Algebra Lessons Research Articles

Development of prospective elementary teachers’ knowledge to teach early algebra through case discussions

AbstractThis study investigated the development of prospective elementary teachers (PETs)’ subject matter and pedagogical content knowledge for teaching early algebra through participation in case discussions. PETs attended five weeks of intervention as part of a method course, which included case discussions. During the intervention, the participants were presented with big ideas of equivalence and equations, generalized arithmetic, and functional thinking as the content of early algebra through text-based classroom cases. Data were collected through one-hour individual interviews before and after the early algebra lessons. The analysis of the pre-interviews indicated that prospective elementary teachers may not be ready to foster algebraic thinking in elementary grades in terms of the required knowledge for teaching. We found that PETs needed more development of required subject matter knowledge to guide algebraic thinking in elementary grades, such as the relational meaning of the equal sign, generalizing, representing, or justifying arithmetic or functional relationships, and reasoning with them. Likewise, PETs were found to lack in pedagogical content knowledge related to students’ conceptions and misconceptions and effective teaching strategies to promote early algebraic thinking. However, we found that specifically designed method courses could help develop PETs’ knowledge of teaching early algebra. Following early algebra lessons centered around case discussions, PETs showed progress in multiple aspects of knowledge across all big ideas, though their improvement in generalized arithmetic was somewhat less pronounced. This progress was observed in their subject matter and pedagogical content knowledge.

Teaching Algebra to High School Students Using Desmos Graphing Calculator

Many high school students find algebra difficult because it requires comprehending abstract concepts, performing symbol and sign manipulation, and solving multi-step problems. Graphing calculators can visualize algebraic expressions and equations, and make learning more interactive. We conducted an experiment in which Desmos online graphing calculator was used to augment the algebra lessons of tenth-standard students. The students were taught about polynomials, linear and quadratic equations, and arithmetic progressions during an intervention period lasting seven weeks. We found that properly integrating the graphing calculator with classroom sessions and homework helped the students to perform better in the post-test and also retain the concepts. The students found the graphing calculator to be easy to use and helpful in learning, which increased their interest in algebra and encouraged them to practice algebra regularly.

Learning to Learn: How to Continuously Teach Humans and Machines.

Curriculum design is a fundamental component of education. For example, when we learn mathematics at school, we build upon our knowledge of addition to learn multiplication. These and other concepts must be mastered before our first algebra lesson, which also reinforces our addition and multiplication skills. Designing a curriculum for teaching either a human or a machine shares the underlying goal of maximizing knowledge transfer from earlier to later tasks, while also minimizing forgetting of learned tasks. Prior research on curriculum design for image classification focuses on the ordering of training examples during a single offline task. Here, we investigate the effect of the order in which multiple distinct tasks are learned in a sequence. We focus on the online class-incremental continual learning setting, where algorithms or humans must learn image classes one at a time during a single pass through a dataset. We find that curriculum consistently influences learning outcomes for humans and for multiple continual machine learning algorithms across several benchmark datasets. We introduce a novel-object recognition dataset for human curriculum learning experiments and observe that curricula that are effective for humans are highly correlated with those that are effective for machines. As an initial step towards automated curriculum design for online class-incremental learning, we propose a novel algorithm, dubbed Curriculum Designer (CD), that designs and ranks curricula based on inter-class feature similarities. We find significant overlap between curricula that are empirically highly effective and those that are highly ranked by our CD. Our study establishes a framework for further research on teaching humans and machines to learn continuously using optimized curricula. Our code and data are available through this link.

Relation between understandings of linear algebra concepts in the embodied world and in the symbolic world

For the use of embodied notions in teaching linear algebra, some studies indicate that it is helpful, whereas other studies indicate that it could be problematic or become an obstacle. Hence, additional research is needed. This study is focused on linear (in)dependence and basis, and investigates the relation between their understandings in the embodied and symbolic worlds. We also examine whether students' conceptions in the embodied world can be improved by the instruction emphasizing geometric images, as our previous studies identified some limitations of students' understanding in the embodied world. To address these issues, we designed four tasks aiming to assess students' conceptions of linear (in)dependence, basis, and dimension, and also designed linear algebra lessons emphasizing geometric images of these concepts. These tasks were conducted during the lessons and the data of 38 engineering students was collected. The analysis for the data showed that conceptions in the embodied world was positively associated with conceptions in the symbolic world; however, students' conceptions in the embodied world were not sufficiently improved by the geometric instruction implemented in this study.

VISUALIZATION OF ALGEBRARY IDENTIFICATIONS IN RXRR2 (PLANE) AT SECONDARY LEVEL

It is known that mathematics has an important effect on the development of science and technology. Identities, which are widely used in many branches of science, form the basis of some mathematics courses such as algebra. However, it appears as one of the learning areas of secondary school 7th grade mathematics and is seen as one of the difficult subjects to learn. The fact that there are many and abstract formulas about identities is difficult for students to remember. Therefore, visualization is important in teaching abstract concepts in a meaningful way and in associating concepts with daily life. Visualization is the use of mathematical and geometric symbols, shapes, diagrams, etc. to make something unseen imaginable. It is the drawing or showing of the shape and picture of an abstract concept with the help of pencil and computer. Visualization and visual proof are central to mathematics. The visualization approach is used in mathematics and mathematics education as a tool, not as a goal in the lessons. Identities are seen as one of the learning areas that we encounter in the solution of problems that we encounter in our daily lives and that are difficult to learn. In this direction, the visual proofs of the identities in R2 are shown in detail to the researchers thanks to the visualization approach, so that the algebraic identities can be learned more easily. Although identities are an important learning area in mathematics and algebra lessons, they are thought to be difficult to understand and learn by students. In line with this idea, students have difficulty in understanding identities as the visual models from which identities are obtained are not adequately expressed by the teachers and are not associated with daily life. For this reason, it is suggested by mathematics teachers that as a solution to this problem, visualizing abstract mathematical concepts in lessons or showing algebraic and visual proofs of some theorems may be beneficial. In this study, which was planned as a theoretical study, Within the scope of descriptive scanning model, document analysis technique was used. Visual proofs of basic algebraic identities are shown to researchers in detail by using geometric figures. In line with this purpose, some suggestions have been made to shed light on this issue for interested researchers.

METHODS OF TEACHING THE SUBJECT OF QUADRATIC EQUATIONS IN 8TH CLASS ALGEBRA LESSONS

METHODS OF TEACHING THE SUBJECT OF QUADRATIC EQUATIONS IN 8TH CLASS ALGEBRA LESSONS

Supporting Teacher Noticing of Moments of Algebraic Potential

A growing body of research in early algebra has illustrated that young children have the capacity to reason algebraically and has recommended introducing algebraic ways of thinking in early-elementary grades. Over the past two decades, there have been great strides in exploring children’s capacity for algebraic thinking and this work has opened the door to designing early algebra curricula and classroom experiences. What is less explored, however, are ways to support teachers in implementing early algebra lessons and supporting students’ thinking about early algebra concepts. In this paper, we explore early algebra teacher education. We introduce an approach to teacher education that leverages a novel algebraic thinking perspective. We demonstrate how using this perspective in video-based professional development can support teachers in identifying and reasoning about students’ nascent algebra thinking, which, we claim, is an important skill in teaching early algebra.

Correction to Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis

Correction to Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis

Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis

This case study explores how 12-13-year-old students encounter proportional reasoning while working with geometric patterning tasks using concrete materials. The focus is on the students’ use of spontaneous concepts when first dealing with such patterns in the context of collaborative work. Based on video recordings of a single lesson, a microgenetic analysis was performed to identify students’ learning trajectories, starting with students familiarizing themselves with pattern structure, followed by engagement in proportional reasoning, and ending with students perceiving a new technique to handle a situation where proportional reasoning did not suffice. While some student groups were able to move along the whole trajectory, most groups, when facing challenges, regressed to simpler techniques. The results provide new insights into students’ learning trajectories, which can be used to support students’ progress in the context of student-teacher interaction.

Развитие естественно-научной грамотности на уроках алгебры основной школы с применением межпредметных связей

The article deals with the issue of natural science literacy formation in teaching mathematics in middle school with the use of intersubject connections. Research methodology and techniques. The research is based on systemic-structural and activity approaches.
 Results. First of all, attention is paid to the consideration of the very concept of natural science literacy; the issue of assessing the level of development of students’ natural science literacy and means of its diagnostics in process of teaching mathematics, for example, at algebra lessons, is considered. Examples of specific tasks of interdisciplinary nature are given, which are used to determine the level of results achievement as a requirement of the Federal State Educational Standard of Basic General Education. Conditions for conducting integrated classes and their application as a means of developing natural science literacy of high school students at algebra lessons are considered. Issues of educational material synchronization of subject areas are considered as a key condition for implementation of intersubject connections between mathematics and physics. 
 Practical significance of the work. The directions of changes in approaches to the implementation of intersubject connections of mathematics with other areas of knowledge as a basis for achieving metasubject educational results as a requirement of the Federal State Standard of Basic General Education are identified. Metasubject assignments that contribute to the achievement of metasubject and subject educational results at algebra and physics lessons have been identified. Particular attention is paid to giving integrated classes as a condition for development of natural science literacy of students, as well as a means of assessing the level of natural science literacy formation. In this regard, examples of tasks in physics are considered that require the use of mathematical literacy, which directly affects the ability to analyze the situation and draw conclusions within the framework of the natural-scientific picture of the world. The main recommendations for the implementation of interdisciplinary relationships to achieve metasubject educational results are formulated.

The role and significance of computer animation in school algebra of complex numbers

The article discusses computer technologies in teaching mathematics, which is relevant in the light of the digitalization of education. The role and importance of computer animation in algebra lessons in the study of complex numbers in the tenth grade in accordance with the current school textbooks are presented. Animation drawings can be used as demonstration drawings when studying the relevant topic, as a tool for testing knowledge, they provide an opportunity for experimentation, support the research style of developing learning. The use of animated drawings effectively replenishes the arsenal of teaching aids for a practicing mathematics teacher. This allows you to eliminate unnecessarily burdensome computational difficulties, to visualize mathematical concepts and statements, turning some of them into obviously correct ones, to facilitate the teacher's work in compiling tasks of the same type with different data. The management of the student's educational and research work on the manufacture of animation-geometric models of objects and phenomena contributes to the successful assimilation of knowledge by students, stimulates an experimental research approach to teaching.

ABOUT TEACHING METHODS FOR SOLVING TEXT TASKS IN THE COURSE OF ALGEBRA

The article is devoted to the solution of text tasks in the course of algebra. Recently, it has been observed that the solution of text tasks causes difficulties not only for schoolchildren, but also there are cases when the solution of such problems causes problems for practicing mathematics teachers. Therefore, in the article, we have considered several types of tasks for which solutions have been proposed, which we consider useful material for preparing existing teachers for algebra lessons when studying the topics of solving text tasks using equations and their systems. Also, reading this article is recommended for students-future teachers of mathematics before or during their teaching practice. The article does not contain solutions to equations and their systems, since the purpose of this article is to teach how to make equations and systems of equations for solving text tasks, and not to solve them. For the design of the task conditions, both drawings, tables, and just writing short conditions are used.

Action Reseach on Universal Design for Learning (UDL) in Higher Education

Atsunori Fujii is a researcher based in the Junior College at the Fukuoka Institute of Technology, Japan, whose work revolves around action research on universal design for learning (UDL) in higher education. Fujii and his team are exploring the potential effectiveness of UDL in the higher education classroom setting. The researchers first offered options based on UDL framework to students in traditional lessons, to investigate how it affects a student's willingness to learn, metacognition, and use of learning strategies. In the next step of the project, they focused on flipped classrooms using ICT. They tested the usefulness of the different types of prior learning content in linear algebra lessons and then conducted a survey to discover the students' thoughts. As a result of that action research, the researchers confirmed that lessons based on UDL had a positive impact on students' motivation, even improving students' attitudes towards learning.

Cognitive instructional principles in elementary mathematics classrooms: a case of teaching inverse relations

ABSTRACT Instructional principles gleaned from cognitive science play a critical role in improving classroom teaching. This study examines how three cognitive instructional principles including worked examples, representations, and deep questions are used in eight experienced elementary teachers’ early algebra lessons in the U.S. Based on the analysis of 32 videotaped lessons of inverse relations, we found that most teachers spent sufficient class time on worked examples; however, some lessons included repetitive examples that also included irrelevant practice problems. Most teachers also situated new teaching in concrete contexts, which were faded into abstract representations. However, connections between concrete and abstract were not always made. The largest challenge was rooted in teachers’ inability to ask deep questions that elicited students’ deep explanations. Some teachers focused on key words and provided students with direct explanations. Implications are discussed.

Instructional practice in algebra: Building from existing practices to inform an incremental improvement approach

Algebra is considered key to success in secondary mathematics, yet instruction remains mostly teacher-centered and procedurally oriented, with limited opportunities for students to develop algebraic understanding. This study draws on a large sample of video-recorded ninth grade U.S. algebra lessons to explore the nature of learning opportunities that may help deepen students’ algebraic understanding. I highlight aspects of opportunities to learn algebraic procedures and describe instruction that can enrich these opportunities. I posit that this holds promise as an incremental improvement approach—relatively small adjustments in teachers’ current practices that can serve as a bridge to more ambitious teaching.

Competing influences that impact preservice teachers

Preservice mathematics teachers are subject to multiple influences when they make in-the-moment instructional decisions in classroom settings. Little is known about which influences make a significant contribution to decisions regarding the mathematical content knowledge (MCK) that preservice teachers decide to enact. The process by which mathematics preservice teachers manage multiple influences when making those decisions also requires particular examination. This study investigated six secondary mathematics preservice teachers’ content-related decisions and actions in ten lower secondary algebra lessons using a combination of lesson observations and stimulated recall interviews. The study uncovered five influences, each comprising a number of influencing elements, that led the preservice teachers to intentionally enact or alternatively to intentionally withhold certain aspects of their MCK during algebra lessons. Combinations of harmonious influences were regularly accommodated by preservice teachers, while other combinations of influences competed for the preservice teachers’ attention at different points in their lesson. The findings reveal that simply possessing content knowledge does not mean that preservice teachers are prepared to impart that knowledge to their students in the classroom context. The multi-faceted nature of content-related decision-making for preservice teachers suggests that content knowledge development should not be attended to in isolation within teacher education programs but should be located in teaching contexts wherever possible.

Students’ mathematical understanding ability using contextual teaching and learning approach

Students’ mathematical understanding ability is one of the students’ compulsory competences in learning mathematics. In the learning process, students commonly hesitated to ask the teacher how to solve mathematical problems even though they did not understand the lesson. This research aimed to improve the mathematical understanding ability by applying Contextual Teaching and Learning (CTL). This study employed a qualitative approach using True Experimental Design. This paper explained that the experimental result of students’ mathematical understanding ability used Contextual Teaching and Learning (CTL). The test of mathematical understanding ability was administered to Year 7 students of one of the junior high school in Sigli, Pidie. The instrument was a mathematical understanding ability test. The data analysis showed that the result of the normality test scores of the students’ mathematical understanding ability of the experimental class and the control class were 0.288 and 0.125. This research found that learning by using CTL indicated to improve the students’ mathematical ability. Mathematics teachers can apply CTL in the learning as an effort of developing students’ mathematical understanding ability in other algebra lessons.

Understanding Students’ Transition from Arithmetic to Algebraic Thinking in the Pre-Algebraic Lesson

The difficulty in learning algebra is mostly blamed to the discrepancy between arithmetic lesson in elementary school and algebra lesson in middle school. Hence, a bridge is needed to connect those two core subjects thorough a pre-algebraic lesson. This study is aimed to generate an understanding of how the students’ shift from arithmetic to algebraic thinking during pre-algebraic lesson. It is a part of a larger study in design research to establish a learning trajectory in pre-algebraic lesson by using pattern investigation. The data were gathered from written works, video registrations and field notes in 32 fifth grade students’ of an elementary school in Palembang. The data were analyzed qualitatively using constant comparative method. The result showed that the students’ shifting from arithmetic to algebraic thinking is highly impacted by their clarity in seeing geometrical structure of a pattern. Hence, it is recommended for the teachers to conduct a pre-algebraic lessons using pattern investigations visualized in geometrical representation for the elementary school students

\u201cLet us discuss math\u201d; Effects of shift\u2010problem lessons on mathematical discussions and level raising in early algebra

We investigated whether early algebra lessons that explicitly aimed to elicit mathematical discussions (shift-problem lessons) invoke more and qualitatively better mathematical discussions and raise students’ mathematical levels more than conventional lessons in a small group setting. A quasi-experimental study (pre- and post-test, control group) was conducted in 6 seventh-grade classes (N = 160). An analysis of the interaction processes of five student groups showed that more mathematical discussions occurred in the shift-problem condition. The quality of the mathematical discussions in the shift-problem condition was better compared to that in the conventional textbook condition, but there is still more room for improvement. A qualitative illustration of two typical mathematical discussions in the shift-problem condition are provided. Although students’ mathematical levels were raised a fair amount in both conditions, no differences between conditions were found. We concluded that shift-problem lessons are powerful for eliciting mathematical discussions in seventh-grade shift-problem early algebra lessons.

The Nature and Quality of Algebra Instruction: Using a Content-Focused Observation Tool as a Lens for Understanding and Improving Instructional Practice

Algebra is a key course in the high school mathematics sequence. Despite its prominence, large-scale examinations of algebra instruction are rare and it is not clear whether and how instructional practices that support students’ learning of algebra content manifest in classroom instruction. Drawing on video from 108 ninth-grade algebra lessons from 5 districts recorded as part of the Measures of Effective Teaching (MET) Project, I describe instruction using scores from a newly developed, algebra-focused observation tool to ground the description of instructional practice in the content being taught. I present the range of instructional formats and the nature and quality of algebra-focused instructional features related to the teaching of procedures and to the ways in which teachers leverage connections to support students’ algebra learning. To illustrate these descriptive results, I present case studies of 2 lessons: 1 rated typical and 1 rated high-quality. I find that although the majority of lessons in the sample follow traditional formats, specific instructional features that benefit student learning in algebra are present to a modest degree across lessons, though not often at high levels of quality. The case lessons demonstrate both glimmers of promise and missed opportunities to engage in instruction that benefits student learning of algebra. I discuss the implications of these results for improving the quality of algebra teaching and argue for the affordances of a content-focused observation tool as a lens for instructional improvement efforts.