Word problems in mathematics education: a survey

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

     

On an SDP relaxation for kissing number

We demonstrate that an earlier semidefinite-programming relaxation for the kissing-number problem cannot provide good upper bounds. Furthermore, we show the existence of an optimal solution for this relaxation that cannot be used as a basis for establishing a good lower bound.     

Approximating max-cut under graph-MSO constraints

We consider the max-cut and max-$k$-cut problems under graph-based constraints. Our approach can handle any constraint specified using monadic second-order (MSO) logic on graphs of constant treewidth. We give a $\frac{1}{2}$-approximation algorithm for this class of problems.     

Chemical sensing based on the plasmonic response of nanoparticle aggregation: Anion sensing in nanoparticles stabilized by amino-functional ionic liquid

We report the synthesis, characterization and modellization of optical anion sensors based on gold nanoparticles (Au NPs) stabilized by amino-functional imidazolium ionic liquids (AFIL). The addition of different salts results in anion exchange of the imidazolium ionic liquid stabilizer leading to a change in the optical response of the original nanoparticle aqueous solution. In all cases except with dodecylbenzenesulfonic acid sodium salt, a sufficient amount of salt concentration (5 times larger than equimolar) leads to the appearance of an absorption band between 600 and 700 nm in the ultravioletvisible (UV-vis) spectrum. The presence of the salt produces aggregation of the particles that localise the optical response and produce a large spectral red shift. Transmission electron microscopy images demonstrated that this optical change was due to the aggregation of the nanoparticles. We simulate the optical response of both situations, before and after salt addition, and interpret the experimental observations in terms of the different response of metallic single nanoparticles and nanoparticle aggregates. Theoretical calculations for single nanoparticle and single nanoparticle dimers demonstrate that the colour change is not due to the enlargement or structural changes of the Au NPs, but due to the formation of NP aggregation. These results show the potential of nanoparticle plasmonics to perform effective chemical sensing.     

Dynamical structure of the turbulent boundary layer on rough surface

The experiments on structure of turbulent boundary layer on the plane rough wall without pressure gradient are presented. Sand roughness of the wall is considered. Measurements are carried out using Time-Resolved PIV technique in planes parallel and perpendicular to the wall. The results on rough wall are compared with the base case of boundary layer on smooth wall. Hairpin vortices have been detected. Topology and typical size of those structures substantially differ in the cases in question. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Total synthesis of gambierol: the generation of the A-C and F-H subunits by using a C-glycoside centered strategy

Gambierol, a representative of the marine ladder toxin family, consists of eight ether rings, 18 stereocenters, and two challenging pyranyl rings having methyl groups that are in a 1,3-diaxial orientation to one another. Herein we describe the generation of gambierol's A-C and F-H ring systems and demonstrate the versatility of the glycosyl anhydride, enol ether-olefin RCM strategy to fused polycyclic ethers. This work has both enabled us to generate sufficient quantities of the gambierol precursors and has enabled us to better understand the chemical transformations that were key to these efforts. Fundamental work included efforts to C-glycosides and C-ketosides, Claisen rearrangements, and enol ether-olefin RCM reactions.     

Improved cell typing by charge-state deconvolution of matrix-assisted laser desorption/ionization mass spectra

Robust, specific, and rapid identification of toxic strains of bacteria and viruses, to guide the mitigation of their adverse health effects and optimum implementation of other response actions, remains a major analytical challenge. This need has driven the development of methods for classification of microorganisms using mass spectrometry, particularly matrix-assisted laser desorption/ionization mass spectrometry (MALDI-MS), that allows high-throughput analyses with minimum sample preparation. We describe a novel approach to cell typing based on pattern recognition of MALDI mass spectra, which involves charge-state deconvolution in conjunction with a new correlation analysis procedure. The method is applicable to both prokaryotic and eukaryotic cells. Charge-state deconvolution improves the quantitative reproducibility of spectra because multiply charged ions resulting from the same biomarker attaching a different number of protons are recognized and their abundances are combined. This allows a clearer distinction of bacterial strains or of cancerous and normal liver cells. Improved class distinction provided by charge-state deconvolution was demonstrated by cluster spacing on canonical variate score charts and by correlation analyses. Deconvolution may enhance detection of early disease state or therapy progress markers in various tissues analyzed by MALDI-MS.     

Experimental study of polarization switching of vertical-cavity surface-emitting lasers as a dynamical bifurcation

We study the role of the bias current sweep rate in measurements of polarization switching (PS) of vertical-cavity surface-emitting lasers (VCSELs). We show that the polarization-resolved L-I (light-intensity) curve depends on the current sweep rate. As the current sweep rate increases, the PS occurs at higher bias currents for upward scans and at lower bias currents for downward scans. We also show that the delay of the dynamical bifurcation follows a power law relationship with the frequency of the ramp, in good agreement with recent theoretical predictions.     

Validation of the Diagnostic Teacher Assessment of Mathematics and Science (DTAMS) Instrument

The Diagnostic Teacher Assessment in Mathematics and Science (DTAMS) was developed to measure the content knowledge and pedagogical content knowledge of middle‐school teachers. Its reliability and validity were initially established by reviewing national standards for content and use of expert question writing teams and reviewers. DTAMS was administered to approximately 1,600 middle‐school mathematics teachers in 17 states. Subsequent analyses using structural equation modeling and item response theory were performed as part of a multistage validation process. This evaluation contributes to the body of work describing the reliability and validity of these assessments. The results of this study confirm trends in middle‐school mathematics teacher preparation and certification and help explain middle‐school student mathematics achievement levels.