Spatial visualizers, object visualizers and verbalizers: their mathematical creative abilities

This paper investigates the relationship between the creative process in mathematical tasks and spatial, object and verbal cognitive styles. A group of 96 prospective primary school teachers completed the Object-Spatial Imagery and Verbal Questionnaire and took a mathematical creativity test. The results of a multiple regression analysis demonstrated that whereas visual cognitive styles (spatial and object imagery) were statistically significant predictors of participants’ creative abilities in mathematics, verbal cognitive style did not predict these abilities. Further analysis of the data indicated that spatial imagery cognitive style was related to mathematical fluency, flexibility and originality. On the other hand, object imagery cognitive style was negatively related to mathematical originality and verbal cognitive style was negatively related to mathematical flexibility. The study also revealed that individuals with a tendency towards different cognitive styles employed different strategies in the creative mathematical tasks.     

How young students communicate their mathematical problem solving in writing

This study investigates young students’ writing in connection to mathematical problem solving. Students’ written communication has traditionally been used by mathematics teachers in the assessment of students’ mathematical knowledge. This study rests on the notion that this writing represents a particular activity which requires a complex set of resources. In order to help students develop their writing, teachers need to have a thorough knowledge of mathematical writing and its distinctive features. The study aims to add to the body of knowledge about writing in school mathematics by investigating young students’ mathematical writing from a communicational, rather than mathematical, perspective. A basic inventory of the communicational choices, that are identifiable across a sample of 519 mathematical texts, produced by 9–12 year old students, is created. The texts have been analysed with multimodal discourse analysis, and the findings suggest diversity in students’ use of images, words, numerals, symbols and layout to organize their texts and to represent their problem-solving process along with an answer to the problem. The inventory and the indication that students have different ideas on how, what, for whom and why they should be writing, can be used by teachers to initiate discussions of what may constitute good communication.     

Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics

Many researchers have investigated flexibility of strategies in various mathematical domains. This study investigates strategy use and strategy flexibility, as well as their relations with performance in non-routine problem solving. In this context, we propose and investigate two types of strategy flexibility, namely inter-task flexibility (changing strategies across problems) and intra-task flexibility (changing strategies within problems). Data were collected on three non-routine problems from 152 Dutch students in grade 4 (age 9–10) with high mathematics scores. Findings showed that students rarely applied heuristic strategies in solving the problems. Among these strategies, the trial-and-error strategy was found to have a general potential to lead to success. The two types of flexibility were not displayed to a large extent in students’ strategic behavior. However, on the one hand, students who showed inter-task strategy flexibility were more successful than students who persevered with the same strategy. On the other hand, contrary to our expectations, intra-task strategy flexibility did not support the students in reaching the correct answer. This stemmed from the construction of an incomplete mental representation of the problems by the students. Findings are discussed and suggestions for further research are made.     

Relationships between students’ learning and their participation during enactment of middle school algebra tasks

This study examines a sequence of four middle school algebra tasks through their enactment in three teachers’ classrooms. The analysis centers on the cognitive demand—the kinds of thinking processes entailed in solving the task—and the participatory demand—the kinds of verbal contributions expected of students—of the task as written in the instructional materials, as set up by the three teachers, and as discussed by the teachers and their students. Relationships between the nature of the task enactments and students’ performance on a pre- and post-test are explored. Findings include the fact that the enacted tasks differed from the written tasks with regard to both the cognitive demand and the participatory demand, which related to students’ lack of success on the post-test. Specifically, cognitive demand declined in the enacted curriculum at different points for different classes, and the participatory demand during enactment tended to involve isolated mathematical terms rather than students verbally expressing mathematical relations.     

Beliefs and engagement structures: behind the affective dimension of mathematical learning

Beliefs influencing students’ mathematical learning and problem solving are structured and intertwined with larger affective and cognitive structures. This theoretical article explores a psychological concept we term an engagement structure, with which beliefs are intertwined. Engagement structures are idealized, hypothetical constructs, analogous in many ways to cognitive structures. They describe complex “in the moment” affective and social interactions as students work on conceptually challenging mathematics. We present engagement structures in a self-contained way, paying special attention to their theoretical justification and relation to other constructs. We suggest how beliefs are characteristically woven into their fabric and influence their activation. The research is based on continuing studies of middle school students in inner-city classrooms in the USA.     

The role of prior knowledge in the development of strategy flexibility: the case of computational estimation

The ability to estimate is a fundamental real-world skill; it allows students to check the reasonableness of answers found through other means, and it can help students develop a better understanding of place value, mathematical operations, and general number sense. Flexibility in the use of strategies is particularly critical in computational estimation. The ability to perform complex calculations mentally is cognitively challenging for many students; thus, it is important to have a broad repertoire of estimation strategies and to select the most appropriate strategy for a given problem. In this paper, we consider the role of students’ prior knowledge of estimation strategies in the effectiveness of interventions designed to promote strategy flexibility across two recent studies. In the first, 65 fifth graders began the study as fluent users of one strategy for computing mental estimates to multi-digit multiplication problems such as 17 × 41. In the second, 157 fifth and sixth graders began the study with moderate to low prior knowledge of strategies for computing mental estimates. Results indicated that students’ fluency with estimation strategies had an impact on which strategies they adopted. Students who exhibited high fluency at pretest were more likely to increase use of estimation strategies that led to more accurate estimates, while students with less fluency adopted strategies that were easiest to implement. Our results suggest that both the ease and accuracy of strategies as well as students’ fluency with strategies are all important factors in the development of strategy flexibility.     

Problems and solutions in students’ reinvention of a definition for sequence convergence

Little research exists on the ways in which students may develop an understanding of formal limit definitions. We conducted a study to (i) generate insights into how students might leverage their intuitive understandings of sequence convergence to construct a formal definition and (ii) assess the extent to which a previously established approximation scheme may support students in constructing their definition. Our research is rooted in the theory of Realistic Mathematics Education and employed the methodology of guided reinvention in a teaching experiment. In three 90-min sessions, two students, neither of whom had previously seen a formal definition of sequence convergence, constructed a rigorous definition using formal mathematical notation and quantification equivalent to the conventional definition. The students’ use of an approximation scheme and concrete examples were both central to their progress, and each portion of their definition emerged in response to overcoming specific cognitive challenges.     

Flexibility across and flexibility within: The domain of integer addition and subtraction

To better understand the role that flexibility plays in students’ success on integer addition and subtraction problems, we examined students’ flexibility when solving open number sentences. We define flexibility as the degree to which a learner uses more than one strategy to solve a single task when prompted, as well as the degree to which a learner changes strategies when solving a range of tasks to accommodate task differences. We introduce the categorizations of flexibility within and flexibility across to distinguish these two ways of operationalizing flexibility. We examined flexibility and performance within and among three groups of students — 2nd and 4th graders who had negative numbers in their numerical domains, 7th graders, and college-track 11th graders. Profiles of five students are shared to provide insight in relation to the quantitative findings.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

The influence of face-to-face communication: a principal-agent experiment

The principal-agent problem is an interesting problem involved in many everyday relationships, such as the one between company owners and their delegates. Our experiment simulates such a relationship, whereby the task of participating pairs is to negotiate labor contracts. Our aim is to find the effects of face-to-face communication and negotiation on contract framing. We argue that including pre-play communication into the principal-agent problem leads to a significant improvement of agent’s effort and results in changes of the compensation scheme. We show that incentives are not the only possibility to induce high effort. We use content analysis of the video-taped negotiations to find some characteristics of the communication process.     

The initial treatment of the concept of function in the selected secondary school mathematics textbooks in the US and China

In order to provide insight into cross-national differences in students’ achievement, this study compares the initial treatment of the concept of function sections of Chinese and US textbooks. The number of lessons, contents, and mathematical problems were analyzed. The results show that the US curricula introduce the concept of function one year earlier than the Chinese curriculum and provide strikingly more problems for students to work on. However, the Chinese curriculum emphasizes developing both concepts and procedures and includes more problems that require explanations, visual representations, and problem solving in worked-out examples that may help students formulate multiple solution methods. This result could indicate that instead of the number of problems and early introduction of the concept, the cognitive demands of textbook problems required for student thinking could be one reason for differences in American and Chinese students’ performances in international comparative studies. Implications of these findings for curriculum developers, teachers, and researchers are discussed.     

Connecting mathematical creativity to mathematical ability

This study aims to investigate whether there is a relationship between mathematical ability and mathematical creativity, and to examine the structure of this relationship. Furthermore, in order to validate the relationship between the two constructs, we will trace groups of students that differ across mathematical ability and investigate the relationships amongst these students’ performance on a mathematical ability test and the components of mathematical creativity. Data were collected by administering two tests, a mathematical ability and a mathematical creativity test, to 359 elementary school students. Mathematical ability was considered as a multidimensional construct, including quantitative ability (number sense and pre-algebraic reasoning), causal ability (examination of cause–effect relations), spatial ability (paper folding, perspective and spatial rotation abilities), qualitative ability (processing of similarity and difference relations) and inductive/deductive ability. Mathematical creativity was defined as a domain-specific characteristic, enabling individuals to be characterized by fluency, flexibility and originality in the domain of mathematics. The data analysis revealed that there is a positive correlation between mathematical creativity and mathematical ability. Moreover, confirmatory factor analysis suggested that mathematical creativity is a subcomponent of mathematical ability. Further, latent class analysis showed that three different categories of students can be identified varying in mathematical ability. These groups of students varying in mathematical ability also reflected three categories of students varying in mathematical creativity.     

Fostering creativity through instruction rich in mathematical problem solving and problem posing

Although creativity is often viewed as being associated with the notions of “genius” or exceptional ability, it can be productive for mathematics educators to view creativity instead as an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population. In this article, it is argued that inquiry-oriented mathematics instruction which includes problem-solving and problem-posing tasks and activities can assist students to develop more creative approaches to mathematics. Through the use of such tasks and activities, teachers can increase their students’ capacity with respect to the core dimensions of creativity, namely, fluency, flexibility, and novelty. Because the instructional techniques discussed in this article have been used successfully with students all over the world, there is little reason to believe that creativity-enriched mathematics instruction cannot be used with a broad range of students in order to increase their representational and strategic fluency and flexibility, and their appreciation for novel problems, solution methods, or solutions.     

The role of symbols in mathematical communication: the case of the limit notation

Symbols play crucial roles in advanced mathematical thinking by providing flexibility and reducing cognitive load but they often have a dual nature since they can signify both processes and objects of mathematics. The limit notation reflects such duality and presents challenges for students. This study uses a discursive approach to explore how one instructor and his students think about the limit notation. The findings indicate that the instructor flexibly differentiated between the process and product aspects of limit when using the limit notation. Yet, the distinction remained implicit for the students, who mainly realised limit as a process when using the limit notation. The results of the study suggest that it is important for teachers to unpack the meanings inherent in symbols to enhance mathematical communication in the classrooms.     

An historical example of mathematical modelling: the trajectory of a cannonball

Classroom considerations of the concept and processes of mathematical modelling can do much to strengthen students’ problem solving skills. A systematic exposure to the techniques of mathematical modelling helps students formulate problems, re‐think those problems in mathematical terms, appreciate possible solution constraints and seek solutions that are realistic within the scope and conditions of the problem. While many mathematical modelling situations can be found in today's world, there are special pedagogical values in examining existing mathematical models that have an historical basis. Such an examination should reveal the mechanics of a modelling situation and how a model evolves or is refined to meet ever increasing human demands for accuracy or practicality. The trajectory of a cannonball provides such a modelling example. This topic captures the imagination of students and supplies the basis for a variety of classroom discussions and problem solving encounters.     

Using components of mathematical ability for initial development and identification of mathematically promising students

Kruteskii's work on the mathematical abilities of school children is a seminal work on the nature of mathematical ability. However, the task of developing methods for the practical application of his work is still a significant problem in mathematics education. The authors have developed a practical application of Kruteskii's approach to the important problem of initially developing components of mathematical ability in student and thereafter identifying mathematically promising students. Examples of problems that were designed to develop ability to generalize, flexibility and reversibility of mental processes are presented. A practical guide for determining the level of development of components of mathematical abilities in individual students, in terms of specified observables, is presented as a set of structured reference tables. The authors set out a practical application protocol that combines use of the tables and sets of specially developed problems for initial development of mathematical abilities prior to identification of mathematically promising students in the general classroom. A significant motivation for this work is the desire to avoid time-consuming and resource intensive practices such as interviews and summer schools which therefore have been used successfully because these practices are now out of reach for all but very wealthy countries or highly ideologically driven systems. On the other hand, special examinations heavily depend on the level of preparedness of the students for the particular examination, and therefore some students with high abilities but with fewer opportunities to prepare could be overlooked.     

Systematic analysis of framing bias in missile defense: Implications toward visualization design

This paper examines the effects of framing on decision making in a homeland missile defense context across three tasks of varying complexity. Mathematically, each task was modeled as abstractions from a common resource allocation task. Logically, therefore, the effects of framing on human subjects should have been consistent across all the tasks. In the first experiment, a simple lottery was used to determine risk postures in a single-attribute case of missile defense. Results showed that, consistent with Prospect Theory, positive framing promotes risk-averse behavior whereas negative framing promotes risk-seeking behavior. In the second experiment, we used the Analytic Hierarchy Process to determine subject rankings in a multi-attribute case of missile defense. Results suggest that subjects’ performances under positive framing were significantly better than performances under negative framing. In the third experiment, we used a human-in-the-loop simulation to elicit human decisions in a missile defense resource allocation task. In comparison to the other experiments, the framing effect in the third experiment was diminished. We submit that decision biases detected in a simple choice task cannot be assumed to carry over to tasks of greater complexity even if the underlying mathematical formulation for all the tasks is the same. Moreover, we submit that the design of the graphical interface has a greater influence on human judgment bias than framing in tasks of higher complexity.     

On the importance of mathematical play

A number of existing theories and proposals for the meaning and characteristics of ‘play’ are considered before the authors suggest six characteristics of mathematical play, including the idea that it is not confined to childhood. Previous studies provide evidence for relating play to cognitive gain while the place of mathematical play in research activities is illustrated by describing a mathematician's approach to a number investigation from the classroom-The Six Circles. The problem-solving process for the Six Circles and observations of students solving calculator and integration problems are analysed in relation to theories of play and cognitive gain and also considered from the perspective of the students' experience. Piaget's theory for the assimilation and accommodation of new information and Davis's view of play as ‘space to support learning’ are reflected in the authors' rationale for suggesting that open questions and mathematical play provide opportunities for students to develop their own conjectures, with no threat of failure, and provide a foundation for mathematical learning. Some difficulties of implementing a ‘play’ approach in the classroom are discussed and further research questions proposed.     

Mathematics teachers’ enactment of cognitively demanding tasks and students’ perception of racial differences in opportunity

This study examined whether secondary students in an urban school district perceived racial differences in opportunity to be successful in mathematics, whether those perceptions differed between students of color and white students, and the relation of those perceptions to teachers’ choice and implementation of mathematical tasks. The results of multi-level regression models based on student survey and teacher observation data revealed two primary findings: (a) students of color were more likely to perceive opportunity differences than were white students; and (b) this difference was greater in classrooms in which teachers attempted to use cognitively demanding tasks but allowed the cognitive demand to decline during the lesson. Implications for both future research and mathematics teacher education are discussed.     

Handheld calculators between instrument and document

The new generations of handheld calculators can be considered either as mathematical tools with opportunities for calculation and representation or as a part of the teachers’ and students’ sets of resources. Framed by the Theory of Didactical Situations and the documentational approach, we take advantage of a particular experiment on introducing complex calculators in scientific classes to investigate the position and the role of this handheld technology both for students and teachers. The results show how different functionalities can be shared among teachers and students, but also how other functionalities remain private and may even conflict with the teacher’s intentions.