Does Conceptual Transparency in Manipulatives Afford Place-Value Understanding in Children at Risk for Mathematics Learning Disabilities?

Transcoding of French numbers for first- and second-language learners in third grade

Transcoding is the process of translating between spoken and written numbers, and it is correlated with other mathematical skills. In the present study, we investigated the link between French number writing of 49 students in the third grade (aged 7-9 years) and their language skills. Transcoding in French is of particular interest because the spoken number language system does not completely correspond to that of the written digits (e.g., quatre-vingt-dix [four-twenty-ten] and 90). We hypothesised that the complex linguistic structure of spoken numbers in French would be challenging for students who are learning to transcode. First and second French-language learners' accuracy and errors were recorded during a writing task of 3- to 7-digit numbers. Children also completed linguistic tests (e.g., receptive vocabulary, receptive syntax). Results showed that first- and second-language learners did not differ in their transcoding accuracy. Number size, decade complexity of stimulus number words in French (i.e., numbers containing a complex decade, operationalized as a number between soixante-dix, 70, and quatre-vingt-dix-neuf, 99), and receptive vocabulary predicted children's French transcoding skills. Students were more likely to produce errors (e.g., 68 or 6018 for 78) when they transcoded complex decade numbers compared with simple decade numbers. When an error was made on the complex decade portion of a number, it was likely a lexical error. In conclusion, third graders, both first- and second-language learners, found complex decade numbers challenging and their performance was related to their general vocabulary skills.

"Interstellar Wanderers:" Digital Support for Teaching Place Value Within the Activity Approach Framework

Background

The majority of mistakes students make while using "well-learned" decimal counting, can be attributed to their miscomprehension of the structure and arrangement of the "positional" numeral system. Digital support may merely serve an illustrative and training function, or it may provide the special environment for locating the problem of positional counting as a part of meaningful actions by the students. Following the Activity approach, we aimed to scaffold the students' own learning actions, in such a way as to reveal the origin of the multi-digit number concept. Thus, we used counting in other-base systems as a way for students to reconsider the reasoning behind familiar operations in the most common base-ten system.

Objective

The purpose of this paper is to present the approach to computer support which we have designed, based on our analysis of the activity content related to the multi-digit number concept, and to discuss some preliminary results of the first training series.

Design

The approach to educational environment design developed within the Learning Activity theory defines the ways in which a computer becomes essential. The computer should provide a transparent interface which allows students to perform transformations with objects which will react accordingly. The additional opportunities to perform concept-mediated orientation procedures should also be scaffolded by digital means. For the purposes of our study, the computer-based educational module "Interstellar wanderers" was designed. Four groups of students from 2nd to 5th grade (8-12 years of age, 20 children in total) participated in the experimental computer-based lessons (over 30 hours); classroom observations, videotaped discussions, and logs of students' individual work in the computer simulation were used for analysis.

Results

The preliminary results of the experimental teaching showed that the computer support which we developed may scaffold students' progress toward acquisition of the multi-digit number concept through a reflective re-thinking of the well-learned decimal system. Yet further research is needed to get a quantitative analysis of students' performance.

Conclusion

The general principles of computer support design based on the Activity approach in education (Galperin, Davydov, & Talyzina) demand a thorough analysis of the origin of the concept being studied, as well as the design of appropriate content and means of students' actions and corresponding contexts and tasks. The digital means which we designed to support students' learning activity, are in demand and bring promising results.