Will learning to solve one-step equations pose a challenge to 8th grade students?

Assimilating multiple interactive elements simultaneously in working memory to allow understanding to occur, while solving an equation, would impose a high cognitive load. Element interactivity arises from the interaction between elements within and across operational and relational lines. Moreover, operating with special features (e.g. negative pronumeral) poses additional challenge to master equation solving skills. In an experiment, 41 8th grade students (girls = 16, boys = 25) sat for a pre-test, attended a session about equation solving, completed an acquisition phase which constituted the main intervention and were tested again in a post-test. The results showed that at post-test, students performed better on one-step equations tapping low rather than high element interactivity knowledge. In addition, students performed better on those one-step equations that contained no special features. Thus, both the degree of element interactivity and the operation with special features affect the challenge posed to 8th grade students on learning how to solve one-step equations.     

Effect of inquiry-based instruction enriched with origami activities on achievement,and self-efficacy in geometry

The purpose of the study was to investigate the effect of inquiry-based instruction enriched with origami activities on 7th grade students’ achievement in reflection symmetry and self-efficacy in geometry. Two classes, instructed by the first author of the paper, were randomly assigned as experimental and control groups. In order to gather data, participants were administered Reflection Symmetry Achievement Test, and Geometry Self-Efficacy Scale as pre-test and post-test. The Analysis of Covariance was performed in order to answer the research questions. Moreover, five participants were interviewed to examine self-efficacy sources which are determinant of the change in self-efficacy levels. Findings revealed that the inquiry-based instruction enriched with origami activities had a significantly positive effect on students’ achievement in reflection symmetry and self-efficacy in geometry. Interviews showed that all four sources were influenced by the intervention.     

Improving senior secondary school students’ attitude towards mathematics through self and cooperative-instructional strategies

This study investigated the effects of self and cooperative-instructional strategies on senior secondary school students’ attitude towards Mathematics. The moderating effects of locus of control and gender were also investigated. The study adopted pre-test and post-test, control group quasi-experimental design using a 3 ×?2 ×?2 factorial matrix with two experimental groups and one control group. Three hundred and fifty SSS II students from six purposively selected secondary schools in Ijebu-North Local Government Area of Ogun State were the subjects. Three instruments were developed, validated and used for data collection. Analysis of Covariance (ANCOVA) and Scheffé post hoc analysis were the statistics used for data analysis. Findings showed that the treatments had significant main effect on students’ attitude towards Mathematics. The participants exposed to self-instructional strategy had the highest post-test mean attitude score. The study found no significant main effects of locus of control and gender on the participants’ attitude towards Mathematics. It was concluded that Mathematics teachers should be trained to use self and cooperative learning packages in the classroom, since the strategies are more effective in improving students’ attitude towards Mathematics than the conventional method.     

Examination of the dynamic software-supported learning environment in data analysis

The Fatih Project in Turkey has improved software in mathematics teaching such as data analysis software. As a result, the need to inquire into the efficiency of computer-supported learning environments has emerged. This study aims to examine the effect of dynamic data analysis software-supported learning environments on secondary school students’ achievement and attitude. The research method employs a quasi-experimental design with a pre-test, post-test control group. Basic topics related to data analysis were introduced through dynamic statistics software in the experimental group while the students were taught with the help of smart boards, course books and exercises in the control group. Data were collected with an achievement test, attitude scale and semi-structured interviews. Also, interviews were conducted with four students from the experimental group in order to get more detailed information from students. The data gained in the study were analysed both quantitatively and qualitatively. The findings revealed that statistics teaching through statistics software is more efficient than the one with the traditional method on achievement and attitudes. In accordance with this result, it is suggested that computer-supported statistics software should be used in statistics teaching.     

Student Strategies Suggesting Emergence of Mental Structures Supporting Logical and Abstract Thinking: Multiplicative Reasoning

For many students, developing mathematical reasoning can prove to be challenging. Such difficulty may be explained by a deficit in the core understanding of many arithmetical concepts taught in early school years. Multiplicative reasoning is one such concept that produces an essential foundation upon which higher‐level mathematical thinking skills are built. The purpose of this study is to recognize indicators of multiplicative reasoning among fourth‐grade students. Through cross‐case analysis, the researcher used a test instrument to observe patterns of multiplicative reasoning at varying levels in a sample of 14 math students from a low socioeconomic school. Results indicate that the participants fell into three categories: premultiplicative, emergent, and multiplier. Consequently, 12 new sublevels were developed that further describe the multiplicative thinking of these fourth graders within the categories mentioned. Rather than being provided the standard mathematical algorithms, students should be encouraged to personally develop their own unique explanations, formulas, and understanding of general number system mechanics. When instructors are aware of their students' distinctive methods of determining multiplicative reasoning strategies and multiplying schemes, they are more apt to provide the most appropriate learning environment for their students.     

A study of factors impacting elementary mathematics preservice teachers: Improving mindfulness,anxiety, self-efficacy,and mindset

Innovation is more imperative now than ever before given the upcoming shortage in prepared teachers and the need to produce students with a strong knowledge of mathematics. A sense of urgency is impacting teacher education/preparation programs as instructional practices need to discover how to arm teachers to increase the number of students to be not only college-ready but also desiring to pursue Science, Technology, Engineering, and Mathematics majors. As such, the purpose of this study, was to determine how the four variables (mindfulness, mathematics anxiety, self-efficacy, and mindset) are interconnected within preservice elementary teachers (PSETs), and how we as teacher educators can better address these variables within our own PSETs. Each semester included three seminars with similar overall foci including the four variables. Participants in this study were recruited from Elementary Education students at an east south central regional university enrolled in a mathematics methods course. Thirty-seven participants were divided into control (N = 20) and treatment (N = 17). In this article, we present both qualitative and quantitative results from our mixed-methods study that considered these questions. With the results of this study revealing an inter-connectedness among the four variables, this research further informs the teacher educator community.     

Exploring Students' Conceptual Understanding of the Averaging Algorithm

Conceptual understanding of arithmetic average includes both an understanding of the computational algorithm and the statistical aspects of the concept. This study focused on the examination of 250 sixth-grade students' understanding of the arithmetic average by assessing their understanding of the computational algorithm. The results of the study showed that the majority of the students knew the “add-them-all-up-and-divide” averaging algorithm, but only about half of the students were able to correctly apply the algorithm to solve a contextualized average problem. Students were able to use various solution strategies and representations to solve the average problem. Those who used algebraic and arithmetic representations were better problem solvers than those who used pictorial and verbal representations. This study not only suggests that the average concept is more complex than the simplicity suggested by the computational algorithm, but also indicates the need for teaching the concept of average, both as a statistical idea for describing and making sense of data sets and as a computational algorithm for solving problems.     

Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions,domain restrictions,and the arcsine function

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers’ instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter.     

A brief historical introduction to solitons and the inverse scattering transform–a vision of Scott Russell

The objective of this study was to evaluate biomathtutor by (i) investigating the impact of biomathtutor on the mathematics skills competencies of bioscience undergraduates, and (ii) assessing students’ and tutors’ reactions to biomathtutor, identifying whether and how tutors might integrate it into their curricula and blend it with more traditional teaching practices to enhance their students’ learning experiences. A multi-method approach was adopted in which a quasi-experiment and non-experimental evaluation of biomathtutor were used to collect both quantitative and qualitative data, using mathematics tests, questionnaires, tutor interviews and student focus groups. Eighty-nine bioscience undergraduates and eight tutors participated in the study. A comparison of student performance in the quasi-experiment, which adopted a pre-test-intervention-post-test methodology, revealed no significant difference between pre-test and post-test scores for either the ‘control’ group (no intervention) or for any of the mathematics learning support interventions used, including biomathtutor. Despite the limitations of the quasi-experiment which are discussed, tutors’ and their students’ reactions towards biomathtutor were very positive, with both groups agreeing that biomathtutor represents a very well designed and useful learning resource that has a valuable role to play in supporting mathematics learning within bioscience curricula. Students felt that using biomathtutor had helped them acquire new biological and mathematical knowledge and had increased their competence and confidence in mathematics, with many students confirming that they would use biomathtutor again. Tutors felt it would be useful to embed biomathtutor, where possible, into their curricula, perhaps linking it to assessment strategies or integrating it with their current more traditional teaching practices. Students indicated that they too would like to see biomathtutor embedded within their curricula, primarily because it would motivate them to use the resource. Modifications to biomathtutor, which may need to be considered in light of any potential further development of this resource, are discussed.     

Are beginning calculus and engineering students adequately prepared for higher education? An assessment of students’ basic mathematical knowledge

Are students transitioning from the secondary level to university studies in mathematics and engineering adequately prepared for education at the tertiary level? In this study, we discuss the prior mathematical knowledge and skills demonstrated by Norwegian engineering (N?=?1537) and calculus (N?=?626) university students by using data from a mathematics assessment administered by the Norwegian Mathematical Council. The assessment examines students’ conceptual understanding, computation skills and problem solving skills on the basis of the mathematics curriculum of lower secondary education. We found that calculus students significantly outperformed engineering students, but both student groups struggled to solve the test, with the calculus and engineering groups scoring an average of 60% and 46%, respectively. Beginning students who fail to master basic skills, such as solving arithmetic and algebra problems, will most likely face difficulties in their further courses. Although few female students enrol in calculus and engineering programmes compared with male ones and are thus underrepresented, male and female students at the same ability level achieved comparable test scores. Furthermore, students reported high levels of intrinsic and extrinsic motivation, and a positive relationship was observed between intrinsic motivation and achievement.     

Individualized additional instruction for calculus

College students enrolling in the calculus sequence have a wide variance in their preparation and abilities, yet they are usually taught from the same lecture. We describe another pedagogical model of Individualized Additional Instruction (IAI) that assesses each student frequently and prescribes further instruction and homework based on the student's performance. Our study compares two calculus classes, one taught with mandatory remedial IAI and the other without. The class with mandatory remedial IAI did significantly better on comprehensive multiple-choice exams, participated more frequently in classroom discussion and showed greater interest in theorem-proving and other advanced topics.     

Developing teacher content understanding by integrating pH and logarithms concepts

Logarithms are notorious for being a difficult concept to understand and teach. Research suggests that learners can be supported in understanding logarithms by making connections between mathematics and science concepts such as pH. This study investigated how an integrated chemistry and mathematics lesson impacted 29 teachers’ understanding of the logarithmic relationship and pH. Pre- and post-test data indicated 23 teachers improved their understanding of logarithms and 28 improved their understanding of pH, suggesting that teacher educators in both science and mathematics context can use this approach to foster better understanding with their teachers and ultimately school students. Our analysis also identified professional development components and teacher characteristics associated with gains in understanding of pH and logarithms, which mathematics and science teacher educators can use to strategically adapt and implement the lesson within other teacher education settings.     

Using math magic to reinforce algebraic concepts: an exploratory study

An exploratory study was conducted to investigate the use of magic activities in a math course for prospective middle-school math teachers. This research report focuses on a lesson using two versions of math magic: (1) the 5-4-3-2-1-½ Magic involves having students choose a secret number and apply six arithmetic operations in sequence to arrive at a resultant number, and the teacher-magician can spontaneously reveal a student’s secret number from the resultant number; and (2) the Everyone-Got-9 Magic also involves choosing a secret number and applying arithmetic operations in sequence, but everyone will end up with the same resultant number of 9. These magic activities were implemented to reinforce students’ understanding of foundational algebra concepts like variables, expressions, and inverse functions. Analysis of students’ written responses revealed that (1) all students who figured out the trick in the first magic activity did not used algebra, (2) most students could apply what they learned in one trick to a similar trick but not to a different trick, and (3) many students were weak in symbolic representations and manipulations. Responses from a survey and a focus group indicate that students found the magic activities to be fun and intellectually engaging.     

Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.     

Establishing the link between the graph of a parametric curve and the derivatives of its component functions

ABSTRACT

This study aimed to explore the effects of an instructional intervention in which GeoGebra and the think-pair-share method were used to teach the relationship between the graph of a parametric curve and the derivatives of its component functions. The participants in the study were 19 prospective mathematics teachers. To assess the understanding of the participants regarding the content of the instruction, two comparable tests were administered as a pre- and post-test. In order to determine whether there was a difference between the students’ performance on the two tests, a paired samples t-test was conducted on the test scores; the results revealed a significant difference in favour of the post-test. Thus, it was concluded that the adopted teaching method, which included the use of GeoGebra, in delivering the focused content had a positive impact on the students’ understanding.     

Seeking mathematics success for college students: a randomized field trial of an adapted approach

Many students enter the Canadian college system with insufficient mathematical ability and leave the system with little improvement. Those students who enter with poor mathematics ability typically take a developmental mathematics course as their first and possibly only mathematics course. The educational experiences that comprise a developmental mathematics course vary widely and are, too often, ineffective at improving students’ ability. This trend is concerning, since low mathematics ability is known to be related to lower rates of success in subsequent courses. To date, little attention has been paid to the selection of an instructional approach to consistently apply across developmental mathematics courses. Prior research suggests that an appropriate instructional method would involve explicit instruction and practising mathematical procedures linked to a mathematical concept. This study reports on a randomized field trial of a developmental mathematics approach at a college in Ontario, Canada. The new approach is an adaptation of the JUMP Math program, an explicit instruction method designed for primary and secondary school curriculae, to the college learning environment. In this study, a subset of courses was assigned to JUMP Math and the remainder was taught in the same style as in the previous years. We found consistent, modest improvement in the JUMP Math sections compared to the non-JUMP sections, after accounting for potential covariates. The findings from this randomized field trial, along with prior research on effective education for developmental mathematics students, suggest that JUMP Math is a promising way to improve college student outcomes.     

Learning the Learning Cycle: The Differential Effect on Elementary Preservice Teachers

This study examined the difficulties and factors that led to understanding the learning cycle teaching strategy. Participants included 83 preservice elementary teachers (PTs) enrolled in multiple sections of a science methods course taught by the same professor, one of the authors of this study. Analysis revealed that there were four categories of PTs, ranging from Enthusiastic to Fearful students, distinguished by their science content background and attitudes to science. High achieving students, successful in science courses, felt confused by the learning cycle that was so different from their previous science learning experiences and formed mindsets against learning it. Average students who expressed disinterest or even fear of science embraced it as their first successful science learning experience. Multiple exposures to the learning cycle were necessary to overcome these mindsets. Most PTs in all categories increased in their understanding of the learning cycle and perceived it as an effective method for allowing students to construct their understanding of science.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).