Teaching children to add and subtract

At a 1980 conference, leading mathematics educators synthesized previous knowledge on children's early understanding of addition and subtraction and proposed central parameters for future research in these areas form a cognitive science perspective. We have, since 1980, increased our knowledge about how children learn to add and subtract, but we need to know more about the best ways for teachers to guide children as they construct knowledge of addition and subtraction.In this article, we review several studies that focus on an enhanced role for teachers in enabling children to learn addition and subtraction. These studies describe efforts that have been made to teach children to use diagrams and mediational representations, number sentences, or algorithms and procedures. The studies report improvement in children's problem-solving performance, but the impact of the efforts described on children's conceptual understanding is less clear. Thus, we analyze this research, pose questions on the relationship of instruction to children's knowledge construction, and propose a research agenda that we believe will enable us to understand how teaching can best help children learn to add and subtract.     

The model method: Students’ performance and its effectiveness

This study describes Singapore students’ (N = 607) performance on two tasks in a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia's NAPLAN and Singapore's PSLE. This study also examines students’ use of the model method to solve the two tasks. The model method is a visual problem-solving heuristic prevalently used in Singapore classrooms. The study found that students who solved the tasks using a visual method predominantly used the model method as a visual problem-solving strategy. Another interesting observation was the hindrance of successful problem solving caused by the persistence of prototypical images of model drawings. Implications include encouraging teachers to get their students to identify problem situations where the model method will both work and not work well, and making the role of the generator in the model method explicit in the mathematics textbooks.     

Children's approaches to area measurement through different contexts

This study focuses on 12 years old children's approaches to area measurement in a project environment. These approaches are not explored through a specific set of mathematical tasks. The tasks, here, are defined through researchers' and children's interactions in a classroom. The children by working in small groups are asked to make a proposal about the location and the form of an area which would be given to them for their leisure activities. This environment defines different contexts where the children act and consider different aspects of the area measurement. These aspects are identified and compared among the three groups of children. The study has shown that the concept of area measurement carries different cultural dimensions for the children. Moreover, the children use those elements of the concept which fit in with their personal experience and the tasks they have to face.     

From Fractions to Rational Numbers of Arithmetic: A Reorganization Hypothesis

Nathan and Arthur, 2 children in a 3-year teaching experiment on children's construction of the rational numbers of arithmetic (RNA), developed their operations for multiplying, dividing, and simplifying fractions over the last 2 years (Grades 4 and 5) of the experiment. The 2 children worked in the context of specially developed computer microworlds with a teacher/researcher for approximately 45 min a week for 50 weeks over the 2-year period. The children's construction of multiplicative fractional schemes was investigated in a retrospective analysis of each of the 50 videotaped teaching episodes. Four distinct modifications of the children's fractional schemes were discerned that contributed to their construction of the RNA. The investigation suggested that the operations and unit types associated with the children's whole-number sequences did not interfere with the reorganization of their fractional schemes but rather contributed to those schemes. The reorganization involved an integration of their whole-number knowledge with their fractional schemes whereby whole-number division was regarded as the same as multiplication by the reciprocal fraction.     

Reflection and Problem Solving: Integrating Methods of Teaching Mathematics and Science

An innovative teacher preparation course which integrates methods of teaching elementary mathematics and science was the context of this study. The course was developed as a prototype for the Teachers As Reflective Problem Solvers model for the preparation of elementary mathematics and science teachers. Data from 35 preservice elementary teachers' performance-portfolios were analyzed to reveal patterns of change in their reflections and problem-solving performance during the semester. Many of the students' reflections changed from task-focused learning to broader teaching applications. No relationship was discerned between changes in students' reflections and changes in their levels of problem-solving performance, although both increased during the semester. A significant correspondence was found, however, between students' perceptions of their problem-solving abilities and their actual performance in solving teaching problems in integrated mathematics and science contexts.     

Children's Perspectives on the Engagement Potential of Mathematical Problem Tasks

This article examines elementary students' perspectives on the engagement potential of particular mathematical problems and students' views on their general classroom problem activities. Third-, fifth-, and seventh-grade children from different reform-oriented classrooms were individually interviewed about (a) how they would improve their classroom problem-solving activities and (b) the problems they find the most and least potentially engaging when presented with a range of routine and nonroutine problems. The children requested more relevant, meaningful, and interesting problem experiences in their classrooms, and the fifth and seventh graders requested more representational materials. The children's criteria for determining potentially engaging and nonengaging problems primarily pertained to problem structure and perceived cognitive demands. The nonroutine examples that focused on important reasoning processes and did not involve computation had the greatest engagement potential, while the computational problems had the least appeal.     

Changing Children's Conceptions of Burning

Relatively few studies have examined the effects of instruction on children's understanding of burning. This study focused on three questions: (a) What are children's views of burning prior to and after instruction? (b) Do children's views become more scientific, that is, more in accord with scientists' views, with instruction, and if so, how? (c) Are the changes in children's understanding of burning correlated to their ages? Data were collected before and after five hours of instruction in a Saturday Science Program, using both a short multiple choice test based on common misconceptions from the literature and "interviews about events." Children were divided into two classes according to their grade in school. A significant difference was found in children's understanding before and after instruction on the multiple‐choice test that was corroborated with interview data. Younger children (ages 8 to 11) made more significant gains than did the older children (ages 11 to 13), with both groups reaching similar levels of understanding after instruction. Although notable gains were made in recognizing the need for oxygen in burning and in distinguishing between decomposition and burning, interviews revealed that few children at any age could explain specifically what was happening on the phenomena level.     

Children's Ideas About Weather: A Review of the Literature

It is generally accepted that children have their own understanding of how the world works prior to receiving formal science instruction. A great deal of research has been done to determine students' misconceptions related to the physical sciences; less has been done to understand children's ideas in the Earth sciences. This paper reports a synthesis of the existing research about children's misconceptions relating to weather, climate and the atmosphere. The scientifically accepted interpretations are presented in tandem with the children's naïve ideas. When possible, a source of the misconception is also presented. In many cases, students' misconceptions are not addressed in the curriculum, allowing them to exist unchallenged.     

An Integrated Research on Children's Construction of Meaningful,Symbolic, Partitioning-related Conceptions and the Teacher's Role in Fostering That Learning

A teaching experiment was conducted with two fourth graders to study the co-emergence of teaching and children's construction of fraction knowledge. The children's learning, i.e., modifications in their fraction schemes, was fostered through working on tasks in a computer microworld. The children advanced from thinking about a unit fraction as one of several equal parts in a whole (the equipartitioning scheme) to operating with a unit fraction as a symbolized, iterable part the magnitude of which is based on the numerosity of the partitioned whole (the partitive fraction scheme). The paper interweaves an analysis of children's construction of partitioning-related symbolic conceptions of fractions with an analysis of the teaching—planning and using tasks—that fosters such an advancement by introducing fraction words and numerals in the context of the children's partitioning activities.     

The use of additive composition in arithmetic: the case of children classified as low attainers

This study investigated the conceptual understanding that low-attaining children have and are able to use in arithmetic. Fifteen 6–7 year old children solved pairs of conceptually-related addition problems. Conceptual relations between equal problems were constructed to reflect aspects of the principle of additive composition. Children's conceptual understanding was explored by examining their capability to use concept-based approaches in related problems, and ability to recognise and explain additive composition relationships. The findings indicate that, when prompted, children who employ only basic calculation procedures have the capability to recognise and use additive composition relations in problem solving. Almost all children showed increased sensitivity to additive composition relations when asked to explain the equality between related problems. Our findings highlight the need to develop pedagogical approaches that instigate low-attaining children's conceptual capabilities and support the operationalisation of these in the kinds of concept-based strategies that are most typically ascribed to high-attaining children.     

Teachers’ metacognitive and heuristic approaches to word problem solving: analysis and impact on students’ beliefs and performance

We conducted a 7-month video-based study in two sixth-grade classrooms focusing on teachers’ metacognitive and heuristic approaches to problem solving. All problem-solving lessons were analysed regarding the extent to which teachers implemented a metacognitive model and addressed a set of eight heuristics. We observed clear differences between both teachers’ instructional approaches. Besides, we examined teachers’ and students’ beliefs about the degree to which metacognitive and heuristic skills were addressed in their classrooms and observed that participants’ beliefs were overall in line with our observations of teachers’ instructional approaches. In addition, we investigated how students’ problem-solving skills developed as a result of teachers’ instructional approaches. A positive relationship between students’ spontaneous application of heuristics to solve non-routine word problems and teachers’ references to these skills in their problem-solving lessons was found. However, this increase in the application of heuristics did not result in students’ better performance on these non-routine word problems.     

The Road To Digitopolis: Perils of Problem Solving

How students solve problems is a topic of central concern both to educational researchers and to math/science teachers: What is the nature of good and poor problem solving? How can students improve their problem-solving capacities? Teachers are in a unique position to witness problem solving in action, and to draw connections between the classroom experiences of their students and the findings of research. This article presents an instance of problem solving (drawn from a popular children's book) annotated with references to current research in cognition and education. The annotations explore issues such as the effect of performance anxiety on problem solving, how problem solvers handle the experience of confusion, and the role of self-monitoring and metacognition in problem solving.     

A Case Study of Three Childrens' Original Interpretations of the Moon's Changing Appearance

A case study of three children was conducted to shed light on the process that children undergo in developing their understanding of physical phenomena. Using the notion of spontaneous construction and its relationship with school learning of scientific concepts, children's early thoughts of the moon's appearance were explored. Research questions were primarily concerned with how children view the moon's appearance, explain how and/or why its appearance changes, quantify the moon's size and its distance to Earth, and explain the moon's illumination. A Piagetian interview was conducted with each child and then each was asked to tell a story about the moon. The external interest of this research study involves when and why do children develop the commonly held Earth's shadow alternative conception as the cause of the moon's phases. The findings show that children have stories and experiences that give meaning to the existence of such things as the moon, stars, sun, and clouds. Similarities were found in the children's interpretations with regard to their natural tendencies to animate celestial objects. Clues were discovered of cultural influence such as family, personal observations and experiences, books, pictures, car travel, and even a strategically placed Palladian window.     

Fractions as Subtraction: An Activity‐Oriented Perspective from Elementary Children

A sample of third‐, fourth‐, and fifth‐grade student responses to the question “What is a fraction?” were examined to gain an understanding of how children in upper elementary grades make sense of fractions. Rather than measure children's understanding of fractions relative to mathematically conventional part–whole constructions of fractions, we attempted to understand children's actions and processes. A small but nontrivial group of children used subtraction (takeaway and removal) as a framework for understanding how fractions were created and written. An analysis of the content of their responses as well as a comparison of the performance of these children with that of children who used other ways of describing fractions suggests that the use of subtraction may be a reasonable (or at least not harmful) way for children to begin to access concepts related to fractions. Also, this study suggests that attention to children's understanding through the lens of children's activity might reveal ways of thinking and insights that are masked when we compare children's thinking in more structured research settings.     

Mathematics in Children's Thinking

The human mind inevitably comprehends the world in mathematical terms (among others). Children's informal and invented mathematics contains on an implicit level many of the mathematical ideas that teachers want to promote on a formal and explicit level. These ideas may be innate, constructed for the purpose of adaptation, or picked up from an environment that is rich in mathematical structure, regardless of culture. Teachers should attempt to uncover the mathematical ideas contained in their students' thinking because much, but not all, of the mathematics curriculum is immanent in children's informal and invented knowledge. This mathematical perspective requires a focus not only on the child's constructive process but also on the mathematical content underlying the child's thinking. Teachers then can use these crude ideas as a foundation on which to construct a significant portion of classroom pedagogy. In doing this, teachers should recognize that children's invented strategies are not an end in themselves. Instead, the ultimate goal is to facilitate children's progressive mathematization of their immanent ideas. Children need to understand mathematics in deep, formal, and conventional ways.     

Adults’ Perceptions of Children's Science Abilities and Interest After Participating in a Family Science Night

The goal of this research was to examine adults’ and children's perceptions of participating in a family science night event, especially in the context of parental belief about children's science abilities. Family science nights are becoming increasingly popular and are used in a wide range of settings. During family science nights, adults and students jointly engaged in a variety of science activities. Results revealed that adults, 90% of whom were parents of attending students, reported learning more about children's interests and abilities in science. Students also agreed that that the adults had learned more about their abilities and interests. Personal characteristics of adults and children, such as gender and ethnicity, were not found to have any relationship to ratings.     

Primary school geometry: some unusual activities†

The lecture given in Lesotho in November 1984 outlines a series of activities which could enliven primary school geometry. It begins with a brief look at the meaning of geometry and how this is related to children's conceptions of space. It puts forward a curriculum which places a high value on perspective and other questions about children's visual acquisition of information. It continues with drawing activities and with exercises in paper folding. It suggests further ways in which geometry is entwined with geography and asks teachers to determine how their children make connections between the conventional primary school geometry they receive and their individual spatial perceptual structures.

     

CHILDREN'S PROPORTIONAL REASONING AND TENDENCY FOR AN ADDITIVE STRATEGY: THE ROLE OF MODELS

In this paper we report on 10 –14 year old children's strategies while solving two versions of ratio and proportion tasks: one ‘with models’ thought to facilitate proportional reasoning and one ‘without’. Rasch methodology was used to develop ‘with’ and ‘without models’ test versions which were given to a linked sample involving 673 children. We examine the pupils’ additive errors, their effect on ratio reasoning and how contingent on ‘model’ presentation this is. First, we provide a single scale on which pupils, item-difficulty and additive errors can be located. We then provide a new scale constructed from the error prone items, which we name the ‘tendency for additive strategy’. The measurement data is supported by qualitative data showing that the presence of ‘models’ can sometimes affect children's strategies, both positively and negatively but rarely makes a significant measurement difference on this, untutored, sample.