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.     

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.     

The use of digital technology in finding multiple paths to solve and extend an equilateral triangle task

Mathematical tasks are crucial elements for teachers to orient, foster and assess students’ processes to comprehend and develop mathematical knowledge. During the process of working and solving a task, searching for or discussing multiple solution paths becomes a powerful strategy for students to engage in mathematical thinking. A simple task that involves the construction of an equilateral triangle is used to present and discuss multiple solution approaches that rely on a variety of concepts and ways of reasoning. To this end, the use of a Dynamic Geometry System (GeoGebra) became instrumental in constructing and exploring dynamic models of the task. These model explorations provided a means to generate novel mathematical results.     

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.     

The role of operating premises and reasoning paths in upper elementary students’ problem solving

The validity of students’ reasoning is central to problem solving. However, equally important are the operating premises from which students’ reason about problems. These premises are based on students’ interpretations of the problem information. This paper describes various premises that 11- and 12-year-old students derived from the information in a particular problem, and the way in which these premises formed part of their reasoning during a lesson. The teacher’s identification of differences in students’ premises for reasoning in this problem shifted the emphasis in a class discussion from the reconciliation of the various problem solutions and a focus on a sole correct reasoning path, to the identification of the students’ premises and the appropriateness of their various reasoning paths. Problem information that can be interpreted ambiguously creates rich mathematical opportunities because students are required to articulate their assumptions, and, thereby identify the origin of their reasoning, and to evaluate the assumptions and reasoning of their peers.     

Prospective teachers anticipate challenges fostering the mathematical practice of making sense

Problem solving has long been a priority in mathematics education, and the first Common Core mathematical practice (SMP1) focuses on this priority through the language of “Make sense of problems and persevere in solving them.” We present findings from a survey about how prospective elementary teachers' (PTs) make sense of potential difficulties with fostering SMP1. Findings suggested that PTs' common anticipated difficulties relate to planning a solution pathway and self monitoring whether the solution makes sense. Moreover, a third of PTs disclosed that their anticipated difficulties are linked to their own personal struggles with aspects of SMP1. An alternative interpretation of SMP1 surfaced in which a small number of PTs described SMP1 as necessitating that a teacher teach multiple solution methods to students, instead of engaging students in productive struggle to develop their own strategies. We present a framework illustrating the connections between SMP 1 and Pólya's problem solving phases, and we discuss how these findings connect to and build on previous research of PTs' experiences with problem solving. We offer implications for the targeted support needed in teacher preparation programs to address these struggles, to prevent them from being replicated in their students.     

Students’ Activity in Computer-Supported Collaborative Problem Solving in Mathematics

The purpose of this study was to analyse secondary school students’ (N = 16) computer-supported collaborative mathematical problem solving. The problem addressed in the study was: What kinds of metacognitive processes appear during computer-supported collaborative learning in mathematics? Another aim of the study was to consider the applicability of networked learning in mathematics. The network-based learning environment Knowledge Forum (KF) was used to support students’ collaborative problem solving. The data consist of 188 posted computer notes, portfolio material such as notebooks, and observations. The computer notes were analysed through three stages of qualitative content analysis. The three stages were content analysis of computer notesin mathematical problem solving, content analysis of mathematical problem solving activity and content analysis of the students’ metacognitive activity. The results of the content analysis illustrate how networked discussions mediated mathematical knowledge and students’ questions, while the mathematical problem solving activity shows that the students co-regulate their thinking. The results of the content analysis of the students’ metacognitive activity revealed that the students use metacognitive knowledge and make metacognitive judgments and perform monitoring during networked discussions. In conclusion, the results of this study demonstrate that working with the networked technology contributes to the students’ use of their mathematical knowledge and stimulates them into making their thinking visible. The findings also show some metacognitive activity in the students’ computer-supported collaborative problem solving in mathematics.     

Why do students not check their solutions to mathematical problems? A field-based hypothesis on epistemological status

ABSTRACT

This study embarks on the question in the title with the construct of epistemological status, which pertains to the solver’s satisfaction with the way and the degree to which their solution had fulfilled their intellectual and psychological needs in a particular problem situation. The construct is used to hypothesize that a solver’s decision to check the solution and the act of checking itself may be shaped by the epistemological status of a developed solution. Driven by the abduction methodology, this hypothesis is supported by two empirical illustrations. The first one comes from the final exam in a large first-year course for non-mathematics majors, where many students accompanied their solutions of a low epistemological status by written checks of final answers as a way to elevate it. No such checks were found in solutions with high epistemological status. The second illustration revisits some previously reported findings to propose that an application of conventional problem-solving methods may not be sufficient for students to endow their solutions with high epistemological status.     

Students’ and Teachers’ Conceptual Metaphors for Mathematical Problem Solving

Metaphors are regularly used by mathematics teachers to relate difficult or complex concepts in classrooms. A complex topic of concern in mathematics education, and most STEM‐based education classes, is problem solving. This study identified how students and teachers contextualize mathematical problem solving through their choice of metaphors. Twenty‐two high‐school student and six teacher interviews demonstrated a rich foundation for these shared experiences by identifying the conceptual metaphors. This mixed‐methods approach qualitatively identified conceptual metaphors via interpretive phenomenology and then quantitatively analyzed the frequency and popularity of the metaphors to explore whether a coherent metaphorical system exists with teachers and students. This study identified the existence of a set of metaphors that describe how multiple classrooms of geometry students and teachers make sense of mathematical problem solving. Moreover, this study determined that the most popular metaphors for problem solving were shared by both students and teachers. The existence of a coherent set of metaphors for problem solving creates a discursive space for teachers to converse with students about problem solving concretely. Moreover, the methodology provides a means to address other complex concepts in STEM education fields that revolve around experiential understanding.     

Teaching mathematical modelling through project work

The paper presents and analyses experiences from developing and running an inservice course in project work and mathematical modelling for mathematics teachers in the Danish gymnasium, e.g. upper secondary level, grade 10~12. The course objective is to support the teachers to develop, try out in their own classes, evaluate and report a project based problem oriented course in mathematical modelling. The in-service course runs over one semester and includes three seminars of 3, 1 and 2 days. Experiences show that the course objectives in general are fulfilled and that the course projects are reported in manners suitable for internet publication for colleagues. The reports and the related discussions reveal interesting dilemmas concerning the teaching of mathematical modelling and how to cope with these through «setting the scene» for the students modelling projects and through dialogues supporting and challenging the students during their work. This is illustrated and analysed on the basis of two course projects.     

Stimulating student aesthetic response to mathematical problems by means of manipulating the extent of surprise

We investigated aesthetic responses of 60 middle school students as they engaged in a pair of similar looking geometry problems in one-on-one semi-structured interviews. The investigation was driven by three predictions. The first two predictions were about the association between the evaluative aesthetic response and surprise stemming from the solution to each problem. The third, main, prediction was that the problem with more surprising solution would be evaluated as more beautiful. The extent of surprise was manipulated by the order in which two problems were given. The third prediction came to be true in 90% of the cases, in which the first two predictions were fulfilled. The findings suggest that school students’ evaluative aesthetic response to mathematical problems can be stimulated in instructional settings. Implications for research and practice are drawn.     

The Effect of Area Measurement Tools on Student Strategies: The Role of a Computer Microworld

This study focuses on the role of tools, provided by a computer microworld (C.AR.ME), on the strategies developed by 14-year-old students for the area measurement of a non-convex polygon. Students' strategies on a transformation and a comparison task were interpreted and classified into categories in terms of the tools used for their development. The analysis of the data shows that an environment providing the students with the opportunity to select various tools and asking them to produce solutions `in any possible way' can stimulate them to construct a plurality of solution strategies. The students selected tools appropriate for their cognitive development and expressed their own individual approaches regarding the concept of area measurement. The nature of tools used affected the nature of solution strategies that the students constructed. Moreover, all students were involved in the tasks and succeeded in completing them with more than one correct solution strategy thereby developing a broader view of the concept, although not all of them realized the same strategies. Three different approaches to area measurement emerged from the strategies which were constructed by the students in this microworld: automatic area measurement, provided by the environment, the operation of area measurement using spatial units and the use of area formulae. Almost all the students experienced qualitative aspects of area measurement through being involved in the process of covering areas using spatial units. Students also managed to use the area formulae meaningfully by studying it in relation to automatic area measurement and to area measurement using spatial units. Through these strategies, the concepts of conservation of area and its measurement as well as area formulae were viewed by the students as interrelated. Finally, some basic difficulties regarding area measurement were overcome in this computer environment.This revised version was published online in September 2005 with corrections to the Cover Date.     

Software Tools for Geometrical Problem Solving: Potentials and Pitfalls

Dynamic geometry software provides tools for students to construct and experiment with geometrical objects and relationships. On the basis of their experimentation, students make conjectures that can be tested with the tools available. In this paper, we explore the role of software tools in geometry problem solving and how these tools, in interaction with activities that embed the goals of teachers and students, mediate the problem solving process. Through analysis of successful student responses, we show how dynamic software tools can not only scaffold the solution process but also help students move from argumentation to logical deduction. However, by reference to the work of less successful students, we illustrate how software tools that cannot be programmed to fit the goals of the students may prevent them from expressing their (correct) mathematical ideas and thus impede their problem solution.This revised version was published online in September 2005 with corrections to the Cover Date.     

Behind the scenes of pseudo-proportionality

In this study, we attempt to put in question students’ spontaneous and uncritical application of the simple and neat mathematical formula of linearity. This is impelled with the help of a written test involving geometrical problems, which is based on an original experimental setting. In this setting, grade 9 and 10 students were instructed first to solve all the geometrical problems and then to select only one problem as the appropriate for a given numerical answer. The difficulty of this choice lied in fact that a superficial handling of each problem would resolve to the same numerical answer as the one given. The results show that students’ choices are systematic and based on the solutions given to the tasks. However, the experimental setting has managed to help students question in some degree the applicability of the linear model in situations that appear to be linear but are not.     

Children’s use of realistic considerations in problem solving: some English evidence

This is the second of two papers exploring children’s responses to an extended version of a division-with-remainder problem intended to elicit general rather than particular realistic considerations during mathematical problem solving. Responses to two problems are analyzed. The first is a ‘realistically’ contextualised item drawn from national tests in England whose ambiguities have been previously discussed (Cooper, 1992); the second is a version of this problem revised to encourage a wider range of realistic responses. In Cooper and Harries (2002), the responses of children at the end of their first year of secondary schooling were analyzed. Here the responses of children at the end of their primary schooling are analyzed and compared with the previous results. It is shown that many children, given suitable encouragement, are willing and able to enter into an extended form of realistic thinking during problem solving, although the original test item renders this invisible.     

The relationship between mathematical problem-solving skills and self-regulated learning through homework behaviours,motivation, and metacognition

Studies highlight that using appropriate strategies during problem solving is important to improve problem-solving skills and draw attention to the fact that using these skills is an important part of students’ self-regulated learning ability. Studies on this matter view the self-regulated learning ability as key to improving problem-solving skills. The aim of this study is to investigate the relationship between mathematical problem-solving skills and the three dimensions of self-regulated learning (motivation, metacognition, and behaviour), and whether this relationship is of a predictive nature. The sample of this study consists of 323 students from two public secondary schools in Istanbul. In this study, the mathematics homework behaviour scale was administered to measure students’ homework behaviours. For metacognition measurements, the mathematics metacognition skills test for students was administered to measure offline mathematical metacognitive skills, and the metacognitive experience scale was used to measure the online mathematical metacognitive experience. The internal and external motivational scales used in the Programme for International Student Assessment (PISA) test were administered to measure motivation. A hierarchic regression analysis was conducted to determine the relationship between the dependent and independent variables in the study. Based on the findings, a model was formed in which 24% of the total variance in students’ mathematical problem-solving skills is explained by the three sub-dimensions of the self-regulated learning model: internal motivation (13%), willingness to do homework (7%), and post-problem retrospective metacognitive experience (4%).     

Grade 3 students’ mathematization through modeling: Situation models and solution models with mutli-digit subtraction problem solving

In considering mathematics problem solving as a model-eliciting activity ( [Lesh and Doerr, 2003], [Lesh and Harel, 2003] and [Lesh and Zawojewski, 2008]), it is important to know what students are modeling for the problems: situations or solutions. This study investigated Grade 3 students’ mathematization process by examining how they modeled different types of multi-digit subtraction situation problems. Students’ modeling processes differed from one problem type to another due to their prior experiences and the complexity of the problems. This study showed that students make their own distinctions between solution and situation models in their mathematization process. Mathematics curricula and teaching should consider these distinctions to carefully facilitate different model development of and support student understanding of a content topic.     

A model of professional competences in mathematics to update mathematical and didactic knowledge of teachers

This paper describes part of a research and development project carried out in public elementary schools. Its objective was to update the mathematical and didactic knowledge of teachers in two consecutive levels in urban and rural public schools of Region de Los Lagos and Region de Los Rios of southern Chile. To that effect, and by means of an advanced training project based on a professional competences model, didactic interventions based on types of problems and types of mathematical competences with analysis of contents and learning assessment were designed. The teachers’ competence regarding the didactic strategy used and its results, as well as the students’ learning achievements are specified. The project made possible to validate a strategy of lifelong improvement in mathematics, based on the professional competences of teachers and their didactic transposition in the classroom, as an alternative to consolidate learning in areas considered vulnerable in two regions of the country.     

Mathematics and molecules: Exploring connections via programming

This article examines the self-directed activity of two students who learned about molecular structure by writing computer programs. The students wrote programs to display the solution of a mathematics problem and then extended their programs to represent several classes of organic molecules. In the course of this activity, the students learned the standard system for naming organic molecules while maintaining a sense of ownership of their project. We discuss ways to enhance the mathematical connections to chemistry education.     

The Cereal Box Problem Revisited

The Cereal Box problem is fascinating for students of ail levels. Rich in mathematical content, this problem offers students the opportunity to collect data, make conjectures, and derive mathematical models. Using Monte Carlo methods, the Cereal Box problem is investigated in this paper, using both an experimental and theoretical framework. This investigation extends previous considerations of the Cereal Box problem. Using empirical data, students can discover patterns and relationships that help them understand the origin of the theoretical solution to the problem. Building on experimental findings, a theoretical model is derived, showing that the expected solution of the Cereal Box problem is formed from the sum of successive geometric series.