Constructing a conceptual framework for elementary algebra through Logo programming

This paper examines the extent to which Logo-experienced children are able to mobilise their Logo-based knowledge to construct meaning for elementary algebraic concepts. It reports the results of an exploratory study which was part of a longitudinal investigation of the mathematical environment created through Logo programming. The study aimed at gauging the influence of children's Logo learning in facilitating their conceptualisation of algebraic variable, and their ability to formalise in a non-computational context.     

Teachers and artificial intelligence. The Logo connection

This article describes a three-phase program for training special education teachers to teach Logo and artificial intelligence. Logo is derived from the LISP computer language and is relatively simple to learn and use, and it is argued that these factors make it an ideal tool for classroom experimentation in basic artificial intelligence concepts. The program trains teachers to develop simple demonstrations of artificial intelligence using Logo. The material that the teachers learn to teach is suitable as an advanced level topic for intermediate- through secondary-level students enrolled in computer competency or similar courses. The material emphasizes problem-solving and thinking skills using a nonverbal expressive medium (Logo), thus it is deemed especially appropriate for hearing-impaired children. It is also sufficiently challenging for academically talented children, whether hearing or deaf. Although the notion of teachers as programmers is controversial, Logo is relatively easy to learn, has direct implications for education, and has been found to be an excellent tool for empowerment-for both teachers and children.     

Reviving the Turtle: Exploring the Use of Logo with Students with Mild Disabilities

In this case study, a group of nine 4th grade children were introduced to the Logo programming language during three 90-minute sessions over a four-week period. They attended a private university-based laboratory school serving students with various learning disabilities. This project demonstrated that a classic version of Logo captured the students’ interest. It was a viable source of interactive challenge and problem-solving experience that provided students with a great deal of pride, intrinsic reward, enjoyment, and sense of ownership of learning. The process of overcoming obstacles while programming with Logo may be especially beneficial for students with mild disabilities.     

Logo as a Strategy for Developing Thinking?

Instruction in the Logo programming language proceeds amid increasing concern about its educational value. Proponents claim that Logo-based instruction increases intelligence, whereas critics contend that children learn little about Logo or about anything else during such instruction. Reasoned argument about the pitfalls and prospects of Logo in the classroom requires a common interpretive framework, which is often obscured by the distinct conceptions of intelligence held by proponents and critics. Accordingly, the implications of two different root metaphors of intelligence for the study of Logo-based learning are traced. Development of ideal educational practices and outcomes for Logo learning within each framework follows, with a selected review of research to support some conjectures and refute others. This presentation includes discussion of weak versus strong problem-solving methods, transfer, "powerful ideas," and discovery-based learning. The article concludes with a partition of cognition into cognitive, metacognitive, and epistemic levels, and it relates these to Logo as a strategy for developing thinking.     

Comparing loops misconceptions in block-based and text-based programming languages at the K-12 level

Novice programmers are facing many difficulties while learning to program. Most studies about misconceptions in programming are conducted at the undergraduate level, yet there is a lack of studies at the elementary school (K-12) level, reasonably because computer science neither programming are regularly still not the part of elementary school curricula’s. Are the misconceptions about loops at elementary school level equal to those at the undergraduate level? Can we “prevent” the misconceptions by using the different pedagogical approach, visual programming language and shifting the programming context toward game programming? In this paper, we tried to answer these questions. We conducted the student misconceptions research on one of the fundamental programming concepts – the loop. The research is conducted in the classroom settings among 207 elementary school students. Students were learning to program in three programming languages: Scratch, Logo and Python. In this paper, we present the results of this research.     

Logo Experience:

This Study investigated whether or not generalized learning experiences with Logo would yield enhanced understanding of relative position and direction for fifth and sixth graders. It was assumed that manipulating the Logo turtle would benefit students in these important geography skills. After one school year of Logo instruction the treatment group outperformed the control group on three of four measures of relative position and direction skills, one of which was statistically different.     

Children's inductive thinking during intrinsic and Euclidean Geometrical activities in a computer programming environment

This paper presents a case study investigating the knowledge constructed by two 12 year-old children working with a geometrical Logo microworld allowing the Logo turtle to measure distances, and turns relative to previously constructed points on the plane. A qualitative analysis of data consisting of everything that the children said, typed and wrote on paper during the 15 hours of the research, provided evidence of the children's developing use of concepts belonging to Plane Geometry. The measuring of angular and length quantities, enabled them to conjecture, reflect on and manipulate triangle properties and relations traditionally associated with Euclidean Geometry. Their developing awareness of the existence of geometrical relations in their work, encouraged an increase in their readiness to use and reflect on them. The paper concludes that the generation of learning environments such as the above may well enhance the opportunity for children to form inductive geometrical understandings.     

A Professional Development Program to Improve Math Skills Among Preschool Children in Head Start

The purpose of this study was to examine the effects on four-year-olds’ knowledge of mathematics by introducing professional development and center-based mathematics activities around four mathematical domains to early educators’ teaching in Head Start programs. Because of the need to provide necessary mathematical experiences to young children to improve their early understanding and skills and provide the foundation for future success in mathematics, we provided the treatment group of early educators with professional development and center-based activities to promote four critical areas in mathematics. By randomly selecting Head Start centers to participate as the treatment group or control group, we were able to examine the effects of the professional development and set of activities on preschool children’s knowledge over a six-month period. We found children in the treatment group were more fluent and flexible with number concepts, were better at solving contextual problems, and had better measurement and spatial abilities than children in the control group.     

Linking logo,levels and language in mathematics

In his book Mindstorms (1980) Papert discusses Turtle Geometry as ego-syntonic or fitting the ways of thinking of the child as a geometric knowledge builder. The van Hiele theory (Wirzup, 1976) looks at geometric knowledge building as occuring through a necessary sequence of levels. Thus if Turtle Geometry is ego-syntonic it would seem that one could apply these van Hiele levels to better describe and understand this form of geometry. It is the first purpose of this paper to provide such an analysis.Using Logo to do Turtle Geometry requires that a child (or learner at any level) do geometry through the deliberate use of language. Thus one ought also to be able to relate theoretically levels of language use to Turtle Geometry. It is a second purpose of this paper to relate a language use framework suggested by the work of Frye (1982) to the language activities of Turtle Geometry.     

Developing language and cognition in young children with Logo

A Logo intervention was implemented to improve the language and problem-solving skills of four young mainstreamed children with developmental disabilities. To ensure that the children had a clear concept of Logo, a specially designed scope and sequence curriculum, with instructional aids, was developed. The children were tested in language and cognitive processing and gain scores were analysed after the intervention. While there were no statistically supportable conclusions attesting to the benefit of Logo in language and problem-solving, a qualitative assessment of gains supported the value of the project.     

Providing a computer based framework for algebraic thinking

This paper describes and presents the findings of a study which aimed to trace the development of pupils' use and understanding of algebraic ideas within a Logo programming context relating this to their use and understanding of similar ideas within a non-computational context. The research consisted predominantly of a three year longitudinal case study of four pairs of pupils (aged 11–14) programming in Logo during their normal school mathematics lessons. The data included video recordings of all the case study pulils' Logo sessions, and individually presented Logo and algebra structured interviews. The overall conclusion of this research is that Logo experience does enhance pupils' understanding of algebraic ideas, but the links which pupils make between Logo and algebra depend very much on the nature and extent of their Logo experience.     

Sociodemographic Inequality in Early Literacy Development: The Role of Teacher Perceptual Accuracy

Previous research has established that student learning is influenced by how accurately teachers perceive student academic ability. But studies rarely investigate the degree to which inaccuracies in teacher perceptions exacerbate demographic inequality in academic ability. Using a sample of almost 14,000 children from the Early Childhood Longitudinal Study, Kindergarten Cohort, we found that children whose literacy skills are overestimated by their teachers typically gain more literacy skills during kindergarten. Conversely, children whose skills are underestimated learn less. It is important to note that the skills of socioeconomically disadvantaged children are on average underestimated. As a result, inequalities in kindergarten literacy development stem in part from the links between teacher misperceptions and student background. We also explored the extent to which these relationships operate through practices associated with ability grouping. We found instructional grouping to be a weak facilitator of the link between teacher perceptions and student learning, suggesting the need for further research that identifies the social and structural classroom characteristics that link teacher perceptual accuracy to student learning.     

The impact of a thinking skills intervention on children's concepts of intelligence

The study reported was part of a large thinking skills intervention for 11–12-year-old children. This paper focuses on the impact of a thinking skills intervention on children's understandings of intelligence. A total of 178 children (n = 86 girls and n = 92 boys) across six schools participated in the study. Children were individually pre-tested in the classroom using written tasks designed to tap concepts of intelligence (definitions, characteristics, causes of intelligence, and the stability of intelligence: entity versus incremental concepts) and a variety of thinking skills. Schools were allocated into one of three intervention conditions: control condition; individual condition; collaborative learning condition. Children in the individual and collaborative learning conditions participated in an 8-week thinking skills intervention. Children in the individual condition worked individually on tasks to apply the thinking skills whereas learners in the collaborative condition applied the thinking skills on tasks in groups of four. Following the thinking skills intervention all children were individually post-tested using the pre-test measures. The results showed that the intervention had an impact on children's understanding of intelligence. In particular, the collaborative learning intervention led to most improvement in concepts of intelligence. The results are discussed with reference to theories of intelligence concepts and thinking skills interventions.     

Programming-languages as a conceptual framework for teaching mathematics

Formal mathematical methods remain, for most high school students, mysterious, artificial and not a part of their regular intuitive thinking. The authors develop some themes that could lead to a radically new approach. According to this thesis, the teaching of programming languages as a regular part of academic progress can contribute effectively to reduce formal barriers. This education can also be used to enable pupils to access an accurate understanding of some key mathematical concepts. In the field of heuristic knowledge for technical problem solving, experience of programming is no less valuable: it lends itself to promote a discussion of relations between formal procedures and the comprehension of intuitive problem solving and provides examples for the development of heuristic precepts (formulating a plan, subdividing the complexities, etc.). The knowledge gained in programming can also be used for the discussion of concepts and problems of classical mathematics. Finally, it can also facilitate the expansion of mathematical culture to topics in biological and physical sciences, linguistics, etc. The authors describe a programming language called ‘Logo’ adapted to objectify an enduring framework of mathematical experimentation.     

Building interfaces for on-line collaborative learning

For several years we have been engaged in the development and research of software environments for collaborative learning, for example in the recently completed CoLabs project (which we presented at the IFIP TC3 WG3.5 working conference in Budapest in 2004, see also http://matchsz.inf.elte.hu/Colabs/), also within lately defended long-range doctoral research, in which the co-author of this paper together with his undergraduate students had developed and evaluated cooperative computer activities for children aged 10 to 18. They observed children when using those environments and studied the influence of different approaches and solutions on the degree of their involvement and will to cooperate. We have also been involved in the London Knowledge Lab pilot project for developing a collaborative layered learning space travel games construction kit. Our department’s prior engagement in the development of collaborative activities also includes publishing a popular on-line journal (developed by A. Hrusecka and D. Lehotska, this on-line journal (in Slovak) can proudly boast up to 250,000 visits per month) for children, which intensively exploits on-line collaboration. This topic attracts us not only as a support for the learning process, but is a challenge for us as developers of educational programming tools as well. In our SuperLogo and Imagine environments we have always tried to provide users (ranging from children to developers) with new and powerful options to foster learning by exploring and developing. Thus we have equipped Imagine with the means for building objects and their behaviours in incremental loops, with parallel independent processes, event-driven programming and complex yet intuitive support for developing on-line environments for collaborative learning. In this paper we place our collaborative applications in the context of other related interfaces reported in literature. We use eight criteria to classify them and conclude that collaborative applications being developed by us and our students—future teachers—are distinguishable from others along two or three of those criteria: they combine in themselves typical features of Logo microworlds and inspiring support for on-line cooperation. We then analyze in detail our collaborative Imagine microworlds along four dimensions of their development. We specify means for establishing and maintaining on-line connection among any number of participants. We study tools for sending and receiving items (data, active characters with their behaviours, instructions etc.). We reflect on what can be shared by two or more participants in a collaborative activity. We examine all possible operations with common and private characters of a participant from the collaboration point of view. Our goals in this research and development are to:

1. Better understand potential the interfaces for on-line collaboration offer to support the learning process,
2. Place our Imagine Logo microworlds into context of other related applications,
3. Build a detailed map of all aspects important for their development (i.e., build a framework for the development),
4. Present the observations from our long-range experimental development and evaluation of the collaborative environments,
5. Point out how simple and natural it is (in the academic surroundings) to develop small and yet powerful collaborative microworlds—built for instance for immediate need in a classroom—with exceptional potential for interaction and openness.

Most of all, however, we want to motivate the endeavour to overcome all obstacles connected with the integration of on-line cooperation into children’s learning.     

Selective spatial working memory impairment in a group of children with mathematics learning disabilities and poor problem-solving skills

This study examines visual and spatial working memory skills in 35 third to fifth graders with both mathematics learning disabilities (MLD) and poor problem-solving skills and 35 of their peers with typical development (TD) on tasks involving both low and high attentional control. Results revealed that children with MLD, relative to TD children, failed spatial working memory tasks that had either low or high attentional demands but did not fail the visual tasks. In addition, children with MLD made more intrusion errors in the spatial working memory tasks requiring high attentional control than did their TD peers. Finally, as a post hoc analysis the sample of MLD was divided in two: children with severe MLD and children with low mathematical achievement. Results showed that only children with severe MLD failed in spatial working memory (WM) tasks if compared with children with low mathematical achievement and TD. The findings are discussed on the basis of their theoretical and clinical implications, in particular considering that children with MLD can benefit from spatial WM processes to solve arithmetic word problems, which involves the ability to both maintain and manipulate relevant information.     

Developmental trajectories of children's spatial skills: Influencing variables and associations with later mathematical thinking

Few studies have examined the long-term relations between children's early spatial skills and their later mathematical abilities. In the current study, we investigated children's developmental trajectories of spatial skills across four waves from age 3–7 years and their association with children's later mathematical understanding. We assessed children's development in a large, heterogeneous sample of children (N = 586) from diverse cultural backgrounds and mostly low-income homes. Spatial and mathematical skills were measured using standardized assessments. Children's starting points and rate of growth in spatial skills were investigated using latent growth curve models. We explored the influence of various covariates on spatial skill development and found that socioeconomic status, language skills, and sex, but not migration background predicted children's spatial development. Furthermore, our findings showed that children's initial spatial skills––but not their rate of growth––predicted later mathematical understanding, indicating that early spatial reasoning may play a crucial role for learning mathematics.     

Connectionism and learning to read: Steps towards a psychologically plausible model

We consider the prospects and need for a psychologically plausible connectionist model of the development of word recognition skills. We present an outline of the Seidenberg and McClelland (1989) distributed, developmental model of word recognition and naming. We emphasize certain incompatibilities between this model and the assumptions underlying it and psychological evidence from studies of children learning to read. In particular, we emphasize the importance of phonological skills as precursors and facilitators of learning to read. We argue that it may be possible to develop a connectionist model of the development of word recognition skills which, by having a built in knowledge of phonology, will be more consistent with evidence from studies of both normal and dyslexic children.     

Parental teaching styles in an open-ended teaching-learning environment

The present study extends earlier work on parental teaching style by making the task to be completed less structured, extending parent-child interaction over a 2 week time period, and equalizing the expertise of the parent relative to the child. Additionally, the study examines the effects of parental intrusiveness on a subsequent task. Subjects were 22 Caucasian middle class parent-child dyads. Audiorecordings were taped over a 2-week period while dyads were learning aspects of Logo computer programming. Even in this prolonged nonspecific learning situation, in which parents were encouraged to proceed at the child's pace, parents were overwhelmingly directive. Additionally, children of directive parents performed best on a subsequent generalization task.