Algebra Learning

As a response to both the implementation of the Common Core State Standards (CCSS) and a recent approval of a change by the New York State Board of Regents to allow multiple pathways for graduation, this curriculum project, which will be referred to as a unit plan throughout the paper, was designed to meet the need for more units of study that apply mathematical modeling in algebra to real world situations that allow students to explore applications of mathematics in careers. The unit plan on rearranging algebraic equations using electrical circuit applications is aligned to the New York State Common Core Learning Standards for mathematics (NYSCCLSM) and addresses mathematical modeling, mathematical practice standard 4. This unit plan, which may provide a method by which algebra and career & technical education (CTE) teachers can continue to work toward the Common Core State Standards Initiative (CCSSI) goal of preparing students for both college and career success, was validated by presenting a single lesson to a small group of students as a pilot study. Responses from the post-lesson student survey indicate that there was a positive change in their average attitude toward rearranging equations with more than one variable. More investigation is required to refine the lessons and prove that this entire unit plan is useful in a larger setting and to a wider student audience.

Images of Molecules are broadly used to foresee atom properties in educating and synthetic exploration. A prepared physicist can undoubtedly infer particle properties by dissecting its construction and assess its useful gatherings. To foresee, for instance, the water dissolvability of a natural compound a scientist would instinctively tally the number of polar gatherings, think about the size of the molecule, and gauge the water/atom communication by tallying the quantity of H straightforwardly available highlights ought to give sufficient data to anticipate the atom's design subordinate properties. To help this postulation, image learning is utilized and assessed in this work. At long last properties of new molecule produced utilizing deep learning were anticipated.

El libro “Álgebra escolar y pensamiento algebraico: aportes para el trabajo en el aula” de Rodolfo Vergel y Pedro Javier Rojas presenta una contribución relevante para la educación matemática y en particular en el área de la enseñanza y aprendizaje del álgebra escolar. Incluye cinco capítulos cuyos contenidos muestran cómo el trabajo de investigación realizado por los autores en sus tesis doctorales en el marco del Doctorado Interinstitucional en Educación -DIE de la Universidad Distrital Francisco José de Caldas, puede ser de gran ayuda para los profesores que enseñan álgebra.

De acuerdo con la investigación realizada en dichas tesis doctorales, los autores proponen una reflexión sobre el álgebra escolar en la que la mirada semiótica juega un papel fundamental ya que consideran que el aprendizaje de las matemáticas esencialmente es una actividad semiótica, por cuanto en toda actividad matemática se recurre a la transformación de signos dentro de sistemas semióticos culturalmente dados. Como referentes teóricos para esta mirada semiótica usan la Teoría de la objetivación y también algunos constructos del Enfoque ontosemiótico de la cognición e instrucción matemática.

El contenido del libro es diverso y aborda los aspectos fundamentales relacionados con lo que significa el proceso de instrucción del álgebra escolar. El material está dirigido a profesores de matemáticas en formación o en servicio ya que presenta herramientas para comprender y reflexionar sobre lo que significa el álgebra escolar en primaria y secundaria. Para ello, se parte primero de una reflexión sobre las relaciones entre la aritmética y el álgebra y se profundiza en la discusión acerca del carácter algebraico de la aritmética. Se continúa con la presentación, de manera concisa, de aspectos generales sobre el álgebra escolar y el currículo, para, a continuación, detallar la propuesta de cambio curricular conocida como Álgebra temprana.

Se sigue, en el capítulo cuatro, con unas tareas que los profesores pueden utilizar en su aula para enseñar el álgebra escolar y se reflexiona sobre ellas básicamente utilizando herramientas de la teoría de la objetivación. Se trata sobre todo de tareas relacionadas con secuencias de patrones y procesos de generalización y simbolización. Los autores consideran la tarea como situación enmarcada en un contexto — intramatemático o extramatemático— que, a partir de su abordaje por parte de un sujeto, pretende desarrollar en él pensamiento matemático. En mi opinión, este capítulo es un buen ejemplo de cómo la investigación en didáctica de las matemáticas puede ser útil para orientar la práctica del profesor en el aula ya que, cuanto mayor sea nuestra capacidad de descripción, comprensión y explicación de los procesos de enseñanza y aprendizaje del álgebra, estaremos en mejores condiciones para conseguir una mejora de su enseñanza.

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Este libro virtual es un borrador, todavía no está publicado y necesita una revisión final para modificaciones necesarias, encontrarán algunos errores por ese motivo. Pero pese a lo anterior sirve para nuestro trabajo y por ello lo ubicamos como libro base.

When designers set out to create a mathematics learning activity, they have a fair sense of its objectives: students will understand a concept and master relevant procedural skills. In reform-oriented activities, students first engage in concrete situations, wherein they achieve situated, intermediary learning objectives (SILOs), and only then they rearticulate their solutions formally. We define SILOs as heuristics learners devise to accommodate contingencies in an evolving problem space, e.g., monitoring and repairing manipulable structures so that they model with fidelity a source situation. Students achieve SILOs through problem-solving with media, instructors orient toward SILOs via discursive solicitation, and designers articulate SILOs via analyzing implementation data. We describe the emergence of three SILOs in developing the activity Giant Steps for Algebra. Whereas the notion of SILOs emerged spontaneously as a framework to organize a system of practice, i.e. our collaborative design, it aligns with phenomenological theory of knowledge as instrumented action.

Teachers use their content and pedagogical content knowledge for teaching algebra. For this reason, the examination of how teachers use this knowledge may help shed light on how students learn algebra, especially in determining why they usually have difficulties. The aim of the current study is to reveal what teachers know, and propose what they actually need to know for teaching the simplification and equivalence of algebraic expressions. The multiple-case study design was used for this study to compare and contrast the two middle school teachers" lesson planning and instruction. The data corpus included lesson plans, actual instruction records, and post-observation interviews. Data analysis was conducted using the Mathematical Knowledge for Teaching (MKT) model. The findings indicated that both teachers had a lack of specialized content knowledge about mathematical representations such as algebra tiles. They did not use algebra tiles effectively and could not link algebraic and geometric representations that underlie the idea of multiplication. It was observed that both teachers generally used unknowns and variables interchangeably indicating the inadequacy of their common content knowledge. In the planning process, the two teachers were able to state the common misconceptions that the students generally had and the ways of addressing them. Through the cases of these two teachers, it was observed that teachers need to have a good conceptual mathematical understanding and also knowledge of students" thinking in order to design effective lessons. Based on the findings, the types of knowledge that the teachers need to have are outlined and the theoretical and practical implications of the study are discussed.

Una tradición curricular muy bien establecida nos ha llevado a pensar que la
enseñanza del álgebra es un asunto de la escuela secundaria. La idea detrás
de esta tradición es que, para entender álgebra, es necesario tener una base
aritmética relativamente sólida. De hecho, muchos —y entre ellos incluimos
no solamente a matemáticos y educadores matemáticos, sino también a filó-
sofos, científicos y sicólogos del desarrollo, como Kant, Newton y Vygotsky,
respectivamente— han considerado el álgebra como una extensión o generalización
de la aritmética. ¿Cómo, dentro de esta línea de pensamiento, podría
pensarse en enseñar álgebra si el estudiante no ha desarrollado suficientemente
su base aritmética?

The Lambert W function is an emerging function, and can be regarded as  im'ortant elementary transcendental function like continued fraction etc.  It has  surpringly rich mathematical structure. Unfortunately in spite of its  aplications in many areas, it does not find as a topic in mathematics  or mathematical  physics, etc.  Lambert W function is  related to infinite tetration (snoring function). Programs in Anaconda Spyder with IPython console are included with interesting properties and detailed calculations are  given to get a feel for this function.

Este curso tiene como propósito introducir a las(os) estudiantes a la disciplina del álgebra desde una perspectiva más abstracta, en la que los objetivos están mayormente dirigidos hacia el estudio de las diferentes estructuras algebraicas, que a la simple resolución de ecuaciones particulares (como es frecuente en los sistemas de enseñanza de bachillerato). Adicionalmente, el curso tiene como objetivo ser un curso auxiliar en el sentido de proporcionar a las(os) estudiantes herramientas teóricas que faciliten su desempeño en otras materias del mapa curricular. Por ejemplo, los principios de la teoría de conjuntos propiciarán la familiaridad con el lenguaje predominante en el que se realizan una gran parte de las demostraciones con las que se enfrentarán a lo largo de su carrera; la presentación de espacios vectoriales, matrices, determinantes y sistemas de ecuaciones generales facilitará el entendimiento de temas posteriores, no sólo en los cursos de álgebra lineal sino también en los de geometría analítica. Por mencionar un último ejemplo, el estudio de las estructuras algebraicas de los números reales y de los números complejos (correspondientes a Álgebra Superior II) sustentarán una gran parte de los resultados que serán presentados en los cursos de cálculo, análisis y variable compleja. Todos los temas del curso se presentan de manera formal pero introductoria, de tal modo que esto le posibilite a la o el estudiante adentrarse posteriormente en las diferentes ramas de la matemática.

Este trabajo tiene como objetivo general analizar, clasificar e interpretar los errores algebraicos que presenta un grupo de 38 estudiantes matriculados en el curso Matemática General de la Universidad Nacional en Costa Rica, que se oferta a diversas carreras. En este estudio, que constituye una réplica de investigación, empleamos un cuestionario centrado en el uso de las letras con diferentes significados, desde etiquetas a variables. Se desarrolló desde el enfoque cuantitativo durante el periodo 2018-2019. Como resultado del análisis se ha detectado que los errores más frecuentes en las respuestas de los ítems responden al uso de la letra como incógnita de valor específico, letra como número generalizado o en el manejo de la letra como variable.

The knowledge of the processes in the history of mathematics is very useful for a greater and complete comprehension of the foundations and nature of mathematics. The al-gebraization of mathematics was a key process in the transformation of the seventeenth century mathematics if we consider two of its relevant features: a new symbolic language and an analytic method for solving problems. The aim of this article, within the framework of the algebraization of mathematics, is to analyze the results of the implementation of a historical activity in a mathematics history course for the bachelor's degree in mathematics, which includes singular geometric constructions for solving equations. Furthermore, the article shows that students gain a better understanding of the role of the relationship between algebra and geometry in the development of mathematics, which helps them to improve their mathematical training.

The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting the postsecondary success of students majoring in STEM fields. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. The present study investigated the connections participants formed between algebraic procedures and basic number properties in the context of rational expressions.

An assessment, given to 107 undergraduate students in precalculus, contained three pairs of closely matched algebraic and numeric rational expressions with the operations of addition, subtraction, and division. The researcher quantitatively analyzed the distribution of scores in the numeric and algebraic context. Qualitative methods were used to analyze the strategies and errors that occurred in the participants‟ written work. Finally, task-based interviews were conducted with eight participants to reveal their mathematical thinking related to numeric and algebraic rational expressions.

Statistical analysis using McNemar‟s test indicated that the undergraduate participants' abilities related to algebraic rational expressions and rational numbers were significantly different, although serious deficiencies were noted in both cases. A small intercorrelation was found in only one of the three pairs of problems, suggesting that the participants had not formed connections between algebraic procedures and basic number properties. The analysis of the participants' written work revealed that the percent of participants who consistently applied the same procedure in the numeric and algebraic items of Problem Sets A, B, and C were 56%, 47%, and 37%, respectively. Correct strategies led to fewer correct solutions in the algebraic context because of a diverse collection of errors. These errors exposed a lack of understanding for the distributive and multiplicative identity properties, as well as the mathematical ideas of equivalence and combining monomials. These fundamental mathematical ideas need to be better developed in primary and secondary education. At the post-secondary level, these ideas should serve as the foundation for interventions that are designed to support underprepared students. The results of the interviews were consistent with the quantitative analyses and the qualitative examination of the strategies used by the participants. The findings in all three areas of the study point to a disconnect between numeric and algebraic contexts in the participants‟ thinking.

This paper gives an account of a teaching experiment on absolute value inequalities, whose aim was to identify characteristics of an approach that would realize the potential of the topic to develop theoretical thinking in students enrolled in prerequisite mathematics courses at a large, urban North American university. The potential is demonstrated in an epistemological analysis of the topic. It is also shown that this potential is not realized in the way the topic is presently taught in prerequisite mathematics courses. Three groups of students enrolled in such courses were each exposed to one of three approaches we conceived for teaching the topic, labeled the procedural (PA), the theoretical (TA), and the visual (VA) approaches. The design of the three lectures was constrained by institutional characteristics of college-level courses, and informed by epistemological and didactical analyses of the topic. It was found that following the VA lecture, which proposed two equally valid mathematical techniques (graphical and analytic), one of which could be used to test the validity of results obtained by the other, students were more likely to engage in some aspects of theoretical thinking. They displayed more reflective and systemic thinking than other groups, and dealt more effectively with the logical intricacies of absolute value inequalities. VA students appeared to have a synthetic grasp of the inequalities, and a flexibility of thought not displayed by PA and TA students. However, without sufficient attention to tasks not easily solved by graphical means, VA approach provided students with a way to avoid the challenges of systemic and analytic thinking, some of which were more apparent in TA students. PA students expectedly behaved more as procedural knowers, but we saw interesting examples of engagement with theoretical thinking while dealing with the procedures proposed in the PA lecture.

En este estudio se analizan las tareas de álgebra temprana en una colección de ocho libros de texto de Educación Primaria chilenos, de amplia difusión. Para la investigación se ha seguido una metodología cualitativa, de carácter descriptivo, a partir de la técnica de análisis de contenido. Los resultados obtenidos muestran una variedad de tareas matemáticas con una distribución no homogénea, predominando las tareas de relaciones basadas en reglas, más específicamente, tareas que involucran la comprensión de patrones, seguidas de tareas de relaciones conocidas-desconocidas y una presencia menor de tareas de relaciones aritméticas situadas. Se concluye que es necesario brindar experiencias de formación al profesorado que les permita, por una parte, profundizar en el análisis de tareas matemáticas que promuevan el desarrollo del pensamiento algebraico temprano para alcanzar una adecuada selección e implementación de dichas tareas y, por otra parte, indagar en la actividad matemática que deben desarrollar los estudiantes para su resolución.

Research findings have established that students often struggle with mathematical symbols including common misconceptions for literal symbolic representations of variables but provide little evidence of when or how these misconceptions arise. This article reports findings from a study of grade 4-6 students' conception(s) for various representations of unknown addends [variables] commonly found in US elementary mathematics textbooks. Specifically, thirty-six US grade 4-6 students participated in two semi-structured task based interviews designed to explore their conception(s) of formal and informal representations of unknown addends as revealed by the types of numbers substituted across core mathematical tasks and task types. Findings showed that participants initially demonstrated a natural number bias and upon further questioning a whole number bias. Participants' number substitutions did not include common established difficulties for literal symbols exhibited by students in algebra courses and higher-level mathematics courses. In addition, even when providing the same types of number substitutions, participants attended to various attributes of the different representations for unknown addends.

This study explored the effects of sequencing students' engagement in mathematical tasks on their level of procedural fluency. The teacher randomly selected two intact classes from a stream of Grade 7 classes. These two classes were randomly assigned as the experimental group and the control group. The teacher employed the conceptual-to-procedural approach in teaching the experimental group while the procedural-to-conceptual approach in teaching the control group. Quantitative quasi-experimental pretest-posttest non-equivalent control group design was employed. Quantitative method was used to examine the students' level of procedural fluency on integer operation. Results showed that conceptual-to-procedural instruction influenced the Grade 7 students' procedural fluency on integer operation more than the procedural-to-conceptual instruction. Thus, the researchers recommend that mathematics teachers may practice the conceptual-to-procedural sequence in teaching mathematics to strengthen the foundation of mathematical concepts, and in turn, develop procedural fluency. Future studies on the effectiveness of these sequences of mathematical tasks are needed to support these research findings.