Metaphor Clusters: Characterizing Instructor Metaphorical Reasoning on Limit Concepts in College Calculus

Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student metaphorical reasoning, we examined 11 college instructors' metaphorical reasoning on limit concepts. This paper focused on previous research of metaphor clusters observed among students to answer the following: (a) Do college instructors use metaphorical reasoning to conceptualize the meaning of a limit? (b) Can we characterize instructor metaphorical reasoning similar to those observed among students? (c) Will an instructor's self‐identification of metaphor clusters be consistent with our metaphor coding? We found that college instructors' perspectives vary, either graphical or algebraic, in their explanations of limit items. All the instructors used metaphors, and instructor metaphorical reasoning aligned with student metaphor clusters. Instructors tended to change their metaphors with respect to the limit item. Instructors were not aware of their use of metaphors, nor were they aware of their inconsistency in their choice of metaphor. We believe that instructor awareness of their own distinct perspectives and metaphors would assist students' understanding of limit concepts.     

Relationships Between Student,Teacher, and School Characteristics and Mathematics Achievement

A variety of factors contributes to student achievement in mathematics, including but not limited to student behaviors and student, teacher, and school characteristics. The purpose of this study was to explore which of these factors have an impact on student mathematics achievement. The target population for this study was North Carolina Algebra II students. Analyses of variance models were examined for group differences and a Three‐level Hierarchical Linear Modeling method was employed to examine individual predictors of student achievement in mathematics. Statistically, significant differences were found between students of different ethnicities, socioeconomic statuses (SES), and parental education levels. No gender effects were statistically significant. All teacher‐level variables investigated were found to be statistically significant, impacting student achievement in mathematics. School size and SES were not found to significantly contribute to student achievement. More research on the relationships between these factors shown to make statistically significant differences on mathematics achievement is needed to further explain several phenomena that this research reveals.     

Assessing the Use of Graphing Calculators in College Algebra: Reflecting on Dimensions of Teaching and Learning

This article considers various aspects of teaching and learning stemming from the integration of graphing calculator use in a semester-long college algebra course. The project examined four class sections, in which two instructors each taught one section using graphing calculators and one section using a traditional approach. Achievement and attitude data showed no significant differences for treatment or instructor; a significant difference in achievement was found for gender. Students in the calculator sections responded to an open-ended questionnaire about their use of the calculator and were generally supportive of the technology. Students were in agreement about specific topics for which the technology was most useful. Overall, findings indicated that the technology use had a positive impact on various dimensions of student learning.     

Beating the System: Course Structure and Student Strategies in a Traditional Introductory Undergraduate Physics Course for Nonmajors

The purpose of this study was to describe relationships among instructor and student goals, course design, and student strategies for learning physics in a traditional introductory undergraduate physics course for nonmajors. The procedures included 16 hours ofnonparticipant observation, during which detailed field notes and photographs were taken and documents were collected. Audiotaped open-ended interviews were given to instructors and students, and all students participated in a limited survey. The data were analyzed using qualitative methods of the Chicago School. Results indicated that both differing goals of students and instructors and the structure of the course inhibited the understanding of physics and were conducive to students engaging in unexpected behaviors designed to help them successfully pass the course. The consequences of the structure of the course and subsequent student behaviors inhibited their understanding and appreciation of physics and deterred them from enrolling in further science courses. The paper includes implications for teaching introductory college physics.     

The impact of instructor pedagogy on college calculus students’ attitude toward mathematics

College calculus teaches students important mathematical concepts and skills. The course also has a substantial impact on students’ attitude toward mathematics, affecting their career aspirations and desires to take more mathematics. This national US study of 3103 students at 123 colleges and universities tracks changes in students’ attitudes toward mathematics during a ‘mainstream’ calculus course while controlling for student backgrounds. The attitude measure combines students’ self-ratings of their mathematics confidence, interest in, and enjoyment of mathematics. Three major kinds of instructor pedagogy, identified through the factor analysis of 61 student-reported variables, are investigated for impact on student attitude as follows: (1) instructors who employ generally accepted ‘good teaching’ practices (e.g. clarity in presentation and answering questions, useful homework, fair exams, help outside of class) are found to have the most positive impact, particularly with students who began with a weaker initial attitude. (2) Use of educational ‘technology’ (e.g. graphing calculators, for demonstrations, in homework), on average, is found to have no impact on attitudes, except when used by graduate student instructors, which negatively affects students’ attitudes towards mathematics. (3) ‘Ambitious teaching’ (e.g. group work, word problems, ‘flipped’ reading, student explanations of thinking) has a small negative impact on student attitudes, while being a relatively more constructive influence only on students who already enjoyed a positive attitude toward mathematics and in classrooms with a large number of students. This study provides support for efforts to improve calculus teaching through the training of faculty and graduate students to use traditional ‘good teaching’ practices through professional development workshops and courses. As currently implemented, technology and ambitious pedagogical practices, while no doubt effective in certain classrooms, do not appear to have a reliable, positive impact on student attitudes toward mathematics.     

Creating discussions with classroom voting in linear algebra

We present a study of classroom voting in linear algebra, in which the instructors posed multiple-choice questions to the class and then allowed a few minutes for consideration and small-group discussion. After each student in the class voted on the correct answer using a classroom response system, a set of clickers, the instructor then guided a class-wide discussion of the results. We recorded the percentage of students voting for each option on each question used in 18 sections of linear algebra, taught by 10 instructors, at 8 institutions, over the course of 5 years, together recording the results of 781 votes on a collection of 311 questions. To find the questions most likely to provoke significant discussions, we identify the six questions for which votes were most broadly distributed. Here we present these questions, we discuss how we used them to advance student learning, and we discuss the common features of these questions, to identify why they were so good at stimulating discussions.     

Student performance and attitudes in a collaborative and flipped linear algebra course

Flipped learning is gaining traction in K-12 for enhancing students’ problem-solving skills at an early age; however, there is relatively little large-scale research showing its effectiveness in promoting better learning outcomes in higher education, especially in mathematics classes. In this study, we examined the data compiled from both quantitative and qualitative measures such as item scores on a common final and attitude survey results between a flipped and a traditional Introductory Linear Algebra class taught by two individual instructors at a state university in California in Fall 2013. Students in the flipped class were asked to watch short video lectures made by the instructor and complete a short online quiz prior to each class attendance. The class time was completely devoted to problem solving in group settings where students were prompted to communicate their reasoning with proper mathematical terms and structured sentences verbally and in writing. Examination of the quality and depth of student responses from the common final exam showed that students in the flipped class produced more comprehensive and well-explained responses to the questions that required reasoning, creating examples, and more complex use of mathematical objects. Furthermore, students in the flipped class performed superiorly in the overall comprehension of the content with a 21% increase in the median final exam score. Overall, students felt more confident about their ability to learn mathematics independently, showed better retention of materials over time, and enjoyed the flipped experience.     

Collaborative statistical investigations in diverse settings

This study presents a continuing investigation of influences on outcomes achieved by students working in groups of three on tasks related to chance and data. Earlier research described final mathematical outcomes and identified 17 factors influencing three types of short-term outcomes for groups working in an isolated setting. The current report documents the 17 factors for groups working in a classroom setting and 1654 events for all groups in both settings are identified and each associated with a factor and a short-term outcome. Consideration is then given to variables that have the potential to influence the factors, the short-term outcomes, and their interaction. The overarching variable is the setting within which the collaboration took place. Within each setting, however, two other variables operated: the task carried out, the age/grade of students, gender balance, or collaborative characteristics. The influences of these variables are described within the two settings before consideration is given to the overall influence of the settings.     

The Relationships Between Parental Influence and Student Achievement in Seventh Grade Mathematics

Parental influence has been identified as an important factor affecting student achievement, but the variables addressed by various parent-school partnership programs vary, and lack literature support. This deficiency was ameliorated in this study through an analysis of a national data base from the Longitudinal Study of American Youth (LSAY). AH parental variables investigated in LSAY were considered in this study, and significant variables of students' achievement in seventh grade mathematics were selected through the Backward Elimination Procedure in the Statistical Analysis System (SAS). The results show that those variables significant at α= .05 have accounted for around 22% variability of the student achievement. A further analysis of the empirical results seems to indicate that parent education and encouragement are strongly related to improved student achievement. However, parents should attempt to understand students' feeling, and expect students to take the full responsibility for their successes in the school.     

Relationships between emotional dispositions and mathematics achievement moderated by instructional practices: analysis of TIMSS 2015

ABSTRACT

This research is a secondary analysis with Korean students’ data collected in the TIMSS 2015 to describe the moderation effects of instructional practices on the relationships between students’ emotional dispositions toward mathematics and mathematics achievement. From the TIMSS 2015 database, we collected mathematics achievement scores, a student-level contextual scale for students’ emotional disposition, and teacher-level contextual scales representing teachers’ instructional practices. We applied hierarchical linear modelling to construct multilevel models. The findings showed that the achievement gap between emotional dispositions – like and dislike – became smaller when teachers more frequently implemented certain instructional practices like asking students to complete challenging exercises, decide their own problem-solving procedures, and express their ideas in class. Students who disliked mathematics were likely to have higher scores as their teachers implemented each of those practices more frequently. Findings provide important implications to teachers regarding: It is important to encourage students to reason through instructional practices like asking them to decide their own problem-solving procedures and to solve challenging problems.     

The Impact of Participation in an Ancillary Science and Mathematics Program (SEMAA) on Engagement Rates of Middle School Students in Regular Mathematics Classrooms

The purpose of this study was to investigate the impact of participation in a federally sponsored, short‐term, cocurricular, mathematics and science program (Science Engineering Mathematics Aerospace Academy, SEMAA) on the engagement rates of sixth‐ and seventh‐grade students in public school mathematics classes. Engagement was measured with the Student Record of Behavior at three time intervals. Results of a 22.3 ANOVA investigating three main effects (participation, level of access to technology, and time) and their primary and secondary interactions reflected no discernable impact of the SEMAA program on student engagement rates. Ancillary programs designed to compensate for deficiencies in daily instructional programs may represent engagement opportunities vastly different from the daily instructional programs they support. Consequently, ancillary programs may not impact engagement in regular classrooms and subsequently improve achievement outcomes, especially when implemented in low‐performing schools and high‐stakes accountability settings. Recommendations include alignment of ancillary programs with the daily instructional programs they support and with ongoing professional development activities and that further study include broadened samples, settings, and variables.     

How do students’ behaviors relate to the growth of their mathematical ideas?

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie-Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school problem-solving sessions. The results suggest that certain types of student behaviors appear to be associated with the growth of ideas and emerge in specific patterns. More specifically, as understanding grows, there is a general shift from behaviors such as students questioning each other, explaining and using their own and others’ ideas toward behaviors involving the setting up of hypothetical situations, linking of representations and connecting of contexts. Recognizing that certain types of student behaviors tend to emerge in specific layers of the Pirie-Kieren model can be important in helping us to understand the development of mathematical ideas in children.     

Dot plots and hat plots: supporting young students emerging understandings of distribution,center and variability through modeling

An important use of statistical models and modeling in education stems from the potential to involve students more deeply with conceptions of distribution, variation and center. As models are key to statistical thinking, introducing students to modeling early in their schooling will likely support the statistical thinking that underpins later, more advanced work with increasingly sophisticated statistical models. In this case study, a class of 10–11 year-old students are engaged in an authentic task designed to elicit modeling. Multiple data sources were used to develop insights into student learning: lesson videotape, work samples and field notes. Through the use of dot plots and hat plots as data models, students made comparisons of the data sets, articulated the sources of variability in the data, sought to minimize the variability, and then used their models to both address the initial problem and to justify the effectiveness of their attempts to reduce induced variation. This research has implications for statistics curriculum in the early formal years of schooling.     

The Effect of a Horseshoe Crab Citizen Science Program on Middle School Student Science Performance and STEM Career Motivation

The purpose of the present quasi‐experimental study was to examine the impact of a horseshoe crab citizen science program on student achievement and science, technology, engineering, and mathematics (STEM) career motivation with 86 (n = 86) eighth‐grade students. The treatment group conducted fieldwork with naturalists and collected data for a professional biologist studying horseshoe crab speciation and a mock survey. The comparison group studied curriculum related to horseshoe crabs in the science classroom. A series of measures related to self‐efficacy, interest, outcome expectations, choice goals, and content knowledge were given to participants before and after the intervention. It was hypothesized that students would report higher motivational beliefs regarding science and show higher levels of achievement following the intervention than the comparison group. Support was shown for most of the hypotheses. In addition, path analyses indicated that students' motivational beliefs influence content knowledge and outcome expectations, which in turn affect their career goals. These results have implications for incorporating authentic fieldwork within a formal school structure as an effective method for promoting student achievement and STEM career motivation.     

Progressive, classical or balanced— A look at mathematical learning environments in Swiss-German lower-secondary schools

In a national supplement to TIMSS, lower-secondary school teachers (N=102) and their students (N=975) reported on mathematics instruction by means of a teacher questionnaire (teaching-learning methods, instructional sub-goals, facilitated student activities, achievement assessment, teacher role) and a student questionnaire (teachers' instructional proficiency, classroom climate). A cluster analysis performed on the ratings of teaching-learning methods yielded a solution with three clusters referred to as progressive, classical, and balanced learning environment. Cluster-related differences in facilitated student activities, achievement evaluation and preferred teacher role were found but not in instructional sub-goals. Students from different learning environments equally approved teachers' instructional proficiency and classroom climate and also had similar TIMSS mathematics scores. The results of this study provide empirical evidence that in addition to classical teacher-centered learning environments there seem to exist more diversified and studentcentered learning environments that address the needs for students to direct their own learning, communicate and work with others, and develop ways of dealing with complex problems. In line with the research literature it was also found that high mathematics achievement is not restricted to a certain type of learning environment.     

Heterogeneous Grouping as an Element of Cooperative Learning in an Elementary Education Science Course

The purpose of this study was to clarify the effectiveness of heterogeneous grouping as an element of cooperative learning. A pretest-posttest control group design was utilized. The independent variable was the method of arranging groups (heterogeneously or homogeneously). Dependent variables were scores from the NABT/NSTA Biology Exam (1987). Students were administered a questionnaire to determine perceptions of their cooperative learning situation. An analysis of covariance was used as the data analysis procedure for the experimental portion of the study. No significant differences in achievement were found between the two treatment conditions. Results from the student perceptions questionnaire did reveal substantial differences between students in the two treatment conditions, with generally more positive responses given by students in the homogeneous condition.     

The use of concrete learning objects taken from the history of mathematics in mathematics education

This study aimed to reveal the effects of teaching with concrete learning objects taken from the history of mathematics on student achievement. Being a quasi-experimental study, it was conducted with two grade 8 classes in a secondary school located in Trabzon. The experimental group consisted of 27 students and the control group consisted of 25. Data were collected by using worksheets, an achievement exam and written opinion forms. The data from the achievement exam were analysed by using the Mann-Whitney U-test while the data from written opinion forms were analysed through content analysis. The Mann–Whitney U-test results showed a significant difference between the mean ranks of the experimental and control groups in favour of the former. Findings from the written opinion forms suggested that the students found the activities to be instructive and fun, enjoyed using concrete models in their classes, and learned from discovering the rules. It was also found that students had previously not engaged in similar activities and had only experienced the history of mathematics through the life stories and works of mathematicians and the representation of ancient numbers at the beginning of each unit.     

Gender Differences in Science Achievement: Differential Effect of Ability,Response Format,and Strands of Learning Outcomes

The purpose of this study was to determine patterns of gender differences in science achievement of fifth graders, taking into account the role of ethnicity, ability, response format, and strands of learning outcomes. The results indicated that the gender differences in science achievement did not depend on ethnicity. However, response format, ability, and strands of learning outcomes had differential effects on student performance. In particular, at the high ability level boys did better than girls on the open-ended format in physical sciences, but there were no gender differences in nature of science, earth and space sciences, and life sciences. There were no gender differences in science achievement for the low and medium ability students. Regardless of gender, the largest achievement gaps between the low, medium, and high ability students were on the open-ended format in physical sciences.     

Using Primary Source Projects to teach undergraduate mathematics content: analysis of instructor implementation and perceptions

ABSTRACT

This study investigates changes in instructor teaching tendencies, instructor’s perception of impact on student learning and dispositions, and methods of implementation of Primary Source Projects (PSPs). PSPs are curricular modules designed to teach core mathematical topics from primary historical sources rather than from standard textbooks. In essence, they are a form of inquiry-based learning that incorporates the history of mathematics through original source texts. We provide an overview of results from two semesters of implementation reports and surveys administered at the beginning and end of the semester by instructors who implemented PSPs in their undergraduate mathematics classes.     

Some Thoughts on Computers and Introductory Statistics

Examination of results of an extensive project in the use of computers in undergraduate statistics reveals that the use is frequently unsuccessful because of insufficient maturity on the part of students. Nevertheless, even when students are not sufficiently mature to warrant hands‐on use, computers may still be used effectively. Indeed, when measured against the possible capabilities, instructor controlled computer usage can be just as effective as student use. An added bonus is economy of operation. Several suggestions for instructor controlled usage include graphical and numerical demonstrations, computer generated problem books with solutions, computer generated visual aids and loose‐leaf files of illustrations, examples and solutions.