Multiple solution methods of factoring trinomials in undergraduate mathematics

The purpose of this study was to investigate the relationship between both 9th-grade and 1st-year undergraduate students’ use of “look back” strategies and problem solving performance in multiple solution methods, the difference in their use of look back strategies and problem solving performance in multiple solution methods, and the role of look back strategies in problem solving in multiple solution methods. Data for this study were comprised of 30 9th-grade and 30 1st-year undergraduate students’ problem solving scores in multiple solution methods and their think-aloud protocols. Based on and expanded from Polya’s (1973) ideas, “look back” in the present study means “examination of what was done or learned previously.” The results of this study indicated that both the 9th-grade and 1st-year undergraduate students who looked back more frequently tended to perform better in multiple solution methods, the 1st-year undergraduate students tended to look back more frequently and perform better than the 9th-grade students in multiple solution methods, and both the 9th-grade and 1st-year undergraduate students tended to review and to compare multiple solution methods in their use of look back strategies.

The experiences of pre-licensure or pre-registration health professional students and their educators in working with intra-professional teams: a systematic review of qualitative evidence protocol

Mastering algebraic concepts and skills has been one of the demands in the Indonesian curriculum for the past few decades. Teaching algebraic concepts and skills is one of the materials taught from the junior high school to university level. Therefore, it is mandatory for pre service mathematics teacher students to master and even fluently use algebraic concepts in order to develop students' algebraic thinking skills. Thus, this study aims to understand and obtain a description of students' algebraic thinking abilities and misconceptions in solving problems. This research using descriptive qualitative. Analysis was conducted using tests and in-depth interviews with students who had taken linear algebra courses in semester 3 of the Mathematics Education Study Program at private university in NTT. Student’s algebraic thinking skills in solving problems were analyzed using Kieran's (2004) indicators consisting of: generational activities, transformational activities, and global meta-level activities. The results of the research show that…. In generational activities, students in the low, medium, and high categories have misconceptions. In transformational activities and global meta-level activities, misconceptions are only carried out by low category students.

Algebraic thinking ability are important for students to master and use algebraic concepts in various situations. Students with good algebraic thinking ability will find it easier to solve real-world problems and understand more abstract mathematical concepts. This study aims to describe students' algebraic thinking ability on linear equation material. Thus, the type of research used is phenomenology with a qualitative descriptive approach. The participants in this study consisted of 37 students; six students with heterogeneous abilities (high, medium, low) were then selected. The research subjects were determined by purposive sampling based on the pattern of answers representing the phenomenon that occurred. Data were collected using algebraic thinking ability tests and interviews to fulfill the validity of the data. The results showed that: 1) students with high algebraic thinking ability can use all indicators of algebraic thinking well. 2) students with medium algebraic thinking ability have difficulty in the indicators of analytical thinking, modeling, and dynamic thinking. 3) students with low algebraic thinking ability have difficulty in all indicators of algebraic thinking, namely generalization, modeling, abstraction, analytical thinking, dynamic thinking, and organization. So, teacher efforts are needed to provide students with an understanding of algebraic thinking so that students can solve math problems well.

Algebra is one of concept that must be learned by pupils. It is because the algebraic concept can be used in all areas of mathematics. One of the ways that is used to develop pupils' algebraic abilities is to think algebraically. While one of the ways to develop pupils' algebraic thinking skills is to adapt pupils with mathematical problem solving. The purpose of this study is describing algebraic thinking profiles of junior high school pupils in mathematical problem solving. The description of pupils' algebraic thinking profiles is explained based on six indicators namely generalization, abstraction, analytic thinking, dynamic thinking, modeling and organization. This research is a qualitative study using test and interview methods. The research subjects consisted of one student in each student with high mathematical abilities, medium mathematical abilities and low mathematical abilities. The results showed that there were differences in algebraic thinking of junior high school pupils in solving mathematics in terms of mathematical abilities. Based on research data shows that pupils with high mathematical abilities always think algebraically in each problems solving are given, while students with medium and low ability do not always think algebraically in each of problem solving.

Multimedia technology impacts and creates interactive learning between students and teachers as well as information technology creates a conducive environment for active learning. According to an assessment done by Trends in International Mathematics and Science Study (2011), cognitive skills such as application and reasoning need to be improved among Malaysian. Hence, it is important to implement these skills among students at the primary school level who will be changing from concrete arithmetic to the symbolic language of algebra as stated in Pelan Pembangunan Pendidikan Malaysia (2013-2025). The rationale of this study is to expose year four students in the field of Algebra to the understanding of variables, expressions, and equations. Algebra thinking courseware is used as a teaching aid to facilitate learning and teaching. This research includes the understanding and mastering Algebra thinking in the learning and teaching approach in Mathematics subject Year 4. The first objective of the study is to design and produce interactive multimedia courseware using the ADDIE model (Analyze, Design, Develop, Implement, and Evaluate). The second objective is to study the effectiveness of interactive multimedia thinking courseware by using the Kieran framework (2016) in improving the thinking concept of Algebra among year 4 students. A quasi-experimental design was applied to the treatment group and the control group. The sample size used was 84 students, who are from SJKT Menglembu School and SJKT Desa Pinji. The findings of the post-test clearly show that learning through Algebra thinking courseware has successfully improved the level of Algebra thinking which the mean for the treatment group was 14.69 (SD:2.694) while the pre-test of the mean for the treatment group was 6.76 (SD:2.750). Studies on Algebra thinking need to be extended by involving a larger sample size and ensure that such studies can be used via information technology media to improve the teaching and learning process at the primary school level.

Elementary school teachers still do not fully utilize effective learning models, especially in teaching algebraic concepts. Lack of understanding of algebraic concepts has caused students to have difficulty mastering algebra, including calculation, representation, and mathematical modeling, as well as recognizing algebraic symbols and variables. The purpose of this study is to answer the research question (RQ) How does the Generative Multi Representation Learning Model Modified Schema Based Instruction (MGMRM-SBI) affect the algebraic thinking skills of elementary school students? In this study, the experimental method was used with a posttest-only control group design, where the sample was selected using Cluster Random Sampling with a total sample of 128 students. The research instrument used was an algebraic thinking ability test in the form of description test questions, and data analysis was carried out using a t-test with the help of SPSS statistical software. The results of the t-test showed a significant difference between the algebraic thinking skills of students in the experimental and control classes, with significance values (2 sides) = 0.000 < sig. = 0.05. Based on the results of the study, it can be concluded that the application of the Generative Multi Representation Learning model Modified Schema Based Instruction has a significant effect on the algebraic thinking ability of elementary school students, and is more effective than the expository model.

Developing algebraic thinking is a key factor in learning mathematics. Despite its importance, many students still struggle with algebraic concepts. This research investigates students’ achievements in algebraic thinking using Demetriou’s test across 7<sup>th</sup> (approximately 12-13 years old), 8<sup>th</sup> (approximately 13-14 years old), and 9<sup>th</sup> (approximately 14-15 years old) grades. The study analyzes performance in different levels of algebraic tasks (i.e., [1] extrapolation of relationships, [2] coordinating simple structures, [3] operating with undefined symbolic structures, and [4] coordination with undefined structures), revealing a significant developmental leap in algebraic abilities during the 9<sup>th</sup> grade. While no statistically significant differences were found in the first level, 9<sup>th</sup> grade students demonstrated superior performance in levels 2, 3, and 4, suggesting cognitive readiness for abstract algebraic concepts around the age of 14. The research unveils a disjointed development in algebraic abilities, indicating a progression from basic arithmetic operations to proportional reasoning before the full integration of algebraic thinking. Notably, tasks involving variables in the third level pose persistent challenges for students. The findings contribute to understanding the optimal age for introducing algebraic concepts and underscore the importance of considering cognitive development in mathematics education. The study proposes implications for educators, such as emphasizing proportional reasoning in earlier grades and employing differentiated instruction based on individual students’ abilities.

This research is a preliminary study that aims to describe the algebraic thinking process of prospective mathematics teachers. This research is a qualitative descriptive study. Subjects were grouped into two categories based on high and low achievement motivation. Data is obtained based on the results of tests conducted in the algebra process. Research subjects (S1) and (S2) with high achievement motivation and subjects (S3) and (S4) with low achievement motivation using different algebraic thought processes. Subjects (S1) are able in the process of thinking algebra until crashing indicators assess understanding with understanding of the concept wrong in solving Higher Order Thinking Skills (HOTS) problems, whereas, (S2) the process of thinking algebra is only capable of chunking information (pieces of information), (S3) able in the process of thinking algebra until indicators of change with wrong answers, and the subject (S4) is able in the process of thinking algebra only until chunking information (pieces of information). Factors that cause subjects S1, S2, S3, and S4 are still unable to solve HOTS questions in algebraic thinking processes are questions of knowledge on HOTS material and difficulty understanding concepts in working on algebra need special handling in improving understanding of concepts in algebra.

There are a large number of students entering college underprepared for college-level mathematics (Carnegie Foundation for the Advancement of Teaching , 2011). While this problem is not new, it has become the focus of national attention because of the impact it has on college completion and workforce development. There is much written in the literature about developmental mathematics; however, there is little research about the experiences of students who find themselves placed into developmental mathematics courses upon entrance into college (Howard & Whitaker, 2011). The purpose of this phenomenological study was to explore and articulate the experiences of students at a community college in the Northeast who were unsuccessful in a developmental mathematics course but took the class in a subsequent semester. The research question the study sought to answer is: What characterizes the experiences and perceptions of students who are unsuccessful in a developmental mathematics course and return to repeat it?Data was collected using the following methods: conducting semi-structured in-depth interviews utilizing open-ended questions, classroom observations, and reviewing artifacts such as placement test scores; class attendance records; and class quizzes, tests, and graded assignments. This provided multiple data points allowing for triangulation of the data to ensure the reliability and validity. Member checking was also employed for this purpose. The researcher used bracketing to eliminate any preconceptions and biases during data collection and analysis. Convenience sampling was used to select participants in the study. Students retaking a developmental mathematics course because they earned below a C in a prior attempt comprised the sample.

Computational Thinking is part of the new curriculum in many countries and this new competence is often combined with Algebraic Thinking. Both types of thinking are part of the core of Mathematics and Computer Science. Algebraic Thinking is linked to acquiring the ability to represent and generalize patterns in any application area. Furthermore, the ability to communicate a mathematical argument, using the necessary language and symbolism, is a skill that is dependent on training in this type of thinking. Although Algebraic Thinking can be developed at different levels, and it is also developed at university levels, more and more countries see it as a basic mode of thought that should be encouraged from early childhood education. Algebraic Thinking has also a close relationship with Computational Thinking, and they are currently united in different situations, such as the international PISA student evaluation tests. We argue in this paper that this is a transversal competence that can be practiced in any subject and at any age. Sometimes combined with the process of teaching Mathematics. It is essential, in our opinion, to strengthen the inclusion of strategies that encourage students to reflect deeply on the concepts, theories, and applications they are learning, giving rise, among others, to number sense and abstraction. In this paper, we present the implementation of these two types of thinking, algebraic and computational, in the pre-university curriculum, particularly in Spain, within a European project. In this project, we seek to create more appropriate learning approaches for those who are often disadvantaged and help them to take advantage of Computational Thinking and Algebraic Thinking and, therefore, STEM knowledge, helping to a stronger and more equal society. We analyze its status and its relationship with the concepts taught in the different courses, although focusing on the subject of Mathematics.

We analyzed how algebraic concepts and representations are introduced and developed in the Chinese, South Korean, and Singaporean elementary curricula, and in selected Russian and U.S. elementary curricula. In all five curricula, the main goal for learning algebraic concepts is to deepen students' understanding of quantitative relationships, but the emphases and approaches to helping students deepen their understanding of quantitative relationships are very different. Based on the analyses of the five curricula, we discuss four issues related to the development of algebraic thinking in earlier grades: (1) To what extent do curricula expect students in early grades to think algebraically? (2) What level of formalism should we expect of students in the early grades? (3) How can we help students make a smooth transition from arithmetic to algebraic thinking? and (4) Are authentic applications necessary for students in early grades?

This action research highlights the experiences of undergraduate students who studied Developmental Mathematics using the Problem-Based Learning(PBL) strategy. They were exposed to fifteen weeks intervention at a Higher Educationinstitution in Trinidad and Tobago called HilltopCollege. A review of the existing literature within the local context indicated that there is a paucityof information about theirexperiences.Consequently, their experiences are critically important since they can be an impetus for the formulation of policy and implementation towards the teaching/learning of Developmental Mathematics inthis country.Thus, it is absolutely necessary that policy makers heed the voices of these students especially when they are formulating curriculum that pertain to Developmental Mathematics. A qualitative case study was conducted to carefully ascertain their experiences and answer the research question: What are students´ experiences with Problem-Based Learning in the study of Developmental Mathematics at Hilltop College? Twenty-four students participated. A structured questionnaire and semi-structured interviews were utilized with four focusgroups. Data were analyzed under six major headings: Approach to teaching, social relationships, resources, pace of teaching, emotional intelligence and the role of the teacher. Recommendations strongly advocating that student-centered strategies be employed when studying Developmental Mathematics were also offered.

Given the growing emphasis on algebraic concepts in primary education curricula and the importance to investigate how students with Autism Spectrum Disorder (ASD) approach mathematical learning, the purpose of this study is to evaluate the effectiveness of an instruction based on multiple representations in promoting early algebraic thinking in students with ASD. Three students with ASD, aged 7, 7, and 9 years, enrolled in the same general education school in Spain participated in the study. The research followed a single-subject, multiple probe across participants design. The results of the study showed that the participants improved their performance on generalization tasks in functional contexts when using multiple representations. These improvements were maintained over time and transferred to everyday situations involving regularities. Social validity data collected from students’ families and teachers indicated that both groups had positive perceptions of students’ attitudes and motivation after the study. The implications of the results for promoting early algebraic thinking in students with ASD are discussed.

The recent “Algebra for all” era has meant “the best of times and the worst of times” in many middle schools. At one extreme, many adolescents delight in the opportunity to study algebra or algebraic thinking and perform well in this course of study. At the other extreme, too many adolescents encounter serious challenges as they delve into fundamental ideas that make up this essential mathematical subject. Instead of viewing algebra as a natural extension of their arithmetic experiences, significant numbers of adolescents do not connect algebraic concepts with previously learned ideas. For instance, data from the Third International Mathematics and Science Study (TIMSS) showed that at the international level, only 47 percent of the seventh graders and only 58 percent of the eighth graders were able to recognize that m + m + m + m was equivalent to 4m (Beaton et al. 1996).