How does imposing a step-by-step solution method impact students’ approach to mathematical word problem solving?

This paper presents the second phase of a larger research program with the purpose of exploring the possible consequences of a gap between what is done in the classroom regarding mathematical word problem solving and what research shows to be effective in this particular field of study. Data from the first phase of our study on teachers’ self-proclaimed practices showed that one-third of elementary teachers from the region of Quebec require their students to follow a specific sequential problem-solving method, known as the ‘what I know, what I look for’ method. These results led us to hypothesize that the observed gap may have an impact on students’ comprehension of mathematical word problems. The use of this particular method was the foundation for us to study, in the second phase, the effect of the imposition of this sequential method on students’ literal and inferential understanding of word problems. A total of 278 fourth graders (9–10 years old) solved mathematical word problems followed by a test to assess their understanding of the word problems they had just solved. The results suggest that the use of this problem solving method does not seem to improve or impair students’ understanding. From a more fundamental point of view, our study led us to the conclusion that the way word problem solving is addressed in the mathematics classroom, through sequential and inflexible methods, does not help students develop their word problem solving competence.     

Computational Construction as a Means to Coordinate Representations of Infinity

In this paper, we describe a design experiment aimed at helping students to explore and develop concepts of infinite processes and objects. Our approach is based on the design and development of a computational microworld, which afforded students the means to construct a range of representational models (symbolic, visual and numeric) of infinity-related objects (infinite sequences, in particular). We present episodes based on four students’ activities, seeking to illustrate how the available tools mediated students’ understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes. We claim that the microworld supported students in the coordination of hitherto unconnected or conflicting intuitions concerning infinity, based on a constructive articulation of different representational forms we name as ‘representational moderation’.

An ‘Emergent Model’ for Rate of Change

Does speed provide a ‘model for’ rate of change in other contexts? Does JavaMathWorlds (JMW), animated simulation software, assist in the development of the ‘model for’ rate of change? This project investigates the transference of understandings of rate gained in a motion context to a non-motion context. Students were 27 14–15 year old students at an Australian secondary school. The instructional sequence, utilising JMW, provided rich learning experiences of rate of change in the context of a moving elevator. This context connects to students’ prior knowledge. The data taken from pre- and post-tests and student interviews revealed a wide variation in students’ understanding of rate of change. The variation was mapped on a hypothetical learning trajectory and interpreted in the terms of the ‘emergent models’ theory (Gravemeijer, Math Think Learn 1(2):155–177, 1999) and illustrated by specific examples from the data. The results demonstrate that most students were able to use the ‘model of’ rate of change developed in a vertical motion context as a ‘model for’ rate of change in a horizontal motion context. A smaller majority of students were able to use their, often incomplete, ‘model of’ rate of change as a ‘model for’ reasoning about rate of change in a non-motion context.     

A changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

Constructing Knowledge Via a Peer Interaction in a CAS Environment with Tasks Designed from a Task–Technique–Theory Perspective

Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task–Technique–Theory (T–T–T) theoretical frame as adapted from Chevallard’s Anthropological Theory of Didactics, we were mindful that a careful task design process was needed in order to promote in students rich and meaningful learning. In this paper, we explore further Lagrange’s (2000) conjecture that the learning of techniques can foster conceptual understanding by investigating at close range the task-based activity of a pair of 10th grade students—activity that illustrates the ways in which the use of symbolic calculators along with appropriate tasks can stimulate the emergence of epistemic actions within technique-oriented algebraic activity.     

Measuring and investigating strategic knowledge about drawing to solve geometry modelling problems

We report on the construction, validation, and implementation of an instrument for measuring students’ strategic knowledge about drawing for geometry modelling problems, namely, the strategic knowledge about drawing scale. We conducted a qualitative study and a quantitative study to validate the proposed construction and interpretation of the scale and to obtain initial findings on students’ strategic knowledge about drawing. Results showed that ninth-grade students in the intermediate achievement track had less elaborated strategic knowledge about drawing than their peers in the high achievement track. Further, strategic knowledge about drawing was found to be related to drawing accuracy and modelling performance even when cognitive abilities and interest were controlled for. The current study suggests that promoting strategic knowledge about drawing might be a means to increase drawing and modelling performance—especially among non-high-achieving students.     

Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.     

New challenges in the 2011 revised middle school curriculum of South Korea: mathematical process and mathematical attitude

The ‘future-oriented middle school mathematics curriculum focused on creativity and personality’ was revised in August of 2011 with the aim of nurturing students’ mathematical creativity and sound personalities. The curriculum emphasizes: contextual learning from which students can grasp mathematical concepts and make connections with their everyday lives; manipulation activities through which students may attain an intuitive idea of what they are learning and enhance their creativity; and reasoning to justify mathematical results based on their knowledge and experience. Since students will not be able to engage in the intended mathematical process with the study-load imposed by the current curriculum, the newly revised curriculum modifies or deletes some parts of the contents that have been traditionally taught mechanically. This paper provides a detailed overview of the main points of the revised curriculum.     

Embodied Semiotic Activities and Their Role in the Construction of Mathematical Meaning of Motion Graphs

This paper examines the relation between bodily actions, artifact-mediated activities, and semiotic processes that students experience while producing and interpreting graphs of two-dimensional motion in the plane. We designed a technology-based setting that enabled students to engage in embodied semiotic activities and experience two modes of interaction: 2D freehand motion and 2D synthesized motion, designed by the composition of single variable function graphs. Our theoretical framework combines two perspectives: the embodied approach to the nature of mathematical thinking and the Vygotskian notion of semiotic mediation. The article describes in detail the actions, gestures, graph drawings, and verbal discourse of one pair of high school students and analyzes the social semiotic processes they experienced. Our analysis shows how the computerized artifacts and the students’ gestures served as means of semiotic mediation. Specifically, they supported the interpretation and the production of motion graphs; they mediated the transition between an individual’s meaning of mathematical signs and culturally accepted mathematical meaning; and they enable linking bodily actions with formal signs.     

Conceptual variation and coordination in probability reasoning

This study investigates students’ conceptual variation and coordination among theoretical and experimental interpretations of probability. In the analysis we follow how Swedish students (12-13 years old) interact with a dice game, specifically designed to offer the students opportunities to elaborate on the logic of sample space, physical/geometrical considerations and experimental evidence when trying to develop their understanding of compound random phenomena.The analytical construct of contextualization was used as a means to provide structure to the qualitative analysis performed. Within the frame of the students’ problem encounters during the game and how they contextualized the solutions of the problems in personal contexts for interpretations, the analysis finds four main forms of appearance, or of limitations in appearance, of conceptual variation and coordination among theoretical and experimental interpretations of probability.     

On the excursion theory for linear diffusions

We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein’s representations that, e.g. the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss the Ornstein–Uhlenbeck processes.     

Measuring in dynamic geometry environments as a tool for conjecturing and proving

This paper sits within the research on the affordances of new technologies in the mathematics classroom and focuses on a specific feature that is available in dynamic geometry environments, i.e. measuring tools, within the context of conjecturing and proving in open geometry problems. We develop a classification of different modalities of measuring, based on our previous work on dragging. The modalities are illustrated through the analysis of 15–16 year-old students’ proving processes, which focuses on how these modalities relate to the moves between the spatio-graphical field and the theoretical field and may either support or hinder the proving process. The classification of the modalities of measuring enables researchers to access students’ cognitive processes and teachers to be aware of the different possible uses and interpretation of measuring, giving them tools to support students when difficulties arise.

Comparing eye movements during mathematical word problem solving in Chinese and German

Language plays an important role in word problem solving. Accordingly, the language in which a word problem is presented could affect its solution process. In particular, East-Asian, non-alphabetic languages are assumed to provide specific benefits for mathematics compared to Indo-European, alphabetic languages. By analyzing students’ eye movements in a cross-linguistic comparative study, we analyzed word problem solving processes in Chinese and German. 72 German and 67 Taiwanese undergraduate students solved PISA word problems in their own language. Results showed differences in eye movements of students, between the two languages. Moreover, independent cluster analyses revealed three clusters of reading patterns based on eye movements in both languages. Corresponding reading patterns emerged in both languages that were similarly and significantly associated with performance and motivational-affective variables. They explained more variance among students in these variables than between the languages alone. Our analyses show that eye movements of students during reading differ between the two languages, but very similar reading patterns exist in both languages. This result supports the assumption that the language alone is not a sufficient explanation for differences in students’ mathematical achievement, but that reading patterns are more strongly related to performance.     

An evolving framework for describing student engagement in classroom activities

Student engagement in classroom activities is usually described as a function of factors such as human needs, affect, intention, motivation, interests, identity, and others. We take a different approach and develop a framework that models classroom engagement as a function of students’ conceptual competence in the specific content (e.g., the mathematics of motion) of an activity. The framework uses a spatial metaphor—i.e., the classroom activity as a territory through which students move—as a way to both capture common engagement-related dynamics and as a communicative device. In this formulation, then, students’ engaged participation can be understood in terms of the nature of the “regions” and overall “topography” of the activity territory, and how much student movement such a territory affords. We offer the framework not in competition with other instructional design approaches, but rather as an additional tool to aid in the analysis and conduct of engaging classroom activities.     

A cross-national comparison of reform curricula in Korea and the US in terms of cognitive complexity: the case of fraction addition and subtraction

The overall level of conceptual understanding and mathematical proficiency of students has been a matter of increasing national interest in South Korea. Recently, a new edition of mathematics textbooks aligned with the amendment of the 7th national mathematics curriculum has become available for all elementary grade levels. To characterize the current reform efforts in South Korea, this study examined the quality of the mathematical problems in the current version of the Korean reform textbooks (KM 2) compared with the previous version (KM 1) and one representative US reform curriculum text (EM). Webb’s (Research monograph No. 18: Alignment of science and mathematics standards and assessments in four states. National Institute for Science Education, Madison, 1999) depth of knowledge framework and Son and Senk’s (Educ Stud Math 74(2):117–142, 2010) cognitive expectation feature were employed to examine the kind and level of students’ opportunities to learn along with the type of word problems presented in the three sets of materials. Analysis revealed that the KM 2 provided better opportunities for students to learn fraction addition and subtraction than the KM 1 in terms of the depth and breadth of cognitive complexity. However, there was little difference in addressing and developing the meaning of fraction addition and subtraction through word problems. Moreover, compared with the US reform curriculum materials, the KM 2 provided more problems requiring lower depth of knowledge levels than the US counterpart. Implications of these findings for curriculum developers, textbook and learning materials developers, teachers and future researchers are discussed.     

Students' performance in reasoning and proof in Taiwan and Germany: Results,paradoxes and open questions

In different international studies on mathematical achievement East Asian students outperformed the students from Western countries. A deeper analysis shows that this is not restricted to routine tasks but also affects students’ performance for complex mathematical problem solving and proof tasks. This fact seems to be surprising since the mathematics instruction in most of the East Asian countries is described as examination driven and based on memorising rules and facts. In contrast, the mathematics classroom in western countries aims at a meaningful and individualised learning. In this article we discuss this “paradox” in detail for Taiwan and Germany as two typical countries from East Asia and Western Europe.     

Recollections of Fred Almgren

While visiting Princeton for the 1997–98 year I have had the privilege of using Fred’s office and helping Jean to go through his papers. It has been moving to see the magnitude of Fred’s helpful correspondence with students and colleagues, some of which Jean and I returned to them with an invitation to contribute short recollections. This article gives some selections from their responses. For more recollections see Mackenzie’s article on “Fred Almgren: Lover of Mathematics, Family, and Life’s Adventures” (Notices AMS 44, 1997, 1102–1106) and the memorial issue ofExperimental Mathematics (Vol. 6, 1997, 1–12).     

Constructs of engagement emerging in an ethnomathematically-based teacher education course

This paper presents a case study, in which we apply and develop theoretical constructs to analyze motivating desires observed in an unconventional, culturally contextualized teacher education course. Participants, Israeli students from several different cultures and backgrounds (pre-service and in-service teachers, Arabs and Jews, religious and secular) together studied geometry through inquiry into geometric ornaments drawn from diverse cultures, and acquired knowledge and skills in multicultural education. To analyze affective behaviors in the course we applied the methodology of engagement structures recently proposed by Goldin and his colleagues. Our study showed that engagement structures were a powerful tool for examining motivating desires of the students. We found that the constructivist ethnomathematical approach highlighted the structures that matched our instructional goals and diminished those related to students’ feelings of dissatisfaction and inequity. We propose a new engagement structure “Acknowledge my culture” to embody motivating desires, arising from multicultural interactions, that foster mathematical learning.     

Instances of mathematical thinking among low attaining students in an ordinary secondary classroom

This paper is a report of a classroom research project whose aim was to find out whether low attaining 14-year-old students of mathematics would be able to think mathematically at a level higher than recall and reproduction during their ordinary classroom mathematics activities. Analysis of classroom interactive episodes revealed many instances of mathematical thinking of a kind which was not normally exploited, required or expected in their classes. Five episodes are described, comparing the students’ thinking to that usually described as “advanced.” In particular, some episodes suggest the power of a type of prompt which can be generalized as “going across the grain.”     

Primal-dual first-order methods with $${\mathcal {O}(1/\epsilon)}$$ iteration-complexity for cone programming

In this paper we consider the general cone programming problem, and propose primal-dual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss first-order methods suitable for solving these reformulations, namely, Nesterov’s optimal method (Nesterov in Doklady AN SSSR 269:543–547, 1983; Math Program 103:127–152, 2005), Nesterov’s smooth approximation scheme (Nesterov in Math Program 103:127–152, 2005), and Nemirovski’s prox-method (Nemirovski in SIAM J Opt 15:229–251, 2005), and propose a variant of Nesterov’s optimal method which has outperformed the latter one in our computational experiments. We also derive iteration-complexity bounds for these first-order methods applied to the proposed primal-dual reformulations of the cone programming problem. The performance of these methods is then compared using a set of randomly generated linear programming and semidefinite programming instances. We also compare the approach based on the variant of Nesterov’s optimal method with the low-rank method proposed by Burer and Monteiro (Math Program Ser B 95:329–357, 2003; Math Program 103:427–444, 2005) for solving a set of randomly generated SDP instances.