Graham A. Jones

A Framework for Assessing and Nurturing Young Children's Thinking in Probability.

Based on a synthesis of the literature and observations of young children over two years, a framework for assessing probabilistic thinking was formulated, refined and validated. The major constructs incorporated in this framework were sample space, probability of an event, probability comparisons, and conditional probability. For each of these constructs, four levels of thinking, which reflected a continuum from subjective to numerical reasoning, were established. At each level, and across all four constructs, learning descriptors were developed and used to generate probability tasks. The framework was validated through data obtained from eight grade three children who served as case studies. The thinking of these children was assessed at three points over a school year and analyzed using the problem tasks in interview settings. The results suggest that although the framework produced a coherent picture of children‘s thinking in probability, there was ‘static’ in the system which generated inconsistencies within levels of thinking. These inconsistencies were more pronounced following instruction. The levels of thinking in the framework appear to be in agreement with levels of cognitive functioning postulated by Neo-Piagetian theorists and provide a theoretical foundation for designers of curriculum and assessment programs in elementary school probability. Further studies are needed to investigate whether the framework is appropriate for children from other cultural and linguistic backgrounds.

A Framework for Characterizing Children's Statistical Thinking.

Based on a review of research and a cognitive development model (Biggs & Collis, 1991), we formulated a framework for characterizing elementary childrens statistical thinking and refined it through a validation process. The 4 constructs in this framework were describing, organizing, representing, and analyzing and interpreting data. For each construct, we hypothesized 4 thinking levels, which represent a continuum from idiosyncratic to analytic reasoning. We developed statistical thinking descriptors for each level and construct and used these to design an interview protocol. We refined and validated the framework using data from protocols of 20 target students in Grades 1 through 5. Results of the study confirm that childrens statistical thinking can be described according to the 4 framework levels and that the framework provides a coherent picture of childrens thinking, in that 80% of them exhibited thinking that was stable on at least 3 constructs. The framework contributes domain-specific theory for characterizing childrens statistical thinking and for planning instruction in data handling.

Beliefs about mathematics teaching held by pre-service teachers involved in a first grade mentorship program

The study compared beliefs about mathematics teaching of four pre-service elementary teachers involved in an intervention experience with those of their non-involved peers. During this intervention, which was based on a socio-constructivist approach to mathematics instruction, the intervention group participated in regular, small-group teaching experiences supported by on-going seminars. The study also examined the relationship between professed beliefs and observed actions for the intervention group.Although most pre-service teachers in this study seemed to attach some importance to children building their own knowledge through social interaction, the intervention group professed significantly stronger beliefs in a socio-constructivist instructional environment than the comparison group. Even though the intervention group strongly espoused socio-constructivist beliefs, they were not uniformly successful in translating these beliefs into instructional actions. Their actions appeared to be most consistent with a socio-constructivist perspective during the initial phase of an instructional episode, but in later phases their actions reflected more traditional beliefs about teaching mathematics.

A framework for assessing middle school students’ thinking in conditional probability and independence

Based on a synthesis of research and observations of middle school students, a framework for assessing students’ thinking on two constructs—conditional probability and independence—was formulated, refined and validated. For both constructs, four levels of thinking which reflected a continuum from subjective to numerical reasoning were established. The framework was validated from interview data with 15 students from Grades 4–8 who served as case studies. Student profiles revealed that levels of probabilistic thinking were stable across the two constructs and were consistent with levels of cognitive functioning postulated by some neo-Piagetians. The framework provides valuable benchmarks for instruction and assessment.

A model for nurturing and assessing multidigit number sense among first grade children

Based on a synthesis of the literature and on the results of a two-year teaching program working with young primary children, a framework was developed, refined and validated for nurturing and assessing multidigit number sense. The major constructs incorporated in this framework were counting, partitioning, grouping, and number relationships. For each of these constructs, four different levels of thinking were established which, in essence, reflected a “learning apprenticeship“ for multidigit number sense. At each level, and across all four constructs, learning indicators were developed and matched to distinctive problem tasks that went beyond the four basic operations.The framework was validated through data obtained from six case studies of grade 1 children. The thinking of these children was assessed and analyzed on the problem tasks for the four constructs and four levels. While the students were at different levels, all but one showed striking consistencies across the four constructs. Moreover, no student was able to solve a problem at a higher level when they had not solved a lower-level problem in the same category. The present framework for multidigit number sense covers only the lower primary grades, but research and instruction would benefit from an extended framework across the elementary grades.

Mathematics Instruction for Elementary Students with Learning Disabilities

Recent research in mathematics instruction requires educators to rethink long-established beliefs about teaching, learning, and assessment. In particular, this research underscores the need for problem solving and higher level thinking in mathematics. Consistent with these recommendations, this article presents and illustrates four promising themes for mathematics instruction that have emerged from research involving students with learning disabilities. These themes—(a) providing a broad and balanced mathematics curriculum; (b) engaging students in rich, meaningful problem tasks; (c) accommodating the diverse ways in which children learn; and (d) encouraging students to discuss and justify their problem-solving strategies and solutions—suggest ways for rethinking the teaching and learning of mathematics in relation to students with learning disabilities.

Probability simulation: What meaning does it have for high school students?

This study investigated high school students’ reasoning and beliefs when confronted with contextual tasks involving the assessment and construction of two-dimensional probability simulations.1 Nine students enrolled in an advanced algebra course, with little formal instruction in probability, engaged in clinical interviews focusing on the simulation tasks. All students showed evidence of being able to recognize or identify an appropriate probability generator to model a contextual problem. However, their thinking in probability simulation was constrained by their inability to deal with two-dimensional trials. In assessing the validity of a given simulation, only one student could identify a flaw that resulted from the use of one-dimensional trials rather than two-dimensional trials. Additionally, when asked to construct a simulation, only two students were able to define an appropriate two-dimensional trial and develop a valid solution. The study also revealed evidence of students’ beliefs about probability simulation—some of which could be helpful in informing instruction, others problematic.RésuméCette étude analyse les raisonnements et les croyances des étudiants et des étudiantes du secondaire lorsqu’ils sont confrontés à des tâches contextuelles qui consistent à envisager des simulations de probabilités à deux dimensions. Dans la première tâche, les élèves doivent évaluer la validité d’une simulation donnée dont le but est de déterminer les probabilités que deux commandes successives de pizza soient identiques. Dans la seconde tâche, ils doivent construire une simulation visant à estimer les probabilités qu’une même pièce musicale soit passée par une certaine station de radio à deux moments précis. L’échantillon se compose de neuf étudiants et étudiantes, expressément choisis parmi les élèves de trois niveaux d’algèbre avancée dans une école secondaire du Midwest américain. Ces étudiants, qui n’avaient guère étudié les probabilités auparavant, ont participé à des entrevues cliniques centrées sur les tâches de simulation.Six des neuf élèves de cette étude ont été en mesure d’identifier un générateur de probabilités adéquat dans la première tâche, et huit dans la seconde. Cependant, leur raisonnement au cours de cette simulation de probabilités s’est trouvé limité par leur incapacité d’affronter une épreuve à deux dimensions. Dans la première simulation, un seul étudiant a pu identifier une faille résultant de l’utilisation d’un système unidimensionnel. De plus, devant la tâche même de construire une simulation, seuls deux élèves sur neuf ont été en mesure de définir une épreuve à deux dimensions appropriée et de trouver une solution valable. Cinq autres ont manifesté l’intention de définir une épreuve à deux dimensions, mais ont été incapables de combiner entre eux les résultats singuliers et ont fini par retourner aux épreuves unidimensionnelles. Les deux autres élèves ont immédiatement transformé la tâche bidimensionnelle en tâche unidimensionnelle. L’incompréhension quant à la nature d’un espace-échantillon à deux dimensions, alliée aux idées erronées sur les simulations de probabilités, semblaient constituer des obstacles importants pour ces sept étudiants. Il n’est pas surprenant de constater que seuls les deux élèves capables d’envisager correctement les deux dimensions étaient en mesure d’expliquer comment calculer les probabilités empiriques de l’événement dans le second problème.Cette recherche a également fait ressortir les idées préconçues des étudiants et des étudiantes sur les simulations en probabilités. Certaines de ces idées ont été considérées comme utiles à la formation, tandis que d’autres constituaient au contraire un problème. Dans la catégories des idées utiles, mentionnons que quatre élèves considéraient les hypothèses comme nécessaires dans un modèle de simulation, que tous croyaient que les probabilités du générateur devaient correspondre aux probabilités contextuelles, et que six d’entre eux prévoyaient que les probabilités empiriques se rapprocheraient des probabilités théoriques. Quant aux idées sources de problèmes, signalons que pour cinq élèves, les simulations ne pouvaient servir de modèle pour des problèmes réels, que quatre élèves manifestaient certains préjugés liés à la représentativité (Tversky et Kahneman, 1974) et que deux élèves croyaient que le résultat visé devait se manifester dès la première tentative, un peu à la manière de la théorie des résultats décrite par Konoid (1991). Ces idées erronées entravaient souvent les raisonnements des étudiants et des étudiantes, en particulier lorsqu’ils affrontaient les complexités d’une épreuve à deux dimensions.Bien que cette étude soit par nature essentiellement exploratoire, elle a permis d’acquérir des données sur les raisonnements et les croyances des étudiants et des étudiantes du secondaire devant les tâches de simulation en probabilités. De telles données s’avéreront certainement fort utiles aux enseignants et aux enseignantes qui voudront créer du matériel didactique centré sur les simulations. En fait, il est souhaitable que d’autres recherches mettent au point des expériences (Cobb, 2000) visant à évaluer la possibilité que ces nouvelles données influencent les pratiques d’enseignement à l’école.

Children's Representation and Organisation of Data

This study investigated how children organised and represented data and also examined relationships between their organisation and representation of data. Two protocols, one involving categorical data and the other involving numerical data, were used to interview 15 students, 3 from each of Grades 1 through 5. Although there were differences between Grade 1 students and the rest, the study suggested that numerical data was significantly harder for children to organise and represent than categorical data. Children beyond Grade 1 could make connections between organising and representing data for categorical data but their connections for numerical data were more tenuous. The process of reorganising numerical data into frequencies was not intuitive for the children in this study but they showed greater readiness in recognising and interpreting data that had already been reorganised as a frequency representation. Given this latter result, a pedagogical approach that asks students to make links between raw data and a frequency representation of it may prepare students to create and construct their own frequency representations.

A MULTIMEDIA STATISTICAL UNIT FOR GENERAL EDUCATION

ABSTRACT This paper describes the development and evaluation of a multimedia statistical unit for a college general education mathematics course. The findings indicate that students in the multimedia instructional program showed significantly higher statistical performance than students in a control group. The multimedia instruction seemed especially well suited to female students who, although initially apprehensive, adapted quickly and effectively to the use of videos, data displays and computer applications incorporated in the program.