The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage

The purpose of this article is to describe how, within the Dutch approach to mathematics education, called Realistic Mathematics Education (RME), models are used to elicit students' growth in understanding of mathematics. First some background information is given about the characteristics of RME related to the role of models in this approach. Then the focus is on the use of the bar model within a longitudinal trajectory on percentage that has been designed for Mathematics in Context, a curriculum for the U.S. middle school. The power of this model is that it develops alongside both the teaching and the students: from a drawing that represents a context related to percentage to a strip for estimation and reasoning to an abstract tool that supports the use of percentage as an operator. This revised version was published online in August 2006 with corrections to the Cover Date.     

Predicting writing skill development with word recognition and preschool readiness skills

In this longitudinal study, the writing skill development of154 Finnish-speaking children was followed from preschool to thethird grade. The focus was on predictive associations betweenpreschool writing readiness skills and later mechanics ofwriting, as well as between word recognition skill, mechanics ofwriting, and composition coherence. In addition, comparisons weremade between boys and girls to see to what extent writing skilldevelopment is gender-specific. Multi-group structural equationmodeling was used for statistical analysis. The results indicatedthat both mechanics of writing and composition coherence could bepredicted from performance on the same skill at an earlier pointin time. Preschool measures of phonological and visual-motorskills predicted later mechanics of writing. Word recognitionworked as a predictor of later mechanics of writing andcomposition coherence, but only starting from second grade, whenthe development of the word recognition skill had becomestabilized at a high enough level. Furthermore, first grademechanics of writing predicted second grade compositioncoherence, but only at this early stage of productive writingwhen there were still difficulties in the mechanics of writing.Girls were better at tasks measuring mechanics of writing andwrote more coherent stories than boys. The gender difference inthe mechanics of writing at the first grade level was explainedby the presented model. Educational implications were discussed.     

Metacognitive knowledge in primary grades: A longitudinal study

The purpose of this longitudinal study was to explore the development of metacognitive knowledge in primary grade children (6–9 years, n=196) from pre-school to third grade. Knowledge about cognitive processes was obtained from children’s oral explanations. The results indicated that the average metacognitive knowledge of the primary grade children developed significantly during the first three school years. The children knew much about simple factors and strategies influencing cognitive activities, but after the first three school years, more complex strategies still remained unfamiliar to them. Moreover, the content analyses of children’s oral explanations revealed interesting differences in children’s metacognitive thinking. There appeared to be two groups of children with either non-developing or developing explanations. In addition, the children understood mental processing better in memory and learning activities, whereas comprehension processes were largely unknown to them.     

Chinese Primary School Mathematics Teachers’ Assessment Profiles: Findings from a Large-Scale Questionnaire Survey

This study investigated Chinese primary school mathematics teachers’ views on assessment in an effort to determine their assessment profiles. A large-scale questionnaire survey with 1101 teachers from 12 Chinese provinces and regions was carried out. The teachers reported to use assessment on a daily or weekly basis for different purposes. They recognized the importance of assessing different types of skills and knowledge and considered assessment useful for improving teaching and learning. To determine teachers’ assessment profiles, we used several latent variable modeling techniques. With exploratory factor analyses, we identified eight factors in the teachers’ responses: general instructional decision-making assessment purposes [1], specific instructional decision-making assessment purposes [2], assessment methods [3], diversity of assessment problem format [4], importance of assessing skills and knowledge [5], importance of assessing extra-curricular skills [6], Perceived usefulness [7], and acceptance of assessment [8]. When these factors were used to interpret the results of a latent class analysis, three distinct assessment profiles could be distinguished. One fifth of the teachers were in the Enthusiastic assessors profile. These teachers not only reported to use assessment frequently [3, 4] and purposefully [1, 2], but also highly endorsed its importance [5, 6] and usefulness [7, 8]. Around half of the teachers were in the Mainstream assessors profile; these teachers scored close to the mean on all factors. The remaining teachers held the relatively negative views on assessment and were therefore in the Unenthusiastic assessors profile. This profile characterization sheds light on Chinese primary school mathematics teachers’ assessment culture.     

Special education students’ use of indirect addition in solving subtraction problems up to 100—A proof of the didactical potential of an ignored procedure

In this study, we examined special education students’ use of indirect addition (subtraction by adding on) for solving two-digit subtraction problems. Fifty-six students (8- to 12-year-olds), with a mathematical level of end grade 2, participated in the study. They were given a computer-based test on subtraction with different types of problems. Although most students had not been taught indirect addition for solving subtraction problems, they frequently applied this procedure spontaneously. The item characteristics were the main prompt for using indirect addition. Context problems that reflect an adding-on situation and problems that have a small difference between the minuend and subtrahend most strongly elicited the use of the indirect addition procedure. Moreover, indirect addition was identified as a highly successful procedure for special education students, and the best predictor of a correct answer was found in combination with a stringing strategy.     

Adult developmental dyslexia in a shallow orthography: Are there subgroups?

The existence and stability of subgroups among adult dyslexic readers of a shallow orthography was explored by comparing three different cluster analyses based on previously suggested combinations of two variables. These were oral reading speed versus accuracy, word versus pseudoword reading speed, and phonological awareness versus rapid naming. The three analyses were conducted with the same group of dyslexic adults. Each analysis produced three subgroups, corresponding to ones previously suggested in the literature. However, the subgroups had only little overlap from one analysis to another. Each clustering produced somewhat different subgroup profiles in phonological processing, reading, intelligence, temporal acuity, and sensory short-term memory. However, the shared difficulties of the solutions in several language-related and sensory tasks suggest the conclusion that developmental dyslexia does not causally consist of subgroups, at least in shallow orthographies. Further, the shared sensory difficulties suggest that impaired temporal acuity and sensory short-term memory may reflect the severity of a primary disorder that dyslexic readers cannot compensate by strategies.     

An Experiment on Using Roles of Variables in Teaching Introductory Programming

Roles of variables is a new concept that captures tacit expert knowledge in a form that can be taught in introductory programming courses. A role describes some stereotypic use of variables, and only ten roles are needed to cover 99% of all variables in novice-level programs.

This paper presents the results of an experiment where roles were introduced to novices learning Pascal programming. Students were divided into three groups that were instructed differently: in the traditional way with no treatment of roles; using roles throughout the course; and using a role-based program animator in addition to using roles in teaching.

The results show that students are not only able to understand the role concept and to apply it in new situations but—more importantly—that roles provide students a new conceptual framework that enables them to mentally process program information in a way demonstrating good programming skills. Moreover, the use of the animator seems to foster the adoption of role knowledge.     

The role of metacognition in the context of integrated strategy intervention

In the present study, the role of metacognition in the context of integrated strategy intervention was examined. The integrated strategy training in reading comprehension, mathematics, and metacognition was carried out over a period of seven months with young 9–10-years-old, 3rd grade elementary school students with learning problems. A pretest-post test design with matched intervention and control groups was applied. Before the intervention, these students and all their classmates had been followed up through 1st to 3rd grade, and tested with multiple cognitive and metacognitive tasks each year. The results showed that early, 1st grade, cognitive-metacognitive differences were strongly associated with later problem solving and reading comprehension proficiency, thus confirming the importance of reading comprehension strategies and metacognition in mathematical problem solving. Further, marked training effects were found. The growth of metacognitive awareness, experiences and self-regulation were observed on the basis of behaviour analyses during the intervention and of post-intervention interviews. However, despite significant training effects, students who were resistant and those who were responsive to training were identified. The results showed that early metacognitive proficiency is closely associated with the responsiveness to training efforts. The nature of metacognitive experiences and the early teaching of metacognitive awareness and self-regulation are emphasised in the conclusions.