investigates the relationship between skills that underpin mathematical word problems and those that underpin numerical operations, such as addition, subtraction, division and multiplication. Sixty children aged 6–7 years were tested on measures of mathematical ability, reading accuracy, reading comprehension, verbal intelligence and phonological awareness, using a mix of standardised and experimenter-designed tests. The experimental hypothesis was that mathematical word problems will call upon cognitive skills that are different and additional to those required by numerical operations, in particular verbal ability and reading comprehension, while phonological awareness and reading accuracy will be associated with both types of mathematical problems. The hypothesis is partly affirmed and partly rejected. Reading comprehension was found to predict performance on mathematical word problems and not numerical operations, and phonological awareness was found to predict performance on both types of mathematics. However, the predictive value of verbal ability and reading accuracy was found to be non-significant.     

Diagnosing word decoding problems: A process-analytic approach

In this article we attempt to demonstrate the theoretical and practical limitations of traditional subgroup classifications of acquired and developmental dyslexia. As an alternative, we suggest a process-analytical approach. A variant of a dual-route model for word processing is presented. The model guided the design of a computerized test battery for assessing strategies in word decoding and processes underlying each strategy. The usefulness of the approach was demonstrated on two cases, which according to traditional criteria, were classified as auditory dyslexics. Although both cases displayed marked deficiencies in the phonological route for word decoding, they differed in terms of underlying processes. In one case the primary problem could be located to phonological synthesis, whereas the other case had poor access of letters (phonological recoding). Such individual processing characteristics are not normally revealed by the traditional subgrouping approach.     

Strategic competence for multistep fraction word problems: an overlooked aspect of mathematical knowledge for teaching

Educational Studies in Mathematics - Prior work on teachers’ mathematical knowledge has contributed to our understanding of the important role of teachers’ knowledge in teaching and...     

Solving word problems: A case of modelling?

According to the researchers who report in this set of papers there are two causes for the phenomenon that primary- and secondary-school students ignore relevant and plausibly familiar aspects of reality in answering word problems. The first cause is the stereotyped character of common word problems. The second cause lies in the classroom climate. In this article, it is argued that the students act sensibly in their situation. Furthermore, it is noted that the use of stereotyped problems and the accompanying classroom climate relate to teacher beliefs about the goals of mathematics education. Therefore, improving the results on problematic word problems will ask for a change in teacher beliefs. Furthermore, a directed effort to change the classroom socio-math norms will be needed. In relation to this, Greer's suggestion for a change towards a modelling perspective is supported. However, what modelling is, is worked out differently. In line with the RME instructional theory, a plea is made for modelling as an activity of organizing, not of translation.     

Problem solving beyond the logic of things: contextual effects on understanding and solving word problems

Arguments are put forward in this paper that classroom word problem solving is more-and also less-than the urgent analysis of a factual structure, in the sense that it is essentially a species of a social-cognitive activity. Word-or story-problems, presented in classroom contexts, represent textual and pragmatic patterms of a certain grammaticality. To present a problem verbally to a student means to organize a fact in some way for the attention of a problem solver. There is not only the structure of the problem text itself by which situations are denoted, but there is also the stimulative nature of the social-pragmatic context which shapes the student's textbook-problem solving behavior over a long period of time.The present paper discusses the results of several studies showing, for example, that subject matter related attitudes towards a problem frequently do not play an important part in the problem solving efforts; that students often solve problems correctly without understanding them; and that false contextual expectations can lead to abstruse errors of understanding and to peculiar solution attempts.The studies indicate that students can become sensitive and skilful in perceiving and capitalizing on subtle textual and contextual signs pointing to the solution and anticipating its pattern. It seems that usual textbook problems let students get accustomed to certain courses of processing where a simple fact, like whether an equation works out evenly or does not, can stop the process or push it further. It is argued that the deeper reason for the observed textual and contextual influences on understanding and problem solving lies in a fundamental weakness of the student's epistemic control behavior. The psychological and instructional significance of the studies is discussed.