Search results for: philosophy of mathematics

Authors: Sibawu Witness Siyepu

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This study explores the effects of instructional design using blended learning in the learning of radian measures among Engineering students. Blended learning is an education programme that combines online digital media with traditional classroom methods. It requires the physical presence of both lecturer and student in a mathematics computer laboratory. Blended learning provides element of class control over time, place, path or pace. The focus was on the use of Khan Academy to supplement traditional classroom interactions. Khan Academy is a non-profit educational organisation created by educator Salman Khan with a goal of creating an accessible place for students to learn through watching videos in a computer assisted computer. The researcher who is an also lecturer in mathematics support programme collected data through instructing students to watch Khan Academy videos on radian measures, and by supplying students with traditional classroom activities. Classroom activities entails radian measure activities extracted from the Internet. Students were given an opportunity to engage in class discussions, social interactions and collaborations. These activities necessitated students to write formative assessments tests. The purpose of formative assessments tests was to find out about the students’ understanding of radian measures, including errors and misconceptions they displayed in their calculations. Identification of errors and misconceptions serve as pointers of students’ weaknesses and strengths in their learning of radian measures. At the end of data collection, semi-structure interviews were administered to a purposefully sampled group to explore their perceptions and feedback regarding the use of blended learning approach in teaching and learning of radian measures. The study employed Algebraic Insight Framework to analyse data collected. Algebraic Insight Framework is a subset of symbol sense which allows a student to correctly enter expressions into a computer assisted systems efficiently. This study offers students opportunities to enter topics and subtopics on radian measures into a computer through the lens of Khan Academy. Khan academy demonstrates procedures followed to reach solutions of mathematical problems. The researcher performed the task of explaining mathematical concepts and facilitated the process of reinvention of rules and formulae in the learning of radian measures. Lastly, activities that reinforce students’ understanding of radian were distributed. Results showed that this study enthused the students in their learning of radian measures. Learning through videos prompted the students to ask questions which brought about clarity and sense making to the classroom discussions. Data revealed that sense making through reinvention of rules and formulae assisted the students in enhancing their learning of radian measures. This study recommends the use of Khan Academy in blended learning to be introduced as a socialisation programme to all first year students. This will prepare students that are computer illiterate to become conversant with the use of Khan Academy as a powerful tool in the learning of mathematics. Khan Academy is a key technological tool that is pivotal for the development of students’ autonomy in the learning of mathematics and that promotes collaboration with lecturers and peers.

Keywords: algebraic insight framework, blended learning, Khan Academy, radian measures

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Authors: Sereen Itani

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As the world aspires for technological innovation, Innovative Robotics Technology-Supported Pedagogic Models in STEAM Education (Science, Technology, Engineering, Arts, and Mathematics) are critical in our global education system to build and enhance the next generation 21st century skills. Thus, diverse international schools endeavor in attempts to construct an integrated robotics and technology enhanced curriculum based on interdisciplinary subjects. Accordingly, it is vital that the globe remains resilient in STEAM fields by equipping the future learners and educators with Innovative Technology Experiences through robotics to support such fields. A variety of advanced teaching methods is employed to learn about Robotics Technology-integrated pedagogic models. Therefore, it is only when STEAM and innovations in Robotic Technology becomes integrated with real-world applications that transformational learning can occur. Robotics STEAM education implementation faces major challenges globally. Moreover, STEAM skills and concepts are communicated in separation from the real world. Instilling the passion for robotics and STEAM subjects and educators’ preparation could lead to the students’ majoring in such fields by acquiring enough knowledge to make vital contributions to the global STEAM industries. Thus, this necessitates the establishment of Pedagogic models such as Innovative Robotics Technologies to enhance STEAM education and develop students’ 21st-century skills. Moreover, an ICT innovative supported robotics classroom will help educators empower and assess students academically. Globally, the Robotics Design System and platforms are developing in schools and university labs creating a suitable environment for the robotics cross-discipline STEAM learning. Accordingly, the research aims at raising awareness about the importance of robotics design systems and methodologies of effective employment of robotics innovative technology-supported pedagogic models to enhance and develop (STEAM) education globally and enhance the next generation 21st century skills.

Keywords: education, robotics, STEAM (Science, Technology, Engineering, Arts and Mathematics Education), challenges

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Authors: Olteanu Constanta, Olteanu Lucian

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Researchers note that algebraic reasoning and sense making is essential for building conceptual knowledge in school mathematics. Consequently, pre-service teachers’ own reasoning and sense making are useful in fostering and developing students’ algebraic reasoning and sense making. This article explores the forms of reasoning and sense making that pre-service mathematics teachers exhibit and use in the process of analysing problem-posing tasks with a focus on first-degree equations. Our research question concerns the characteristics of the problem-posing tasks used for reasoning and sense making of first-degree equations as well as the characteristics of pre-service teachers’ reasoning and sense making in problem-posing tasks. The analyses are grounded in a post-structuralist philosophical perspective and variation theory. Sixty-six pre-service primary teachers participated in the study. The results show that the characteristics of reasoning in problem-posing tasks and of pre-service teachers are selecting, exploring, reconfiguring, encoding, abstracting and connecting. The characteristics of sense making in problem-posing tasks and of pre-service teachers are recognition, relationships, profiling, comparing, laddering and verifying. Beside this, the connection between reasoning and sense making is rich in line of flight in problem-posing tasks, while the connection is rich in line of rupture for pre-service teachers.

Keywords: first-degree equations, problem posing, reasoning, rhizomatic assemblage, sense-making, variation theory

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Authors: Sevket Ceyhun Cetin, Mehmet Oren

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According to existing educational research studies, students learn better with high teacher quality. Therefore, professional development has become a crucial way of increasing the quality of novices and veteran in-service teachers by providing support regarding content and pedagogy. To answer what makes PD effective, researchers have studied different PD models and revealed some critical elements that need to be considered, such as duration of a PD and the manner of delivery (e.g., lecture vs. engaging). Also, it has been pointed out that if PDs are prepared as one-size-fits-all, they most likely be ineffective in addressing teachers’ needs toward improving instructional quality. Instead, teachers’ voices need to be heard, and the foci of PDs should be determined based on their specific needs. Thus, this study was conducted to identify professional development needs of middle school mathematics teachers based on their self-evaluation of their performances in light of teaching standards. This study also aimed to explore whether the PD needs with respect to years of teaching experience (novice vs. veteran). These teachers had participated in a federally-funded research grant, which aimed to improve the competencies of 6-9 grade-level mathematics teachers in pedagogy and content areas. In the research project, the participants had consistently videoed their lessons throughout a school year and reflected on their performances, using Teacher Advanced Program (TAPTM) rubric, which was based on the best practices of teaching. Particularly, they scored their performances in the following areas and provided evidence as the justifications of their scores: Standards and Objectives, Presenting Instructional Content, Lesson Structure and Pacing, Activities and Materials, Academic Feedback, Grouping Students, and Questioning. The rating scale of the rubric is 1 through 5 (i.e., 1=Unsatisfactory [performance], 3=Proficient, and 5=Exemplary). For each area mentioned above, the numerical scores of 77 written reports (for 77 videoed lessons) of 24 teachers (nnovices=12 and nveteran=12) were averaged. Overall, the average score of each area was below 3 (ranging between 2.43 and 2.86); in other words, teachers judged their performances incompetent across the seven areas. In the second step of the data analysis, the lowest three areas in which novice and veteran teachers performed poorly were selected for further qualitative analysis. According to the preliminary results, the lowest three areas for the novice teachers were: Questioning, Grouping Students, and Academic Feedback. Grouping Students was also one of the lowest areas of the veteran teachers, but the other two areas for this group were: Lesson Structure & Pacing, and Standards & Objectives. Identifying in-service teachers’ needs based on their reflective practitioner experiences provides educators very crucial information that can be used to create more effective PD that improves teacher quality.

Keywords: mathematics teacher, professional development, self-reflection, video data

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Authors: Wedad Albalawi

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The mathematical inequalities have been the core of mathematical study and used in almost all branches of mathematics as well in various areas of science and engineering. The inequalities by Hardy, Littlewood and Polya were the first significant composition of several science. This work presents fundamental ideas, results and techniques and it has had much influence on research in various branches of analysis. Since 1934, various inequalities have been produced and studied in the literature. Furthermore, some inequalities have been formulated by some operators; in 1989, weighted Hardy inequalities have been obtained for integration operators. Then, they obtained weighted estimates for Steklov operators that were used in the solution of the Cauchy problem for the wave equation. They were improved upon in 2011 to include the boundedness of integral operators from the weighted Sobolev space to the weighted Lebesgue space. Some inequalities have been demonstrated and improved using the Hardy–Steklov operator. Recently, a lot of integral inequalities have been improved by differential operators. Hardy inequality has been one of the tools that is used to consider integrity solutions of differential equations. Then dynamic inequalities of Hardy and Coposon have been extended and improved by various integral operators. These inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Some inequalities have been appeared involving Copson and Hardy inequalities on time scales to obtain new special version of them. A time scale is defined as a closed subset contains real numbers. Then the inequalities of time scales version have received a lot of attention and has had a major field in both pure and applied mathematics. There are many applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, and population dynamics. This study focuses on double integrals to obtain new time-scale inequalities of Copson driven by Steklov operator. They will be applied in the solution of the Cauchy problem for the wave equation. The proof can be done by introducing restriction on the operator in several cases. In addition, the obtained inequalities done by using some concepts in time scale version such as time scales calculus, theorem of Fubini and the inequality of H¨older.

Keywords: time scales, inequality of Hardy, inequality of Coposon, Steklov operator

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Authors: Zahra Alikhani Koopaei

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In the fifth lesson of the third-grade mathematics book, students are introduced to the concept of multiplication. However, some students showed a lack of interest in learning this topic. To address this, a simple electronic multiplication table was designed with the aim of making the concept of multiplication entertaining and engaging for students. It provides them with moments of excitement during the learning process. To achieve this goal, a device was created that produced a bell sound when two wire ends were connected. Each wire end was connected to a specific number in the multiplication table, and the other end was linked to the corresponding answer. Consequently, if the answer is correct, the bell will ring. This study employs interactive and engaging methods to teach mathematics, particularly to students who have previously shown little interest in the subject. By integrating game-based learning and critical thinking, we observed an increase in understanding and interest in learning multiplication compared to before using this method. This further motivated the students. As a result, the intelligent multiplication table was successfully designed. Students, under the instructor's supervision, could easily construct the device during the lesson. Through the implementation of these operations, the concept of multiplication was firmly established in the students' minds. Engaging multiple intelligences in each student enhances a more stable and improved understanding of the concept of multiplication.

Keywords: intelligent multiplication table, design, construction, education, increased interest, students

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Authors: Anandit Sachdev

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Auroville, a settlement in Tamil Nadu, India, is based on the principles of ‘human unity’as defined by Indian philosopher Sri Aurobindo. The settlement was conceptualized on these principles by Sri Aurobindo’s spiritual partner Mirra Alfassa, known as ‘The Mother’ to the Aurovillians. In common perception, the settlement is an experiment in achieving ‘human unity’ through sustainable living. Since its inception in late 1960s, the settlement has attracted people from a variety of nationalities, each understanding, seeking, and rendering ‘human unity’ in their own unique way. This multiplicity of inhabitation has created and continues to create complex and layered human and more-than-human geographies, which are collectively understood as Auroville. This essay builds on these multiple narratives of local metaphysical and every inhabitation of spiritual and philosophical ideas of Sri Aurobindo as rendered in materiality by the Mother. The research aims to assess how theseforms of everyday spirituality conflict, interact, and engage with the principles of Auroville. The research further aims to understands how, if at all, the diverse landscapes of social, cultural, and infrastructural conflictssynthesizewhen perceived through the lens of spirituality. The research does so by detailing the different forms of the built environment which evoke the transcendental and its underlying processes. While doing so, it aims to understand how different manifestations of interiority within the Aurovillian landscape tie back to the self and its entanglements. By analysing the settlement through a spiritual lens, the research ultimately ties together questions relating to the built environment and ontology and asks how each facilitates a continuous synthesis with the other. Lastly, the paper enquires if these ongoing processes of synthesis of built space and ontological entanglements are what can be conceptualized as ‘human unity’ as perceived by Sri Aurobindo himself.

Keywords: sacrality, sacred, spirituality, philosophy, Indian philosophy, auroville, India

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Authors: Joanna Wielochowska, Aneta Wielochowska

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Whilst the relationship between scientific areas such as cognitive psychology, neurobiology and philosophy of mind has been emphasized in recent decades of scientific research, concepts and discoveries made in both fields overlap and complement each other in their quest for answers to similar questions. The object of the following case study is to describe, analyze and illustrate the nature and characteristics of a certain cognitive experience which appears to display features of synaesthesia, or rather high-level synaesthesia (ideasthesia). The following research has been conducted on the subject of two authors, monozygotic twins (both polysynaesthetes) experiencing involuntary associations of identical nature. Authors made attempts to identify which cognitive and conceptual dependencies may guide this experience. Operating on self-introduced nomenclature, the described phenomenon- multi-dimensional processing of textual and visual information- aims to define a relationship that involuntarily and immediately couples the content introduced by means of text or image a sensation of appearing in a certain place in the mind’s eye. More precisely: (I) defining a concept introduced by means of textual content during activity of reading or writing, or (II) defining a concept introduced by means of visual content during activity of looking at image(s) with simultaneous sensation of being allocated to a given place in the mind’s eye. A place can be then defined as a cognitive representation of a certain concept. During the activity of processing information, a person has an immediate and involuntary feel of appearing in a certain place themselves, just like a character of a story, ‘observing’ a venue or a scenery from one or more perspectives and angles. That forms a unique and unified experience, constituting a background mental landscape of text or image being looked at. We came to a conclusion that semantic allocations to a given place could be divided and classified into the categories and subcategories and are naturally linked with an individual’s semantic knowledge-base. A place can be defined as a representation one’s unique idea of a given concept that has been established in their semantic knowledge base. A multi-level structure of selectivity of places in the mind’s eye, as a reaction to a given information (one stimuli), draws comparisons to structures and patterns found in botany. Double-flowered varieties of flowers and a whorl system (arrangement) which is characteristic to components of some flower species were given as an illustrative example. A composition of petals that fan out from one single point and wrap around a stem inspired an idea that, just like in nature, in philosophy of mind there are patterns driven by the logic specific to a given phenomenon. The study intertwines terms perceived through the philosophical lens, such as definition of meaning, subjectivity of meaning, mental atmosphere of places, and others. Analysis of this rare experience aims to contribute to constantly developing theoretical framework of the philosophy of mind and influence the way human semantic knowledge base and processing given content in terms of distinguishing between information and meaning is researched.

Keywords: information and meaning, information processing, mental atmosphere of places, patterns in nature, philosophy of mind, selectivity, semantic knowledge base, senses, synaesthesia

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Authors: Melanie Tang

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According to Nietzsche, the eternal recurrence is his most important idea. However, it is perhaps his most cryptic and difficult to interpret. Early readings considered it as a cosmological hypothesis about the cyclic nature of time. However, following Nehamas’s ‘Life as Literature’ (1985), it has become a widespread interpretation that the eternal recurrence never really had any theoretical dimensions, and is not actually a philosophy of time, but a practical thought experiment intended to measure the extent to which we have mastered and perfected our lives. This paper endeavours to challenge this line of thought becoming scholarly consensus, and to carry out a more complex analysis of the eternal recurrence as it is presented in Thus Spoke Zarathustra. In its wider scope, this research proposes that Thus Spoke Zarathustra — as opposed to The Birth of Tragedy — be taken as the primary source for a study of Nietzsche’s Aesthetics, due to its more intrinsic aesthetic qualities and expressive devices. The eternal recurrence is the central philosophy of a work that communicates its ideas in unprecedentedly experimental and aesthetic terms, and a more in-depth understanding of why Nietzsche chooses to present his conception of time in aesthetic terms is warranted. Through hermeneutical analysis of Thus Spoke Zarathustra and engagement with secondary sources such as those by Nehamas, Karl Löwith, and Jill Marsden, the present analysis challenges the ethics of self-perfection upon which current interpretations of the recurrence are based, as well as their reliance upon a linear conception of time. Instead, it finds the recurrence to be a cyclic interplay between the self and the world, rather than a metric pertaining solely to the self. In this interpretation, time is found to be composed of an intertemporal rather than linear multitude of will to power, which structures itself through tensional cycles into an experience of circular time that can be seen to have aesthetic dimensions. In putting forth this understanding of the eternal recurrence, this research hopes to reopen debate on this key concept in the field of Nietzsche studies.

Keywords: Nietzsche, eternal recurrence, Zarathustra, aesthetics, time

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Authors: Faiza Abdul Latip

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One of the ultimate goals of any nation is to produce competitive manpower and this includes Philippines. Inclination in the field of Mathematics has a significant role in achieving this goal. However, Mathematics, as considered by most people, is the most difficult subject matter along with its topics to learn. This could be manifested from the low performance of students in national and international assessments. Educators have been widely using learning style models in identifying the way students learn. Moreover, it could be the frontline in knowing the difficulties held by each learner in a particular topic specifically concepts pertaining to fractions. However, as what many educators observed, students show difficulties in doing mathematical tasks and in great degree in dealing with fractions most specifically in the district of Datu Odin Sinsuat, Maguindanao. This study focused on the Datu Odin Sinsuat district grade six pupils’ learning styles along with their achievements and difficulties in learning concepts on fractions. Five hundred thirty-two pupils from ten different public elementary schools of the Datu Odin Sinsuat districts were purposively used as the respondents of the study. A descriptive research using the survey method was employed in this study. Quantitative analysis on the pupils’ learning styles on the Kolb’s Learning Style Inventory (KLSI) and scores on the mathematics diagnostic test on fraction concepts were made using this method. The simple frequency and percentage counts were used to analyze the pupils’ learning styles and their achievements on fractions. To determine the pupils’ difficulties in fractions, the index of difficulty on every item was determined. Lastly, the Kruskal-Wallis Test was used in determining the significant difference in the pupils’ achievements on fractions classified by their learning styles. This test was set at 0.05 level of significance. The minimum H-Value of 7.82 was used to determine the significance of the test. The results revealed that the pupils of Datu Odin Sinsuat districts learn fractions in varied ways as they are of different learning styles. However, their achievements in fractions are low regardless of their learning styles. Difficulties in learning fractions were found most in the area of Estimation, Comparing/Ordering, and Division Interpretation of Fractions. Most of the pupils find it very difficult to use fraction as a measure, compare or arrange series of fractions and use the concept of fraction as a quotient.

Keywords: difficulties in fraction, fraction, Kolb's model, learning styles

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Authors: Kathaleeya Chanda

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The schools under the Project Initiated by H.R.H Princess in Maha Chakri Sirindhorn in Nakhonnayok Educational Service Area Office are the small schools, situated in a remote and undeveloped area.Thus, the school-age youth didn’t have or have fewer opportunities to study at the higher education level which can lead to many social and economic problems. This study aims to solve these educational issues of the schools, under The Project Initiated by H.R.H Princess in Maha Chakri Sirindhorn, Nakhonnayok Educational Service Area Office, by the development of teachers, so that teachers could develop teaching and learning system with the ultimate goal to increase students’ academic achievement, increase the educational opportunities for the youth in the area, and help them learn happily. 154 in-service teachers from 22 schools and 4 different districts in Nakhonnayok participated in this teacher training. Most teachers were satisfied with the training content and the trainer. Thereafter, the teachers were given the test to assess the skills and knowledge after training. Most of the teachers earned a score higher than 75%. Accordingly, it can be concluded that after attending the training, teachers have a clear understanding of the contents. After the training session, the teachers have to write a lesson plan that is integrated or adapted to the Sufficiency Economy Philosophy. The teachers can either adopt intradisciplinary or interdisciplinary integration according to their actual teaching conditions in the school. Two weeks after training session, the researchers went to the schools to discuss with the teachers and follow up the assigned integrated lesson plan. It was revealed that the progress of integrated lesson plan could be divided into 3 groups: 1) the teachers who have completed the integrated lesson plan, but are concerned about the accuracy and consistency, 2) teachers who almost complete the lesson plan or made a great progress but are still concerned, confused in some aspects and not fill in the details of the plan, and 3), the teachers who made few progress, are uncertain and confused in many aspects, and may had overloaded tasks from their school. However, a follow-up procedure led to the commitment of teachers to complete the lesson plan. Regarding student learning assessment, from an experiment teaching, most of the students earned a score higher than 50 %. The rate is higher than the one from actual teaching. In addition, the teacher have assessed that the student is happy, enjoys learning, and providing a good cooperates in teaching activities. The students’ interview about the new lesson plan shows that they are happy with it, willing to learn, and able to apply such knowledge in daily life. Integrated lesson plan can increases the educational opportunities for youth in the area.

Keywords: sufficiency, economy, philosophy, integrated education syllabus

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Authors: Verónica Arredondo, Miguel Martínez-Panero, Teresa Peña, Victoriano Ramírez

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The Mexican Chamber of Deputies is composed of 500 representatives: 300 of them elected by relative majority and another 200 ones elected through proportional representation in five electoral clusters (constituencies) with 40 representatives each. In this mixed-member electoral system, the seats distribution of proportional representation is not independent of the election by relative majority, as it attempts to correct representation imbalances produced in single-member districts. This two-fold structure has been maintained in the successive electoral reforms carried out along the last three decades (eight from 1986 to 2014). In all of them, the election process of 200 seats becomes complex: Formulas in the Law are difficult to understand and to be interpreted. This paper analyzes the Mexican electoral system after the electoral reform of 2014, which was applied for the first time in 2015. The research focuses on contradictions and issues of applicability, in particular situations where seats allocation is affected by ambiguity in the law and where asymmetrical treatment of political parties arises. Due to these facts, a proposal of electoral reform will be presented. It is intended to be simpler, clearer, and more enduring than the current system. Furthermore, this model is more suitable for producing electoral outcomes free of contradictions and paradoxes. This approach would allow a fair treatment of political parties and as a result an improved opportunity to exercise democracy.

Keywords: electoral mathematics, electoral reform, Mexican electoral system, political asymmetry, proportional representation

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Authors: Noa Shriki, Atara Shriki

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Lifelong learning is considered as essential for teacher's professional development, which in turn has implications for the improvement of the entire education system. In recent years, many programs designed to support teachers' professional development are criticized for not achieving their goal. A variety of reasons have been proposed for the purpose of explaining the causes of the ineffectiveness of such programs. In this study, we put to test the possibility that teachers do not change as a result of their participation in professional programs due to a gap between the contents and approaches included in them and teacher's beliefs about teaching and learning. Eighteen elementary school mathematics teachers participated in the study. These teachers were involved in collaborating with their students in inquiring mathematical ideas, while implementing action research. Employing educational theories, the results indicated that this experience had a positive effect on teacher's professional development. In particular, there was an evident change in their beliefs regarding their role as mathematics teachers. However, while employing a different perspective for analyzing the data, the lens of Kurt Lewin's theory of re-education, we realized that this change of beliefs must be questioned. Therefore, it is suggested that analysis of educational processes should be carried out not only through common educational theories, but also on the basis of social and organizational theories. It is assumed that both the field of education and the fields of social studies and organizational consulting will benefit from the multifocal viewpoint

Keywords: educational theories, professional development, re-education, teachers' beliefs

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Authors: Jaber Shurrab, Matloub Hussain

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The rapid pace of changes in the construction industry, technological advancements, and rising costs present tremendous challenges for project managers. Project managers are under severe pressure to minimize the waste, improve the efficiency of the entire operations and the philosophy of ‘lean thinking’ so that ‘more could be achieved with less’ is becoming very popular. Though, lean management has strong roots in manufacturing industry and over the last decade lean philosophy has started gaining attention in the service industry as well. However, little has been known in the context of waste minimization and lean implementation in the construction industry and this paper deals with this important issue. The primary objective of this paper is to propose a conceptual framework for the exploration of appropriate lean techniques applicable to medium and large construction companies and measure their impact on the competitiveness and economic performance of construction companies of United Arab Emirates (UAE). To this end, a comprehensive literature review and interviews with eight project managers of medium and large construction companies of UAE have been conducted. It has been found that competitive, reduce waste and costs are critical to the construction industry. This is an ongoing research in lean management, giving project managers a practical framework for improving the efficiency of their project through various lean techniques. Originality/value: Research significance emphasizes increasing the effectiveness of the construction industry, influences the development of lean construction framework which improves lean construction practices using the lean techniques. This contributes to the effort of applying lean techniques in the construction industry. Limited publications were done in the construction industry mainly in United Arab Emirates (UAE) compared to the lean manufacturing. This research will recommend a systematic approach for the implementing of the anticipated framework within a cyclical look-ahead period and emphasizes the practical implications of the proposed approach.

Keywords: construction, lean, lean manufacturing, waste

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Authors: J. R. Bhalla, Gautam, Gurcharan Singh, Sanjeev Naval

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Mathematics is everywhere behind the every things on the earth as well as in the universe. Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. Now a day’s we can made and apply an equation on a complex geometry through applied mathematics. Here we work and focus on to create a formula which apply in the field of civil engineering in new concrete technology. In this paper our target is to make a formula which is applied on “BAUT” Metal Aggregate. In this paper our approach is to make formulation and equation on special “BAUT” Metal Aggregate by Applied Mathematical Study Case 1. BASIC PHYSICAL FORMULATION 2. ADVANCE EQUATION which shows the mechanical performance of special metal aggregates for concrete technology. In case 1. Basic physical formulation shows the surface area and volume manually and in case 2. Advance equation shows the mechanical performance has been discussed, the metal aggregates which had outstandingly qualities to resist shear, tension and compression forces. In this paper coarse metal aggregates is 20 mm which used for making high performance concrete (H.P.C).

Keywords: applied mathematical study case, special metal aggregates, concrete technology, basic physical formulation, advance equation

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Authors: Roselyn Alejandro-Ymana

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The study was an attempt to assess the MTAP Saturday Class Program along its eight components namely, modules, instructional materials, scheduling, trainer-teachers, supervisory support, administrative support, financial support and educational facilities, the results of which served as bases in developing policy guidelines to enhance the MTAP Saturday Class Program. Using a descriptive development method of research, this study involved the participation of twenty-eight (28) schools with MTAP Saturday Class Program in the Division of Dasmarinas City where twenty-eight school heads, one hundred twenty-five (125) teacher-trainer, one hundred twenty-five (125) pupil program participants, and their corresponding one hundred twenty-five (125) parents were purposively drawn to constitute the study’s respondent. A self-made validated survey questionnaire together with Pre and Post-Test Assessment Test in Mathematics for pupils participating in the program, and an unstructured interview guide was used to gather the data needed in the study. Data obtained from the instruments administered was organized and analyzed through the use of statistical tools that included the Mean, Weighted Mean, Relative Frequency, Standard Deviation, F-Test or One-Way ANOVA and the T-Test. Results of the study revealed that all the eight domains involved in the MTAP Saturday Class Program were practiced with the areas of 'trainer-teachers', 'educational facilities', and 'supervisory support' identified as the program’s strongest components while the areas of 'financial support', 'modules' and 'scheduling' as being the weakest program’s components. Moreover, the study revealed based on F-Test, that there was a significant difference in the assessment made by the respondents in each of the eight (8) domains. It was found out that the parents deviated significantly from the assessment of either the school heads or the teachers on the indicators of the program. There is much to be desired when it comes to the quality of the implementation of the MTAP Saturday Class Program. With most of the indicators of each component of the program, having received overall average ratings that were at least 0.5 point away from the ideal rating 5 for total quality, school heads, teachers, and supervisors need to work harder for total quality of the implementation of the MTAP Saturday Class Program in the division.

Keywords: mathematics achievement, MTAP program, policy guidelines, program assessment

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Authors: Ashley DelCorno

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GEM Anscombe’s 1958 essay Modern Moral Philosophy and the tradition of virtue ethics that followed has given rise to the restoration (or, more plainly, the resurrection) of Aristotle as something of an authority figure. Alisdair MacIntyre and Martha Nussbaum are proponents, for example, not just of Aristotle’s relevancy but also of his apparent implicit authority. That said, it’s not clear that the schema imagined by virtue ethicists accurately describes moral life or that it does not inadvertently work to impoverish genuine decision-making. If the label ‘virtue’ is categorically denied to some groups (while arbitrarily afforded to others), it can only turn on itself, thus rendering ridiculous its own premise. Likewise, as an inescapable feature of virtue ethics, Aristotelean binaries like ‘virtue/vice’ and ‘voluntary/involuntary’ offer up false dichotomies that may seriously compromise an agent’s ability to conceptualize choices that are truly free and rooted in meaningful criteria. Here, this topic is analyzed through a feminist lens predicated on the known paradoxes of patriarchy. The work of feminist theorists Jacqui Alexander, Katharine Angel, Simone de Beauvoir, bell hooks, Audre Lorde, Imani Perry, and Amia Srinivasan serves as important guideposts, and the argument here is built from a key tenet of black feminist thought regarding scarcity and possibility. Above all, it’s clear that though the philosophical tradition of virtue ethics presents itself as recovering the place of agency in ethics, its premises possess crippling limitations toward the achievement of this goal. These include, most notably, virtue ethics’ binding analysis of history, as well as its axiomatic attachment to obligatory clauses, problematic reading-in of Aristotle and arbitrary commitment to predetermined and competitively patriarchal ideas of what counts as a virtue.

Keywords: feminist history, the limits of utopic imagination, curatorial creation, truth, virtue, freedom

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Authors: Jacqueline Jasmine Kumar

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This paper seeks to challenge the predominate Western-centric paradigm concerning empathy by conducting an exploration of its presence within both Western and Eastern philosophical traditions. The primary focus of this inquiry is the examination of the Indian yogic tradition, encompassing the four yogas: bhakti (love/devotion), karma (action), jnāna (knowledge), and rāja (psychic control). Through this examination, it is demonstrated that empathy does not exclusively originate from Western philosophical thought. Rather than superimposing the Western conceptualization of empathy onto the tenets of Indian philosophy, this study endeavours to unearth a distinct array of ideas and concepts within the four yogas, which significantly contribute to our comprehension of empathy as a universally relevant phenomenon. To achieve this objective, an innovative approach is adopted, delving into various facets of empathy, including the propositional, affective/intuitive, perspective-taking, and actionable dimensions. This approach intentionally deviates from conventional Western frameworks, shifting the emphasis towards lived morally as opposed to engagement in abstract theoretical discourse. While it is acknowledged that the explicit term “empathy” may not be overly articulated within the yogic tradition, a scrupulous examination reveals the underlying substance and significance of this phenomenon. Throughout this comparative analysis, the paper aims to lay a robust foundation for the discourse of empathy within the contexts of the human experience. By assimilating insights gleaned from the Indian yogic tradition, it contributes to the expansion of our comprehension of empathy, enabling an exploration of its multifaceted dimensions. Ultimately, this scholarly endeavour facilitates the development of a more comprehensive and inclusive perspective on empathy, transcending cultural boundaries and enriching our collective repository of knowledge.

Keywords: Bhakti, Yogic, Jnana, Karma

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Authors: Samitharathana R. Wadigala

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The peace will not be established until the self-consciousness would reveal in the human beings. In this nuclear age, the establishment of a lasting peace on the earth represents the primary condition for the preservation of human civilization and survival of human beings. Nothing perhaps is so important and indispensable as the achievement and maintenance of peace in the modern world today. Peace in today’s world implies much more than the mere absence of war and violence. In the interdependent world of today the United Nations needs to be representative of the modern world and democratic in its functioning because it came into existence to save the generations from the scourge of war and conflict. Buddhism is the religion of peaceful co-existence and philosophy of enlightenment. Violence and conflict from the perspective of the Buddhist theory of interdependent origination (Paṭiccasamuppāda) are same with everything else in the world a product of causes and conditions. Buddhism is totally compatible with the congenial and peaceful global order. The canonical literature, doctrines, and philosophy of Buddhism are the best suited for inter-faith dialogue, harmony, and universal peace. Even today Buddhism can resurrect the universal brotherhood, peaceful co-existence and harmonious surroundings in the comity of nations. With its increasing vitality in regions around the world, many people today turn to Buddhism for relief and guidance at the time when peace seems to be a deferred dream more than ever. From a Buddhist point of view the roots of all unwholesome actions (Conflict) i. e. greed, hatred and delusion are viewed as the root cause of all human conflicts. Conflict often emanates from attachment to material things: pleasures, property, territory, wealth, economic dominance or political superiority. Buddhism has some particularly rich resources for deployment in dissolving conflict. Buddhism addresses the Buddhist perspective on the causes of conflict and ways to resolve conflict to realize world peace. The world has enough to satisfy every body’s needs but not every body’s greed.

Keywords: Buddhism, conflict-violence, peace, self-consciousness

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Authors: Irena Smetackova

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The important part of school mathematics are word problems which represent the connection between school knowledge and life reality. To find the reasons why students consider word problems to be difficult, it is necessary to take into consideration the motivational settings, besides mathematical knowledge and reading skills. Our goal is to identify whether the familiar or unfamiliar context of math word problem influences solving success rate and if so, whether the reasons are motivational or cognitive. For this purpose, we conducted three steps study in group of fifty pupils 9-10 years old. In the first step, we asked pupils to create ‘the best’ word problems for entered numerical formula. The set of 19 word problems with different contexts were selected. In the second step, pupils were asked to evaluate (without solving) how they like each item and how easy it is for them. The 6 word problems with low preference and low estimated success rate were selected and combined with other 6 problems with high preference and success rate. In the third step, the same pupils were asked to solve the word problems. The analysis showed that pupils attitudes and solving toward word problems varied by the context. The strong gender patterns both in preferred contexts and in estimated success rates were identified however the real success rate did not differ so strongly. The success gap between word problems with and without preferred contexts were stronger than the gap between problems with and without real experience with the context. The hypothesis that motivational factors are more important than cognitive factors was confirmed.

Keywords: mathematics, context of reality, motivation, cognition, word problems

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Authors: P. Nafisi Poor, P. Javid

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Religion and divinity have always held important meaning to humans, and therefore it affects different aspects of life including art and architecture. Numerous works of art are related to religion whether supporting or denying it. Religion and religious scholars have influenced and changed art throughout history. This paper focuses on Islam and Christianity because these two religions have been the most discussed and most popular of all time, starting from the birth of Jesus to the arrival of Mohammad. Based on this popularity, these religions have influenced the arts and especially architecture. Islam on one hand changed Iranian and Arabian architecture and they applied it in different places around the world. From the appearance of Islam at 622 AD to this day, Islamic architecture has been evolving; however, one of the most important periods for this style was between 1501 AD and 1736 AD in Iran. Christianity, on the other hand, changed European architecture especially between 1150 AD and 1450 AD or the so-called "Gothic" era, which begins at medieval time and reaches its peak at International Gothic ages. At both of these periods, designing buildings based on spiritual concepts and divine statements reached its peak, and architects were considering God and religion as their center of attention. This article studies the focus on the religions of Islam and Christianity in terms of architecture and presents a general philosophy of both styles to comprehend the idea behind each one, followed by an analysis of their geometry and architectural aspects derived from the best examples, all to understand the purpose of each style and to realize, which one was more successful in reaching their purpose. Subsequently, a comprehensive review of each building is provided including 3D visualizations to help achieve the goal of the article. These studies can support diverse inquiries about both Islamic and Gothic architecture and can be used as a resource to support studies and research towards designing based on religion or for divine purposes.

Keywords: architecture, Gothic, Islamic, religion

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Authors: Meshandren Naidoo

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Personhood is an important philosophical and ethical device that belies many major ethical and legal issues. The concept of African personhood is often overlooked, however, given the decolonization projects occurring in Africa, it is important to consider this view. African personhood, as opposed to Western personhood, is not individualistic in nature. The latter is predominantly Kantian and based on the notion that all persons have equal moral due to their capacity for a reason, whereas communitarianism is central to an African conception of personhood.

Keywords: African philosophy, bioethics, ethics, personhood

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Authors: F. Longo, G. Granello, G. Tecchio, F. Da Porto

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Unreinforced masonry (URM) infill walls are widely used throughout the world, also in seismic prone regions, as partitions in reinforced concrete building frames. Even if they do not represent structural elements, they can dramatically affect both strength and stiffness of RC structures by acting as a diagonal strut, modifying shear and displacements distribution along the building height, with uncertain consequences on structural safety. In the last decades, many refined models have been developed to describe infill walls effect on frame structural behaviour, but generally restricted to in-plane actions. Only very recently some new approaches were implemented to consider in-plane/out-of-plane interaction of URM infill walls in progressive collapse simulations. In the present work, a particularly promising macro-model was adopted for the progressive collapse analysis of infilled RC frames. The model allows to consider the bi-directional interaction in terms of displacement and strength capacity for URM infills, and to remove the infill contribution when the URM wall is supposed to fail during the analysis process. The model was calibrated on experimental data regarding two different URM panels thickness, modelling with particular care the post-critic softening branch. A frame specimen set representing the most common Italian structures was built considering two main normative approaches: a traditional design philosophy, corresponding to structures erected between 50’s-80’s basically designed to support vertical loads, and a seismic design philosophy, corresponding to current criteria that take into account horizontal actions. Non-Linear Static analyses were carried out on the specimen set and some preliminary evaluations were drawn in terms of different performance exhibited by the RC frame when the contemporary effect of the out-of-plane damage is considered for the URM infill.

Keywords: infill Panels macromodels, in plane-out of plane interaction, RC frames, URM infills

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Authors: Sylvia Huang

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Contemporary Buddhist music has been emerged at the new forms of large-scale public Buddhist ritual ceremonies that may involve up to 10,000 participants at a time. Since 2007, the Buddha’s Birthday ceremony (Sanskrit, Vesak) by the Buddhist Tzu Chi Foundation has being held at major cities in Taiwan and many affiliated Tzu Chi offices around the world. Analysis of this modern and technologically-dependent ceremony sheds new light on the significance of music in contemporary Buddhist ritual, and also on recently enhanced and increasingly intimate connections between music and Buddhism. Through extensive ethnographic research of ten years (2007-2017), the research explores how the form of contemporary Buddhist music relates to the role of music in participants’ experience of the ritual and the way in which they construct meaning. The theoretical approach draws on both ethnomusicology and Buddhist teachings, Dharma. As soundscape is defined as the entire sonic energy produced by a landscape, the concept of soundscape is utilised to examine the contemporary ritual music in the Tzu Chi Vesak ceremony. The analysis opens new territory in exploring how analysis of Buddhist music can benefit from incorporating Buddhist philosophy within the methodological approach. Main findings are: 1) music becomes a method for Buddhist understanding through a focus in particular on how the ceremonial program is followed by music, and 2) participants engage with each other and entrain with music in the Vesak ceremony. As Buddhist sounding, such as scripture reading, liturgical chanting, and ceremonial music singing, is a sonic epistemological knowing of the conditions in which Buddhism is practiced, experienced, and transmigrated, the research concludes by showing that studies of Buddhist music have the potential to reveal distinctively Buddhist concepts, meaning, and values. Certain principles of Buddhist philosophy are adopted within ethnomusicological analysis to further enhance understandings of the crucial function of music within such a ritual context. Finally, the contemporary Buddhist music performance in the ceremony is possessed as a means of direct access to the spiritual experience in Buddhism.

Keywords: buddhist music, Taiwan, soundscape, Vesak ceremony

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Authors: M. Tomaschko

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The integration of technology in educational settings has gained a lot of interest. Especially the use of mobile devices and accompanying mobile applications can offer great potentials to complement traditional education with new technologies and enrich students’ learning in various ways. Nevertheless, the usability of the deployed mathematics application is an indicative factor to exploit the full potential of technology enhanced learning because directing cognitive load toward using an application will likely inhibit effective learning. For this reason, the purpose of this research study is the identification of possible usability issues of the mobile GeoGebra Graphing Calculator application. Therefore, eye tracking in combination with task scenarios, think aloud method, and a SUS questionnaire were used. Based on the revealed usability issues, the mobile application was iteratively redesigned and assessed in order to verify the success of the usability improvements. In this paper, the identified usability issues are presented, and recommendations on how to overcome these concerns are provided. The main findings relate to the conception of a mathematics keyboard and the interaction design in relation to an equation editor, as well as the representation of geometrical construction tools. In total, 12 recommendations were formed to improve the usability of a mobile graphing calculator application. The benefit to be gained from this research study is not only the improvement of the usability of the existing GeoGebra Graphing Calculator application but also to provide helpful hints that could be considered from designers and developers of mobile math applications.

Keywords: GeoGebra, graphing calculator, math education, smartphone, usability

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Authors: John E. Boberg, Steven J. Bourgeois

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The decline in student engagement and motivation through the middle years is well documented and clearly associated with a decline in mathematics achievement that persists through high school. To combat this trend and, very often, to meet high-stakes accountability standards, a growing number of parents, teachers, and schools have implemented various methods to incentivize learning. However, according to Self-Determination Theory, forms of incentivized learning such as public praise, tangible rewards, or threats of punishment tend to undermine intrinsic motivation and learning. By focusing on external forms of motivation that thwart autonomy in children, adults also potentially threaten relatedness measures such as trust and emotional engagement. Furthermore, these controlling motivational techniques tend to promote shallow forms of cognitive engagement at the expense of more effective deep processing strategies. Therefore, any short-term gains in apparent engagement or test scores are overshadowed by long-term diminished motivation, resulting in inauthentic approaches to learning and lower achievement. The current study focuses on the relationships between student trust, engagement, and motivation during these crucial years as students transition from elementary to middle school. In order to test the effects of controlled motivational techniques on achievement in mathematics, this quantitative study was conducted on a convenience sample of 22 elementary and middle schools from a single public charter school district in the south-central United States. The study employed multi-source data from students (N = 1,054), parents (N = 7,166), and teachers (N = 356), along with student achievement data and contextual campus variables. Cross-sectional questionnaires were used to measure the students’ self-regulated learning, emotional and cognitive engagement, and trust in teachers. Parents responded to a single item on incentivizing the academic performance of their child, and teachers responded to a series of questions about their acceptance of various incentive strategies. Structural equation modeling (SEM) was used to evaluate model fit and analyze the direct and indirect effects of the predictor variables on achievement. Although a student’s trust in teacher positively predicted both emotional and cognitive engagement, none of these three predictors accounted for any variance in achievement in mathematics. The parents’ use of incentives, on the other hand, predicted a student’s perception of his or her controlled motivation, and these two variables had significant negative effects on achievement. While controlled motivation had the greatest effects on achievement, parental incentives demonstrated both direct and indirect effects on achievement through the students’ self-reported controlled motivation. Comparing upper elementary student data with middle-school student data revealed that controlling forms of motivation may be taking their toll on student trust and engagement over time. While parental incentives positively predicted both cognitive and emotional engagement in the younger sub-group, such forms of controlling motivation negatively predicted both trust in teachers and emotional engagement in the middle-school sub-group. These findings support the claims, posited by Self-Determination Theory, about the dangers of incentivizing learning. Short-term gains belie the underlying damage to motivational processes that lead to decreased intrinsic motivation and achievement. Such practices also appear to thwart basic human needs such as relatedness.

Keywords: controlled motivation, student engagement, incentivized learning, mathematics achievement, self-determination theory, student trust

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Authors: Araceli Martinez Ortiz, Gillian Bayne, Solomon Abraham

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This paper describes research led by faculty from three American universities and four Ethiopian universities on the delivery of professional leadership development for early-career female Ethiopian university instructors in the Science, Technology, Engineering, and Mathematics (STEM) fields. The objective was to carry out a case study focused on the impact of an innovative intervention program designed to assist in the empowerment and leadership development related to teaching effectiveness, scholarly activity participation, and professional service participation by female instructors. This research was conducted utilizing a case study methodology for the weeklong intervention and a survey to capture the voices of the leadership program participants. The data regarding insights into the challenges and opportunities for women in these fields is presented. The research effort project expands upon existing linkages between universities to support professional development and research effort in this region of the world. Findings indicate the positive reception of this kind of professional development by the participating women. Survey data also reflects the particular cultural challenges professional women in STEM education face in Ethiopia as well as the global challenges of balancing family expectations with career development.

Keywords: Ethiopian women, STEM leadership, professional development, gender equity

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Authors: Atara Shriki, Ilana Lavy

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Teaching for the first time an online course dealing with the history of mathematics, we were struggling with questions related to the design of a proper learning environment (LE). Thirteen high school mathematics teachers, M.Ed. students, attended the course. The teachers were engaged in independent reading of mathematical texts, a task that is recognized as complex due to the unique characteristics of such texts. In order to support the learning processes and develop skills that are essential for succeeding in learning online (e.g. self-regulated learning skills, meta-cognitive skills, reflective ability, and self-assessment skills), the LE comprised of three components aimed at “scaffolding” the learning: (1) An online "self-feedback" questionnaires that included drill-and-practice questions. Subsequent to responding the questions the online system provided a grade and the teachers were entitled to correct their answers; (2) Open-ended questions aimed at stimulating critical thinking about the mathematical contents; (3) Reflective questionnaires designed to assist the teachers in steering their learning. Using a mixed-method methodology, an inquiry study examined the learning processes, the learners' difficulties in reading the mathematical texts and on the unique contribution of each component of the LE to the ability of teachers to comprehend the mathematical contents, and support the development of their learning skills. The results indicate that the teachers found the online feedback as most helpful in developing self-regulated learning skills and ability to reflect on deficiencies in knowledge. Lacking previous experience in expressing opinion on mathematical ideas, the teachers had troubles in responding open-ended questions; however, they perceived this assignment as nurturing cognitive and meta-cognitive skills. The teachers also attested that the reflective questionnaires were useful for steering the learning. Although in general the teachers found the LE as supportive, most of them indicated the need to strengthen instructor-learners and learners-learners interactions. They suggested to generate an online forum to enable them receive direct feedback from the instructor, share ideas with other learners, and consult with them about solutions. Apparently, within online LE, supporting learning merely with respect to cognitive aspects is not sufficient. Leaners also need an emotional support and sense a social presence.

Keywords: cognitive and meta-cognitive skills, independent reading of mathematical texts, online learning environment, self-regulated learning skills

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Authors: Ozgul Kartal, Wade Tillett, Lyn D. English

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Despite its global significance in curricula, mathematical modeling encounters persistent disparities in recognition and emphasis within regular mathematics classrooms and teacher education across countries with diverse educational and cultural traditions, including variations in the perceived role of mathematical modeling. Over the past two decades, increased attention has been given to the integration of mathematical modeling into national curriculum standards in the U.S. and other countries. Therefore, the mathematics education research community has dedicated significant efforts to investigate various aspects associated with the teaching and learning of mathematical modeling, primarily focusing on exploring the applicability of modeling in schools and assessing students', teachers', and preservice teachers' (PTs) competencies and engagement in modeling cycles and processes. However, limited attention has been directed toward examining potential resistance hindering teachers and PTs from effectively implementing mathematical modeling. This study focuses on how PTs, without prior modeling experience, resist and/or embrace mathematical modeling and its pedagogy as they learn about models and modeling perspectives, navigate the modeling process, design and implement their modeling activities and lesson plans, and experience the pedagogy enabling modeling. Model eliciting activities (MEAs) were employed due to their high potential to support the development of mathematical modeling pedagogy. The mathematical modeling module was integrated into a mathematics methods course to explore how PTs embraced or resisted mathematical modeling and its pedagogy. The module design included reading, reflecting, engaging in modeling, assessing models, creating a modeling task (MEA), and designing a modeling lesson employing an MEA. Twelve senior undergraduate students participated, and data collection involved video recordings, written prompts, lesson plans, and reflections. An open coding analysis revealed acceptance and resistance toward teaching mathematical modeling. The study identified four overarching themes, including both acceptance and resistance: pedagogy, affordance of modeling (tasks), modeling actions, and adjusting modeling. In the category of pedagogy, PTs displayed acceptance based on potential pedagogical benefits and resistance due to various concerns. The affordance of modeling (tasks) category emerged from instances when PTs showed acceptance or resistance while discussing the nature and quality of modeling tasks, often debating whether modeling is considered mathematics. PTs demonstrated both acceptance and resistance in their modeling actions, engaging in modeling cycles as students and designing/implementing MEAs as teachers. The adjusting modeling category captured instances where PTs accepted or resisted maintaining the qualities and nature of the modeling experience or converted modeling into a typical structured mathematics experience for students. While PTs displayed a mix of acceptance and resistance in their modeling actions, limitations were observed in embracing complexity and adhering to model principles. The study provides valuable insights into the challenges and opportunities of integrating mathematical modeling into teacher education, emphasizing the importance of addressing pedagogical concerns and providing support for effective implementation. In conclusion, this research offers a comprehensive understanding of PTs' engagement with modeling, advocating for a more focused discussion on the distinct nature and significance of mathematical modeling in the broader curriculum to establish a foundation for effective teacher education programs.

Keywords: mathematical modeling, model eliciting activities, modeling pedagogy, secondary teacher education

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Authors: Juan Torralba

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Student digital artifacts were investigated, utilizing a qualitative exploratory research design to understand the ways in which students represented their knowledge of seventh-grade proportionality concepts as they participated in maker-based activities that culminated in the creation of digital 3-dimensional models of their dream homes. Representations of the geometric and numeric dimensions of proportionality were analyzed in the written, verbal, and visual data collected from the students. A directed content analysis approach was utilized in the data analysis, as this work aimed to build upon existing research in the field of maker-based STEAM Education. The results from this work show that students can represent their understanding of proportional reasoning through open-ended written responses more accurately than through verbal descriptions or digital artifacts. The geometric and numeric dimensions of proportionality and their respective components of attributes of similarity representation and percents, rates, and ratios representations were the most represented by the students than any other across the data, suggesting a maker-based instructional approach to teaching proportionality in the middle grades may be promising in helping students gain a solid foundation in those components. Recommendations for practice and research are discussed.

Keywords: learning through making, maker-based education, maker education in the middle grades, making in mathematics, the maker movement

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