Search

Evaluation has become a common, and even an expected, practice across undergraduate mathematics tutoring centres in the USA, UK and other countries. However, these evaluation efforts could benefit greatly from leveraging organizational theory research. In this study, we situate mathematics tutoring centres as non-profit organizations (NPOs) to consider how an organization performance assessment framework can be adapted to study mathematics tutoring centre data and characteristics. We use qualitative and quantitative data, collected from six mathematics tutoring centres and adapt Lee & Nowell's (2015, "Am. J. Eval.," 36, 299-319) NPO performance framework to situate our study. Using thematic analysis, the research team underwent iterative cycles of data collection and analysis to code for alignment with Lee and Nowell's framework. By adapting Lee and Nowell's framework to mathematics centres, each of the six centres was given a more relevant lens to consider its performance. Regardless of the university setting, previous evaluations for centres have focused primarily on outputs (e.g., number of visits), behavioural change outcomes (e.g., correlating visits to grades) and client satisfaction outcomes (e.g., student surveys) that ignore the particular context of a centre. However, Lee and Nowell's framework takes into account additional performance indicators that provide a more nuanced understanding of a centre's performance by bringing to light the interplay among its various dimensions. Lee and Nowell's framework allows centres to look beyond outputs and outcomes to understand why these outputs and outcomes come to be. The use of this adapted performance framework, for the six mathematics centres in this study, allows an interpretation on a variety of dimensions using relevant data while indicating possible areas for change for each centre.

This study analyses a standards-based textbook series to investigate how slope is presented across seven sequential textbooks in a secondary mathematics curriculum. Analysis of the expository content across the textbooks shows that slope is covered in all seven books, is frequently described as a "Constant Parameter" or "Ratio," is frequently used in real-world contexts, and is often described alongside covariational language of how inputs and outputs change in relation to one another. Results indicate that "Determining Property" and "Steepness" components of slope are less prevalent and disproportionately unlikely to be accompanied by covariational language and applications compared with the other slope components. Although many consistencies were found across the textbook series, differences in emphases from one textbook to another are also discussed and connected to the recommendation that future work around slope investigate its development across the secondary curriculum. Results are discussed relative to students' preparation for specific calculus topics that build on prior knowledge of slope. Implications for both teachers and researchers are provided.

In this case study, we use organizational development and change theory to look at how the University of Oklahoma (OU) has attended to the mathematical pathway leading to STEM fields. Using multiple sources of data, we are able to create an informative narrative that provides the particular context of the situation and in-depth picture of the campus response, including key changes and ongoing challenges that were noted over a five-year period. Through the use of both internal and external funding, OU has been able to improve its teaching and learning in the STEM mathematical pathway. OU has also been able to develop a First-Year Mathematics program based on national recommendations. We report how efforts have directly impacted hiring, placement, pedagogical methods, course coordination, TA training, as well as the collection and use of data in decision making.

In this study, we investigate a sample of in-service teachers' professional noticing of slope. The analysis focuses on which elements of students' statements about slope teachers attend to, and how teachers interpreted those elements. Study results suggest that teachers attended to the vocabulary and mathematical focus of students' statements. Teachers' interpretations were, for the most part, rather consistent across teachers and in line with the researchers' anticipations for elicited slope components. In fact, even for vague student statements expected to elicit a variety of interpretations by teachers, there was a noted lack of diversity in teachers' interpretations. Results for teachers' interpretations suggest they place great academic value on nonvisual slope reasoning and attribute the ability to solve real-world problems to students who used "rate of change" terminology. Other real-world interpretations of slope were interpreted as disconnected from the mathematical meaning of slope or only trivially linked to contextual situations. Findings from this study imply that teachers should be provided with professional development opportunities involving slope. Specifically, teachers need experiences interacting with tasks that build robust notions of steepness through explicit connections to a variety of physical contexts by allowing students to develop imagery and mathematical terminology in rich and meaningful ways. In addition, professional development should help uncover the meaning behind students' vocabulary rather than having them focus on buzz words and catch phrases, which may be used without understanding.

Slope is a fundamental mathematics concept in middle and high school that transcends to the university level. An understanding of slope is needed at the university level since slope plays an important role in understanding problems involving variation and change. In this study Mexican curricula documents were examined to determine which conceptualizations of slope are addressed in the intended mathematics curriculum. To explain the results, we use conceptualizations of slope identified in previous research. Our findings reveal that, to a certain extent, the conceptualizations proposed in the Mexican intended mathematics curriculum differ slightly in terms of the emphasis and timing of instruction from what others have identified in the U.S., with slope as a geometric ratio receiving less emphasis in the Mexican curriculum. There was also noted discontinuity within the Mexican curriculum in introducing slope in grade 9 and subsequently introducing of linear functions in grade 10 without explicit mention of slope. Suggestions are made for future studies, both to consider the conceptualizations of slope promoted in the Mexican textbooks and the impact they have in classroom instruction and student learning of slope.

This article provides an initial comparison of the Principles and Standards for School Mathematics and the Common Core State Standards for Mathematics by examining the fundamental notion of slope. Each set of standards is analyzed using eleven previously identified conceptualizations of slope. Both sets of standards emphasize Functional Property, Real World Situation, and Linear Constant conceptualizations of slope and describe a similar instructional sequence during the middle grades. However, the elementary and high school standards include differences that reflect alternative approaches to covering key prerequisite notions related to slope and in extending ideas of slope to non-linear functions. Both documents are examined in light of their respective purposes, with careful attention to potentially unintended consequences on the treatment of slope across the curriculum. The findings warrant careful consideration of other key mathematical topics to understand the curricular implications of adopting a new set of standards.

In the field of mathematics education, understanding teachers' content knowledge (Grossman, 1995; Hill, Sleep, Lewis, & Ball, 2007; Munby, Russell, & Martin, 2001) and studying the relationship between content knowledge and instructional decisions (Fennema & Franke, 1992; Raymond, 1997) are both crucial. Teachers need a robust understanding of key mathematical topics and connections to make informed choices about which instructional tasks will be assigned and how the content will be represented (Ball & Bass, 2000, Fennema & Franke, 1992). Ma (1999) described this "profound understanding of fundamental mathematics" as how accomplished teachers conceptualize key ideas in mathematics with a deep and flexible understanding so that they are able to represent those ideas in multiple ways and to recognize how those ideas fit into the preK-16 curriculum. Slope is a fundamental topic in the secondary mathematics curricula. Unit rate and proportional relationships introduced in sixth grade prepare students for interpreting equations such as y = 2x-3 as functions with particular, linear behavior in eighth grade (National Governors Association [NGA] Center for Best Practices & Council of Chief State School Officers [CSSO]), 2010; National Council of Teachers of Mathematics [NCTM], 2006). The focus on relationships with constant rate of change leads to distinctions between linear and non-linear functions (Yerushalmy, 1997) and the idea of average rate of change in high school (NGA Center for Best Practices & CCSSO, 2010). Ultimately, these ideas prepare students for instantaneous rates of change and the concept of a derivative in calculus. The diversity of conceptualizations and representations of slope across the secondary mathematics curriculum presents a challenge for secondary teachers. These teachers must work flexibly and fluently with various representations in many contexts in order for their students to build a coherent, connected conceptualization of slope. Since secondary mathematics teachers need a deep understanding of slope to mediate students' conceptual development of this key topic, the study reported here investigates both how teachers think about and present slope.

Historically, the teaching of design theory in an engineering curriculum has been relegated to a senior capstone design experience. Presently, however, engineering design concepts and courses can be found through the entirety of most engineering programs. Educators have recognized that engineering design provides a foundational platform that can be used to develop educational strategies for a wide array of engineering science principles. More recently, educators have found that product archaeology provides an effective platform to develop scalable learning materials, strategies, and educational innovations across these design courses. This paper presents and discusses how product archaeology has been incorporated at a large research university in two design-related courses for mechanical engineering students: (1) a sophomore-level course and (2) a senior-level class. More specifically, details are reported regarding how and how easily global, societal, economic, and environmental factors were emphasized in the curricula of these courses. Next, the paper shares the qualitative and quantitative assessment tools and methods used to determine the impact of incorporating a product archaeology paradigm in the courses. Finally, the results are reported which demonstrate a significant increase in the students' perceptions across a number of skill and knowledge areas related to ABET-required Outcome h without negatively impacting other important academic areas. Results demonstrate a significant increase in student perception across a number of skill and knowledge areas critical to the next generation of engineers.

Undergraduate mathematics tutoring centres are prevalent in many countries; however, there is limited research-based evidence on effective organizational structures for these centres. In this study, we consider two research questions. First, how can the quantitative and qualitative data from 10 mathematics tutoring centres be organized for research purposes? Second, what hypotheses do expert mathematics tutoring centre leaders generate about characteristics of effective centres given data from a sample of ten centres? We collected quantitative data from over 26,000 students taking mathematics courses at ten institutions. Data collected included college entrance exam scores, high school grade point average, number of student visits to the centre per eligible student and course letter grade. We used exploratory data analysis to look for relationships between visits to the tutoring centre, student grades and other variables. Qualitative centre characteristics that were considered include: specialist–generalist tutoring system, tutoring capacity, physical layout, relationships between tutors and mathematics instructors and extent of tutor training. We used the Delphi process to generate testable hypotheses from the data, such as the following: (1) The more courses a tutor is responsible for tutoring the more likely it is that the tutor will struggle to answer student questions, when the difficulty level of the courses is roughly the same. (2) Centres with more specialized tutor models have more visits per student than centres with generalized tutor models. The preceding two hypotheses, along with the other generated hypotheses, have been identified by the experts participating in this study as plausible based on professional experience, exploratory data analysis and inferences based on prior research on tutoring. This study has not rigorously shown the validity of these hypotheses; rather it lays the groundwork for future investigations to determine what combination of features characterize an effective tutoring centre. [ABSTRACT FROM AUTHOR]

This paper brings together research on slope from mathematics education and research on line of best fit from statistics education by considering what knowledge of slope students transfer to a novel task involving determining the placement of an informal line of best fit. This study focuses on two students who transitioned from placing inaccurate to accurate lines of best fit during a task-based interview. The analysis focuses on describing shifts in slope reasoning that accompanied the change to accurately placed lines, and investigates factors that may have influenced the shift in reasoning. The results have implications for the teaching of both slope and the line of best fit.

This study focuses on how students in different postsecondary mathematics courses perform on domain and range tasks regarding graphs of functions. Students often seem to focus on notable aspects of a graph and fail to see the graph in its entirety. Many students struggled with piecewise functions, especially those involving horizontal segments. Findings indicate that Calculus I students performed better on domain tasks than students in lower math courses; however, they did not outperform students in lower math courses on range tasks. In general, student performance did not provide evidence of a deep understanding of domain and range.

As college students enter engineering, they face challenges when they navigate across various transitions. These challenges impact whether a student can successfully adapt to the rigorous curricular requirements of an engineering degree and to the norms and expectations that are particular to engineering. This article focuses on the transitions between disciplinary literacies, primarily addressing how ideas are communicated and which types of semiotic representations are valued, required, and ignored by the different cultures that come into play both prior and during a student's engineering education. Specifically, it addresses navigation: (1) Across secondary to postsecondary institutional expectations, contrasting common curricular experiences of secondary students and postsecondary curricular expectations; (2) Between and within disciplines (including mathematics and science), which often have different ways of communicating and representing a single concept, especially in theoretical applications and practical calculations; and (3) From the traditional pedagogies of classrooms to the different methods and expectations of laboratory settings (for example, collaborative engagement in planning, experimentation, an understanding of experimental error, and an understanding of how all three are communicated and represented).

In this article, we provide a commentary on the current state of gamification in higher education by reviewing the research literature, noting a lack of research focused on the design of gamified postsecondary courses. To address this issue, we discuss four potential pitfalls instructors may encounter when designing gamified learning activities. We conclude with suggestions related to how instructors can avoid these common stumbling blocks.

In two dimensions (2D), representations associated with slopes are seen in numerous forms before representations associated with derivatives are presented. These include the slope between two points and the constant slope of a linear function of a single variable. In almost all multivariable calculus textbooks, however, the first discussion of slopes in three dimensions (3D) is seen with the introduction of partial derivatives. The nature of the discussions indicates that authors seem to assume that students are able to naturally extend the concept of a 2D slope to 3D and correspondingly it is not necessary to explicitly present slopes in 3D. This article presents results comparing students that do not explicitly discuss slopes in 3D with students that explicitly discuss slopes in 3D as a precursor to discussing derivatives in 3D. The results indicate that students may, in fact, have significant difficulty extending the concept of a 2D slope to a 3D slope. And that the explicit presentation of slopes in 3D as a precursor to the presentation of derivatives in 3D may significantly improve student comprehension of topics of differentiation in multivariable calculus.

Reviewing numerous textbooks, we found that in both differential and integral calculus textbooks the authors commonly assume that: (i) students can generalize associations between representations in two dimensions to associations between representations of the same mathematical concept in three dimensions on their own; and (ii) explicit discussions of these representations are not necessary. For example, in the presentation of partial derivatives, textbook presentations assume that students will understand the slope in a specified direction in without it ever being explicitly presented. Our preliminary interviews indicated that such assumptions may be erroneous, so we created and tested materials that explicitly use representations commonly omitted in textbook presentations of the differential and integral calculus of functions of two and three variables. This paper discusses the classroom activities that were created to include these missing representations, as well as the results when they were implemented in classroom instruction.

A test project at the University of Puerto Rico in Mayagüez used GeoGebra applets to promote the concept of multirepresentational fluency among high school mathematics preservice teachers. For this study, this fluency was defined as simultaneous awareness of all representations associated with a mathematical concept, as measured by the ability to pass seamlessly among verbal, geometric, symbolic, and numerical representations of the same mathematical object. The preservice teachers in this study attended a seminar where they were introduced to the underlying concepts and the pedagogical advantages of multirepresentational fluency. For select topics, this idea was reinforced with interactive GeoGebra applets that allowed preservice teachers to alter a parameter and simultaneously view how it changes all four associated representations simultaneously. A qualitative study found that this approach appeared to (a) promote the use of multirepresentational fluency in problem solving approaches used among preservice teachers, (b) change preservice teachers' perceptions of what it means for a student to understand a concept, and (c) change the nature of evaluations that preservice teachers felt were appropriate for high school students.

While digital environments can offer convenient, viable options for preservice and inservice teachers to engage in or continue their studies, little is known about teachers' experiences with and perceptions of various existing online learning spaces. This paper describes an initial investigation using data from a group of preservice and in-service mathematics teachers who interacted by posting their reflections regarding online learning spaces to an asynchronous, electronic discussion board. Inductive qualitative techniques were incorporated to determine which online learning spaces study participants had experienced as well as their perceptions of each. Results are reported in light of possible implications for teacher education, including specific suggestions for additional study of online learning spaces.

The authors argue for a broader view of what it means to be literate to include meaning making with both linguistic and visual representations.

This study considers tertiary calculus students' and instructors' conceptualizations of slope. Qualitative techniques were employed to classify responses to 5 items using conceptualizations of slope identified across various research settings. Students' responses suggest that they rely on procedurally based conceptualizations of slope, showing little evidence of covariational reasoning. In contrast, instructors' responses demonstrated a multi-dimensional understanding of slope as a functional property, which applies to real-world situations and plays an integral role in the development of key calculus concepts. While relatively diverse, the instructors' responses seldom reported determining increasing or decreasing trends of a line from its slope. This conceptualization was used frequently by students and could help them better understand how slope ties to positive and negative derivatives. The most frequently used conceptualizations for students in this study align with past research findings on the emphasis of the secondary mathematics curriculum, supporting the possibility of cultural influences (academic and geographic) on individuals' conceptualizations of slope. Thus, this study provides valuable insight into conceptualizations of slope and provides direction for future research on slope and the broader topic of cultural influences on mathematical meaning.

All teachers, especially high school teachers, face the challenge of ensuring that students have opportunities to relate and connect the various representations and notions of mathematics concepts developed over the course of the pre-K-12 mathematics curriculum. NCTM's (2000) Representation Standard emphasizes the importance of students being able to represent a concept in various forms--for example, verbally, numerically, graphically, and analytically--but students' recognition that each representation depicts the same underlying concept is of equal importance. When given one ordered pair and a constant rate of change between two variables, students should be able to graph a line, identify the parameters of a linear equation, and complete a table. However, being able to use each representation independently is not sufficient; students need to connect the representations and recognize how each indicates a specific relationship between two simultaneously varying quantities. Thus, when teaching slope, the emphasis should be not only on using multiple representations but also on connecting various representations to form a coherent and complete conception of this mathematical idea. A student activity sheet in the appendix provides a series of tasks that can be used to engage students in a sense-making process by using GeoGebra or other similar dynamic geometry software. These activities encourage students to apply their prior knowledge of slope and steepness as they reason about new problem situations. Such activities give students a chance to make connections between the two ideas and, ultimately, to come to a deeper understanding of slope (Pirie and Martin 2000).

This paper conceptualizes spatial literacy as consisting of three overlapping domains: visualization, reasoning, and communication. By considering these domains, this study explores different aspects of spatial literacy to better understand how a group of mathematics teachers reasoned about spatial tasks. Seventy-five preservice and inservice teachers worked on problems that involved spatial objects, their properties, and their relationships. Teachers' responses suggested that their spatial literacy skills were underdeveloped with deficiencies most evident on problems that were solvable by dimensional reasoning. Poor vocabulary and misconceptions hindered teachers' performance. Teachers who used multiple reasoning strategies were more likely to solve a problem correctly. (Contains 6 tables and 9 figures.)

What does it mean to think like a mathematician? One of the great paradoxes of mathematics education is that, although mathematics teachers are immersed in mathematical work every day of their professional lives, most of them nevertheless have little experience with the kind of work that research mathematicians do. Their ideas of what doing mathematics looks like are based mainly on their own experiences as students. Creating opportunities for students to engage in the kind of flexible thinking that is characteristic of mathematicians' practice can be a challenge for all teachers. For their classrooms to become venues where students experience mathematical activities that foster flexible thinking, teachers need to change how they think about and do mathematics. Teachers at all levels should make the mechanisms of problem posing explicit and draw attention to how they can be used to navigate through open-ended problems. Through engagement in such mathematical activities, teachers and their students might come to view mathematics differently. By extension, these teachers might come to teach mathematics in a manner that is more authentic to actual mathematical practice. (Contains 2 figures.)

This paper reports on a study that introduces and applies the "K[subscript 5]Connected Cognition Diagram" as a lens to explore video data showing teachers' interactions related to the partitioning of regions by axes in a three-dimensional geometric space. The study considers "semiotic bundles" (Arzarello, 2006), introduces "semiotic connections," and discusses the fundamental role each plays in developing individual understanding and communication with peers. While all teachers solved the problem posed, many failed to make or verbalize connections between the types of semiotic resources introduced during their discussions. (Contains 8 figures and 2 tables.)

Since slope is a fundamental topic that is embedded throughout the U.S. secondary school curriculum, this study examined standards documents for all 50 states to determine how they address the concept of slope. The study used eleven conceptualizations of slope as categories to classify the material in the documents. The findings indicate that all of the slope conceptualization categories were evidenced in the standards documents with the vast majority of state documents addressing five or more of the eleven conceptualizations. Results are reported on the most and least commonly documented conceptualizations of slope. The findings provide evidence that there tends to be consensus among the states as to the conceptualizations of slope that should be addressed. Suggestions are made for future work both to consider the initial conceptualizations of slope that students hold or most readily form when introduced to the concept as well as to consider if (or how) slope instruction takes into account students' initial conceptualizations of slope.

It is common for both algebra and calculus instructors to use power functions of various degrees as well as exponential functions to examine and compare rates of growth. This can be done on a chalkboard, with a graphing calculator, or with a spreadsheet. Instructors often are careful to connect the symbolic and graphical (and occasionally the tabular) representations of the functions. However, the graphs that are typically used for this are static. The most recent versions of Microsoft[R] Excel[R] (Excel) allow instructors to illustrate the connections between the symbolic, tabular, and graphical representations of the equations through quick generation of the function graphs. This requires only minimal input including three components: the equation of the function, its starting point, and the incremental changes between independent variable values. By formatting the spreadsheet to depend on these three things, the input values (and the calculated output values) are easily manipulated, allowing for changes in scale. (Contains 11 figures.)