Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence‐level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence‐level and text integration processes. (...) Working memory variables measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K‐3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process‐oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships. (shrink)

ABSTRACTIn a bilingual educational setting, even when mathematical word problems are presented in one’s first language, students may still perform poorly if cognitive constraints such as working me...

The central question underlying this study was whether metacognition training could enhance the two metacognition components—knowledge and skills—and the mathematical problem-solving capacities of normal children in grade 3. We also investigated whether metacognitive training had a differential effect according to the children's mathematics level. A total of 48 participants took part in this study, divided into an experimental and a control group, each subdivided into a lower and a normal achievers group. The training programme took an interactive (...) approach in accordance with Schraw's (1998) recommendation and was carried out over five training sessions. Results indicated that children in the training group had significantly higher post-test metacognitive knowledge, metacognitive skills, and mathematical problem-solving scores. In addition, metacognitive training was particularly beneficial to the low achievers. Thus metacognitive training enabled the low achievers to make progress and solve the same number of problems on the post-test as the normal achievers solved on the pre-test. (shrink)

Mathematics word problems provide students with an opportunity to apply what they are learning in their mathematics classes to the world around them. However, students often neglect their knowledge of the world and provide nonsensical responses (e.g., they may answer that a school needs 12.5 buses for a field trip). This study examined if the question design of word problems affects students' mindset in ways that affect subsequent sense‐making. The hypothesis was that rewriting standard word problems to (...) introduce inherent uncertainty about the result would be beneficial to student performance and sense‐making because it requires students to reason explicitly about the context described in the problem. Middle school students (N = 229) were randomly assigned to one of three conditions. In the standard textbook condition, students solved a set of six word problems taken from current textbooks. In the modified yes/no condition, students solved the same six problems rewritten so the solution helped answer a “yes” or “no” question. In the disfluency control condition, students solved the standard problems each rewritten in a variety of fonts to make them look unusual. After solving the six problems in their assigned condition, all students solved the same three “problematic” problems designed to assess sense‐making. Contrary to predictions, results showed that students in the modified yes/no condition solved the fewest problems correctly in their assigned condition problem set. However, consistent with predictions, they subsequently demonstrated more sense‐making on the three problematic problems. Results suggest that standard textbook word problems may be able to be rewritten in ways that mitigate a “senseless” mindset. (shrink)

The practice of mathematical word problem is ubiquitous and thought to impact academic achievement. However, the underlying neural mechanisms are still poorly understood. In this study, we investigate how lexical consistency of word problem description is modulated in adults' brain responses during word problem solution. Using functional magnetic resonance imaging methods, we examined compare word problems that included relational statements, such as “A dumpling costs 9 dollars. A wonton is 2 dollars less (...) than a dumpling. How much does a wonton cost?” and manipulated lexical consistency and problem operation. We found a consistency by operation interaction in the widespread fronto-insular-parietal activations, including the anterior insula, dorsoanterior cingulate cortex, middle frontal gyrus, and intraparietal sulcus, such that inconsistent problems engaged stronger activations than consistent problems for addition, whereas the consistency effect was inverse for subtraction. Critically, these results were more salient in the less successful problem solvers than their more successful peers. Our study is the first to demonstrate that lexical consistency effects on arithmetic neural networks are modulated during reading word problem that required distinct arithmetic operations. More broadly, our study has strong potentials to add linkage between neuroscience and education by remediating deficits and enhance instruction design in the school curriculum. (shrink)

We have two polynomial time results for the uniform word problem for a quasivariety Q: The uniform word problem for Q can be solved in polynomial time iff one can find a certain congruence on finite partial algebras in polynomial time. Let Q* be the relational class determined by Q. If any universal Horn class between the universal closure S and the weak embedding closure S̄ of Q* is finitely axiomatizable then the uniform word (...) class='Hi'>problem for Q is solvable in polynomial time. This covers Skolem's 1920 solution to the uniform word problem for lattices and Evans' 1953 applications of the weak embeddability property for finite partial V algebras. (shrink)

In previous papers (see Van Bendegem [1993], [1996], [1998], [2000], [2004], [2005], and jointly with Van Kerkhove [2005]) we have proposed the idea that, if we look at what mathematicians do in their daily work, one will find that conceiving and writing down proofs does not fully capture their activity. In other words, it is of course true that mathematicians spend lots of time proving theorems, but at the same time they also spend lots of time preparing the ground, if (...) you like, to construct a proof. (shrink)

Humans have long been characterized as poor probabilistic reasoners when presented with explicit numerical information. Bayesian word problems provide a well-known example of this, where even highly educated and cognitively skilled individuals fail to adhere to mathematical norms. It is widely agreed that natural frequencies can facilitate Bayesian reasoning relative to normalized formats (e.g. probabilities, percentages), both by clarifying logical set-subset relations and by simplifying numerical calculations. Nevertheless, between-study performance on “transparent” Bayesian problems varies widely, and generally remains (...) rather unimpressive. We suggest there has been an over-focus on this representational facilitator (i.e. transparent problem structures) at the expense of the specific logical and numerical processing requirements and the corresponding individual abilities and skills necessary for providing Bayesian-like output given specific verbal and numerical input. We further suggest that understanding this task-individual pair could benefit from considerations from the literature on mathematical cognition, which emphasizes text comprehension and problem solving, along with contributions of online executive working memory, metacognitive regulation, and relevant stored knowledge and skills. We conclude by offering avenues for future research aimed at identifying the stages in problem solving at which correct versus incorrect reasoners depart, and how individual difference might influence this time point. (shrink)

A pseudo-natural algorithm for the word problem of a finitely presented group is an algorithm which not only tells us whether or not a word w equals 1 in the group but also gives a derivation of 1 from w when w equals 1. In [13], [14] Madlener and Otto show that, if we measure complexity of a primitive recursive algorithm by its level in the Grzegorczyk hierarchy, there are groups in which a pseudo-natural algorithm is arbitrarily (...) more complicated than an algorithm which simply solves the word problem. In a given group the lowest degree of complexity that can be realised by a pseudo-natural algorithm is essentially the derivational complexity of that group. Thus the result separates the derivational complexity of the word problem of a finitely presented group from its intrinsic complexity. The proof given in [13] involves the construction of a finitely presented group G from a Turing machine T such that the intrinsic complexity of the word problem for G reflects the complexity of the halting problem of T, while the derivational complexity of the word problem for G reflects the runtime complexity of T. The proof of one of the crucial lemmas in [13] is only sketched, and part of the purpose of this paper is to give the full details of this proof. We will also obtain a variant of their proof, using modular machines rather than Turing machines. As for several other results, this simplifies the proofs considerably. MSC: 03D40, 20F10. (shrink)

Over the last decades, understanding the sources of the difficulty of Bayesian problem solving has been an important research goal, with the effects of numerical format and individual numeracy being widely studied. However, the focus on the comprehension of probability numbers has overshadowed the relational reasoning demand of the Bayesian task. This is particularly the case when the statistical data are verbally described since the requested quantitative relation (posterior ratio) is misaligned with the presented ones (prior and likelihood (...) ratios). In this regard, here I develop the proposal that research on Bayesian reasoning might improve by considering the notational alignment framework of mathematical problem-solving. Specifically, this framework can help to understand the sources of the main difficulties underlying Bayesian inferences based on verbal descriptions. In essence, the present proposal supports the general claim in math education regarding the need to foster relational comprehension to avoid misleading alignments and improve problem solving. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that (...) the method of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

ABSTRACTWhile metaphor researchers commonly use the word “mapping” in explanations of various types of figurative language, there is a lack of recognition that the term is itself metaphorical. In fact, the term has two metaphor-based working definitions, the more commonly cited being that relating to mathematical set theory and the less common definition originating in cognitive neuroscience. Perhaps not coincidentally, terminological inconsistencies relating to mapping have led to theoretical problems both for single-domain theories of metonymy and attempts to (...) examine Lakoff’s invariance hypothesis. This article will assert that expressing metonymic connectivity using the neuroscience term “binding” will both eliminate unnecessary theoretical confusion for cognitive linguists and provide common terminology to facilitate productive communication between cognitive linguists and cognitive neuroscientists. (shrink)

In this paper I study the activity of mathematical problem-solving in scientific practice, focussing on enquiries in mathematical social science. I identify three salient phases of mathematical problem-solving and adopt them as a reference frame to investigate aspects of applications that have not yet received extensive attention in the philosophical literature.

This study examined the effects of opportunity to learn or the content coverage in mathematics on student mathematics anxiety, problem-solving performance, and mathematics performance. The pathways examining the influences of OTL on student problem-solving performance and mathematics performance via mathematics anxiety were also tested. A sample of 1,676 students from Shanghai-China, and a sample of 1,511 students from the United States who participated in the Programme for International Student Assessment 2012 were used for the analyses. The (...) results from multilevel models and path models supported our hypotheses that OTL not only showed significant direct effects on student mathematics anxiety, problem-solving performance, and mathematics performance, but also presented indirect effects on student problem-solving performance and mathematics performance via mathematics anxiety in both Shanghai-China and United States, controlling for student gender, grade, and socioeconomic status. The practical implications of the current results were also discussed. (shrink)

Of course, words aren’t magic. Neither are sextants, compasses, maps, slide rules and all the other paraphenelia which have accreted around the basic biological brains of homo sapiens. In the case of these other tools and props, however, it is transparently clear that they function so as to either carry out or to facilitate computational operations important to various human projects. The slide rule transforms complex mathematical problems (ones that would baffle or tax the unaided subject) into simple tasks (...) of perceptual recognition. The map provides geographical information in a format well-suited to aid complex planning and strategic military operations. The compass gathers and displays a kind of information that (most) unaided human subjects do not seem to command. These various tools and props thus act to generate information, or to store it, or to transform it, or some combination of the three. In so doing, they impact our individual and collective problem- solving capacities in much the same dramatic ways as various software packages impact the performance of a simple pc. (shrink)

In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem’s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical to a graph structure (...) with nodes and links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in mentally supported space, actively constructing possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)

Word problems are notoriously difficult to solve. We suggest that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language. We tested this hypothesis by (1) requiring children to recall problems either before or after solving them, (2) requiring them to generate f'mal questions to incomplete word problems, and (3) modeling performance pattems using a computer simulation. Solution performance was found to be systematically related to recall (...) and question generation performance. Correct solutions were associated with accurate recall of the problem structure and with appropriate question generation. Solution "errors" were found to be correct solutions to miscomprehended problems. Word problems that contained abstract or ambiguous language tended to be miscomprehended more often than those using simpler language, and there was a great deal of systematicity in the way these problems were miscomprehended. Solution error pattems were successfully simulated by manipulating a computer model’s language comprehension strategies, as opposed to its knowledge of logical set relations. o was.. (shrink)

In this paper the reader is asked to engage in some simple problem-solving in classical pure number theory and to then describe, on the basis of a series of questions, what it is like to solve the problems. In the recent philosophy of mind this “what is it like” question is one way of signaling a turn to phenomenological description. The description of what it is like to solve the problems in this paper, it is argued, leads to (...) several morals about the epistemology and ontology of classical pure mathematical practice. Instead of simply making philosophical judgments about the subject matter in advance, the exercise asks the reader to briefly engage in a mathematical practice and to then reflect on the practice. (shrink)

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure (...) texts – mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive. (shrink)

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach (...) applies methods from computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way. (shrink)

Although to live is to face problems, the general concept of a problem has been significantly understudied. So much so, that the publication of Polya’s delightful How to Solve It caused quite a stir. And, although the concept of a conceptual problem is philosophical because it is deep and occurs across fields, from mathematics to politics, no philosophers have produced any memorable studies of it. Moreover, the word ‘problem’ is absent from most philosophical reference works. There (...) are plenty of texts on particular problems such as those of evil and induction, but none on the general concept of a conceptual problem—as if it were not a problematic concept. Worse, we are seldom told that doing original mathematics, science or technology is to wrestle with original problems. Nor have we been told that the hardest problems are likely to be inverse, in that they go from effect to cause, or from sketch to implementation, or from conclusion to premises—as in diagnosing disease from symptoms, looking for a westward passage from Europe to Asia, and designing public policies to attack social issues. The existence of this significant gap in the philosophical literature justifies the publication of the present study. (shrink)

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure (...) texts – mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive. (shrink)

Following Marr’s famous three-level distinction between explanations in cognitive science, it is often accepted that focus on modeling cognitive tasks should be on the computational level rather than the algorithmic level. When it comes to mathematical problem solving, this approach suggests that the complexity of the task of solving a problem can be characterized by the computational complexity of that problem. In this paper, I argue that human cognizers use heuristic and didactic tools and (...) thus engage in cognitive processes that make their problem solving algorithms computationally suboptimal, in contrast with the optimal algorithms studied in the computational approach. Therefore, in order to accurately model the human cognitive tasks involved in mathematical problem solving, we need to expand our methodology to also include aspects relevant to the algorithmic level. This allows us to study algorithms that are cognitively optimal for human problem solvers. Since problem solving methods are not universal, I propose that they should be studied in the framework of enculturation, which can explain the expected cultural variance in the humanly optimal algorithms. While mathematical problem solving is used as the case study, the considerations in this paper concern modeling of cognitive tasks in general. (shrink)

Integrative systems biology is an emerging field that attempts to integrate computation, applied mathematics, engineering concepts and methods, and biological experimentation in order to model large-scale complex biochemical networks. The field is thus an important contemporary instance of an interdisciplinary approach to solving complex problems. Interdisciplinary science is a recent topic in the philosophy of science. Determining what is philosophically important and distinct about interdisciplinary practices requires detailed accounts of problem-solving practices that attempt to understand how specific (...) practices address the challenges and constraints of interdisciplinary research in different contexts. In this paper we draw from our 5-year empirical ethnographic study of two systems biology labs and their collaborations with experimental biologists to analyze a significant problem-solving approach in ISB, which we call adaptive problem solving. ISB lacks much of the methodological and theoretical resources usually found in disciplines in the natural sciences, such as methodological frameworks that prescribe reliable model-building processes. Researchers in our labs compensate for the lack of these and for the complexity of their problems by using a range of heuristics and experimenting with multiple methods and concepts from the background fields available to them. Using these resources researchers search out good techniques and practices for transforming intractable problems into potentially solvable ones. The relative freedom lab directors grant their researchers to explore methodological options and find good practices that suit their problems is not only a response to the complex interdisciplinary nature of the specific problem, but also provides the field itself with an opportunity to discover more general methodological approaches and develop theories of biological systems. Such developments in turn can help to establish the field as an identifiably distinct and successful approach to understanding biological systems. (shrink)

The principles of sexual selection were used as an organizing framework for interpreting cross-national patterns of sex differences in mathematical abilities. Cross-national studies suggest that there are no sex differences in biologically primary mathematical abilities, that is, for those mathematical abilities that are found in all cultures as well as in nonhuman primates, and show moderate heritability estimates. Sex differences in several biologically secondary mathematical domains are found throughout the industrialized world. In particular, males consistently outperform (...) females in the solving of mathematical word problems and geometry. Sexual selection and any associated proximate mechanisms influence these sex differences in mathematical performance indirectly. First, sexual selection resulted in greater elaboration in males than in females of the neurocognitive systems that support navigation in three-dimensional space. Knowledge implicit in these systems reflects an understanding of basic Euclidean geometry, and may thus be one source of the male advantage in geometry. Males also use more readily than females these spatial systems in problem-solving situations, which provides them with an advantage in solving word problems and geometry. In addition, sex differences in social styles and interests, which also appear to be related in part to sexual selection, result in sex differences in engagement iii mathematics-related activities, thus further increasing the male advantage in certain mathematical domains. A model that integrates these biological influences with sociocultural influences on the sex differences in mathematical performance is presented in this article. (shrink)

People spontaneously discover new representations during problem solving. Discovery of a mathematical representation is of special interest, because it shows that the underlying structure of the problem has been extracted. In the current study, participants solved gear‐system problems as part of a game. Although none of the participants initially used a mathematical representation, many discovered a parity‐based, mathematical strategy during problem solving. Two accounts of the spontaneous discovery of mathematical strategies were (...) tested. According to the automatic schema abstraction hypothesis, experience with multiple, unique problem exemplars facilitates extraction of the parity relation. According to the comparison‐based abstraction hypothesis, explicitly comparing gear pathways that have different number, but the same parity, should result in extraction of parity. An event history analysis showed that accumulation of experiences with different‐number, same‐parity comparisons predicted discovery of parity; accumulation of unique exemplars did not. Results suggest that comparison‐based abstraction processes can lead to the discovery of a mathematical relation. (shrink)

Over the past few decades, the question regarding the proper understanding of Diophantus’s method has attracted much scholarly attention. “Modern algebra”, “algebraic geometry”, “arithmetic”, “analysis and synthesis”, have been suggested by historians as suitable contexts for describing Diophantus’s resolutory procedures, while the category of “premodern algebra” has recently been proposed by other historians to this end. The aim of this paper is to provide arguments against the idea of contextualizing Diophantus’s modus operandi within the conceptual framework of the ancient analysis (...) and to examine the few instances, in the preserved books of the Arithmetica, which might be regarded as linked to practices belonging to the field of analysis. (shrink)

Word problems in mathematics seem to constantly pose learning difficulties for all kinds of students. Recent work in math education (for example, [Lakoff, G. & Nuñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books]) suggests that the difficulties stem from an inability on the part of students to decipher the metaphorical properties of the language in which such problems are cast. A 2003 pilot study [Danesi, M. (2003a). Semiotica, (...) 145, 71–83] confirmed this hypothesis in an anecdotal way. This paper reviews the implications of that study and of a follow-up one that is described here as well, in the light of how the metaphorical analysis of word problems allows learners to overcome typical difficulties in word problem-solving by teaching them how to flesh out the underlying concepts and convert them into appropriate representations. (shrink)

Context: The paper utilizes a conceptual analysis to examine the development of abstract conceptual structures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC influenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? Problem: The paper documents how Ernst von Glasersfeld’s view of mental representation can (...) be illustrated in the context of mathematical problem solving and used to explain the development of conceptual structure in mathematical problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internalization and interiorization of solution activity. Method: A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particularly her emerging and developing reflections about her solution activity. The interview was videotaped and written transcripts of the solver’s verbal responses were prepared. Results: The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly abstract levels of conceptual knowledge demonstrated by the solver. Implications: The results suggest a framework for an explanation of problem solving that is activity-based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed. (shrink)

In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem‘s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical to a graph structure (...) with nodes and links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in this mentally supported space, actively construction possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)

This paper begins with an analysis of global problems shaping education, particularly as they impact upon learning and life chances. In addressing these problems a range of philosophical positions and controversies are considered, including: traditional romantic and institutional views of schooling; and more recent maximalist, neo-liberal, emancipatory and post-modern-perspectives of lifelong learning. In this paper we argue that these do not represent 'the last word' on the provision of learning and the enhancement of life chances and instead we put (...) forward an alternative view based upon the notion of adopting an inclusive problem-solving approach, which we believe may be better suited to offering solutions to the various global problems that have been identified or that are still emerging. We believe that such an approach addresses and expands the individual, social, economic and political features of education and conduces to the improvement and expansion of learning and life chances. (shrink)

Experimental research by social and cognitive psychologists has established that cooperative groups solve a wide range of problems better than individuals. Cooperative problem solving groups of scientific researchers, auditors, financial analysts, air crash investigators, and forensic art experts are increasingly important in our complex and interdependent society. This comprehensive textbook--the first of its kind in decades--presents important theories and experimental research about group problem solving. The book focuses on tasks that have demonstrably correct solutions within (...) class='Hi'>mathematical, logical, scientific, or verbal systems, including algebra problems, analogies, vocabulary, and logical reasoning problems.The book explores basic concepts in group problem solving, social combination models, group memory, group ability and world knowledge tasks, rule induction problems, letters-to-numbers problems, evidence for positive group-to-individual transfer, and social choice theory. The conclusion proposes ten generalizations that are supported by the theory and research on group problem solving. Group Problem Solving is an essential resource for decision-making research in social and cognitive psychology, but also extremely relevant to multidisciplinary and multicultural problem-solving teams in organizational behavior, business administration, management, and behavioral economics. (shrink)

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: Problem solving is an enormous field of study, where so-called “problems” can end up having very little in common. One of the least studied categories of problems is open-ended mathematical modeling research. Cifarelli and Sevim’s framework - although not developed for this purpose - (...) may be a useful lens for studying the development of mathematical modelers and researchers in applied mathematics. (shrink)

Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a (...) theoretical account of this multiple strategy effect and have begun to formulate this theory in an ACT-R computer model. We show why students may reach common impasses in the use of written algebra, and how subsequent or concurrent use of informal strategies leads to better problem-solving performance. Formal strategies facilitate computation because of their abstract and syntactic nature; however, abstraction can lead to nonsensical interpretations and conceptual errors. Reapplying the formal strategy will not repair such errors; switching to an informal one may. We explain the multiple strategy effect as a complementary relationship between the computational efficiency of formal strategies and the sense-making function of informal strategies. (shrink)

In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in (...) class='Hi'>mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable (...) permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non‐periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non‐commutative and non‐Boolean c.e. ring. (shrink)

Any part of the world can be viewed and modelled in terms of its chosen qualitative and/or quantitative properties, OR its structure. The former approach has been used by nearly the whole of ‘human intellectual endeavor’, i.e conventional science of physics, the arts etc. Development of the latter or the ‘systemic view’ is the subject matter of the current work. The Purpose of Change is Problem Solving suggests that the ‘structural view’ is empirical, pervasive throughout experience and as (...) such results in a single domain as opposed to conventional science which consists of many domains like mechanics, electricity etc. Thus, a unique approach is required which is based on ‘general principles of systems’ translated into operational form by the symbolism of processed natural language called ‘linguistic modelling of scenarios’ which can carry mathematics and uncertainties. To model scenarios with complex structure, a description or story in natural language is expressed in terms of homogenous language of one – and two – place sentences, the ‘elementary constituents’ of which complex structures can be constructed [like a variety of buildings from bricks]. To correspond to the single domain, based on the logic of causation, a single scheme of ‘Management/producers – Product – User/consumer’ is proposed which is immediately applicable to structuring scenarios and guides their detailed linguistic modelling or design. The approach, subject to debate, can have significant impact on society and education, especially that of engineering which lacks a ‘comprehensive theory of structure’ of problematic scenarios. (shrink)