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.     

Defining and demonstrating an equivalence way of thinking in enumerative combinatorics

Counting problems offer opportunities for rich mathematical thinking, and yet there is evidence that students struggle to solve counting problems correctly. There is a need to identify useful approaches and thought processes that can help students be successful in their combinatorial activity. In this paper, we propose a characterization of an equivalence way of thinking, we discuss examples of how it arises mathematically in a variety of combinatorial concepts, and we offer episodes from a paired teaching experiment with undergraduate students that demonstrate useful ways in which students developed and leverage this way of thinking. Ultimately, we argue that this way of thinking can apply to a variety of combinatorial situations, and we make the case that it is a valuable way of thinking that should be prioritized for students learning combinatorics.     

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.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

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.     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

On high-school students' use of graphic calculators in mathematics

When students are working with hand held technology, such as graphic calculators, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve the problem. In this paper we report on two case studies that follow the progress of students as they solve mathematical problems. We use software that works in the background of the graphic calculator capturing the students' keystrokes as they use the calculator. The aim of the research studies described in this paper was to provide insights into the working styles of these students. Through a detailed analysis of their graphic calculator keystrokes, interviews and associated written solutions we will discuss the effectiveness of their solution strategies and the efficiency of their use of the technology and identify some barriers to the use of graphic calculators in mathematical problem solving.     

A Modeling Perspective on the Teaching and Learning of Mathematical Problem Solving

This study analyzed the processes used by students when engaged in modeling activities and examined how students' abilities to solve modeling problems changed over time. Two student populations, one experimental and one control group, participated in the study. To examine students' modeling processes, the experimental group participated in an intervention program consisting of a sequence of six modeling activities. To examine students' modeling abilities, the experimental and control groups completed a modeling abilities test on three occasions. Results showed that students' models improved as they worked through the sequence of problem activities and also revealed a number of factors, such as students' grade, experiences with modeling activities, and modeling abilities that influenced their modeling processes. The study proposes a three-dimensional theoretical model for examining students' modeling behavior, with ubsequent implications for the teaching and learning of mathematical problem solving.     

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.     

A three-year perspective on conceptions of and orientations to learning mathematics of prospective teachers and first year university students

The paper reports a compilation of results from three studies conducted over three years to determine students' conceptions of mathematics, and orientations they follow in learning the subject. Respondents were 459 first year mathematics students from four universities and one teacher college. Results indicated that more than half the sample reported mathematics to be a subject made of numbers and formulae that could be memorized. This suggests a shallow emphasis when learning the subject, with no intention to understand. However, most students passed their examinations. It was concluded that there was no statistically significant relationship between examinations results and students' learning orientations. It is recommended that lecturers should foster students' meta-learning capabilities and an awareness of different learning strategies.     

Learning the Algebraic Method of Solving Problems

This study of students' attempts to formulate and solve algebra word problems shows that the logic underlying algebraic problem solving methods is little understood. Students' prior experiences with solving problems in arithmetic gives them a compulsion to calculate which is manifested in the meaning they give to “the unknown” and how they use letters, their interpretation of what an equation is, and the methods they choose to solve equations. At every stage of the process of solving problems by algebra, students were deflected from the algebraic path by reverting to thinking grounded in arithmetic problem solving methods.     

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.     

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.     

Solving methods of combinatorial geometric problems

There is considerable experience of organization and management of mathematical contests and interest groups in Latvia. It is necessary to analyse solutions of different mathematical challenges in out-door activities for to develop students' skills of solving non-standard problems. For this reason collections of thematically related problems with references to applicable methods are useful as valuable manuals for teachers and also as a source of original ideas for the students' independent work. Frequently students' attention is attracted by problems which do not require complicated mathematical formulae. For instance the combinatorial geometric problems deal with systems of geometric objects requiring to estimate dimensional quantities, or to investigate the features of decomposition as well as covering or colouring of geometric figures. Although these problems seem rather simple, various methods are used in their solutions, where general geometric regularities are supplemented with results from different fields of mathematics in combination with general thinking methods such as mathematical induction, the method of invariants and the mean value method. An effective auxiliary in problem solution is the mean value method, which allows making qualitative estimates of given objects by dealing with their quantitative properties.     

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.