Students’ understanding of the relation between tangent plane and directional derivatives of functions of two variables

APOS Theory is applied to study student understanding of directional derivatives of functions of two variables. A conjecture of mental constructions that students may do in order to come to understand the idea of a directional derivative is proposed and is tested by conducting semi-structured interviews with 26 students. The conjectured mental construction of directional derivative is largely based on the notion of slope. The interviews explored the specific conjectured constructions that student were able to do, the ones they had difficulty doing, as well as unexpected mental constructions that students seemed to do. The results of the empirical study suggest specific mental constructions that play a key role in the development of student understanding, common student difficulties typically overlooked in instruction, and ways to improve student understanding of this multivariable calculus topic. A refined version of the genetic decomposition for this concept is presented.     

The unit circle approach to the construction of the sine and cosine functions and their inverses: An application of APOS theory

We use Action-Process-Object-Schema (APOS) Theory to analyze the mental constructions made by students in developing a unit circle approach to the sine, cosine, and their corresponding inverse trigonometric functions. Student understanding of the inverse trigonometric functions has not received much attention in the mathematics education research literature. We conjectured a small number of mental constructions, (genetic decomposition) which seem to play a key role in student understanding of these functions. To test and refine the conjecture we held semi-structured interviews with eleven students who had just completed a traditional college trigonometry course. A detailed analysis of the interviews shows that the conjecture is useful in describing student behavior in problem solving situations. Results suggest that students having a process conception of the conjectured mental constructions can perform better in problem solving activities. We report on some observed student mental constructions which were unexpected and can help improve our genetic decomposition.     

Requirements in mathematics textbooks: a five-dimensional analysis of textbook exercises and examples

Mathematics textbooks play a very important role in mathematics education and textbook tasks are used by students for practice to a large extent. Since the nature of the tasks may influence the way students think it is important that the textbooks provide a balance of a variety of tasks. The analyses of the requirements in textbook tasks contain the usual dimensions of content, cognitive demands, question type and contextual features. The aim of this study is to embed a new fifth dimension into the framework: mathematical activities. This addresses the question of what a student should do in a particular textbook task: to represent, to compute, to interpret or to use argumentation. The analysis encompassed more than 22,000 tasks from the most commonly used Croatian mathematics textbooks in the 6th, 7th and 8th grade. The results show that the textbooks do not provide a full range of task types. There is an emphasis on computation, while argumentation and interpretation activities, reflective thinking and open answer tasks are underrepresented. The study revealed that incorporating mathematical activities into the multidimensional framework of textbook tasks may help to better understand the opportunities to learn which are afforded students by using mathematics textbooks.     

模糊层次分析法在矩阵论教材评价方面的应用

基础学科的教材直接影响学生的基本功,尤其是几乎每个学科都会涉及的数学类教材,如矩阵论.矩阵论是研究生的基础课程,在对学生以后的学术道路有举足轻重的作用.所以选择一本合适的教材,对学生和教师来说都有不小的帮助.然而,对教材的评价而言,不能单单从一个方面入手,因此将模糊综合分析法与层次分析法结合,在矩阵论教材的评价方面建立评价体系,为高校选择合适的教材提供依据.     

Students' understanding of quadratic equations

Action–Process–Object–Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.     

Learning difficulties with solids of revolution: classroom observations

The study aims to identify areas of difficulty in learning about volumes of solids of revolution (VSOR) at a Further Education and Training college in South Africa. Students’ competency is evaluated along five skill factors which refer to knowledge skills required to succeed in performing tasks relating to applications of the definite integral, in particular to VSOR. The paper reflects on reasons for the difficulties that students experience in this topic. The study reveals that many students are not competent in drawing graphs and in interpreting the region bounded by the given graphs. If the graphs are given, students have difficulty in selecting the representative strip that is used in approximating the bounded region. Although many students are able to produce the correct formula to calculate the volume, be it a disc, washer or shell, they find it problematic to draw the three-dimensional (3D) representation of the rotated strip and the generated solid of revolution. Students seem to succeed better with tasks requiring simple manipulation skills. The study illustrates how a measure (the skill factors) can be put into practice for establishing exactly where the problems lie when students under-perform in the topic of VSOR. The results can serve as guide on how conclusions can be drawn by assessing the problematic situation through breaking it down along the framework of skill factors.     

The Advantages of an STS Approach Over a Typical Textbook Dominated Approach in Middle School Science

Two sections of middle school science were taught by two longtime teachers where one used an STS approach and the other followed the more typical textbook approach closely. Pre‐ and post assessments were administered to one section of students for each teacher. The testing focused on student concept mastery, general science achievement, concept applications, use of concepts in new situations, and attitudes toward science. Videotapes of classroom actions were recorded and analyzed to determine the level of the use of STS teaching strategies in the two sections. Information was also be collected that gave evidence of and noted changes in student creativity and the continuation of student learning and the use of it beyond the classroom. Major findings indicate that students experiencing the STS format where constructivist teaching practices were used to (a) learn basic concepts as well as students who studied them directly from the textbook, (b) achieve as much in terms of general concept mastery as students who studied almost exclusively by using a textbook closely, (c) apply science concepts in new situations better than students who studied science in a more traditional way, (d) develop more positive attitudes about science, (e) exhibit creativity skills more often and more uniquely, and (f) learn and use science at home and in the community more than did students in the textbook dominated classroom.     

A preliminary genetic decomposition for conceptual understanding of the indefinite integral

This paper reports on the use of APOS theory to investigate conceptual understanding of the indefinite integral amongst undergraduate students at the University of KwaZulu-Natal, South Africa. We present a Preliminary Genetic Decomposition (PGD) for the indefinite integral, which predicts the mental constructions and mechanisms that may facilitate conceptual understanding. In this pilot study, the analysis of students’ written responses to the research instrument suggested that more than half of the participants lacked the prerequisite knowledge of the concepts of function, derivative of functions and the chain rule. The findings confirmed that students experience greater difficulty dealing with transcendental functions than with algebraic functions. The analysis indicated that the cognitive mechanism of reversal, to recognise the inverse nature of differentiation and integration, was inconsistent or absent in many students. Interview data from a subset of participants was employed for triangulation with the document analysis. The empirical data suggested refinement of the PGD and modification of the research instrument, and further fine-grained interviews with study participants to investigate their conceptual understanding more deeply, and for the purposes of data triangulation.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

The Washer Method

<正>You've learned the disk method to use integral finding the volume of solid revolution.You'll learn how to use other method for finding the volume of a special type of solids of revolution.Solids of revolution are used commonly in industry.Sometimes,slicing a solid of revolution results in disks with holes in the middle.It's called washer method.The washer is formed by revolving a rectangle about an axis,as shown in Figure 1&2.     

Opportunity to learn problem solving in Dutch primary school mathematics textbooks

In the Netherlands, mathematics textbooks are a decisive influence on the enacted curriculum. About a decade ago, Dutch primary school mathematics textbooks provided hardly any opportunities to learn problem solving. In this study we investigated whether this provision has changed. In order to do so, we carried out a textbook analysis in which we established to what degree current textbooks provide non-routine problem-solving tasks for which students do not immediately have a particular solution strategy at their disposal. We also analyzed to what degree textbooks provide ‘gray-area’ tasks, which are not really non-routine problems, but are also not straightforwardly solvable. In addition, we inventoried other ways in which present textbooks facilitate the opportunity to learn problem solving. Finally, we researched how inclusive these textbooks are with respect to offering opportunities to learn problem solving for students with varying mathematical abilities. The results of our study show that the opportunities that the currently most widely used Dutch textbooks offer to learn problem solving are very limited, and these opportunities are mainly offered in materials meant for more able students. In this regard, Dutch mainstream textbooks have not changed compared to the situation a decade ago. A textbook that is the Dutch edition of a Singapore mathematics textbook stands out in offering the highest number of problem-solving tasks, and in offering these in the materials meant for all students. However, in the ways this textbook facilitates the opportunity to learn problem solving, sometimes a tension occurs concerning the creative character of genuine problem solving.     

A semiotic interpretation of the derivative concept in a textbook

Differential and integral calculus textbooks are widely used as the main resource for teaching. They appear in a variety of forms and adopt various approaches to present the content. In this paper, we turn our attention to one chapter of a calculus textbook and our focus is on the introduction of the derivative concept. With the purpose of examining the presentation of the derivative concept in the textbook, we give a view of Peirce’s semiotics, in particular of his classification of sign-vehicles. The analysis allows us to point out that the sign-vehicle in relation to the derivative concept may be iconic, indexical, or symbolic. These do not constitute mutually exclusive kinds of signs, but they are interrelated in such a way that we can identify iconicity in indexicality and indexicality in symbolicity. We conclude that the textbook has the potential to enable students to conceptualize the derivative. However, in some aspects, the book may constrain students’ conceptualization and it could be improved to meet the students’ needs to make meaning of the derivative concept.     

Students' understanding of the notion of function in dynamic geometry environments

To make optimal use of computational environments, one must understand how students interact with the environments and how students' mathematical thinking is reflected and affected by their use of the environments. Similarly, to make sense of research on students' thinking and learning, one must understand how the environments and contexts used in the research may affect the conclusions one derives.The research on students' learning of functions has approached the topic in terms of symbols and graphs (see, for example, Leinhardt et al. (1990) for a review of work up to that point; Harel and Dubinsky (1992) for a collection of research; and Dugdale et. al. (1995), for some recent thinking about implications for curriculum reform using technology). Dynamic geometry environments (DGEs) like Cabri Geometry or Geometer's Sketchpad, offer us an opportunity to get a new perspective on these old and important issues. DGEs let students build geometrical constructions and then drag certain objects around the screen in a continuous manner while observing how the entire construction responds dynamically. In this way DGEs model functional relationships that are not specified by symbols or represented by graphs.Based on interviews with undergraduate mathematics majors, this paper presents preliminary observations that confirm some old results and raise some new questions about students' notions of function.     

The textbook-in-use: students’ utilization schemes of mathematics textbooks related to self-regulated practicing

This paper presents a qualitative study on how students make use of their mathematics textbooks for practicing. The study was carried out in two German secondary schools with 74 students (44 in 6th and 30 in 12th grade). Students’ utilization of textbooks for practicing is analyzed using the theoretical framework of instrumental genesis. The results indicate that students’ choices of contents from the book for practicing can be categorized into three utilization schemes: position-dependent practicing, block-dependent practicing, and salience-dependent practicing. In terms of position-dependent practicing the relative position of the textbook’s contents to teacher-mediated sections guides the students’ choice. Block-dependent practicing relates to the use of contents from the book that belong to particular blocks. Finally, salience-dependent practicing is a utilization scheme of the book where students’ choice is guided by perceptual salience of the book contents. These findings both show how textbook users are influenced by the way mathematics is presented in textbooks and provide insights into students’ conceptions of practicing mathematics.     

The volume of a solid of revolution in calculus

<正>On the last time,you have known what a right circular cylinder is in calculus.Right circular cylinders are just a part of solids of revolution.Today you will learn solids of revolution arent right circular cylinders.In general way,a solid is bounded by the region under the curve y=f(x)by rotating a-     

The State of Proof in Finnish and Swedish Mathematics Textbooks—Capturing Differences in Approaches to Upper-Secondary Integral Calculus

Students’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish textbooks than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.     

Reasoning-and-Proving in School Mathematics Textbooks

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.     

An investigation of 6th graders’ solutions of Cartesian product problems and representation of these problems using arrays

Two hour-long interviews were conducted with each of 14 sixth-grade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students’ reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007).     

From the Written to the Enacted Curricula: The Intermediary Role of Middle School Mathematics Teachers in Shaping Students' Opportunity to Learn

In this paper is reported the extent of textbook use by 39 middle school mathematics teachers in six states, 17 utilizing a textbook series developed with funding from the National Science Foundation (NSF‐funded) and 22 using textbooks developed by commercial publishers (publisher‐generated). Results indicate that both sets of teachers placed significantly higher emphasis on Number and Operation, often at the expense of other content strands. Location of topics within a textbook represented an oversimplified explanation of what mathematics gets taught or omitted. Most teachers using an NSF‐funded curriculum taught content intended for students in a different (lower) grade, and both sets of teachers supplemented with skill‐building and “practice” worksheets. Implications for documenting teachers' “fidelity of implementation” ( National Research Council, 2004 ) are offered.     

Envisioning the infinite by projecting finite properties

We analyze interviews with 24 post-secondary students as they reason about infinite processes in the context of the tricky Tennis Ball Problem. By metaphorically projecting various properties from the finite states such as counting and indexing, participants envisioned widely varying final states for the infinite process. Depending on which properties they attended to, some participants recognized contradictions arising from these infinite projections and sought to resolve the contradictions in non-dismissive ways. The findings indicate that increasing their meta-level awareness of the features of their mental constructions may help students overcome common difficulties with the infinite.