Unlocking Complex Vector Calculus Concepts For Engineering Students Using Geogebra

There is an increasing drive to exploit the power of technology to improve students mathematical conceptual understanding. This work is motivated by the authors research presented at the SEFI 2022 conference which reported on students experienced difficulties with the double integral, a concept central to vector calculus. Some of the difficulties included visualising and sketching three dimensional surfaces and regions of integration and changing coordinate systems from rectangular to polar. Vector calculus is a crucial subject for engineering students, but its abstract concepts can be challenging to grasp. This curriculum proposal is a response to improve visualisation and conceptual understanding and is part of a larger project to develop an innovative, engaging and effective way for undergraduate engineering students at the University of Cape Town to learn vector calculus concepts supported by GeoGebra. The choice was made in favour of the easy to use, freely downloadable mathematical software, GeoGebra which presents a creative, visual and integrative way to experience and understand mathematical concepts. Informing this curriculum development initiative is Vygotsky’s social constructivist perspectives with an emphasis on inclusivity, diversity and participant interactions. In this paper we discuss the above theoretical underpinnings with case studies on how to teach the double integral concept in GeoGebra for conceptual understanding. Additionally the benefits of using GeoGebra including its ability to engage students, promote critical thinking, and increase motivation will be discussed. This research will be of interest to those intending to use GeoGebra to improve the teaching and learning of vector calculus concepts

The WOZ Recognizer: A Tool For Understanding User Perceptions of Sketch-Based Interfaces

Sketch recognition has the potential to be an important input method for computers in the coming years; however, designing and building an accurate and sophisticated sketch recognition system is a time consuming and daunting task. Since sketch recognition is still at a level where mistakes are common, it is important to understand how users perceive and tolerate recognition errors and other user interface elements with these imperfect systems. A problem in performing this type of research is that we cannot easily control aspects of recognition in order to rigorously study the systems. We performed a study examining user perceptions of three pen-based systems for creating logic gate diagrams: a sketch-based interface, a WIMP-based interface, and a hybrid interface that combined elements of sketching and WIMP. We found that users preferred the sketch-based interface and we identified important criteria for pen-based application design. This work exposed the issue of studying recognition systems without fine-grained control over accuracy, recognition mode, and other recognizer properties. In order to solve this problem, we developed a Wizard of Oz sketch recognition tool, the WOZ Recognizer, that supports controlled symbol and position accuracy and batch and streaming recognition modes for a variety of sketching domains. We present the design of the WOZ Recognizer, modeling recognition domains using graphs, symbol alphabets, and grammars; and discuss the types of recognition errors we included in its design. Further, we discuss how the WOZ Recognizer simulates sketch recognition, controlling the WOZ Recognizer, and how users interact with it. In addition, we present an evaluative user study of the WOZ Recognizer and the lessons we learned. We have used the WOZ Recognizer to perform two user studies examining user perceptions of sketch recognition; both studies focused on mathematical sketching. In the first study, we examined whether users prefer recognition feedback now (real-time recognition) or later (batch recognition) in relation to different recognition accuracies and sketch complexities. We found that participants displayed a preference for real-time recognition in some situations (multiple expressions, low accuracy), but no statistical preference in others. In our second study, we examined whether users displayed a greater tolerance for recognition errors when they used mathematical sketching applications they found interesting or useful compared to applications they found less interesting. Participants felt they had a greater tolerance for the applications they preferred, although our statistical analysis did not positively support this. In addition to the research already performed, we propose several avenues for future research into user perceptions of sketch recognition that we believe will be of value to sketch recognizer researchers and application designers

Pen-based Methods For Recognition and Animation of Handwritten Physics Solutions

There has been considerable interest in constructing pen-based intelligent tutoring systems due to the natural interaction metaphor and low cognitive load afforded by pen-based interaction. We believe that pen-based intelligent tutoring systems can be further enhanced by integrating animation techniques. In this work, we explore methods for recognizing and animating sketched physics diagrams. Our methodologies enable an Intelligent Tutoring System (ITS) to understand the scenario and requirements posed by a given problem statement and to couple this knowledge with a computational model of the student\u27s handwritten solution. These pieces of information are used to construct meaningful animations and feedback mechanisms that can highlight errors in student solutions. We have constructed a prototype ITS that can recognize mathematics and diagrams in a handwritten solution and infer implicit relationships among diagram elements, mathematics and annotations such as arrows and dotted lines. We use natural language processing to identify the domain of a given problem, and use this information to select one or more of four domain-specific physics simulators to animate the user\u27s sketched diagram. We enable students to use their answers to guide animation behavior and also describe a novel algorithm for checking recognized student solutions. We provide examples of scenarios that can be modeled using our prototype system and discuss the strengths and weaknesses of our current prototype. Additionally, we present the findings of a user study that aimed to identify animation requirements for physics tutoring systems. We describe a taxonomy for categorizing different types of animations for physics problems and highlight how the taxonomy can be used to define requirements for 50 physics problems chosen from a university textbook. We also present a discussion of 56 handwritten solutions acquired from physics students and describe how suitable animations could be constructed for each of them

Syntactic Generation of Research Thesis Sketches Across Disciplines Using Formal Grammars

A part of the prerequisites for granting a degree in higher education institutions, students at postgraduate levels normally carry out research, which they do report in the form of theses or dissertations. Study has shown that students tend to go through difficulties in writing research thesis across all disciplines because they do not fully comprehend what constitutes a research thesis. This project proposes the syntactic generation of research thesis sketches across disciplines using formal grammars. Sketching is a synthesis technique which enables users to deliver high-level intuitions into a synthesis snag while leaving low-level details to synthesis tools. This work extends sketching to document generation for research thesis documents. Context-free grammar rules were designed and implemented for this task. A link to 10,000 generated thesis sketches was presented

An Investigation of Students' Learning of Integral Calculus with Maple Software and Paper-Pencil Strategies in the Western Region of Ghana.

The goal of the research was to look into the impact of Maple software instruction on senior high school students' understanding of integral calculus. The study adopted a mixed-method design comprising qualitative and quantitative research designs. The researcher used both purposive and simple random sampling techniques to select one hundred (100) participants: fifty (50) participants for the control group and fifty (50) participants for the experimental group. The data collection instruments used in the study were an interview, pre-test and post-test. Data analysis was carried out using descriptive statistics and an Independent Samples t-test. The study found that 7(7%) participants found it difficult to execute correct substitution of the lower and upper limits of definite integral questions. Moreover, most of the participants, 35(35%), omitted the constant of integration after responding to the indefinite integral test item of the pre-test. It was noted that 18(18%) of the participants could not correctly integrate the polynomial or quadratic function administered to them. The independent samples t-test analysis of the post-test scores for the experimental and control groups revealed a statistically significant difference between the experimental group (M = 24.80; SD = 9.48) and the control group (M = 20.65; SD = 7.67). The estimated t-statistic was (t = 2.986; p = 0.005). This shows that Maple Software's experimental group outperformed the control group using the paper and pencil strategy. The analysis of the interview data indicated that Maple Software has contributed to the success of students’ achievement in the integral calculus by arousing and sustaining the student’s interest. The Maple Software also made it easier for students to follow the calculus instruction. The findings recommended that technology and mathematical software should be used in the teaching and learning of integration at schools

Integrated technologies instructional method to enhance bilingual undergraduate engineering students

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonMathematics permeates almost every aspect of human life and it is a skill much needed by the increasingly complex technological world. It is necessary that this essential skill must be properly developed among students to prepare them for future academic and professional careers. An assessment of the research-based instructional strategies blending with old traditional methods with the modern technological development is a must. Due to the complexity of mathematics learning and the varied learning styles of learners, an integration of appropriate multiple instructional strategies into mathematics education will positively impact mathematical achievement of students. The purpose of this research was to examine the effects of the use of Integrated Technologies Instructional Method (ITIM) as a supplement to the traditional lecture method on mathematics achievement of the Integral Calculus students at the College of Engineering, University of Ha'il, Saudi Arabia. The ITIM includes the four instructional strategies such as the use of the Computer-Supported Collaborative Learning, the collaborative learning, the bilingual support and the study support. Different types of academic supports have been used to examine their effects on students achievement in mathematics. Mathematics, the bedrock of science and engineering, is considered a very important indicator of a student's academic success in professional higher education. Undergraduate engineering students' low achievement in the first year mathematics is an issue demands much attention. The study was undertaken to address students' weak background in mathematics and particularly their high failure rates in this particular course. A total of 218 undergraduate engineering students, comprising of both the experimental and the control groups, were involved in this experimental design study. The control group was taught by the traditional lecture method whereas the experimental group was exposed to the ITIM as a supplement to the traditional lecture method. Apart from the effects of the use of ITIM, students' performance in the previous courses (covariates) such as mathematics, computer, and the English language were compared with their final grades of the Integral Calculus course. The final grades of students were taken as the dependent variable and the ITIM and students' scores in the previous courses as the independent variables. It has been noticed from the literature review that the application of only one instructional strategy does not address the needs of the diverse learning styles of students. A mixed mode method, quantitative and qualitative, was used to collect and analyse data. The quantitative data instruments included students' final exam grades and the student questionnaires. Interviews with students were used as qualitative tools of data collection. An independent t-test, ANOVA, univariate analysis and the stepwise multiple regression analysis were performed to determine the overall statistical significance. The study concluded that there was a statistically significant difference in the performance of the experimental group of students' in terms of their end-of-course grades compared to that of the control group. The regression model revealed significance of covariates on the dependent variable. However, no significant relationship was found between the mathematics achievement and attitudes towards the use of ITIM. The study was an attempt to demonstrate the suitability of the instructional strategies on the bilingual Arab undergraduate engineering students; however, they can probably be applicable to other bilingual students

An investigation on the framework of dressing virtual humans

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Realistic human models are widely used in variety of applications. Much research has been carried out on improving realism of virtual humans from various aspects, such as body shapes, hair, and facial expressions and so on. In most occasions, these virtual humans need to wear garments. However, it is time-consuming and tedious to dress a human model using current software packages [Maya2004]. Several methods for dressing virtual humans have been proposed recently [Bourguignon2001, Turquin2004, Turquin2007 and Wang2003B]. The method proposed by Bourguignon et al [Bourguignon2001] can only generate 3D garment contour instead of 3D surface. The method presented by Turquin et al. [Turquin2004, Turquin2007] could generate various kinds of garments from sketches but their garments followed the shape of the body and the side of a garment looked not convincing because of using simple linear interpolation. The method proposed by Wang et al. [Wang2003B] lacked interactivity from users, so users had very limited control on the garment shape.This thesis proposes a framework for dressing virtual humans to obtain convincing dressing results, which overcomes problems existing in previous papers mentioned above by using nonlinear interpolation, level set-based shape modification, feature constraints and so on. Human models used in this thesis are reconstructed from real human body data obtained using a body scanning system. Semantic information is then extracted from human models to assist in generation of 3 dimensional (3D) garments. The proposed framework allows users to dress virtual humans using garment patterns and sketches. The proposed dressing method is based on semantic virtual humans. A semantic human model is a human body with semantic information represented by certain of structure and body features. The semantic human body is reconstructed from body scanned data from a real human body. After segmenting the human model into six parts some key features are extracted. These key features are used as constraints for garment construction.Simple 3D garment patterns are generated using the techniques of sweep and offset. To dress a virtual human, users just choose a garment pattern, which is put on the human body at the default position with a default size automatically. Users are allowed to change simple parameters to specify some sizes of a garment by sketching the desired position on the human body.To enable users to dress virtual humans by their own design styles in an intuitive way, this thesis proposes an approach for garment generation from user-drawn sketches. Users can directly draw sketches around reconstructed human bodies and then generates 3D garments based on user-drawn strokes. Some techniques for generating 3D garments and dressing virtual humans are proposed. The specific focus of the research lies in generation of 3D geometric garments, garment shape modification, local shape modification, garment surface processing and decoration creation. A sketch-based interface has been developed allowing users to draw garment contour representing the front-view shape of a garment, and the system can generate a 3D geometric garment surface accordingly. To improve realism of a garment surface, this thesis presents three methods as follows. Firstly, the procedure of garment vertices generation takes key body features as constraints. Secondly, an optimisation algorithm is carried out after generation of garment vertices to optimise positions of garment vertices. Finally, some mesh processing schemes are applied to further process the garment surface. Then, an elaborate 3D geometric garment surface can be obtained through this series of processing. Finally, this thesis proposes some modification and editing methods. The user-drawn sketches are processed into spline curves, which allow users to modify the existing garment shape by dragging the control points into desired positions. This makes it easy for users to obtain a more satisfactory garment shape compared with the existing one. Three decoration tools including a 3D pen, a brush and an embroidery tool, are provided letting users decorate the garment surface by adding some small 3D details such as brand names, symbols and so on. The prototype of the framework is developed using Microsoft Visual Studio C++,OpenGL and GPU programming

The Inclusive Classroom: Mathematics and Science Instruction for Students with Learning Disabilities

This online book explores the unique educational needs of students with learning disabilities and provides strategies for meaningfully engaging all students in mathematics and science curricula. Drawing on key principles of inclusion, special education, multicultural education, and standards-based reform, the book guides K-12 teachers in creating optimal learning environments where diverse learners can thrive. In addition to offering instructional strategies, the book suggests ways to adapt textbooks to make texts and materials more manageable for students, and describes a variety of assessments that enable diverse learners to demonstrate their knowledge. Educational levels: Graduate or professional