General Education Mathematics Research Articles

Concept-Focused and Procedure-Focused Instruction on the Algebra Performance of Grade 9 Students With and Without Mathematics Difficulty.

Developing both conceptual and procedural knowledge is important for students' mathematics competence. This study examined whether Grade 9 general education mathematics teachers' self-reported use of concept-focused instruction (CFI) and procedure-focused instruction (PFI) were associated differently with ninth graders' algebra achievement after 2.5 years, depending on students' mathematics difficulty (MD) status. Data for this study were drawn from the High School Longitudinal Study for the years 2009-2010 and 2011-2012 (N = 19,104). Multiple regression analyses indicated that students with MD who participated in Grade 9 mathematics classrooms where teachers self-reported the use of less CFI and more PFI were more positively associated with having higher algebra achievement after 2.5 years. Conversely, students without MD in classrooms where mathematics teachers self-reported the use of more CFI and less PFI were positively associated with having higher algebra achievement after 2.5 years. However, this study's findings do not suggest that teachers should disregard CFI and provide only PFI when teaching students with MD. Because the data set did not include any variable to discover whether teachers provided sufficient support (i.e., evidence-based practices) for students with MD, who have more constraints in their cognitive skills compared to students without MD, to benefit from CFI, the findings of this study should be interpreted cautiously. Directions for future research and practical implications are discussed.

Using Mixed Methods to Evaluate Modified Schema-Based Instruction in General Education Classrooms

In this convergent mixed methods design study, single-subject and qualitative data were collected concurrently to provide an in-depth picture of the impact of a modified schema-based instructional intervention. The intervention was delivered using instructional trials embedded across general education math lessons and a modified concrete-semi-concrete-abstract instructional sequence. This study investigated the impact of the intervention on the word-problem-solving, strategy use, and concept acquisition of three students with extensive support needs. The paraprofessional-delivered intervention was implemented in elementary general education mathematics classrooms using embedded instruction and focused on teaching students to solve addition and subtraction word-problems. Single-subject data indicated that all three students learned to solve word-problems given concrete materials but needed more time to master the use of semi concrete supports. Qualitative data indicated that students used taught and untaught strategies to solve word-problems, and mastered addition word-problems before subtraction when they were taught simultaneously. Data were integrated in narrative format to explore how strategy use and concept acquisition related to student word-problem-solving performance. Limitations and implications for research are discussed.

Predicting Student Success in Co-remediated General Education Mathematics Courses at a Large Urban Public University

Placement and support of students in first-year mathematics courses at institutions of higher education has long been a consequential issue. In a bid to address it, many systems and institutions of higher learning have elected to implement a co-remediation framework in place of pre-remediation, due in large part to the prohibitive cost of the latter, both in terms of financial resource, as well as student academic progress. Accompanying this evolution has been the expansion of the introductory mathematics curriculum beyond algebra to include statistics and quantitative reasoning. The present study discusses three distinct introductory mathematics courses at a large urban public M.S. granting institution in the U.S., with the goal of identifying the characteristics that correlate with success in each. Conditional probability and non-completion rate analyses were implemented to compare student performance in each course. Predictive models were then trained and validated, building insight concerning the differential relationships of demographic and academic covariates with course completion.

Enhancing access to general education mathematics instruction through special education preteaching: special education-general education teacher collaboration

The study described in this paper investigated a unique collaboration among a special education and two general educators that sought to prepare students for success in the general education mathematics curriculum and that consisted of coordinated, reform-based instructional design across classroom and pull-out services provided by the special educator including preteaching. During preteaching, the special educator delivered instruction on key mathematics and collaboration skills that special education students would need a few days before they would receive instruction on the same content in their general education classrooms. Their collaboration produced high levels of teacher satisfaction, substantive teacher change, and increases in student engagement, self-efficacy, and achievement.

Teachers’ Views of the Mathematical Capabilities of Students With Disabilities: A Mixed Methods Study

Background/Context: Most students with disabilities receive the majority of their instruction in general education classrooms. Yet, general education teachers persistently describe feeling unprepared to academically support students with disabilities in those spaces. Because disabled students are typically excluded from mathematics education research, and because special education researchers typically describe mathematics teaching and learning in ways that are incongruent with ambitious mathematics instruction, there is arguably a lack of guidance for these teachers. In the absence of clear guidance, teachers may turn to the well-established mathematical ability hierarchy, which positions disabled students (among others) as less capable. Purpose/Objective/Research Question/Focus of Study: The purpose of this study was to uncover teachers’ talk about the mathematical capabilities of students with (and without) disabilities. Existing coding schemes (perhaps inadvertently) treat teachers’ views as uniform across students despite evidence that teachers hold different views of different students, in part because of the multiple and varied identities that students bring to the classroom. By using an adapted interview protocol, which yielded more (and more nuanced) analytic categories, I foregrounded students’ disability status as a factor that could relate to differences in teachers’ conceptions of who they view as mathematically capable. Research Design: I interviewed general education mathematics teachers ( N = 20) about their students ( n = 407) using an adapted version of Jackson et al.’s (2017) semi-structured protocol that focused on uncovering teachers’ usages of diagnostic and prognostic frames. I used open and concept coding to develop an expanded version of Jackson et al.’s coding scheme and then applied the new coding framework to the entire data set. I used student demographic data to compare within-group percentages, noticing to what degree students with disabilities were represented within particular qualitative categories in relation to their representation within the entire data set. I also used transformed data to estimate two multinomial logistic regressions: one that used diagnostic frames as the outcome variable, and one that used prognostic frames as the outcome variable. Both models used students’ disability status and teacher dummy codes as predictor variables. Conclusions/Recommendations: The majority of teachers in this sample explained mathematical struggle in unproductive terms and said they would aim instructional adjustment at unproductive outcomes for students with and without disabilities. However, students with disabilities were overrepresented in unproductive categories and underrepresented in productive categories in relation to both diagnostic and prognostic frames. Regression analyses indicated that a student was statistically less likely to get a productive diagnostic or prognostic frame if they had a disability label. Findings from this study highlight the necessity of including teachers’ views of their students’ mathematical capabilities in instructional improvement efforts. Second, they indicate that student-level factors, such as disability status, relate to qualitatively and quantitatively meaningful differences in teachers’ views of their students, suggesting the importance of attending to broader narratives around constructs that may be associated with teachers’ views, and the subsequent enactment of those views, in mathematics instruction.

Novice mentors versus mentees: Mentoring experiences in mathematics at General Education and Training phase

In the South African educational system, student teachers are deployed to schools for practical experience, where they are monitored by lecturers from their universities. Student teachers are also mentored by teachers allocated to them by the school principal. Some of these mentoring teachers are themselves newly qualified and may have little or no teaching experience. This study analysed the relationship between these various role players during the teaching of mathematics in General Education and Training (GET) phase, at secondary schools in the Western Cape. The theoretical framework for the study was provided by Lave and Wenger’s Communities of Practice. An ethnographic qualitative research design was used for collecting data from classroom observations and semi-structured interviews. The selected participants comprised four novice mathematics teachers, four mathematics student teachers in the GET phase, two lecturers and one school principal. The purposive selection method was used to select these participants. The findings revealed that novice mentor teachers were challenged by facing (1) no or little communication and collaboration between themselves and lecturers, (2) limited cooperation between mentor and mentee in the teaching of mathematics in GET phase, (3) limited mathematics content knowledge by student teachers and (4) limited mentoring skills of novice mentors. It is recommended that universities create a sound educational partnership with mentor teachers. Universities should also consider the voices of novice mentor teachers in their mentoring of student teachers.

Postsecondary general education mathematics: theory and practice

Many students who take a mathematics course as part of their post-secondary education are not enrolled in a mathematically-intensive degree program. This poses a challenge to mathematics departments that, for example, value and privilege more traditional mathematics curriculum: should students in such courses be taught differently than those in more traditional mathematics courses? What should constitute the curricula of these courses and who should teach them? How might the local institutional context influence the courses that are ultimately offered? This paper sketches a theory intended to help frame responses to these questions. We then present an example of this theory in action, in the form of a general education mathematics course for first-year, underprepared students. The ongoing design, teaching, and evaluation of this course might inspire further revisions to general education in mathematics at the undergraduate level. We present the motivation for the design of the course, a summary of what was ultimately enacted, and our reflections of this ongoing event. Our intention here is not to present our course and an evaluation of its ‘success' – however that might be conceptualised – but rather to display a rigorous vision for post-secondary general education mathematics.

Theory to Practice: Reducing Student Attrition in Online Undergraduate Math

Researchers have developed numerous effective theory-based practices for teaching undergraduate general education mathematics (UGEM); however, many universities have struggled for decades to close the gap from theory to practice. This gap contributes to the national lack of skilled STEM workers. Recently, an all-online university closed the gap by shifting their philosophical framework for UGEM from traditionalist methodology to a synthesis of seminal theories and practices. This paper disseminates the implementation of theory-based practices in UGEM toward reducing student attrition (withdraw or fail), including: Rationale for theory identification, construction of a philosophical framework, collection of stakeholder input, implementation, evaluation, post-implementation maintenance and communication, and institutional socialization of the new paradigmatic shift. These efforts yielded an attrition reduction from 17.5% to 4.7% in Quantitative Reasoning 1 (QR1) and from 13.9% to 4.0% in QR2. A key reported outcome is a blueprint for an institution to similarly close the gap.

Reflection on the Complexity of Mathematical Objects in the Initial Training of Teachers

This work aims to know the perception of future mathematics teachers about the complexity of mathematical objects and their possible application in teaching practice, in order to improve the teaching and learning of mathematics in Basic General Education (GBS) and the Unified General Baccalaureate (BGU). For this, 19 future professors were asked about the different meanings of some mathematical objects, and they were proposed to raise contextualized problems in which a certain meaning had to be applied in their resolution. The results show that encouraging reflection on the complexity of mathematical objects in the initial training of teachers and their relationship with the design of contextualized problems affects their way of understanding mathematical competence

A study on the improvement of basic general mathematics according to changes in the basic mathematical ability of first-year college students in the fields of science and engineering

The objective of this study is to examine changes in the Mathematics Curriculum and the College Scholastic Ability Test that affect the changes in the Basic Mathematical Ability of students enrolled in Science and Engineering departments and to compare and analyze the educational contents of basic general mathematics completed by first-year students in science and engineering fields in order to provide improvement plans for basic general mathematics. In Korea, whenever the mathematics curriculum is revised, the content covered in the previous curriculum is deleted or weakened in order to optimize the educational content, and ‘Calculus’ and ‘Geometry’ are required for science and engineering students to choose in the college scholastic ability test. Thus, the basic academic ability of mathematics is lowered, and it is difficult to complete the basic general mathematics taught in the first year of university. In order to solve this problem, it is necessary to analyze the mathematics and curriculum to include deleted or weakened contents in the university's basic general mathematics, to understand the students' basic academic ability in mathematics, and to operate basic general mathematics by level or operate basic mathematics courses. In addition, when revising the mathematics both in curriculum and the university scholastic ability test, the opinions of professors in charge of basic general mathematics education at universities should be sufficiently reflected to minimize changes in key factors for students to complete basic general mathematics, and it is necessary to provide policy support at the national level for the development and operation of the curriculum for general mathematics education at a college level.

Using Number Talks to Support Students With High-Incidence Disabilities in Mathematics

Number talks are increasingly used in general education mathematics classes to engage students. Yet despite the potential benefits, number talks are given limited attention for students with high-incidence disabilities in special education settings. This article presents special education teachers with both the why and, more important, the how for implementing number talks to support students with high-incidence disabilities in special education settings. Specifically, the authors address how number talks can serve as both a formative assessment and an intervention for fluency and activating students’ background knowledge to be successful in general education settings. The article also provides suggestions for implementing number talks with fidelity and flexibility (e.g., use of manipulatives, pictorial representations, and teacher explicit instruction of numerical strategies).

Development of Mathematical Practices Through Word Problem–Solving Instruction for Students With Autism Spectrum Disorder

The Common Core State Standards for Mathematics highlight the importance of not only content standards for mathematics but also mathematical practices such as communication, representation, and reasoning, skills that are often difficult for students with autism spectrum disorder (ASD). Through a single-case multiple-probe-across-participants design, this study found modified schema-based instruction (MSBI) to be an effective strategy to increase the use of mathematical practices for middle school students with ASD when solving multiplicative word problems. Four students eligible for special education services under the area of autism enrolled in sixth-grade general education mathematics classes increased their use of mathematical practices for two problem types (multiplicative comparison and proportion) and maintained the use of mathematical practices 4 to 8 weeks after intervention. Additionally, all participants generalized their use of mathematical practices to novel multiplicative comparison problems containing extraneous information, and three of the participants generalized mathematical practice skills to proportion problems containing extraneous information. Implications for practice are discussed.

Enhancing Secondary Students’ Goal Attainment and Self-Determination in General Education Mathematics Classes Using the Self-Determined Learning Model of Instruction

Almost every state includes content standards that are closely related to skills associated with self-determination (e.g., decision making, planning, problem solving, goal setting and attainment). Additionally, research in both general and special education has demonstrated a close relationship between the promotion of self-determination and positive school (e.g., academic goal attainment) and post-school (e.g., employment) outcomes. Therefore, this study examines implementation of an evidence-based intervention to promote self-determination in inclusive, general education classrooms. This study utilized multilevel linear modeling to examine changes in student self-determination in secondary mathematics classes after a year of exposure to The Self-Determined Learning Model of Instruction (SDLMI). Participants included were 81 students ranging from 10th to 12th grade learning in secondary mathematics classes (Algebra II, Pre-Calculus, and Advanced Placement Calculus). The findings demonstrated a small, positive effect on student self-determination and that goal attainment during the first semester predicted second semester goal attainment. Specifically, the inclusion of goal attainment as a predictor explained 10% of the variability in self-determination observed, corresponding to a small effect. Overall, students who attained at higher level of goal attainment had higher self-determination at the conclusion of the study and were expected to grow in their self-determination at a higher rate than those with lower levels of goal attainment. The results from this study suggest that implementing the SDLMI to support all students in secondary mathematics classes positively impacts student attainment of skills that impact self-determination and has the potential to impact mathematical learning in future research.

Технология проектирования элективных курсов по математике при подготовке к ЕГЭ с целью развития вероятностного стиля мышления обучающихся

In today’s rapidly changing socio-economic conditions, people often have to face the problems, the solution of which is non-standard. A potentially possible means of the creative problem solving is a probabilistic style of thinking, which implies a departure from stereotypes. It is often connected with a refusal to act according to strictly determined rules, and a deviation of the negative attitude to the randomness of many phenomena and processes of the world. The order is born out of chaos as a result of self-organization. The article proposes a solution to the problem of achieving the quality and productivity of education through the development of schoolchildren’s probabilistic thinking style while training them for the Unifi ed State Examination in mathematics of the profi le level. A substantial component of the elective course “Probabilistic nature of mathematical problems of the Unifi ed State Exam” in mathematics in the system of general education is proposed. A group of tasks is presented which are done with the use of operations: the analysis and synthesis, combination, transposition, evaluation. The actualization of the components of the probabilistic style of thinking is aimed at improving the effi ciency of the learning process, improving the entire holistic system of development and self-development of the student’s personality. The main task of the informative fi lling of the elective course in mathematics in general education is to form a unifi ed scientifi c picture of the world in students, to develop a new culture of thinking, to identify interdisciplinary connections and to combine diff erent activities in the learning process. The reported study was funded by RFBR according to the research project № 18-313-20002 “Theoretical and methodical ensuring the fractal formation and development of the probabilistic style of thinking in the conditions of global informatization of education (on the example of teaching mathematics)”.

Promoting Creativity in General Education Mathematics Courses

This paper describes the development and implementation of course modules intended to encourage creative thinking in an undergraduate general education mathematics course. The modules were designed to address the characteristics of creativity outlined in the literature through explorations of mathematics relevant to non-mathematics majors. A general description of the modules, as well as specifics about two lessons is provided. Student work, including journal entries, surveys, and class assignments, provide evidence that this intentional focus on creativity in mathematics challenged students’ conceptions about mathematics, allowed them to reconsider the mathematics familiar to them in new ways, and engaged them in meaningful collaborations. Interviews of instructors revealed that they, too, engaged in creative processes as they planned the course and thought about innovative ways to engage students with mathematics. The course modules have been shared with other faculty teaching the course and will eventually be adapted for use in other mathematics courses, including those for STEM majors. Implications and future avenues for implementation and research are discussed.

Recommendations for Inclusive Educational Practices in Mathematics for Students With Extensive Support Needs

Abstract There is very limited research on inclusive mathematics instruction for students with extensive support needs (i.e., moderate to severe intellectual disability, autism, multiple disabilities) and this is a critical area of need. The purpose of this article is to propose a framework for mathematics instruction for students with extensive support needs in inclusive settings using practices that have been found effective in more restrictive settings, combined with the limited findings on effective inclusive practices for students with extensive support needs. A conceptual model for mathematics instruction for students with extensive support needs is illustrated and described. The proposed model provides guidance for collaborative teams and key stakeholders when planning for mathematics instruction in inclusive settings. Barriers to including students with extensive support needs in general education mathematics settings are discussed and recommendations for overcoming these barriers are described.

Using Plan-Do-Study-Act Cycles and Interdisciplinary Conversations to Transform Introductory Mathematics Courses

We describe how faculty and staff used improvement science design Plan-Do-Study-Act (PDSA) cycles and engaged in interdisciplinary conversations in an effort to improve students’ performance in introductory mathematics courses. This initiative started with College Algebra a course with a large enrollment of students that had a high failure rate. We engaged in PDSA cycles over a period of 4 years. From 2011–2015, we used data from student assessment scores, course evaluations, lab surveys, and instructors’ observations to inform decisions to transform introductory mathematics courses. The transformative efforts positively enhanced students’ performance. Furthermore, to reflect on our PDSA cycles and the broader implications of institutionalized change to mathematics courses, we engaged in frequent interdisciplinary conversations to discuss complexities and challenges. These meetings provided an opportunity to build a common vision about the nature and delivery of the mathematics taught to better serve students The themes of the conversations were on curriculum, assessment, instruction, the complexities of the change process, cultivating a growth mindset, and the need to engage in research in general education mathematics courses. Therefore, in this article we provide insights as to how interdisciplinary conversations were used to implement evidence-based teaching practices and also pedagogical approaches that can be used to facilitate students learning.

Measuring Income Inequality in a General Education or Calculus Mathematics Classroom

Income inequality is a central social justice concern, and hence excellent motivation for real-world applications in mathematics classrooms at every level. We describe the Mathematics for Social Justice course at Saint Michael’s College, giving a specific example of one of the typical social justice projects for the course, and showing how projects can be adapted to other courses such as Calculus. The projects described focus on the Gini coefficient, a commonly used measure of income inequality. The original lesson used the trapezoid rule and Microsoft Excel to estimate the Gini coefficient for a country, while the project developed for a Calculus I course uses Maple to fit a power function to data and then integration to calculate the Gini coefficient. We also include readings on the Gini coefficient’s role in policy formulation and advocacy.

Implementing a RtI tier 2 mathematics lab in a middle school

Response to intervention (RtI) in secondary schools is a viable option for mathematics, but limited research exists. In the study, students who received just Tier 1 (i.e., general education mathematics class) and students who received both Tier 1 and Tier 2 (i.e., a math lab) completed a pretest and posttest of grade-level content. The results suggest no statistically significant difference between the overall mean assessment gain scores of the two groups. The Tier 2 math lab intervention as implemented did not further exacerbate mathematical differences between struggling students and their peers but it did not work to decrease the mathematics deficits of students needing intervention either. The efficacy of a Tier 2 math lab as a RtI system at the secondary level can be questioned.

Puzzles as a Didactic Tool for Development of Mathematical Abilities of Junior Schoolchildren in Basic and Additional Mathematical Education

Pedagogical science has always faced the issue of finding effective means for achieving educational results. This problem is especially urgent today, when in the rapidly changing world the tools, which yesterday could be used to support the interest of schoolchildren in study of mathematics and could provide an opportunity for the development of their mathematical abilities, quickly become obsolete. Today it is very important to search for new means that foster the development of students with the help of mathematics and mechanisms for including mathematics in the educational process. Thus, the aim of the article is to analyze puzzles as a didactic tool and study the possibilities of using puzzles in the process of teaching junior schoolchildren mathematics, both in the classroom and extra-curricular activities. The leading method here is the modeling of the methodical training system in general and additional mathematical education of schoolchildren, with the inclusion of a new didactic tool that fosters the students’ interest to the subject, develops individual mathematical abilities: logical thinking, abstraction, combining, operating spatial images, critical thinking, mathematical memory, etc. As a result of the research, the authors have determined the place, features and methodological aspects of the inclusion of puzzles in the process of teaching mathematics in general and additional school education. They can be used in the system of classical and creative math lessons and in extra-curricular activities of students: a mathematical club, a system of mathematical competitions, a mathematical camp, etc. The practical use of this model makes it possible to reduce the lack of tools in teaching for the development of students’ mathematical abilities, which in its turn, makes it possible to speak of purposefully high results in students’ mathematical activities, which is confirmed by the conducted experimental research.