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"Numbers, Rational Social aspects."
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Which Type of Rational Numbers Should Students Learn First?
Many children and adults have difficulty gaining a comprehensive understanding of rational numbers. Although fractions are taught before decimals and percentages in many countries, including the USA, a number of researchers have argued that decimals are easier to learn than fractions and therefore teaching them first might mitigate children's difficulty with rational numbers in general. We evaluate this proposal by discussing evidence regarding whether decimals are in fact easier to understand than fractions and whether teaching decimals before fractions leads to superior learning. Our review indicates that decimals are not generally easier to understand than fractions, though they are easier on some tasks. Learners have similar difficulty in understanding fraction and decimal magnitudes, arithmetic, and density, as well as with converting from either notation to the other. There was too little research on knowledge of percentages to include them in the comparisons or to establish the ideal order of instruction of the three types of rational numbers. Although existing research is insufficient to determine the best sequence for teaching the three rational number formats, we recommend several types of research that could help in addressing the issue in the future.
Journal Article
Teachers' professional practice conducting mathematical discussions
This paper seeks to identify actions that can be regarded as building elements of teachers' classroom practice in mathematical discussion and how these actions may be combined to provide fruitful learning opportunities for students. It stands on a framework that focuses on two key elements of teaching practice: the tasks that teachers propose to students and the way teachers handle classroom communication. Tasks are appraised concerning their level of challenge. Teachers' actions in discussions are classified as informing/suggesting, guiding, and challenging. The methodology is qualitative with data collected from video recording of the classroom. The analysis of classroom episodes dealing with rational numbers but with different agendas, such as providing students opportunities for learning about representations, concepts, connections, and procedures and for developing reasoning suggests that some degree of challenge promotes fruitful learning situations. However, such situations tend to require preparation and follow-up with guiding and even informing/suggesting actions so that the students can learn what has been set in the teacher's agenda.
Journal Article
THE RELATION BETWEEN LEARNERS’ SPONTANEOUS FOCUSING ON QUANTITATIVE RELATIONS AND THEIR RATIONAL NUMBER KNOWLEDGE
Many difficulties learners have with rational number tasks can be attributed to the \"natural number bias\", i.e. the tendency to inappropriately use natural number properties in rational numbers tasks (Van Hoof, 2015). McMullen and colleagues found a relevant source of individual differences in the learning of those aspects of rational numbers that are susceptible to the natural number bias, namely Spontaneous Focusing On quantitative Relations (SFOR) (McMullen, 2014). While McMullen and colleagues showed that SFOR relates to rational number knowledge as a whole, we studied its relation with several aspects of the natural number bias. Additionally, we 1) included test items addressing operations with rational numbers and 2) controlled for general mathematics achievement and age. Results showed that SFOR related strongly to rational number knowledge, even after taking into account several control variables. Results are discussed for each of the three aspects of the natural number bias separately.
Journal Article
Exploring the way rational expressions trigger the use of \mental\ brackets by primary school students
When a number sentence includes more than one operation, students are taught to follow the rules for the order of operations to get the correct result. In this context, brackets are used to determine the operations that should be calculated first. However, it seems that the written format of an arithmetical expression has an impact on the way students evaluate this expression. It also seems that a connection exists between this way of evaluation and an understanding of structure. Both issues are examined in this paper. A number of arithmetical expressions in a rational form were given to primary school students from Greece and Sweden. The collected findings strengthen our hypothesis that this rational form of the arithmetical expressions was of critical importance for the students' decision on how to evaluate these expressions. They temporarily put aside their knowledge about the rules for the order of operations, bastead, the way they evaluated the expressions indicates an implicit use of what we call in this paper \"mental\" brackets. It is very likely that the use of these \"mental\" brackets is closely connected with students' structure sense.
Journal Article
The Role of the Inhibition of Natural Number Based Reasoning and Strategy Switch Cost in a Fraction Comparison Task
Previous research amply showed the importance of a good fraction understanding but also people's lack of fraction understanding. It is therefore important to investigate the cognitive processes that underlie reasoning with fractions. The present study investigated the role of inhibition and switch costs in fraction comparison tasks. Participants solved a fraction comparison task that alternated between 4 items congruent and 4 items incongruent with natural number reasoning. This allowed to not only investigate congruency switch effects, but also inhibition, given that inhibition was experimentally increased by the prolonged exposure to incongruent trials. Based on data of seventh graders, the present study showed that inhibition does not only play a role in learners' general mathematics achievement, but also in specific areas of mathematics, such as fractions. Moreover, a switch cost was found in the lower accuracy rates and higher reaction times needed to correctly solve switch items compared to non-switch items.
Journal Article
Irrational gap
We investigate how students make sense of irrational exponents. The data comprise 32 interviews with university students, which revolved around a task designed to examine students’sensemaking processes involved in predicting and subsequently interpreting the shape of the graph of
f
(
x
)
=
x
2
. The task design and data analysis relied on the concept of sensemaking trajectories, blending the notions of sensemaking and (hypothetical/actual) learning trajectories. The findings present four typical sensemaking trajectories the participants went through while coping with the notion of an irrational exponent, alongside associated reasoning themes that seemed to have guided these trajectories. In addition to irrational exponents, the analysis revealed the participants’ reasoning and sensemaking of related mathematical ideas, such as rational exponents, approximations of irrational numbers by rational numbers, even/odd functions and numbers, and the meaning of exponentiation in general. The findings provide a step towards a better understanding of students’ conceptual development of irrational exponents, which could in turn be used for the refinement of tasks aimed at promoting students’ comprehension of the topic.
Journal Article
Angles et Grandeur
The series is devoted to the study of scientific and philosophical texts from the Classical and the Islamic world handed down in Arabic. Through critical text editions and monographs, it provides access to ancient scientific inquiry as it developed in a continuous tradition from Antiquity to the modern period. All editions are accompanied by translations and philological and explanatory notes.
eBook
Professional School Counselors Using Choice Theory to Meet the Needs of Children of Prisoners
The prison population in the United States has increased significantly. Children of prisoners experience academic and social challenges. Professional school counselors are in an ideal position to provide theory-based interventions to support children of prisoners. This article (a) describes challenges experienced by children of prisoners, (b) advances choice theory as a theoretical framework to meet their needs, and (c) describes a case study that details the effective use of choice theory with children of prisoners.
Journal Article