Visual spatial representation in mathematical problem solving by deaf and hearing students

This research examined the use of visual-spatial representation by deaf and hearing students while solving mathematical problems. The connection between spatial skills and success in mathematics performance has long been established in the literature. This study examined the distinction between visual-spatial "schematic" representations that encode the spatial relations described in a problem versus visual-spatial "pictorial" representations that encode only the visual appearance of the objects described in a problem. A total of 305 hearing (n = 156) and deaf (n = 149) participants from middle school, high school, and college participated in this study. At all educational levels, the hearing students performed significantly better in solving the mathematical problems compared to their deaf peers. Although the deaf baccalaureate students exhibited the highest performance of all the deaf participants, they only performed as well as the hearing middle school students who were the lowest scoring hearing group. Deaf students remained flat in their performance on the mathematical problem-solving task from middle school through the college associate degree level. The analysis of the students' problem representations showed that the hearing participants utilized visual-spatial schematic representation to a greater extent than did the deaf participants. However, the use of visual-spatial schematic representations was a stronger positive predictor of mathematical problem-solving performance for the deaf students. When deaf students' problem representation focused simply on the visual-spatial pictorial or iconic aspects of the mathematical problems, there was a negative predictive relationship with their problem-solving performance. On two measures of visual-spatial abilities, the hearing students in high school and college performed significantly better than their deaf peers.     

中美学生数学问题解决的差异比较

“问题解决”是国际数学教育研究的一个热点,本文从中美两国学生在问题解决的过程和结果的比较研究入手,比较和研究了中美两国学生在问题解决中提出问题的能力及解决问题的策略两方面的差异。     

Vision problems among students in schools and programs for deaf children. A survey of teachers of deaf students.

Vision problems are more common among deaf people than among the general population. Eight percent of the students in schools and programs for deaf children can be expected to have vision problems. We sent questionnaires to institutions and programs for deaf children, requesting information about the extent of vision problems among their students and about their teachers' levels of preparation in the area of vision impairment. Responses were received from 490 teachers. The data confirm that significant numbers of deaf students also have vision problems, and that teachers do not receive adequate training and information to meet the needs of this population.     

Impact of authenticity on sense making in word problem solving

The study presented in this paper seeks to investigate the impact of authenticity on the students’ disposition to make necessary real world considerations in their word problem solving. The aim is also to gather information about the extent to which different reasons for the students’ behaviors are responsible for not providing solutions that are consistent with the ‘real’ situations described in the word problems. The study includes both written solutions to word problems and interview data from 161 5th graders. The results show an impact of authenticity on both the presence of ‘real life’ considerations in the solution process and on the proportion of written solutions that were really affected by these considerations. The students’ frequent use of superficial solution strategies and their beliefs about mathematical word problem solving were found to be the main reasons for providing solutions that are inconsistent with the situations described in the word problems.     

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.     

Mathematics achievement and gender: A longitudinal study of selected cognitive and affective variables [grades 6–12]

This study examined the relationship of selected cognitive and affective variables to mathematics achievement for a random sample of 60 students as they progressed through 6th, 8th,10th and 12th grades. A consistent gender difference was found for stereotyping mathematics as a male domain. No consistent significant gender difference between means was found for spatial skills, verbal skill or mathematics achievement. Confidence, verbal skill and spatial visualization were each consistently positively correlated with mathematics achievement for both males and females. No gender difference was found for these correlations. However, spatial skills alone were found to be consistent significant predictors of mathematics achievement for females each year of the study, but not for males. Verbal skill was a consistent significant predictor of mathematics achievement for males, but not for females. The results of this study could lead to a reevaluation of the hypothesis that spatial skills help boys achieve in mathematics.     

Fostering analogical transfer: The multiple components approach to algebra word problem solving in a chemistry context

Holyoak and Koh (1987) and Holyoak (1984) propose four critical tasks for analogical transfer to occur in problem solving. A study was conducted to test this hypothesis by comparing a multiple components (MC) approach against worked examples (WE) in helping students to solve algebra word problems in chemistry classes. The MC approach incorporated multiple components (symbolic equations, symbols, categorization, hint) in the source, or target, or both, to address the four analogical tasks. Different combinations of the components were tested in a series of four experiments. Symbolic equations (main component) fostered a mental construction of the problem in its solution mode. Categorization enabled an identification of the problem category. A hint in the target directed the learners to the source problem. The interaction between these components facilitated the mapping of the symbolic equations in the source onto the target, resulting in the superiority of the MC approach in fostering analogical transfer. Neither the main component alone nor the main component plus one sub-component was sufficient for analogical transfer. Hence for analogical transfer to occur, at least the main component (symbolic equations) and two sub-components (categorization and hint) are required. However, symbols may not have additional effects for transfer to occur.     

解决妨碍大学生创业政策执行问题的对策

事实表明,解决好妨碍大学生创业政策执行这一问题,能促进大学生创业政策体系的完善,有效促进大学生积极创业,从而有效解决当前大学生就业难的问题;此外,还可为国家改进其他相关创业政策提供有效参考,对进一步优化我国创业环境、完善整个社会的创业政策体系、落实科学发展观、加快我国创新体系的形成均具有极大的推动作用.本文着重从提高大学生创业政策的可执行性、提高政策执行人员素质水平等方面,就解决好妨碍大学生创业政策执行这个问题进行了较为深入的分析和探讨.     

Intensive scaffolding in an intelligent tutoring system for the learning of algebraic word problem solving

The term intensive scaffolding refers to any set of conceptual scaffolding strategies that always allow the user to find the solution to a problem. Despite the many benefits of scaffolding, some negative effects have also been reported. These are mainly related to the possibility that a student solves the problems without actually engaging in their content. In this paper, we have used an intelligent tutoring system (ITS) to analyze the effect that intensive scaffolding has on the learning of algebraic word problem solving (AWPS). Two different versions of the ITS, which differ in the amount of scaffolding that they provide, have been created. These two versions were used by two groups of students in Secondary Education, in a quasi‐experimental study. The comparison of pre‐ and posttest scores shows a significant increase of the competence in AWPS in the group that used the ITS with intensive scaffolding.     

Problem solving in science lessons: How students explore the problem space

This report examines, from a constructivist framework, how students presented with discrepant event problems during science lessons begin the problem solving process by exploration of the problem space. Three discrepant events were presented to five classes of eleven to thirteen year-olds using three different teaching strategies identified from the literature. The teaching strategies used teacher demonstrations followed by students asking the teacher questions, teacher demonstrations followed by an explanation by the teacher, and small groups of students under the guidance of the teacher. Data were collected from field notes, video tapes of the lessons, and student interviews using a stimulated recall technique to elicit the students' thinking during the lessons. The extent to which students could explore the problem space was found to be determined by the teaching strategy used, as the first and last strategies encouraged students to find their own explanations. Different information sources were also available in each of the teaching strategies. That is, exploration of the problem space was inextricably linked to the social context, partially determined by the teaching strategy. Because of constraints imposed by each teaching strategy none of those used was considered entirely satisfactory. An alternative strategy is suggested from a combination of aspects of the strategies trialled.     

Language-related longitudinal predictors of arithmetic word problem solving: A structural equation modeling approach

We investigated the longitudinal relations between cognitive skills, specifically language-related skills, and word-problem solving in 340 children (6.10–9.02 years). We used structural equation modeling to examine whether word-problem solving, computation skill, working memory, nonverbal reasoning, oral language, and word reading fluency measured at second grade were associated with performance on measures of word-problem solving in fourth grade. Results indicated that prior word-problem solving, computation skill, nonverbal reasoning, and oral language were significantly associated with children’s later word-problem solving. Multi-group modeling suggested that these relations were not significantly different for boys versus girls. Implications of these findings are discussed.     

Cognitive development and problem solving of Afro-American students in chemistry

The aims of this study were to investigate the level of cognitive development of Afro-American students enrolled in general chemistry courses at the college level and to determine the strategies used by both successful and unsuccessful Afro-American students in solving specific types of stoichiometric problems. It was found that the choice of a strategy is not significantly related to cognitive development of the student in specific types of stoichiometric problems. However, the following trend was noted: Students who are formal-operational in thought are more likely to be successful when solving mole-volume problems and complex mole-mole problems than are their concrete-operational counterparts. Additionally, a systematic strategy proved to be successful for the students, regardless of the cognitive development, when balancing simple and complex chemical equations. Also, algorithmic/reasoning strategies were needed to solve the mole-volume problem. A higher level of cognitive development and reasoning may be crucial factors in solving the more sophisticated types of problems in stoichiometry.     

A survey of school psychologists' practices for identifying mentally retarded students

Surveys containing questions about identifying mentally retarded students were mailed to 500 randomly selected school psychologists, of whom 160 (32%) returned completed, usable forms. The Wechsler scales were the most frequently used tests for deriving IQ scores, which together with adaptive behavior scale scores were rated as most influential in identification-placement decisions. The Vineland Adaptive Behavior Scales were rated as the most used instruments for assessing adaptive delays. School psychologists were in less accord on issues such as the use of alternative norms for minority and lower SES students (35% confirmed their use), the application of an adaptive behavior standard score of 70 (47.7% indicated such use), and whether an established IQ score, such as 70, could be treated flexibly (27.2% stated they sometimes identified students with scores above the cut-off value). Suggestions are made for future research and current practice.     

Incidental word learning in a hearing child of deaf adults

It is unclear how children develop the ability to learn words incidentally (i.e., without direct instruction or numerous exposures). This investigation examined the early achievement of this skill by longitudinally tracking the expressive vocabulary and incidental word-learning capacities of a hearing child of Deaf adults who was natively learning American Sign Language (ASL) and spoken English. Despite receiving only 20% of language input in spoken English, the child's expressive vocabularies at 16 and 20 months of age, in each language, were similar to those of monolingual age-matched peers. At 16 months of age, the child showed signs of greater proficiency in the incidental learning of novel ASL signs than she did for spoken English words. At 20 months of age, the child was skilled at incidental word learning in both languages. These results support the methodology as it applies to examining theoretical models of incidental word learning. They also suggest that bilingual children can achieve typical vocabulary levels (even with minimal input in one of the languages) and that the development of incidental word learning follows a similar trajectory in ASL and spoken English.