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Resumen El presente trabajo tiene como objetivo caracterizar el razonamiento probabilístico de cinco estudiantes de secundaria con diagnóstico de Trastorno del Espectro Autista (TEA), tras analizar los argumentos que emplean al comparar probabilidades simples. Se consideran, en el análisis, argumentos subjetivistas u objetivistas, así como distintos tipos de estrategias empleadas para la construcción de los segundos. Los estudiantes resolvieron seis ítems en línea con lo anterior, todos ellos diseñados con una estructura similar y en un contexto de bolas en cajas: entre las dos cajas posibles, debían elegir aquella en que la probabilidad del suceso sacar una bola negra era mayor Los resultados obtenidos advierten que los argumentos objetivistas determinan las elecciones de los cinco estudiantes y que, mayoritariamente, conducen a elecciones correctas. Además, en relación con los argumentos objetivistas se han reconocido distintas estrategias, siendo predominante la denominada comparación aditiva de casos favorables y desfavorables. Se han observado, también, ciertas dificultades por parte de algunos estudiantes en el reconocimiento de situaciones de equiprobabilidad. Se discuten posibles relaciones del desempeño observado con algunos rasgos cognitivos característicos de personas con TEA.

Purpose This paper aims to study the strategies used by ten students diagnosed with autism when solving multiplication and division problems because these operations are rarely studied in students with this condition. Design/methodology/approach This study conducted an exploratory study with ten students diagnosed with autism to explore and describe the strategies used in solving equal group problems. The authors also describe in detail the case of a student whom the authors deem to be representative because of the reasoning the student employed. Findings The informal strategies that they used are described, as well as the difficulties observed in the various problems, depending on the operation required to solve them. The strategies used include direct modeling with counting and others that relied on incorrect additive relationships, with strategies based on multiplication and division operations being scarce. Difficulties were observed in several problems, with measurement division being particularly challenging for the study participants. Practical implications The detailed description of the strategies used by the students revealed the meanings that they associate with the operations they are executing and brought to light potential difficulties, which can help teachers plan their instruction. Originality/value This research supplements other studies focusing on mathematical problem-solving with autistic students.

This study examines relationships between mathematical problem-solving performance (in terms of strategies used and accuracy) and the main cognitive domains associated with mathematical learning (i.e. executive functions, verbal comprehension and social perception) of children with and without autism spectrum disorder (ASD and non-ASD resp.). The study involved 26 ASD and 26 non-ASD children without intellectual disabilities, between 6 and 12 years old, matched by sex, age and school (grade and classroom). The results show a higher percentage of ASD children with problem solving difficulties than non-ASD (57% vs. 23% resp.). Poor performing ASD children showed comparatively lower scores in inhibition, theory of mind and verbal comprehension. Implications for the design of mathematical interventions for ASD students are discussed.

The importance of students being acquainted with algebraic ideas before secondary education has been revealed in the research literature. It is therefore essential that prospective elementary teachers (PTs) be prepared to instill an early algebra perspective in their teaching. However, PTs often show difficulties in algebra content knowledge, which need to be diagnosed aiming to assist them in developing the required knowledge to teach according to that perspective. This study aims to understand what aspects of functional thinking Spanish and Portuguese elementary PTs exhibit at the beginning of their teacher education program. The findings show that although PTs from both countries use different strategies to generalize functional relationships, the occurrence of successful strategies is low. Also, most participants provide local approaches in their interpretation of relationships between variables and reveal difficulties in understanding and connecting different representations of functions. These difficulties show that PTs lack important knowledge about functional thinking. By providing a framework concerning the functional thinking required for PTs to teach within an early algebra perspective, we shed light on a necessary step for teacher education programs to diagnose PTs? functional thinking and to assist them in developing the needed mathematical knowledge to teach accordingly. Pentingnya pengenalan ide-ide aljabar siswa sebelum pendidikan menengah telah terungkap dalam penelitian literatur ini. Oleh karena itu, calon guru sekolah dasar (CG) harus siap untuk menanamkan perspektif aljabar awal dalam pengajaran mereka. Namun, CG sering kali menunjukkan kesulitan dalam pengetahuan konten aljabar, yang perlu didiagnosis dengan tujuan membantu mereka dalam mengembangkan pengetahuan yang diperlukan untuk mengajar sesuai dengan perspektif itu. Penelitian ini bertujuan untuk memahami aspek-aspek berpikir fungsional apa yang diperlihatkan oleh, CG Spanyol dan Portugis di awal program pendidikan guru mereka. Temuan menunjukkan bahwa meskipun CG dari kedua negara menggunakan strategi yang berbeda untuk menggeneralisasi hubungan fungsional, kejadian strategi yang berhasil rendah. Selain itu, sebagian besar peserta memberikan pendekatan lokal dalam interpretasi mereka tentang hubungan antar variabel dan mengungkapkan kesulitan dalam memahami dan menghubungkan representasi fungsi yang berbeda. Kesulitan ini menunjukkan bahwa CG kurang memiliki pengetahuan penting tentang berpikir fungsional. Dengan memberikan kerangka kerja mengenai pemikiran fungsional yang diperlukan untuk CG untuk mengajar dalam perspektif aljabar awal, kami menjelaskan langkah yang diperlukan untuk program pendidikan guru untuk mendiagnosis pemikiran fungsional PT dan untuk membantu mereka dalam mengembangkan pengetahuan matematika yang dibutuhkan untuk mengajar sesuai. This work was supported by the Ministerio de Economía y Competitividad. Madrid. Spain [projects EDU2017-84276-R and PID2019-105677RB-I00].

Students with autism spectrum disorder frequently face challenges when learning mathematical concepts. For example, they may have difficulties solving mathematical word problems, in particular Cartesian product problems. This research is a case study with a multiple probe design in which the participants were three students diagnosed with autism spectrum disorder. A conceptual model-based problem-solving approach adapted to the characteristics of the participants was used to teach them how to solve Cartesian product word problems introduced sequentially in the instruction (first multiplication, then division, and finally both operations). The results show a functional relationship between the intervention and the students’ performance. The three participants generalized their learning to two-operation Cartesian product problems (an addition and a multiplication). Moreover, two of them retained what they learned 6 weeks after completing the instruction. The implications for teaching this and other mathematical content to students with autism are discussed.

En este artículo se examina la efectividad de un enfoque de Instrucción Basada en Esquemas para mejorar el rendimiento matemático de resolución de problemas verbales aditivos por un estudiante con trastorno del espectro autista. Se llevó a cabo una intervención adaptada a las necesidades educativas del alumno. Este mostró una mejoría en la resolución de problemas verbales aditivos de cambio con incógnita en las tres cantidades (inicial, cambio y final) en dos aspectos: la identificación de la operación aritmética y la obtención del resultado numérico correcto. El estudiante abandonó sus primeras estrategias informales, que fueron sustituidas por hechos numéricos aditivos. Además, generalizó las habilidades adquiridas a problemas verbales de dos operaciones. Los logros se mantuvieron cuatro semanas después de finalizar la instrucción. In this study, the effectiveness of a SBI instructional approach to improve the mathematical word-problemsolving performance of one student diagnosed with autism spectrum disorder was examined. An intervention adapted to the educational needs of the student was implemented. The student improved his performance when solving addition and subtraction change word problems with the unknown in the three quantities (initial, change and final amount) in two aspects: identifying the arithmetic operation needed to solve the problem and providing the correct numerical answer to the problem. The student replaced his initial informal strategies by additive numerical facts. The effects of the instruction were generalized to two-step addition and subtraction word problems. The results were maintained four weeks after instruction. Este trabajo se ha realizado en colaboración con Alicia Bruno y ha recibido apoyo de los proyectos con referencias EDU2017-84276-R y PID2019-105677RB-I00.

Background & aims In recent years, there has been an increased interest in analyzing the mathematical performance of students with learning difficulties in order to provide them with teaching methods adapted to their needs. In particular, the importance of studying the type of informal strategy that students use when solving problems has been highlighted. Observing how these strategies emerge and develop in children with learning difficulties is crucial, as it allows us to understand how they develop a subsequent understanding of arithmetic operations. In this paper we study the effect of explicit instruction in addition strategies, focusing on the minimum addend strategy, and analyze the difficulties that arise during this process. Methods An adapted multiple-probe design across students with a microgenetic approach was employed to assess the effectiveness of the teaching instruction and the acquisition of the minimum addend strategy while solving addition word problems. The participants were three primary-school children (two boys and one girl) with learning difficulties, one of them diagnosed with autism spectrum disorder. The instruction on the minimum addend strategy was sequenced into levels of abstraction based on the addends represented with and without manipulatives. Results The results show that the three participants were able to acquire the minimum addend strategy and transfer it to two-step problems. They all showed difficulties during the instructional process, with quantity comparison difficulties predominating. The instruction provided to address these and other difficulties is detailed for each participant. Conclusions The teaching of the minimum addend strategy has proven effective, and all three students acquired it throughout the instruction. The results concerning the student diagnosed with autism spectrum disorder are especially interesting given the lack of studies that focus on the strategies employed by students with this disorder to solve arithmetic problems. In this sense, the use of the microgenetic approach was especially useful to observe the type of spontaneous strategies used by this participant, and how they varied in response to the instruction. Implications Each study participant faced different difficulties and needed different periods of time to assimilate the new strategy. Conclusions are drawn for educators to help children with learning difficulties advance to more sophisticated strategies, so they can acquire these and subsequent mathematical concepts.

Los estudiantes diagnosticados con Trastornos del Espectro Autista (TEA) suelen desarrollar un interés especial sobre áreas que no son habituales en cuanto a su intensidad o temática. Este trabajo profundiza en comprender la influencia que tienen las áreas de interés especial en los procesos de resolución de problemas aritméticos verbales de multiplicación y división, mediante un estudio de caso único con un estudiante de 11 años diagnosticado con TEA y discapacidad intelectual. Se utiliza un cuestionario formado por 15 problemas enunciados en tres tipos de contextos: de interés especial, familiar y no familiar. Siguiendo una metodología cualitativa, se clasifican las estrategias informales y el éxito en la obtención de la solución. Los resultados muestran que las áreas de interés especial han supuesto una mayor implicación del estudiante, pero no han logrado una mejora efectiva respecto de los contextos familiares, ya que en ambos contextos el estudiante ha resuelto los problemas de multiplicación y de división-agrupamiento pero no ha logrado resolver los problemas de división-reparto. En el contexto no familiar no ha resuelto ningún problema. Estos hallazgos contribuyen a completar la literatura existente sobre la utilidad educativa de las áreas de interés especial en estudiantes diagnosticados con TEA. Students diagnosed with Autism Spectrum Disorders (ASD) usually de-velop interest in areas that is unusual in its intensity or focus. This paper focuses on the influence of the special interest areas in the problem solving processes of multiplication and division word arithmetic problems, through a single case study with an 11-year-old student diagnosed with ASD and intellectual disability. A questionnaire consisting of 15 problems contextualized in three types of contexts ?special interest context, a familiar context and an unfamiliar context? was applied. Following a qualitative method, infor-mal strategies and success in obtaining the solution were registered. The results show that both, the special interest context and the familiar context, determined the success in obtaining the problem solution; the student did not solved any problem in the unfa-miliar context; the special interest context led to a greater involvement of the student in the search of the solution, but it did not represent an effective improvement with respect to familiar contexts, since in both contexts the student solved the multiplication and division-grouping problems, but did not solve the partitive-division problems. These findings contribute to completing the existing literature on the educational potential of special interest areas in students diagnosed with ASD. Este trabajo ha sido financiado por los proyectos de investigación con referencias EDU 2017-84276-R y PID2019-105677RB-I00

Motivated by some string and plaster models dating back from the late 19th and early 20th century, this note recalls some of the early history of the classification of plane cubic curves over the real numbers. Examples of different classifications are provided, showing their connection with some of the models in the Schilling collection.

The recent increase in the number of mathematics museums has given rise to a need for tools with which to design and assess mathematics educational proposals in this non-formal context. This study proposes the use of the van Hiele model, a benchmark in mathematics instruction, for the design of interactive museum models focusing on mathematics content in general. By way of example, the model is characterised for application to the Pythagorean theorem. An interactive module designed around the aforementioned characterisation for teaching the theorem at the Museo Didáctico e Interactivo de Ciencias [interactive science instruction museum, MUDIC] may serve as a reference for future interactive modules and workshops in museums dealing with any manner of mathematics content.

En los últimos años ha aumentado notablemente el interés por analizar el rendimiento académico de los estudiantes con trastorno del espectro autista (TEA). Entre las materias escolares, las matemáticas son uno de los grandes obstáculos que encuentran estos estudiantes. Por consiguiente, es fundamental mejorar nuestro conocimiento sobre el modo en que los estudiantes con TEA aprenden diferentes conceptos matemáticos para luego proporcionarles métodos de enseñanza adaptados a sus necesidades. Este documento explora las estrategias y los errores que un estudiante de 11 años diagnosticado con TEA muestra al resolver problemas aritméticos verbales de división. Se diseñó una secuencia de enseñanza compuesta por problemas en dos formatos diferentes: con y sin material manipulativo. Se recogieron datos durante 15 sesiones de una hora en las cuales el estudiante resolvió un total de 49 problemas. Los resultados muestran una clara preferencia por la estrategia de reparto por múltiplos para los problemas en los que dispone de material manipulativo, mientras que recurre principalmente a la estrategia de reparto uno a uno cuando no dispone de material. Se identifica un conjunto de errores relacionados con los significados de las nociones de partición, equidad y representatividad, necesarios para resolver con éxito problemas aritméticos verbales de división partitiva In recent years there has been an increasing interest in studying the academic performance of students with Autism Spectrum Disorder (ASD). Among school subjects, mathematics is one of the great obstacles that face students with ASD. It is therefore crucial to go in depth into the understanding that they develop on mathematical concepts, to later provide learning instructions adapted to their needs. This paper explores the strategies and errors that an 11-year old student diagnosed with ASD shows when solving partitive division word problems. A teaching sequence has been designed that includes problems in two different formats: with and without support material. The data was collected during 15 one-hour sessions in which the student solved a total of 49 problems. Results show a clear preference for the one-to-many correspondence strategy in the problems with support material whereas the student mainly resorted to the sharing one-by-one strategy when he did not have the material. A list of errors has been identified related to the meaning of the notions of partition, equity and representativeness, required in partitive division word problems.

Considering polyhedra as particular cases of objects known as polytopes, which are the analogies of polyhedra in spaces with dimensions greater than three, makes it possible to discover characteristics that were not immediately apparent before. The method used by Alicia Boole Stott to discover the sections of four-dimensional polytopes relied on these analogies. The method of sections also makes it possible to discover a great number of connections between the theory of polytopes and other branches of mathematics.

Alicia Boole Stott (1860–1940) was an amateur mathematician who worked on four-dimensional geometry. She is remembered for finding all three dimensional sections of the four- dimensional polytopes (that is, the analogues of the three-dimensional Platonic solids), and for the discovery of many of the semi-regular polytopes. In this text we give a short biography of her and explain her method to calculate the three-dimensional sections of the four-dimensional polytopes. We illustrate her results by showing pictures of her original models and drawings.

B etween 1850 and 1852, the Swiss mathematician Ludwig Schlafli developed a theory of geometry in n-dimensions. In Theorie der vielfachen Kontinuitat ([4]), he defined the n-dimensional sphere, introduced the concept of four-dimensional polytopes, which he called polychemes, and proved that there are exactly six regular polytopes in four dimensions but only three in dimensions higher than four. Unfortunately, his work was not accepted for publication, and only fragments were published some years later. The entire manuscript did not appear until 1901. Thus, mathematicians in the second half of the century were unaware of Schlafli’s discoveries. Between 1880 and 1900 the six regular polytopes were independently rediscovered by, among many others, Stringham in 1880 [5], Gosset in 1900 [3] and Boole Stott in 1900 [1]. The work of Stringham and Gosset is well known today, but that of Boole Stott, an amateur mathematician, has remained almost unnoticed. In this note we present her original algorithmic approach, together with her original drawings and models. We emphasize that, as a woman born in the mid-nineteenth century, Boole Stott never received any formal mathematical training. Her discoveries came from an extraordinary capacity to visualize the fourth dimension. Rigorous mathematical proofs can therefore not be expected in herwork, but instead we find a watershed of surprising and original ideas. Alicia Boole was born near Cork (Ireland) in 1860, the third daughter of the famous logician George Boole. He died when Alicia was four years old, and her mother became an innovative educator. The amateur mathematician Howard Hinton, a frequent guest in their home, was deeply interested in the fourth dimension. He taught the children to visualize four-dimensional shapes with small cubes; this may (or may not) have inspired Alicia’s later research. Whatever the inspiriation, Alicia Boole Stott (she married in 1890) rediscovered the six four-dimensional polytopes by computing their three-dimensional sections. In 1895 she was introduced to the Dutch geometer P. H. Schoute. They collaborated for more than 20 years, combining Schoute’s analytical methods with her unusual visualization ability; in 1914, after Schoute’s death (1913), the University of Groningen awarded Boole Stott an honorary doctorate. After that, she was isolated from the mathematical community until about 1930, when her nephew, G. I. Taylor, introduced her to H. S. M. Coxeter. Despite the nearly 50 year difference in their ages, Boole Stott and Coxeter collaborated productively until her death in 1940. (For more details, see [2].) To clarify her approach to four-dimensional polytopes, we first apply Boole Scott’s method to the five regular polyhedra (Figure 1). Constructing the parallel two-dimensional sections of any polyhedron (i.e., the sections parallel to one of its faces) is quite elementary. To compute, for example, the sections of the cube, we intersect the plane containing a given face of the cube with the cube itself. This intersection is, of course, the face of the cube; that is, the parallel section is a square. Translating the plane towards the center of the cube, we see that all parallel sections are isometric squares. Similarly, parallel sections of the tetrahedron are decreasing triangles, triangles and hexagons for the octahedron, pentagons and decagons for the dodecahedron and triangles, hexagons and dodecagons for the icosahedron. Diagonal sections of a regular polyhedron P are sections H \ P, where H is a plane perpendicular to the segment OV joining the center of the polyhedron with a vertex. We can visualize a regular solid by unfolding it to a planar net. Roughly speaking, this means ‘‘cutting’’ certain edges of the polyhedron and mapping it to a two-dimensional space. The well-known net for the cube is shown in Figure 2. Note that to recover the three-dimensional cube from the unfolded version, one must identify certain edges. This allows us to describe the parallel sections of the cube in a very easy way. Namely, one parallel section could be one of the squares in Figure 2, for example, the middle square (call it MS). In order to obtain the other sections (which will be parallel to the square MS after folding the net) one just needs to move the four edges of MS in the unfolded cube parallel towards the remaining squares. In each case, one obtains a square isometric to the square MS (after necessary identification of end points of the edges).

Closed form solutions for transforming 3D Cartesian to geodetic coordinates reduce the problem to finding the real solutions of the fourth degree latitude equation or variations of it. By using symbolic tools (Sturm–Habicht coefficients and subresultants mainly) we study the methods (and polynomials) proposed by Vermeille and Borkowski to solve this problem. For Vermeille approach, the region where it cannot be applied is completely characterized. For those points it is shown how to transform 3D Cartesian to geodetic coordinates and a new method for solving Vermeille equation for those cases not yet covered is introduced. Concerning Borkowski’s approach, the symbolic analysis produces a complete characterization of the singular cases (i.e. where multiple roots appear).

In this paper we present the life and work of Alicia Boole Stott, an Irish woman who made a significant contribution to the study of four-dimensional geometry. Although she never studied mathematics, she taught herself to “see” the fourth dimension and developed a new method of visualizing four-dimensional polytopes. In particular, she constructed three-dimensional sections of these four-dimensional objects, which resulted in a series of Archimedean solids. The presence in the University of Groningen of an extensive collection of these three-dimensional models, together with related drawings, reveals a collaboration between Boole Stott and the Groningen professor of geometry, P.H. Schoute. This collaboration lasted more than 20 years and combined Schoute's analytical methods with Boole Stott's unusual ability to visualize the fourth dimension. After Schoute's death in 1913 Boole Stott was isolated from the mathematical community until about 1930, when she was introduced to the geometer H.S.M. Coxeter, with whom she collaborated until her death in 1940.

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by L. Schläfli in 1858. Using this, explicit examples of surfaces of every possible type are given.

New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a `modern' treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. A conceptual proof is presented of a result of Rohn concerning curves in $\mathbb{P}^1\times \mathbb{P}^1$ of bi-degree $(2,2)$. The string models of Series XIII (of some ruled quartic surfaces) are based on Rohn's result., 35 pages, including 3 figures

It is well known that every Del Pezzo surface of degree 5 defined over k is parametrizable over k. In this paper we give an efficient construction for parametrizing, as well as algorithms for constructing examples in every isomorphism class and for deciding equivalence., Comment: 15 pages

Extending a geometric construction due to Sederberg and to Bajaj, Holt, and Netravali, an algorithm is presented for parameterizing a nonsingular cubic surface by polynomials of degree three. The fact that such a parametrization exists is classical. The present algorithm is, by its purely geometric nature, a very natural one. Moreover, it contains a practical way of finding all lines in an implicitly given cubic surface. Two explicit examples are presented, namely the classical Clebsch diagonal surface and the cubic Fermat surface. (C) 2009 Elsevier B.V. All rights reserved.