A new hypothesis concerning children’s fractional knowledge

The basic hypothesis of the teaching experiment, The Child’s Construction of the Rational Numbers of Arithmetic (Steffe & Olive, 1990) was that children’s fractional schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because when a new scheme is established by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. In that case where children’s fractional schemes do emerge as accommodations in their numerical counting schemes, I regard the fractional schemes as superseding their earlier numerical counting schemes. If one scheme supersedes another, that does not mean the earlier scheme is replaced by the superseding scheme. Rather, it means that the superseding scheme solves the problems the earlier scheme solved but solves them better, and it solves new problems the earlier scheme didn’t solve. It is in this sense that we hypothesized children’s fractional schemes can supersede their numerical counting schemes and it is the sense in which we regarded numerical schemes as constructive mechanisms in the production of fractional schemes (Kieren, 1980).     

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.     

Reinforcing key combinatorial ideas in a computational setting: A case of encoding outcomes in computer programming

Counting problems are difficult for students to solve, and there is a perennial need to investigate ways to help students solve counting problems successfully. One promising avenue for students’ successful counting is for them to think judiciously about how they encode outcomes – that is, how they symbolize and represent the outcomes they are trying to count. We provide a detailed case study of two students as they encoded outcomes in their work on several related counting problems within a computational setting. We highlight the role that a computational environment may have played in this encoding activity. We illustrate ways in which by-hand work and computer programming worked together to facilitate the students’ successful encoding activity. This case demonstrates ways in which the activity of computation seemed to interact with by-hand work to facilitate sophisticated encoding of outcomes.     

Third- to fourth-grade students’ conceptions of multiplication and area measurement

In this study, 34 children were evaluated in order to elucidate their multiplicative thinking and interpretation of the area formula of a rectangle, and to determine what roles these factors play in solving area measurement problems. One-on-one interviews and problem-solving tasks were employed to explore the problem-solving skills of the children regarding these two concepts. This study also explored how the associations changed throughout two consecutive phases, from the third to the fourth grades. The results indicated that in the third grade, multiplicative thinking was associated with the solving of area measurement problems. Third-grade children who understood the meaning of the multiplication symbol “p × q” in models (e.g., the set model and arrays) outperformed children who understood only partial multiplicative concepts or additive thinking; however, the association between multiplicative thinking and solving area measurement problems was not significant in the fourth grade. In contrast, children’s ability to interpret the area formula of a rectangle was associated with their performance at solving area measurement problems throughout the third and fourth grades. The way of interpreting the area formula was associated with the extent to which the children understood multiplication, area measurement, and the spatial concepts embedded in rectangular figures. The instructional implications of the study are discussed in terms of developing child abilities to solve area measurement problems by connecting multiplication and area measurement.     

Supervision of teachers based on adjusted arithmetic learning in special education

This article reports on 20 children's learning in arithmetic after teaching was adjusted to their conceptual development. The report covers periods from three months up to three terms in an ongoing intervention study of teachers and children in schools for the intellectually disabled and of remedial teaching in regular schools. The researcher classified each child's current counting scheme before and after each term. Recurrent supervision, aiming to facilitate the teachers’ modelling of their children's various conceptual levels and needs of learning, was conducted by the researcher. The teaching content in harmony with each child's ability was discussed with the teachers. This approach gives the teachers the opportunity to experience the children's own operational ways of solving problems. At the supervision meetings, the teachers theorized their practice together with the researcher, ending up with consistent models of the arithmetic of the child. So far, the children's and the teachers’ learning patterns are promising.     

CHILDREN INVENTING SIGNS FOR MENTAL NUMBER STRATEGIES

The research study described here was conducted with a small group of 5- and 6-year-old children in a 35-pupil rural school in the Langdale Valley (Lake District, UK) over a period of 14 weeks. It considers the theoretical implications of children inventing their own signs for their counting actions instead of using the culturally inherited signs of conventional arithmetic. It also raises questions concerning ‘transformational addition’ in which ordinals (‘position numbers’) are distinguished from cardinals (‘size numbers’), and suggests that the invented signs may have a consistency and validity comparable with the signs of conventional arithmetic.     

Part-whole number knowledge in preschool children

Ability to reflect on a number as an object of thought, and to isolate its constituent parts, is basic to a deep knowledge of arithmetic, as well as much practical and applied mathematical problem solving. Part-whole reasoning and counting are closely related in children’s numerical development. The mathematical behavior of young children in part-whole problem settings was examined by using a dynamic problem situation, in which a small set of items was partitioned such that one of the subsets remained perceptually inaccessible. Issues addressed include the problem solving strategies successful children used, adaptations children make in response to successive administrations of the task over time, and characterizations of children’s mathematical thinking based on their responses to the task.     

Understanding the Development of Flexibility in Struggling Algebra Students

The ability to flexibly solve problems is considered an important outcome for school mathematics and is the focus of this paper. The paper describes the impact of a three-week summer course for students who struggle with algebra. During the course, students regularly compared and contrasted worked examples of algebra problems in order to promote flexible use of solution strategies. Assessments were designed to capture both knowledge and use of multiple strategies. The students were interviewed in order to understand their rationales for choosing particular strategies as well as their attitudes toward instruction that emphasized multiple strategies. Findings suggest that students gained both knowledge of and appreciation for multiple strategies, but they did not always use alternate strategies. Familiarity, understandability, efficiency, and form of the problem were all considerations for strategy choice. Practical and theoretical implications are discussed.     

“Can you help me count these pennies?”: Surfacing preschoolers’ understandings of counting

ABSTRACT

Capturing the breadth and variety of children’s understanding is critical if studies of children’s mathematical thinking are to inform policy and practice in early childhood education. This article presents an investigation of young children’s counting. Detailed coding and analyses of assessment interviews with 476 preschoolers revealed understandings that would be overlooked by solely assessing the accuracy of their responses. In particular, many children demonstrated understandings of counting principles on a challenging task that were not captured by other, simpler tasks. We conclude that common approaches to capturing young children’s mathematical understanding are likely underestimating their capabilities. This study contributes to researchers’ understanding of what making sense of counting looks and sounds like for preschool age children (3–5 years), the development and relations among counting principles (one-to-one, cardinal, and patterns of the number sequence), and the affordances of challenging, open-ended tasks. We close by considering the implications of recognizing and building from what children know and can do for researchers, practitioners, and policymakers.     

Mapping and development of intuitive proportional thinking

The purpose of this study was two-fold. First, to find out students’ informal understanding of proportional problems, and discuss their solution strategies. Second, to investigate how the intuitions developed by students influence their strategies to solve proportional problems. To this end, we interviewed 16 students in Grades 4 and 5, while they were solving proportional problems. It was found that students intuitively used the unit-rate strategy indicating an attempt to transfer the knowledge resulted by their experience with solving simple multiplicative problems. Fourth and fifth graders tended to shift from the unit-rate strategy to other strategies if there was no easy way to calculate the unit-value directly from the context of the problems. Since fifth graders were more comfortable than fourth graders in calculating the unit-value, they felt less the need to invent other solution strategies.     

Are middle school mathematics teachers able to solve word problems without using variable?

Many people consider problem solving as a complex process in which variables such as x,?y are used. Problems may not be solved by only using ‘variable.’ Problem solving can be rationalized and made easier using practical strategies. When especially the development of children at younger ages is considered, it is obvious that mathematics teachers should solve problems through concrete processes. In this context, middle school mathematics teachers' skills to solve word problems without using variables were examined in the current study. Through the case study method, this study was conducted with 60 middle school mathematics teachers who have different professional experiences in five provinces in Turkey. A test consisting of five open-ended word problems was used as the data collection tool. The content analysis technique was used to analyze the data. As a result of the analysis, it was seen that the most of the teachers used trial-and-error strategy or area model as the solution strategy. On the other hand, the teachers who solved the problems using variables such as x, a, n or symbols such as Δ, □, ○, * and who also felt into error by considering these solutions as without variable were also seen in the study.     

Tracing Robert's use of representations to solve counting problems: A 16-year case study

This study describes how Robert used his external representations (formulae, notations, sketches, models, and figures) to solve progressively more challenging counting tasks over a 16-year period. In his explorations of counting tasks, Robert discovered intricate connections between solutions to problems that looked different on the surface. Using video data from Robert's problem solving, analyses of his solutions are presented that shed light on how he built new ideas from existing ideas and how he modified external representations to make new mathematical discoveries and provide justifications for his solutions.     

Students' Verification Strategies for Combinatorial Problems

We focus on a major difficulty in solving combinatorial problems, namely, on the verification of a solution. Our study aimed at identifying undergraduate students' tendencies to verify their solutions, and the verification strategies that they employ when solving these problems. In addition, an attempt was made to evaluate the level of efficiency of the students' various verification strategies in terms of their contribution to reaching a correct solution. 14 undergraduate students, who had taken at least 1 course in combinatorics, participated in the study. None of the students had prior direct learning experience with combinatorial verification strategies. Data were collected through interviews with individual or pairs of participants as they solved, 1 by 1, 10 combinatorial problems. 5 types of verification strategies were identified, 2 of which were more frequent and more helpful than others. Students' verifications proved most efficient in terms of reaching a correct solution when they were informed that their solution was incorrect. Implications for teaching and learning combinatorics are discussed.     

Analyzing the word-problem performance and strategies of students experiencing mathematics difficulty

The purpose of this study was to examine the word-problem performance and strategies utilized by 3rd-grade students experiencing mathematics difficulty (MD). We assessed the efficacy of a word-problem intervention and compared the word-problem performance of students with MD who received intervention (n = 51) to students with MD who received general education classroom word-problem instruction (n = 60). Intervention occurred for 16 weeks, 3 times per week, 30 min per session and focused on helping students understand the schemas of word problems. Results demonstrated that students with MD who received the word-problem intervention outperformed students with MD who received general education classroom word-problem instruction. We also analyzed the word-problem strategies of 30 randomly-selected students from the study to understand how students set up and solve word problems. Students who received intervention demonstrated more sophisticated word-problem strategies than students who only received general education classroom word-problem instruction. Findings suggest students with MD benefit from use of meta-cognitive strategies and explicit schema instruction to solve word problems.     

To teach or not to teach algorithms

Second, third, and fourth graders in 12 classes were individually interviewed to investigate the effects of teaching computational algorithms such as those of “carrying.” Some of the children had been encouraged to invent their own procedures and had not been taught any algorithms in grades 1 and 2, or in grades 1–3. Others had been taught the conventional algorithms prescribed by textbooks. The children were asked to solve multidigit addition and multiplication problems and to explain how they got their answers. It was found that those who had not been taught any algorithms produced significantly more correct answers. If children made errors, the incorrect answers of those who had not been taught any algorithms were much more reasonable than those found in the “Algorithms” classes. It was concluded that algorithms “unteach” place value and hinder children's development of number sense.     

Control decisions and personal beliefs: their effect on solving mathematical problems

The main purpose of this paper is to discuss how college students enrolled in a college level elementary algebra course exercised control decisions while working on routine and non-routine problems, and how their personal belief systems shaped those control decisions. In order to prepare students for success in mathematics we as educators need to understand the process steps they use to solve homework or examination questions, in other words, understand how they “do” mathematics. The findings in this study suggest that an individual’s belief system impacts how they approach a problem. Lack of confidence and previous lack of success combined to prompt swift decisions to stop working. Further findings indicate that students continue with unsuccessful strategies when working on unfamiliar problems due to a perceived dependence of solution strategies to specific problem types. In this situation, the students persisted in an inappropriate solution strategy, never reaching a correct solution. Control decisions concerning the pursuit of alternative strategies are not an issue if the students are unaware that they might need to make different choices during their solutions. More successful control decisions were made when working with familiar problems.     

Making Sense by Building Sense: Kindergarten Children’s Construction and Understanding of Adaptive Robot Behaviors

This study explores young children’s ability to construct and explain adaptive behaviors of a behaving artifact, an autonomous mobile robot with sensors. A central component of the behavior construction environment is the RoboGan software that supports children’s construction of spatiotemporal events with an a-temporal rule structure. Six kindergarten children participated in the study, three girls and three boys. Activities and interviews were conducted individually along five sessions that included increasingly complex construction tasks. It was found that all of the children succeeded in constructing most such behaviors, debugging their constructions in a relatively small number of cycles. An adult’s assistance in noticing relevant features of the problem was necessary for the more complex tasks that involved four complementary rules. The spatial scaffolding afforded by the RoboGan interface was well used by the children, as they consistently used partial backtracking strategies to improve their constructions, and employed modular construction strategies in the more complex tasks. The children’s explanations following their construction usually capped at one rule, or two condition-action couples, one rule short of their final constructions. With respect to tasks that involved describing a demonstrated robot’s behavior, in describing their constructions, explanations tended to be more rule-based, complex and mechanistic. These results are discussed with respect to the importance of making such physical/computational environments available to young children, and support of young children’s learning about such intelligent systems and reasoning in developmentally-advanced forms.     

Students' Verification Strategies for Combinatorial Problems

We focus on a major difficulty in solving combinatorial problems, namely, on the verification of a solution. Our study aimed at identifying undergraduate students' tendencies to verify their solutions, and the verification strategies that they employ when solving these problems. In addition, an attempt was made to evaluate the level of efficiency of the students' various verification strategies in terms of their contribution to reaching a correct solution. 14 undergraduate students, who had taken at least 1 course in combinatorics, participated in the study. None of the students had prior direct learning experience with combinatorial verification strategies. Data were collected through interviews with individual or pairs of participants as they solved, 1 by 1, 10 combinatorial problems. 5 types of verification strategies were identified, 2 of which were more frequent and more helpful than others. Students' verifications proved most efficient in terms of reaching a correct solution when they were informed that their solution was incorrect. Implications for teaching and learning combinatorics are discussed.     

Undergraduate students’ initial conceptions of factorials

Counting problems offer rich opportunities for students to engage in mathematical thinking, but they can be difficult for students to solve. In this paper, we present a study that examines student thinking about one concept within counting, factorials, which are a key aspect of many combinatorial ideas. In an effort to better understand students’ conceptions of factorials, we conducted interviews with 20 undergraduate students. We present a key distinction between computational versus combinatorial conceptions, and we explore three aspects of data that shed light on students’ conceptions (their initial characterizations, their definitions of 0!, and their responses to Likert-response questions). We present implications this may have for mathematics educators both within and separate from combinatorics.