First-graders’ knowledge of multiplicative reasoning before formal instruction in this domain

Our study investigated children’s knowledge of multiplicative reasoning (multiplication and division) at the end of Grade 1, just before the start of formal instruction on multiplicative reasoning in Grade 2. A large sample of children (N = 1176) was assessed in a relatively formal test setting, using an online test with 28 multiplicative problems of different types. On average, the children correctly answered more than half (58%) of the problems, including several bare number problems. This indicates that before formal instruction on multiplicative reasoning, children already have a considerable amount of knowledge in this domain, which teachers can build on when teaching them formal multiplication and division. Using analysis of variance and cross-classified multilevel regression analysis, we identified several predictors of children’s pre-instructional multiplicative knowledge. With respect to the characteristics of the multiplicative problems, we found that the problems were easiest to solve when they included a picture involving countable objects, and when the multiplicative situation was of the equal groups semantic structure (e.g., 3 boxes of 4 cookies). Regarding student characteristics, pre-instructional multiplicative knowledge was higher for children with higher-educated parents. Finally, the mathematics textbook used in school appeared to have influenced children’s pre-instructional multiplicative knowledge.     

Multiplicative reasoning: teaching primary pupils in ways that focus on functional relations

This paper examines a problem described as widespread and long-standing in mathematics education: supporting pupils into multiplicative reasoning, a form of reasoning that has been noted as central to large tracts of secondary mathematics and beyond. Also noted, however, is a persistent perception of multiplicative situations only in terms of repeated addition – a perception held not only primary pupils, but also among primary teachers and curriculum developers. The focus of this paper is to synthesize literature on multiplicative reasoning as a conceptual field together with a sociocultural discussion of the role of mediating artifacts in the development of this conceptual field. Bringing MR into the primary classroom can then be achieved, I propose, through a pedagogy oriented toward model-eliciting and teacher appropriation of pupils’ models as pedagogic tools with the subsequent re-appropriation of refined models by pupils. This pedagogy is illustrated through the analysis of two vignettes from a teaching experiment which demonstrate the beginnings of MR as an emergent conceptual field in the classroom. The paper concludes that it is possible to move primary teaching and learning toward understanding the functional aspects of multiplicative reasoning, but that any such moves requires attention to teachers’ pedagogic and content knowledge.     

Understanding the concepts of proportion and ratio constructed by two grade six students

The purpose of this study was to construct an understanding of two grade six students' proportional reasoning schemes. The data from the clinical interviews gives insight as to the importance of multiplicative thinking in proportional reasoning. Two mental operations, unitizing and iterating play an important role in student's use of multiplicative thinking in proportion tasks. Unitizing a composite unit and iterating it to its referent point enables one to preserve the invariance of a ratio. Proportions involved the coordination of two number sequences, keeping the ratio unit invariant under the iteration. In the iteration process, one needed to explicitly conceptualize the iteration action of the composite ratio unit to make sense of ratio problems and to have sufficient understanding of the meaning of multiplication and division and its relevance in the iteration process. One needed to have constructed multiplicative structures and iteration schemes in order to reason proportionally. This revised version was published online in July 2006 with corrections to the Cover Date.     

Deaf children's informal knowledge of multiplicative reasoning

Multiplicative reasoning is required in different contexts in mathematics: it is necessary to understand the concept of multipart units, involved in learning place value and measurement, and also to solve multiplication and division problems. Measures of hearing children's multiplicative reasoning at school entry are reliable and specific predictors of their mathematics achievement in school. An analysis of deaf children's informal multiplicative reasoning showed that deaf children under-perform in comparison to the hearing cohorts in their first two years of school. However, a brief training study, which significantly improved their success on these problems, suggested that this may be a performance, rather than a competence difference. Thus, it is possible and desirable to promote deaf children's multiplicative reasoning when they start school so that they are provided with a more solid basis for learning mathematics.     

Educating to Use Evidence in Thinking About Education

There is an increasing emphasis on evidence‐based education, and the sciences of learning are progressing rapidly. But are reports, guidelines, and outreach enough to disseminate this knowledge and affect educational practice? In fact, policy makers and the public often resist evidence‐based recommendations about education. This article suggests that some of the problems lie in well‐known difficulties in everyday reasoning and decision making, especially in situations that involve probability and uncertainty. We should communicate and educate so as to address these difficulties.     

Mental Calculation: its place in the development of numeracy

Recent curricular initiatives in the UK have been emphasising the importance of mental calculation in the achievement of improved numeracy standards. This article argues that while the emphasis on mental calculation is to be welcomed, the construct itself needs to be understood in all its complexity. This means recognising what is involved in mental calculation and, further, recognising the developmental shift from additive to multiplicative reasoning which is necessary for a mature conceptualisation of number. It is suggested that this shift is difficult to achieve because it involves reconceptualising the essential meaning of ‘a number’. Furthermore, the pedagogical practices which could possibly support children in their transition from additive to multiplicative reasoning must be more than helping children to develop an ever increasing repertoire of mental strategies.     

Prospective mathematics teachers’ sense making of polynomial multiplication and factorization modeled with algebra tiles

This study is about prospective secondary mathematics teachers’ understanding and sense making of representational quantities generated by algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations—referent preserving versus referent transforming compositions—acting on these quantities. Although multiplicative structures can be modeled by additive structures, they have their own characteristics inherent in their nature. I situate my analysis within a framework of unit coordination with different levels of units supported by a theory of quantitative reasoning and theorems-in-action. Data consist of videotaped qualitative interviews during which prospective mathematics teachers were asked problems on multiplication and factorization of polynomial expressions in x and y. I generated a thematic analysis by undertaking a retrospective analysis, using constant comparison methodology. There was a pattern which showed itself in all my findings. Two student–teachers constantly relied on an additive interpretation of the context, whereas three others were able to distinguish between and when to rely on an additive or a multiplicative interpretation of the context. My results indicate that the identification and coordination of the representational quantities and their units at different categories (multiplicative, additive, pseudo-multiplicative) are critical aspects of quantitative reasoning and need to be emphasized in the teaching–learning process. Moreover, representational Cartesian products-in-action at two different levels, indicators of multiplicative thinking, were available to two research participants only.     

Natural-Born Arguers: Teaching How to Make the Best of Our Reasoning Abilities

We summarize the argumentative theory of reasoning, which claims that the main function of reasoning is to argue. In this theory, argumentation is seen as being essentially cooperative (people have to listen to others' arguments and be ready to change their mind) but with an adversarial dimension (their goal as argument producers is to convince). Consistent with this theory, the experimental literature shows that solitary reasoning is biased and lazy, whereas reasoning in group discussion produces good results, provided some conditions are met. We formulate recommendations for improving reasoning performance, mainly, to make people argue more and better by creating felicitous conditions for group discussion. We also make some suggestions for improving solitary reasoning, in particular to maximize students' exposure to arguments challenging their positions. Teaching people about the value of argumentation is likely to improve not only immediate reasoning performance but also long-term solitary reasoning skills.     

Effects of playing mathematics computer games on primary school students’ multiplicative reasoning ability

This study used a large-scale cluster randomized longitudinal experiment (N = 719; 35 schools) to investigate the effects of online mathematics mini-games on primary school students’ multiplicative reasoning ability. The experiment included four conditions: playing at school, integrated in a lesson (E_school), playing at home without attention at school (E_home), playing at home with debriefing at school (E_home-school) and, in the control group, playing at school mini-games on other mathematics topics (C). The mini-games were played in Grade 2 and Grade 3 (32 mini-games in total). Using tests at the end of each grade, effects on three aspects of multiplicative reasoning ability were measured: knowledge of multiplicative number facts, skills in multiplicative operations, and insight in multiplicative number relations and properties of multiplicative operations. Through path analyses it was found that the mini-games were most effective in the E_home-school condition, where both students’ skills and their insight were positively affected as compared to the control group (significant ds ranging from 0.22 to 0.29). In the E_school condition, an effect was only found for insight in Grade 2 (d = 0.35), while in the E_home condition no significant effects were found. Students’ gameplay behavior (time and effort put in the games) was in some cases, but not always, related to their learning outcomes.     

Characteristics of 5th Graders' Logical Development Through Learning Division with Decimals

When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter level, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school be clarified. In this study we focus on the teaching and learning of “division with decimals” in a 5th grade classroom, because it is well known to be difficult for children to understand the meaning of division with decimals, caused by certain conceptions which children have implicitly or explicitly. In this paper we discuss how children develop their logical reasoning beyond such difficulties/misconceptions in the process of making sense of division with decimals in the classroom setting. We then suggest that children's explanations based on two kinds of reversibility (inversion and reciprocity) are effective in overcoming the difficulties/misconceptions related to division with decimals, and that they enable children to conceive multiplication and division as a system of operations.     

Building Conceptual Understanding of Multiplicative Reasoning Content in Third Graders Struggling to Learn Mathematics: A Feasibility Study

This formative study of a multiplicative reasoning (MR) intervention explored the intervention's potential for improving the ability of third-grade struggling students’ ability to reason with multiplicative concepts and procedures. The feasibility of the study was examined in a school setting before a randomized control trial was conducted. Students who scored between the 10th and 35th percentile on a district-administered math screening test received the MR intervention from their teachers. We developed intervention units to build a conceptual foundation in a student-centered approach to Tier 2 instruction that included opportunities for students to engage in critical thinking as they generalized big ideas, participated in classroom discourse, and modeled multiplicative relationships with multiple representations. Preliminary data demonstrate the potential of the intervention to promote students’ MR skills. Instructional implications are discussed in terms of opportunities for these students to engage in grade-level mathematics content.     

Working with words

Abstract

Despite the rhetoric that students with learning difficulties are adequately supported within schools, the evidence suggests that they continue to experience school failure with devastating consequences. Students with learning difficulties are disproportionately represented as juvenile delinquents, as the unemployed and in mental health statistics. However, the defining of this group remains confused and imprecise and has not been a national priority. This has repercussions for both secondary schools and for the students themselves. This paper highlights research related to teaching practices, policies and school structure and their effects on the academic outcomes and emotional well being of students with learning difficulties. Finally, it makes a number of recommendations to change the status quo for these students.     

不同类型推理偏差和困难研究综述

心理学对推理困难的研究主要体现在演绎推理、归纳推理和类比推理等方面。不同类型的推理困难研究有自己的特点,但都有深厚的逻辑学渊源。国内学者对推理困难的研究比较零散,主要针对引起困难的原因进行探讨。文章从研究内容和研究方法等方面分析了推理困难研究的不足,对研究的发展趋势进行了预测。     

Students’ Ideas about How and Why Chemical Reactions Happen: Mapping the conceptual landscape

Research in science education has revealed that many students struggle to understand chemical reactions. Improving teaching and learning about chemical processes demands that we develop a clearer understanding of student reasoning in this area and of how this reasoning evolves with training in the domain. Thus, we have carried out a qualitative study to explore students reasoning about chemical causality and mechanism. Study participants included individuals at different educational levels, from college to graduate school. We identified diverse conceptual modes expressed by students when engaged in the analysis of different types of reactions. Main findings indicate that student reasoning about chemical reactions is influenced by the nature of the process. More advanced students tended to express conceptual modes that were more normative and had more explanatory power, but major conceptual difficulties persisted in their reasoning. The results of our study are relevant to educators interested in conceptual development, learning progressions, and assessment.     

Some multiplicative structures in elementary education: a view from relational paradigm

Educational Studies in Mathematics - The multiplicative reasoning that students should develop in elementary school is a key area of research in contemporary mathematics education. Researchers...     

2021年PISA数学——来自CCR的分析

数学是理解世界、公民身份和经济增长的基石。为了满足全社会对教育的需求,21世纪教育应该注重对知识理解的深度和多样性的培养。PISA关于数学能力的测试中,最重视学生运用数学推理来解决问题的能力。我们建议扩展数学过程的描述(表述、应用、解释、评估),并在PISA数学框架内确定这些处理过程为数学建模的主要组成部分,其中有七个最常用于寻找正确推理方法的推理工具:比较、比例推理、应用乘法量表、拆分、归并、由简入繁、概率推理和逻辑推理。PISA数学的素养领域涉及形状与空间、变化与关系、不确定性与数据、数量等,还要特别注意创造性思维能力、品格和元认知技能的培养。     

Designing a sequence of activities to build reasoning about data and visualization

Our complex world requires multivariate reasoning to make sense of reality. Within this paper, we offer a sequence of activities designed to develop multivariate reasoning by explicitly connecting data and visualization. The activities were designed based on a hypothetical learning trajectory we conjectured for students with limited experience with multivariate visualizations. Drawing from evidence collected using these activities in a series of professional development sessions with in-service teachers, we find that the activities functioned as intended, and thus we promote these activities for developing students' multivariate reasoning at the secondary and post-secondary level. We detail specific challenges the teachers faced, and based on these results, offer our reflections and recommendations for curricula and teaching.     

RELATIONSHIPS BETWEEN FRACTIONAL KNOWLEDGE AND ALGEBRAIC REASONING: THE CASE OF WILLA

To investigate relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The students were interviewed twice, once to explore their quantitative reasoning with fractions and once to explore their solutions of problems that required explicit use of unknowns to write equations. As a part of the larger study, the first author conducted a case study of a seventh grade student, Willa. Willa’s fractional knowledge—specifically her reversible iterative fraction scheme and use of fractions as multipliers—influenced how she wrote equations to represent multiplicative relationships between two unknown quantities. The finding indicates that implicit use of powerful fractional knowledge can lead to more explicit use of structures and relationships in algebraic situations. Curricular and instructional implications are explored.     

Beyond additive and multiplicative reasoning abilities: how preference enters the picture

While previous studies mainly focused on children’s additive and multiplicative reasoning abilities, we studied third to sixth graders’ preference for additive or multiplicative relations. This was investigated by means of schematic problems that were open to both types of relations, namely arrow schemes containing three given numbers and a fourth missing one. In study 1, children had to fill out the missing number, while in study 2, children had to indicate all possibly correct answers among a set of given alternatives. Both studies explicitly showed the existence of a preference for additive relations in some children, while others preferred multiplicative relations. Mainly younger children preferred additive relations, whereas mainly children in upper primary education preferred multiplicative relations. Number ratios also impacted children’s preference, especially in fifth grade. Moreover, the results of study 2 provided evidence for the strength of children’s preference and showed that calculation skills do not coincide with preference, and hence, that preference and calculation skills are two distinct child characteristics. The results of both studies using these open problems resembled previous research results using classical multiplicative or additive word problems. This supports the hypothesis that children’s preferred type of relations may be at play in solving classical word problems as well—besides their abilities—and may hence be an additional factor explaining the mistakes that children make in those word problems. This research line thus seems promising for further research as well as educational practice.