Singaporean students' mathematical thinking in problem solving and problem posing: an exploratory study

This study explored Singaporean fourth, fifth, and sixth grade students' mathematical thinking in problem solving and problem posing. The results of this study showed that the majority of Singaporean fourth, fifth, and sixth graders are able to select appropriate solution strategies to solve these problems, and choose appropriate solution representations to clearly communicate their solution processes. Most Singaporean students are able to pose problems beyond the initial figures in the pattern. The results of this study also showed that across the four tasks, as the grade level advances, a higher percentage of students in that grade level show evidence of having correct answers. Surprisingly, the overall statistically significant differences across the three grade levels are mainly due to statistically significant differences between fourth and fifth grade students. Between fifth and sixth grade students, there are no statistically significant differences in most of the analyses. Compared to the findings concerning US and Chinese students' mathematical thinking, Singaporean students seem to be much more similar to Chinese students than to US students.     

Teacher attention to number choice in problem posing

This study examined how teachers used number choice in contextualized word problems as a pedagogical approach for meeting instructional goals. By collecting and analyzing the contextualized word problems posed by 20 teachers along with their rationales, I identified several means by which teachers used number choice. Additionally, results indicate and characterize a progression for using number choice from no attention to purposeful attention. Implications for decomposing the teaching practice of posing problems to children are discussed.     

Pre-service teachers' free and structured mathematical problem posing

This exploratory study examined how pre-service teachers (PSTs) pose mathematical problems for free and structured mathematical problem-posing conditions. It was hypothesized that PSTs would pose more complex mathematical problems under structured posing conditions, with increasing levels of complexity, than PSTs would pose under free posing conditions, because the structured posing condition would guide PSTs to more closely consider the mathematical relationships in a posing situation. Sixty-five PSTs – 61 participating in a written assessment and 4 participating in task-based interviews – responded to problem-posing tasks under free or structured posing conditions. Two-way independent samples t-tests and chi-square tests were used to test the hypothesis, along with a qualitative analysis of the task-based interviews. We found that while the task format had limited impact on the complexity of problems posed, PSTs in the structured-posing condition may have more closely attended to the mathematical concepts in each task, and may have also impacted their process of posing problems than those in the free posing condition.     

On the role of kinesthetic thinking in computational geometry

Computational geometry is a new (about 30 years) and rapidly growing branch of knowledge in computer science that deals with the analysis and design of algorithms for solving geometric problems. These problems typically arise in computer graphics, image processing, computer vision, robotics, manufacturing, knot theory, polymer physics and molecular biology. Since its inception many of the algorithms proposed for solving geometric problems, published in the literature, have been found to be incorrect. These incorrect algorithms rather than being ‘purely mathematical’ often contain a strong kinesthetic component. This paper explores the relationship between computational geometric thinking and kinesthetic thinking, the effect of the latter on the correctness and efficiency of the resulting algorithms, and their implications for education.     

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.     

An exploratory framework for handling the complexity of mathematical problem posing in small groups

The paper introduces an exploratory framework for handling the complexity of students’ mathematical problem posing in small groups. The framework integrates four facets known from past research: task organization, students’ knowledge base, problem-posing heuristics and schemes, and group dynamics and interactions. In addition, it contains a new facet, individual considerations of aptness, which accounts for the posers’ comprehensions of implicit requirements of a problem-posing task and reflects their assumptions about the relative importance of these requirements. The framework is first argued theoretically. The framework at work is illustrated by its application to a situation, in which two groups of high-school students with similar background were given the same problem-posing task, but acted very differently. The novelty and usefulness of the framework is attributed to its three main features: it supports fine-grained analysis of directly observed problem-posing processes, it has a confluence nature, it attempts to account for hidden mechanisms involved in students’ decision making while posing problems.     

Revisiting mathematical problem solving and posing in the digital era: toward pedagogically sound uses of modern technology

The availability of sophisticated computer programs such as Wolfram Alpha has made many problems found in the secondary mathematics curriculum somewhat obsolete for they can be easily solved by the software. Against this background, an interplay between the power of a modern tool of technology and educational constraints it presents is discussed. Using topics from algebra (equations) and elementary number theory (summation of powers of integers), the paper suggests ways of developing problems that are both technology-immune and technology-enabled in the sense that whereas software can facilitate problem solving, its direct application is not sufficient for finding an answer. Stemming from the author's work with secondary mathematics teacher candidates, this paper highlights the appropriate use of technology as support system for multiple ways of knowing and knowledge construction in the modern classroom.     

Engaging in problem posing activities in a dynamic geometry setting and the development of prospective teachers’ mathematical knowledge

In the present study we explore changes in perceptions of our class of prospective mathematics teachers (PTs) regarding their mathematical knowledge. The PTs engaged in problem posing activities in geometry, using the “What If Not?” (WIN) strategy, as part of their work on computerized inquiry-based activities. Data received from the PTs’ portfolios reveals that they believe that engaging in the inquiry-based activity enhanced both their mathematical and meta-mathematical knowledge. As to the mathematical knowledge, they deepened their knowledge regarding the geometrical concepts and shapes involved, and during the process of creating the problem and checking its validity and its solution, they deepened their understanding of the interconnections among the concepts and shapes involved. As to meta-mathematical knowledge, the PTs refer to aspects such as the meaning of the givens and their relations, validity of an argument, the importance and usefulness of the definitions of concepts and objects, and the importance of providing a formal proof.     

Systems thinking and mathematical problem solving

Problem solving lies at the core of engineering and remains central in school mathematics. Word problems are a traditional instructional mechanism for learning how to apply mathematics to solving problems. Word problems are formulated so that a student can identify data relevant to the question asked and choose a set of mathematical operations that leads to the answer. However, the complexity and interconnectedness of contemporary problems demands that problem‐solving methods be shaped by systems thinking. This article presents results from three clinical interviews that aimed at understanding the effects that traditional word problems have on a student’s ability to use systems thinking. In particular, the interviews examined how children parse word problems and how they update their answers when contextual information is provided. Results show that traditional word problems create unintended dispositions that limit systems thinking.     

On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

On the cancellation problem

Let k be an algebraically closed field. For every n ≥ 8 we give examples of Zariski open, dense, affine subsets of the affine space A ⁿ (k) which do not have the cancellation property. Dedicated to Professor Mikhail Zaidenberg. The author was partially supported by the grant of Polish Ministry of Science, 2006–2009.     

On the Schur-Cohn problem

A new criterion (in terms of determinant inequalities) is obtained for all the roots of a real polynomial to lie inside the unit circle, i.e., a criterion of stability of periodic motions. In contrast to the Schur-Cohn criterion, the number of determinants used by us is by four times smaller.