Using psychometric technology in educational assessment: The case of a schema-based isomorphic approach to the automatic generation of quantitative reasoning items

This article deals with the investigation of the psychometric quality and constructs validity of algebra word problems generated by means of a schema-based version of the automatic min–max approach. Based on review of the research literature in algebra word problem solving and automatic item generation this new approach is introduced as a theory-based top–down method of automatic item generation featuring a quality control framework aimed to minimize the construct unrelated variance in the item parameters. The first study deals with the evaluation of an initial set of items. The results are replicated in the second study using a larger item set which also allows the investigation of the construct representation of the generated item. Since construct unrelated variance components (e.g. reading comprehension) have been controlled for in the item generation phase the results revealed some interesting insights into the cognitive processes of the actual mathematization phase of algebra word problem solving. The third study investigated the nomothetic span is using hierarchical confirmatory factor analysis. The results argue for the convergent and discriminant validity of the automatically generated items. Taken together, the results indicate that the automatic generation of construct valid algebra word problems at a high psychometric level is viable. The discussion is thus concerned with the implications of this new approach to item generation for theory development and evaluation as well as practical benefits for educational assessment and the development of intelligent tutoring systems.     

Learning from comparison in algebra

Mastery of algebra is an important yet difficult milestone for students, suggesting the need for more effective teaching strategies in the algebra classroom. Learning by comparing worked-out examples of algebra problems may be one such strategy. Comparison is a powerful learning tool from cognitive science that has shown promising results in prior small-scale studies in mathematics classrooms. This study reports on a yearlong randomized controlled trial testing the effect of an Algebra I supplemental comparison curriculum on students’ mathematical knowledge. 141 Algebra I teachers were randomly assigned to either implement the comparison curriculum as a supplement to their regular curriculum or to be a ‘business as usual’ control. Use of the supplemental curriculum was much less frequent than requested for many teachers, and there was no main effect of condition on student achievement. However, greater use of the supplemental curriculum was associated with greater procedural student knowledge. These findings suggest a role for comparison in the algebra classroom but also the challenges of supporting teacher integration of new materials into the curriculum.     

浅谈新课程改革中的中学美术教育

课程改革已成为21世纪全球教育改革的一个热点,新世纪的美术教育既不是"应试教育",也不是专业知识教育,而是汇集文化百科,使知识具有广度、深度及综合性的"智能式"素质教育。由于种种原因,我国的美术新课程改革是一个很棘手的问题,本文试从几方面对我国中学美术教育在改革过程中出现的一些问题进行分析并提出一些看法。     

Does Early Algebraic Reasoning Differ as a Function of Students’ Difficulty with Calculations versus Word Problems?

According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 second‐grade students, we administered: (1) measures of calculations and word problems in the fall and (2) an assessment of prealgebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word‐problem measures, we placed 148 students into one of four difficulty status categories: typically performing, calculation difficulty, word‐problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word‐problem difficulty is more strongly associated with difficulty with prealgebraic reasoning. As an indicator of later algebra difficulty, word‐problem difficulty may be a more useful predictor than calculation difficulty, and students with word‐problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.     

The progress of disadvantaged pupils in cognitive and affective processes by means of a specially tailored curriculum in physical science

This is the first time in Israel that an inclusive physical science curriculum, based on problems and phenomena of everyday life of school children, has been devised and tried. Through this curriculum we have succeeded both in teaching scientific concepts and in bringing about improvement in certain capacities of scientific thinking and in the attitudes to science among disadvantaged children in school. It is our belief that the approach exemplified by our curriculum can contribute to raising the level of personal and cultural achievement in this population of school children.     

Substituting one mystery for another: the role of self-explanations in solving algebra word-problems

Explanations students provide themselves (self-explanations) in the course of learning or problem-solving have been shown to be positively associated with performance. However, the role self-explanation plays in problem solving has not been fully elaborated. This study aims to analyze the role of self-explanation in solving algebra word problems. We argue that self-explanation may function as verbal mediation that supports the transformation between different external representations of the problem. Our analysis of the problem solving protocols aims to illustrate this point through a multiple case studies design. Specifically we illustrate the way a particular kind of self-explanation (categorical explanation) mediates students' transformation from the sentential representation of the problem to the tabular one.     

Through a glass darkly: A naturalistic study of students' understanding of mathematical word problems

A father is now 20 years older than his son. In 8 years, the father's age will be 5 years more than twice the son's age. Find their present age.Word problems, such as this one, are a perennial source of difficulty for students of school mathematics. Unfortunately, very little is known about why they are problematic, mainly because so little is known about how students understand such problems or the strategies they use in their efforts to solve them. Traditional research into word problems has shed precious little light on this question owing, in no small part, to its almost singular preoccupation with results of pupils' activities—as expressed in some sort of test score, and to its tendency to all but ignore what students actually do when confronted with problems of this kind.This study was carried out as one facet of a larger research project designed to gain more insight into some of the ways in which students understand school mathematics. It focuses on the efforts of one pupil, a twelve-year-old girl in grade seven, to come to terms with solving word problems using an algebraic approach. Strategies associated with both the structured and the unstructured clinical interview were used in order to reveal what was involved in her attempts to make sense of the word problems in her grade seven mathematics textbook.Based on the information gained in the interview, a rational reconstruction of the student's problem-solving strategy is proposed, and compared with the strategies normally prescribed in contemporary school mathematics textbooks. What emerges from this comparison is the finding that, while there appear to be systematic and fundamental differences between the procedures prescribed by the text and those actually used by the pupil in working through certain problems, these differences are undetectable in the finished product; either in the answer itself or in the rough or finished work. What this suggests, among other things, is that if, as educators and/or researchers, we limit our attempts to understand how students go about learning to solve word problems (or how they approach any other part of the school mathematics curriculum, for that matter) to examining what they commit to paper, we are apt to be seriously misled concerning what they genuinely understand and what they fail to understand. In short, if we are to learn more about why pupils experience difficulties with word problems we must begin to pay serious attention to what they say and do as they work their way through them.This study was supported by a Research and Development Grant from the Faculty of Education, Queen's University.     

Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving

In Singapore, 6–12 year-old students are taught to solve algebra word problems with a mix of arithmetic and pre-algebraic strategies; 13–17 year-olds are typically encouraged to replace these strategies with letter-symbolic algebra. We examined whether algebra problem-solving proficiency amongst beginning learners of letter-symbolic algebra is correlated with the ability to inhibit intrusions from the earlier arithmetic strategies. Similar to typical school practice in Singapore, we asked 14 year-old students (N = 157) to use only letter-symbolic algebra to solve 9 algebra word problems. After having controlled for algebraic knowledge, working memory, and intelligence, better inhibitory ability still predicted fewer arithmetic intrusions and higher problem solving accuracy. Path analysis revealed 2 types of inhibition. Inhibition-of-reified-processes predicted accuracy through arithmetic intrusions. Inhibition-of-recently-learned-associations predicted accuracy through intelligence. Findings suggest establishing pedagogical links between arithmetic and algebraic methods may facilitate students' transition to letter-symbolic algebra.     

Extending Responsiveness to Intervention to Mathematics at First and Third Grades

Responsiveness to intervention (RTI) is an innovative approach to the identification of learning disabilities (LD). The central assumption is that RTI can differentiate between two explanations for low achievement: poor instruction versus disability. If the child responds poorly to validated instruction, then the assessment eliminates instructional quality as a viable explanation for poor academic growth and instead provides evidence of a disability. For children who do respond nicely, RTI serves a critical prevention function. Most of RTI research has been focused on early reading. In this article, we describe two ongoing programs of research on RTI in the area of mathematics: one on a comprehensive mathematics curriculum at first grade and the other focused on word problems at third grade. For each research program, we describe the sample, explain how students are identified as at risk for mathematics disability, provide an overview of the interventions to which responsiveness is gauged, and describe some results to date.     

代数应用题图式研究概述

代数应用题图式是学生在代数应用题学习过程中,对学习材料概括的基础上形成的、存储在长时记忆中的、具有一定框架结构的陈述性知识.模板图式、家族图式、概念图式和类别图式是心理学家研究数学学习心理常用的4种图式类型,也是代数应用题学习环境的主要设计元素.当今代数应用题图式的一个重要应用就是基于图式的代数应用题学习环境设计.     

Skill-oriented task sequencing in an intelligent tutor for basic algebra

As part of a project to develop an intelligent computer tutor for basic algebra, we have been investigating task sequencing. In this paper we present an approach to task sequencing that is based on a component-skills view of intelligence and learning. We postulate that tutors use inferences about past and present student performance to determine a current skill set that will be the new target for learning. The skill set is then used as a basis for generating tasks that should elicit those skills. Current skill sets are modified slowly over time so that lessons appear coherent and well-planned. We first describe the approach at a general level, where it can be viewed as a cognitive model of human task sequencing. Then we discuss the implementation of the model in our intelligent algebra tutoring system.     

初中数学新课程代数学习策略研究

新《标准》的颁布，对初中数学在内容上作了重新定位，因而代数的学习也应相应地改变，如何更好地学习代数以达到事半功倍的效果．就需要一定的策略。本在分析学习条件的基础上对代数的学习提出了相应的策略，并指出了值得注意的问题。     

"我的图文"伴我成长 --聋校语文校本课程的初步探讨

我国的课程政策改革方向正从原来单一的国家课程走向国家、地方、学校三级课程模式,这给予学校办学极大的自主性,为此我校在切实执行国家课程的同时,开发和实施校本课程,构建我校的新课程体系.本文从开发"我的图文"的起因、意义、作用及"我的图文"的目标和内容、开发的形式和方法、管理及评价、应注意的问题七个方面论述了对聋校语文校本课程的探讨与研究,旨在通过校本课程"我的图文"的开发激发学生学语文的兴趣,提高学生的语文素养.     

Rethinking literacy: communication,representation and text

In this article I want to consider shifts in the use of the word ‘literacy’ and the implications for classroom work with texts, particularly the implications of the rapid and radical emergence of new relationships between different modes of representation and communication ( Kress, 2003; Raney, 1996; Unsworth, 2001 ). My concern is to argue that any approach to classroom literacy needs not only to recognise the new forms of text which children meet every day but to give multimodal texts a firm place in the curriculum. Further, if the text experience of young learners about these new combinations of modes of representation are to be realised in the classroom, then we need a framework for describing those texts. An approach which takes account of the rhetoric of design may be a way forward.     

美国《成功的基础》评介及其对数学课程改革的启示

美国没有全国统一的数学课程标准与教材,标准的制订与教材的编写都由各州或学区自己来完成.美国数学课程改革的特点有:聚焦"代数基础",提前渗透代数概念和思想,寻找平衡点等.《成功的基础》的启示是:应将有关模式内容从低年级开始重视,同时把代数思想以非正式的形式逐步渗透.     

Meta-Representation in an Algebra I Classroom

We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different interpretations of the equal sign) previously reported in the literature. Moreover, the teacher and her students had trouble understanding one another's proposed algebraic models of problem situations due to differences in the criteria that each applied, and this impeded learning. These findings (a) extend an accumulating body of evidence for the role meta-representation plays in mathematics and science learning and (b) add a new dimension to researchers' growing understanding of what teachers must know in order to teach algebra and other complex mathematics and science topics effectively.     

The New ‘Functional Mathematics’ for Learners with Down Syndrome: Numeracy for a Digital World

ABSTRACT

This paper explores the challenge of determining the mathematics curriculum for students with Down syndrome in our digital age where the tools of mathematics including electronic calculators and smartphone applications are readily available. Aspects once considered essential for ‘functional mathematics’ such as written calculation, using cash and telling time can be undertaken with easily accessible devices. Deeper concepts such as generalising through algebra, making decisions about money, and engaging in mathematical problem solving and applications are now a possibility. This paper explores the challenge through the theoretical perspectives of Educational Quality of Life and Numeracy Development, presenting five principles for mathematics curriculum planning. The possibility exists of using year level appropriate mathematics curriculum to build lifelong numeracy – ‘functional’ mathematics for a new age.     

也谈新课程改革之代价——与孙天华、张济洲同志商榷

如何看待知识在新课程中的地位成为当前学界讨论的热点问题,面对诸多对新课程改革有“轻视知识”倾向的讨论,有学者试图用代价论的观点来遮蔽这场讨论的实质。然而,新课程改革中出现的问题与困难能等同于改革的必然代价吗?且须以“轻视知识”为代价吗?居于我国国情能承受这种代价之痛吗?应如何正确看待知识与能力的关系?新课程改革能大破大立吗?在这些问题的启发下,新课程改革要分清课程改革中面临的问题与所谓的改革代价,认真权衡改革所可能产生的利弊,理性对待改革过程中所出现的问题与困难,认清改革中存在的矛盾与误区,摒弃将知识与能力二者机械分割的观点,树立正确的课程改革发展观,以继续推进新课程改革的健康发展。     

核心素养视角下初中数学高效课堂构建策略探究

随着新课改的不断推进,核心素养逐步成为教学的关键词。初中作为九年义务教育的重要阶段,当前仍存在不少不符合核心素养的问题。对学生而言,初中数学具有不可替代的基础作用。文章首先简单介绍核心素养对构建初中高效课堂的意义,随后给出高效课堂构建中存在的问题及其解决对策,以此供相关人士进行交流、参考。