Algebraic Thinking

Several components shows that the process of algebraic thinking on students in solving problems. One of these components is the process of using algebra as a tool for doing mathematical modelling. This study aims to reveal how the students' competency in making mathematical modelling for solving an algebraic problem. The descriptive qualitative study design was used to obtain an overview of the mathematical modelling competencies of nine students (15-16 years old) in Pekanbaru, Riau Province. Each participant received a test that contains an algebraic problem. The results of this study show that when students do mathematical modelling there are several errors as follows: (1) failure in making suppositions for the problem and simplifying the given situation; (2) failure in recognizing quantities that influence the given situation, to identify the key variable; (3) failure in construct relation between variable; and (4) failure in solving question within mathematical modelling.

We are engaged in a project named Mathematics and patterns in elementary schools: perspectives and
classroom experiences of students and teachers. Our aim is to analyze the impact of an intervention
centered on the study of patterns in the learning of mathematics concepts and on the development of
communication and development of higher order thinking skills. In this paper we present part of an
ongoing research with pre-service teachers concerning the development of teachers’ algebraic thinking,
in particular how they move through pattern tasks involving generalization. We will present some of the
tasks used in the didactical experience and some preliminary conclusions of its implementation in the
mathematics didactics classes of a mathematics elementary teachers’ course of a School of Education.

In this month’s problem, students are presented with a growing geometric pattern for table configurations at a birthday party. The students must verbally describe how to extend the pattern and use their verbal description to generate an expression for the number of tables for different configurations. Each month, elementary school teachers receive a problem along with suggested instructional notes and are asked to use the problem in their own classrooms and submit solutions, strategies, and reflections to the journal.

Özet Bu çalışmanın amacı; 8. sınıf öğrencilerinin genellemeleri formüle etme, sembolleri ve cebirsel ilişkileri kullanma ve çoklu gösterimlerden yararlanma şeklinde sıralanan cebirsel düşünme becerilerini SOLO Taksonomisi ile incelemektir. Nitel araştırma yönteminin kullanıldığı bu çalışma 2011-2012 Eğitim-Öğretim yılı ikinci döneminde Bursa ili İnegöl ilçesindeki ilköğretim okullarından birinde 15 tane 8. sınıf öğrencisi ile gerçekleştirilmiştir. Katılımcılar 8. sınıf öğrencileri arasından maksimum çeşitlilik örneklemesine uygun olacak şekilde seçilmiştir. Çalışmada uzman görüşler doğrultusunda araştırmacı tarafından hazırlanan sekiz problem kullanılmış ve öğrencilerle bu problemler üzerinde klinik görüşmeler yürütülmüştür. Yazıya dökülen klinik görüşme, video kayıtları ve araştırmacı notları çalışmanın veri toplama araçlarını oluşturmaktadır. Toplanan veriler betimsel yöntemle analiz edilmiştir. Veri analizinde Miles ve Huberman (1994) tarafından tanımlanan çift-kodlama yöntemi kullanılmıştır. Araştırma sonucunda öğrencilerin çoğunluğunun SOLO Taksonomisine göre ilişkilendirilmiş yapı (İY) seviyesinin altında yer aldığı görülmüştür. Öğrencilerin en çok zorlandıkları beceri sembolleri ve cebirsel ilişkileri kullanma becerisi olmuştur. Öğrencilerin akademik başarılarına göre yapılan analizde ders notu yüksek öğrencilerin cebirsel düşünme becerilerinin diğer öğrencilere göre daha yüksek olduğu görülmüştür. Anahtar Kelimeler: Cebirsel Düşünme, SOLO Taksonomisi, İlköğretim Matematik Eğitimi Abstract The purpose of this study was to investigate 8th grade students' algebraic thinking skills listed in the form of formulizing generalizations, using symbols and algebraic relations and utilizing multiple representations with SOLO Taxonomy. Following the qualitative research methods, this study was carried out during the spring semester of 2011-2012 academic year on 15 students at eighth grade whom are chosen from a school in İnegöl the town of Bursa. Participants were chosen from 8th grade students according to maximum diversity sampling. 8 problems were prepared for data

The introduction of algebraic reasoning in primary education is a subject of interest for research and curricular innovation in mathematics education, which supposes an extended vision of the nature of school algebra. In this paper we propose a way to conceive of algebraic reasoning based on the types of mathematical objects and processes introduced in the onto-semiotic approach to mathematical knowledge. In particular, considering a mathematical practice as algebraic is based on the intervention of generalization and symbolization processes, along with other objects usually considered as algebraic, such as binary relations, operations, functions and structures. This way to conceive of elementary algebra is based on and compared to the characterizations given by other authors. We also propose a typology of algebraic configurations that allows defining degrees of algebrization of mathematical activity.

Aunque son diversas las investigaciones y propuestas curriculares que sugieren la introducción de formas de pensamiento algebraico en el nivel de Educación Primaria, las fronteras entre lo que se puede considerar claramente algebraico y lo que no es algebraico son todavía difusas. En este sentido y con el fin de facilitar el reconocimiento de los rasgos característicos del razonamiento algebraico en los niveles elementales se incluyeron, en este trabajo, ejemplos de actividades
matemáticas en las que se distingue el carácter aritmético del algebraico a través de un análisis basado en los niveles de algebrización. La propuesta teórica de los niveles de algebrización permite identificar una actividad matemática que no incorpora rasgos algebraicos, de otra que se puede considerar como propiamente algebraica.

RESUMEN En la primera parte de este artículo se realiza una revisión de las principales causas que han dado lugar a la vinculación exclusiva del álgebra con el lenguaje simbólico y los niveles medios y superiores de escolarización: se argumenta que la nomenclatura tradicional usada para denominar los conocimientos de naturaleza algebraica (lógica, lógica matemática o razonamiento lógico-matemático), junto con la aparición de la matemática moderna, ha provocado la ausencia de un bloque de contenidos de álgebra en los primeros niveles educativos. En la segunda parte se describe el cambio de tendencia que se ha producido en países como Estados Unidos, Singapur, Australia o Nueva Zelanda y se justifica la presencia del álgebra temprana en los currículos de Educación Infantil. Desde este prisma, se presenta una propuesta de secuenciación de contenidos de álgebra temprana de 3 a 6 años y se describe un itinerario didáctico de enseñanza para favorecer el pensamiento algebraico de los niños y el desarrollo profesional del profesorado de Educación Infantil. Palabras clave: álgebra temprana; pensamiento algebraico; secuenciación de contenidos; itinerario de enseñanza; Educación Infantil. ABSTRACT Firstly, the main causes that explain the exclusive connection of algebra with symbolic language and middle and superior degrees are reviewed: traditional nomenclature used to refer to algebraic knowledge (logic, mathematical logic or logical-mathematical reasoning), together with the presence of modern mathematics, has caused the absence of a standard of algebra in the first educational levels. In the second part, the change of trend in the curricula of countries such as the United States, Singapore, Australia or New Zealand is described and the explicit presence of early algebra in the Early Childhood curriculum is defended. From this point of view, a proposal of the sequencing of early algebra from 3 to 6 years is presented and a teaching itinerary to empower both children's algebraic thinking and to promote the professional development of Early Childhood Education' teachers is described.

Basic algebraic notions are introduced which can be used to describe and to transform program-data complexes for traditional languages, restricted computation, non-numeric computation and modeling. General algebraic program systems (GAPS)  are free of assumptions  what computing  (programming) systems have as their particular constructs.

First of all we argue why current technique is sometimes not adequate.  Then we show how to interpret notions of abstract algebra (groupoid structure) as super-von Neumann computational structure and how to express  many useful structures and notions in an algebra. Then GAPS are introduced and basic mathematical results are stated including the precise criterion when  a given system of actions over programs can be added to given programming language. Various examples of GAPS are given. And at last we show possible primitives of `structured algebraic modeling' (programming)

In this study, it was aimed to determine the views of graduate students on mathematics learning with dynamic mathematics software GeoGebra. Case study method was used in qualitative research patterns. The study group consists of 7 graduate students who were doing their degrees at the Education Faculty of a state university in Turkey. Data were collected with observations, models on the GeoGebra screen made by the students, dynamic worksheets and a semi-structured questionnaire consisting of open and closed ended questions developed by the researcher. The findings show that

En el presente escrito, exponemos los avances del trabajo que estamos realizando, el cual trata acerca de la equivalencia de expresiones algebraicas, en la situación concreta de resolución de secuencias numéricas, que pueden ser estudiadas a través de la visualización. Presentamos y analizamos los resultados de una actividad propuesta a 30 estudiantes de tercer grado de secundaria, que nos permite suponer la posibilidad de dotar de significado a la equivalencia de expresiones algebraicas a partir del análisis visual y nos permite plantearnos la pregunta ¿cómo el análisis visual influye en la construcción de significado de la equivalencia de distintas expresiones algebraicas que surgen durante la resolución de secuencias numéricas visuales? Como resultado preliminar podemos mencionar que a partir de éste tipo de análisis los estudiantes son capaces de construir distintas expresiones algebraicas para una secuencia numérica; pero la equivalencia de dichas expresiones algebraicas no se discute.
Trabajo presentado en el III Coloquio de Doctorado del DME Cinvestav (2015).

– Algebra is generally considered as manipulating symbols, while algebraic thin king is about generalization. Patterns can be used for generalizat ion to develop early graders' algebraic thinking. In the generalization of pattern context, the purpose of this study is to investigate middle school students' reasoning and strategies at different grades when their algebraic thin king begin s to develop. First, 6 open-ended linear growth pattern problems as numeric, pictorial, and tabular representations were asked to 154 middle g rade students. Next, two students from each grade (6 th , 7 th , and 8 th grade) were interviewed to investigate how they interpret the relationship in different represented patterns, and which strategies they use. The findings of this study showed that students tended to use algebraic symbolis m as their grade level was increased. However, the students' conceptions about 'variable' we re troublesome.

The efficient use of generalization in textbooks, which are regarded as a supplementary teaching material and constitute a substantial place in the processes of learning and teaching, is important for effective mathematics learning and teaching, as well as for developing mathematical and algebraic thinking. This research seeks to understand how and to what extent middle school mathematics textbooks include both components of generalization in terms of developing algebraic thinking. This study uses the document analysis approach, a qualitative research method, for collecting data. The tasks and exercises in the textbooks have been analyzed using a coding matrix based on the two basic dimensions of the research' s analytical framework. As a result of the research, the tasks and exercises in the textbooks are seen to sufficiently support the component of Generalizing Arithmetic and Quantitative Reasoning, while partially ignoring the component of Patterns and Functional Relationships/Variables. Moreover, the tasks and exercises that are expected to allow students to explore and express generalizations are seen to have been mostly ignored. In addition, generalizations are observed to have been generally associated with everyday life, in parallel with the curriculum.

Resumen Aunque son diversas las investigaciones y propuestas curriculares que sugieren la introducción de formas de pensamiento algebraico en el nivel de Educación Primaria, las fronteras entre lo que se puede considerar claramente algebraico y lo que no es algebraico son todavía difusas. En este sentido y con el fin de facilitar el reconocimiento de los rasgos característicos del razonamiento algebraico en los niveles elementales se incluyeron, en este trabajo, ejemplos de actividades matemáticas en las que se distingue el carácter aritmético del algebraico a través de un análisis basado en los niveles de algebrización. La propuesta teórica de los niveles de algebrización permite identificar una actividad matemática que no incorpora rasgos algebraicos, de otra que se puede considerar como propiamente algebraica. Palabras clave: pensamiento aritmético, pensamiento algebraico, niveles de algebrización, actividades matemáticas.

espanolEl objetivo de este articulo es presentar las primeras orientaciones didacticas para desarrollar el razonamiento algebraico en Educacion Infantil a traves del pensamiento computacional, usando la robotica como recurso. A partir de los vinculos entre estos aspectos y el analisis de una experiencia con robots educativos programables para trabajar los patrones en 3-4 anos, se establecen cinco recomendaciones iniciales en el marco de la educacion STEAM: 1) plantear fenomenos relevantes, basados en la resolucion de problemas; 2) fomentar procesos de razonamiento mediante buenas preguntas; 3) impulsar la interaccion, la negociacion y el dialogo; 4) vincular conocimientos de distinta naturaleza; 5) plantear la representacion como medio para comprender, estructurar, capturar y transferir conceptos. EnglishThe aim of this article is to present some first didactic orientations to develop the algebraic reasoning in Pre-school Education through computational thinking, using robotics as a...

Teachers’ understanding of elementary mathematics is an important factor contributing to students’ success with mathematics. However, how can this knowledge be nurtured within the time limitations of an 18-week Bachelor of Education program? This paper addresses this question by examining changes in prospective elementary teachers’ knowledge of rational numbers after completing a 9-week technology-enhanced elective course and viewing supplemental web-based video clips. The paper describes the course design and video clip format, offers empirical evidence including the impact of moderator variables and proposes implications for teaching practice.

This paper reports on a design experiment conducted to explore the construction of meanings by 17-year-old students, emerging from their interpretations and uses of algebraic-like formalism. The students worked collaboratively in groups of two or three, using MoPiX, a constructionist computational environment with which they could create concrete entities in the form of Newtonian models by using equations and animate them to link the equations’ formalism to its visual representation. Some illustrative examples of two groups of students’ work indicate the potential of the activities and tools for expressing and reflecting on the mathematical nature of the available formalism. We particularly focused on the students’ engagement in reification processes, i.e. making sense of structural aspects of equations, involved in conceptualising them as objects that underlie the behaviour of the respective models.

We are engaged in a project named Mathematics and patterns in elementary schools: perspectives and classroom experiences of students and teachers. Our aim is to analyze the impact of an intervention centered on the study of patterns in the learning of mathematics concepts and on the development of communication and development of higher order thinking skills. In this paper we present part of an ongoing research with pre-service teachers concerning the development of teachers’ algebraic thinking, in particular how they move through pattern tasks involving generalization. We will present some of the tasks used in the didactical experience and some preliminary conclusions of its implementation in the mathematics didactics classes of a mathematics elementary teachers’ course of a School of Education.

Este trabalho foi desenvolvido no seio de um projeto de iniciação científica, no curso de Licenciatura em Matemática da Universidade Federal Rural de Pernambuco. O objetivo desta pesquisa é identificar organizações matemáticas, em torno do tema Relações de Igualdade e Princípios de Equivalência, no uso do recurso digital "Explorador Básico de Igualdade"-disponível na plataforma Physics Education Technology-PhET Interactive Simulations. Para tanto, tomamos como suporte teórico a Teoria Antropológica do Didático (TAD), com ênfase nas noções de praxeologia matemática, ostensivos e não ostensivos. A abordagem da pesquisa é qualitativa e do tipo documental. A coleta de dados ocorreu a partir de informações gerais disponíveis na PhET, tais como: descrições do recurso e manual de instrução; além do manuseio do "Explorador Básico de Igualdade", para o levantamento dos tipos de tarefas e suas respectivas técnicas e tecnologias. Dentre os resultados da pesquisa, destacamos três ostensivos disponíveis no recurso digital. Apresentamos ainda, no bloco prático-técnico, seis tipos de tarefas e seis técnicas; bem como, cinco tecnologias que as justificam. Por fim, reconhecemos a TAD como uma possível abordagem teórica-metodológica para a análise de recursos digitais.

Function plays a central role in middle years mathematics, including modeling with tables, graphs, equations, and contexts and linking these representations to explore patterns of change. In this article, I describe a strategy for generating a table that makes the link between the context, table, and equation more explicit for students.

Keywords The purpose of this study, to determine problem-posing skills and algebraic thinking levels and whether there is a relationship between problem-posing skills and algebraic thinking levels. In this research, correlation was used from quantitative research methods. In the fall semester of the 2016-2017 academic year, 308 students (151 girls, 157 boys) attending 7th and 8th grade in three secondary schools in the province of Konya were participated in the research. In the research, to determine the algebraic thinking levels of students, developed by Hart et al. (1998) and "Algebraic Thinking Level Test" adapted to Turkish by Altun (2005) was used. In addition, to measure the problem-posing skills of the students, Problem Posing Test was used as a data collection tool. The Spearman's Rank Correlation coefficient technique was used to determine the relationship between problem-posing skills and algebraic thinking levels. As a result of the research, while there is an accumulated at level 0 and level 1 among 7th grade students, it is accumulated at level 2 and level 4 among 8th grade students. In addition, it was determined that there is a strong correlation between students' algebraic thinking level and problem-posing skills in the positive direction. Math education Problem posing Algebraic thinking level

Uma Equação Diofantina (ED) _e uma equação algébrica de uma ou mais variáveis para a qual se inquerem soluções inteiras. EDs do tipo ax + by = c com a; b; c 2 Z, são as Equações Diofantinas (EDs) mais estudadas em livros-texto de Aritmética. Para obtenção de soluções (x; y) nos inteiros de EDs deste tipo, são importantes os conceitos de divisibilidade, Algoritmo de Euclides e Máximo Divisor Comum (MDC). A estratégia comumente empregada e que compõe a maioria dos livros-texto sobre EDs consiste da aplicação sucessiva do Algoritmo de Euclides e escrita do mdc(a; b) como uma combinação linear de a e b, fornecendo assim uma solução nos inteiros para a equação ax + by = mdc (a; b). Neste texto apresentamos e discutimos alguns resultados preliminares para o estudo de EDs e detalhamos um desencadeamento operatório para calcular a solução particular de ax + by = mdc(a; b), denominado Abordagem por Substituição Progressiva (ASP).

This article explores howdifferences in problem representations change both the performance
and underlying cognitive processes of beginning algebra students engaged
in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in
mathematics education, students were more successful solving simple algebra story
problems than solving mathematically equivalent equations. Contrary to some views
of situated cognition, this result is not simply a consequence of situated world knowledge
facilitating problem-solving performance, but rather a consequence of student
difficulties with comprehending the formal symbolic representation of quantitative
relations. We draw on analyses of students’ strategies and errors as the basis for a
cognitive process explanation of when, why, and how differences in problem representation
affect problem solving. We conclude that differences in external representations
can affect performance and learning when one representation is easier to comprehend
than another or when one representation elicits more reliable and
meaningful solution strategies than another.

In this month’s problem scenario, students are presented with a problem-solving task using the Hundreds Chart. They must determine the relationship between two numbers on the Hundreds Chart. Each month, elementary school teachers receive a problem along with suggested instructional notes and are asked to use the problem in their own classrooms and submit solutions, strategies, and reflections to the journal.

Research findings have established common student misconceptions for literal symbolic representations of variables but lack corresponding findings of when or how these misconceptions arise. This article reports findings from an exploratory study of U.S. grade 4?6 students? conception(s) for various representations of unknown addends commonly found in U.S. elementary mathematics textbooks. Thirty-six U.S. grade 4?6 students participated in two semistructured task-based interviews designed to explore their conception(s) of conventional and nonconventional representations of unknown addends as revealed by their number substitutions across task types and core mathematical tasks. Results showed that participants initially demonstrated a bias toward positive integers, upon further questioning a bias toward non-negative integers, potential conflict factors related to rational-number substitutions, and did not demonstrate common difficulties with literal symbols exhibited by students in algebra and higher-level mathematics courses.

ABSTRACT Function plays a central role in middle years mathematics, including modeling with tables, graphs, equations, and contexts and linking these representations to explore patterns of change. In this article, I describe a strategy for generating a table that makes the link between the context, table, and equation more explicit for students.