
A Rewarding Journey of Connecting Mathematics and Technology to Applications




Authors: 
Philip Todd 
Affiliations: 
Saltire Software 
The fact that one directed line bisects the angle between two directed
lines may be expressed (modulo $2\pi$) as a linear equation in the
directions of the three lines. The coefficients of the equation are
2, 1, 1 and its right hand side is 0. The external angle bisector
differs only in that its right hand side is $\pi$. The direction
of the base of an isosceles triangle is the same as the external bisector
of the angle between its other sides. The relationship between the
chord of a circle and the lines joining its center to the endpoints
of the chord is that between the base and the sides of an isosceles
triangle. Hence a variety of angle theorems are in fact angle bisection
theorems. In this paper we consider theorems where a collection of
angle bisection relationships determine an angle of the figure. We
present a graph theoretical representation of such theorems and develop
conditions on the graphs for the theorem to be true.
Abstract for 21869
Wavelet Neural Network Prediction of Stock Performance
Author: 
EnBing Lin 
Affiliations: 
Central Michigan University, IEEE, AMS 
We incorporate wavelet methods and artificial neural networks to examine the forecasting performance of the daily closing price of the Microsoft stock, NASDAQ:MSFT. The predictions achieve good performance by choosing the appropriate level of wavelet resolution and an optimal number of hidden layers for the neural network. Our model could be used by investors, financial managers, or others to enhance their ability to select desired stocks.
Abstract for 21875
Understanding Geometric Pattern and its Geometry Part 6 – Using Geometer’s Sketchpad for designing sizeable geometric projects
Authors: 
Miroslaw Majewski 
Affiliations: 
New York Institute of Technology, Abu Dhabi Campus 
This document aims to explain a few aspects of dealing with geometric objects in Geometer’s Sketchpad that are not entirely related to geometry. We discuss how one can reduce the amount of data needed to create bulky geometric constructions without losing the quality and accuracy of the final design. We use a real complex geometric pattern to demonstrate our conclusions.
Abstract for 21877
Locus of Antipodal Projection When Fixed Point is Outside a Curve or Surface
Authors: 
WeiChi Yang, Antonio Morante 
Affiliations: 
Radford University, and UASLP, San Luis Potosi (SLP), Mexico 
We continue the investigations regarding the locus problems for certain curves and quadric surfaces, which are discussed in [6] and [7] in 2D and 3D cases. We explore the intersecting curves when the fixed point A is outside the surface, and when A is at an infinity. The discoveries from explorations with the help of technological tools in this paper will assist learners to conduct further research in the area of projective geometry and beyond.
Abstract for 21878
Mathematical Patterns formed by the Spheres, the Cylinders and the Disks
Authors: 
Jenchung Chuan 
Affiliations: 
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300 
G.H. Hardy once said: “A mathematician, like a painter or a poet, is a maker of patterns.” After many years of experimenting with the concrete objects constructed with Cabri 3D, the visual patterns created have enriched my repertoire in Recreational Mathematics presentations. In this talk, we shall focus on three classes of 3D objects: the Spheres, the Cylinders and the Disks. In order to reduce the latency and the network congestion, the reader is advised to enhance the computer with the Edge Computing infrastructure (such as the Apple Photos/iCloud or the Google Photos/Google Cloud) to view the animations linked in this paper.
Abstract for 21885
Technologies and laboratorial activities for a robust understanding approach: from closed to open laboratories
Authors: 
Daniela Ferrarello, Maria Flavia Mammana, Mario Pennisi, Eugenia Taranto 
Affiliations: 
Department of Mathematics and Computer Science, University of Catania Italy, Department of Agriculture, Food and Environment, University of Catania  Italy, Department of Mathematics and Computer Science, University of Catania – Italy 
The aim of this paper is to show the evolution of a metamethodology adopted by the authors in some of the activities they proposed over the last 10 years to students and teachers. All the activities, called laboratories, involve the use of technology and are based on laboratorial methodology aiming to a robust understanding of mathematics. We describe, as example, five laboratories and provide a discussion on our metamethodology as researchers on such laboratories finding a metamethodological evolution. In particular, by analysing this evolution, we detect an initial attitude towards “closed” laboratories (rigidly structured, letting no freedom of choice for students and teachers) evolving into an attitude to “open” laboratories (letting freedom of choice to students and/or teachers).
Abstract for 21886
Flowgorithm: A possible passage from algorithmic reasoning to creatively founded mathematical reasoning
Authors: 
Weng Kin Ho, Chee Kit Looi, Wendy Huang, Peter Seow, Shiau Wei Chan, Longkai Wu 
Affiliations: 
Nanyang Technological University 
This paper argues that algorithm design in the sense of computational thinking (CT) does not involve only routinized procedural applications void of deep conceptual understanding of mathematics. By using the programming language Flowgroithm, we demonstrate how classroom tasks centred around algorithm design may be used to activate creatively founded mathematical reasoning (CMR) in mathematics students.
Abstract for 21887
Some affordances and opportunities of advanced scientific calculators for learning introductory calculus
Authors: 
Barry Kissane 
Affiliations: 
Murdoch University 
Advanced scientific calculators, developed in recent years, include more sophisticated capabilities than original models, including some of direct relevance to elementary calculus. In this paper, some relevant calculator capabilities are described and their potential use for students and teachers of elementary calculus are illustrated and evaluated, against the framework of a model for learning mathematics with calculators. Despite the name of such devices, calculators derive their educational value from processes other than mere calculation. Key ideas of continuity, convergence, limits, differentiation and integration can be both represented and gainfully explored on an advanced scientific calculator, supporting student learning in powerful ways not otherwise available to many students in Asian countries.
Abstract for 21889
Computational Discovery with Newton Fractals, Bohemian Matrices, & Mandelbrot Polynomials
Authors: 
Neil Calkin, Eunice Chan, Rob Corless 
Affiliations: 
Western University, University of Waterloo and Clemson University 
The authors have been using a largely algebraic form of "computational discovery"
in various undergraduate classes at their respective institutions for some decades now to
teach pure mathematics, applied mathematics, and computational mathematics. This
paper describes what we mean by "computational discovery," what good it does for the
students, and some specific techniques that we used.
Abstract for 21890
On the locus related to chords of conic sections
Author: 
Pavel Pech 
Affiliations: 
Faculty of Education, University of South Bohemia, Czech Republic 
The paper deals with the locus of points related to chords of conic sections. Firstly the
locus is explored using dynamic geometry software, particularly for displaying it, secondly,
using elimination in computer algebra software, the locus equation is derived. However this
elimination leads to the zero elimination ideal. It is shown how to compute the searched
equation in such a case. Further the locus is applied in the proof of the theorem which is
related to the Fregier point. Finally, connection between the original formulation of the
locus and the formulation by an envelope is demonstrated.
By solving the problem we mainly use dynamic geometry software GeoGebra and computer
algebra program CoCoA and Singular.
Abstract for 21891
The Mathematics of QR Codes
Authors: 
Adam Downs, Neil Sigmon, Rick Klima

Affiliations: 
Appalachian State University, Radford University, USA 
Quick Response (QR) codes are twodimensional bar codes that have become a common medium for easily accessing information such as URLs, phone numbers, and small amounts of text. The creation of QR codes requires placing data in a prescribed format so that it can be displayed on a surface and then detected by common QR code scanner software. Like other physical means of storing data, QR codes are prone to errors when their data is interpreted in a digital format. ReedSolomon codes provide a mechanism for ensuring that QR code scanners can reliably process information when errors occur. This involves sending information in the form of polynomial coefficients using finite field arithmetic. Utilizing ReedSolomon codes allows logos and emblems to be embedded within QR codes to advertise their purpose. In this paper we will present how QR codes are constructed, and how ReedSolomon codes are incorporated into them to provide error correction. To assist in demonstrating this, technology involving Maplets will be used.
Abstract for 21894
Approaching Cesàro's inequality through GeoGebra Discovery
Authors: 
Zoltán Kovács, Tomás Recio, M. Pilar Vélez 
Affiliations: 
The Private University College of Education of the Diocese of Linz, Austria, Universidad Antonio de Nebrija, Universidad Antonio de Nebrija, Madrid, Spain 
We illustrate the possibility of developing, using the automated reasoning tools implemented in the dynamic geometry program GeoGebra Discovery, a certain parallelism with Michael de Villiers' reflection on the ''discovery function'' of proof, as described in his 2012 paper concerning the formulation by one student of a certain geometric conjecture (Clough's conjecture).
Abstract for 21895
Blending Knowledge and Technology to Construct Steiner Chains
Author: 
JeanJacques Dahan 
Affiliations: 
IRES of Toulouse, France 
The aim of this paper is to show the crucial role of knowledge to tackle some geometric problems. We have chosen to show in detail how to tackle the problem of construction of Steiner chains, that is to say, closed chains of tangent circles also tangent to two given nested circles (to simplify the presentation and get understandable figures). Accordingly, a great part of the paper is a reminder of the necessary knowledge used for the techniques of construction of such chains: notion of harmonic division, notion of bundle of circles, notion of polar of a point with respect to a circle and above all the notion of inversion and the link between this transformation and the previous notions. The paper shows how these techniques can be used with some dynamic geometry software such as the New Cabri (containing the tool “macro” but not the tool “inversion”), Cabri 3D (containing the tool “inversion” with respect to spheres) or TINSpire (less efficient for complex figures because the tools “macro” and “inversion” are not available). Finally, we will perform very precise figures mostly with the New Cabri where Steiner chains associated to two given nested circles are generated and can be animated when their conditions of existence are satisfied.
Abstract for 21903
Mathematics Learning Strategy Scales for Junior High School Students: Scale development, validation and intelligent application
Authors: 
Guangming Wang, Mingyu Su, Xia Chen 
Affiliation: 
Tianjin Normal University 
Mathematics learning strategies are an important part of the psychological structure of highquality mathematics students. How to measure and evaluate students'' mathematics learning strategies Level is of great significance in related quantitative research. Based on the existing research on mathematics learning strategies, a set of mathematics learning strategy scales suitable for Chinese junior high school students and with mathematics learning characteristics were compiled. This research has done 4 times of data collection (a total of 959 valid questionnaires), item analysis, confirmatory factor analysis and exploratory factor analysis, which proved the validity of the scale in the measurement of mathematics learning strategies.In this study, the mathematics learning strategies of junior high school students were divided into 3 main aspects: Mathematics Cognitive Strategies, Mathematics Metacognitive Strategies, Mathematics Resource Management Strategies; 10 kinds of subdimensions: Retelling, Elaboration, Organize, Planning, Monitoring, FeedbackAdjustment, Timemanagement, Environmentmanagement, Moodmanagement, Helpseeking. According to the operational definition, test results show that the questionnaire has good reliability and validity, which can be used as an effective measurement tool for mathematics learning strategies of junior high school students. The study has written the scale into an intelligent batch assessment system of mathematics learning quality for primary and secondary school students, providing an accurate and convenient test of mathematics learning strategies for a wide range of junior secondary school students in an automated and intelligent manner, and providing accurate measurement reports and personalized improvement strategies for each tested student.
Abstract for 21912
Designing Performance Tasks in Mathematics Using Technological Tools
Authors: 
Ma. Louise Antonette De Las Penas, Debbie Marie B. Verzosa, Maria Alva Q. Aberin, Agnes D. Garciano, Jumela F. Sarmiento, Mark Anthony C. Tolentino 
Affiliations: 
Ateneo de Manila University, University of Southern Mindanao, Ateneo de Manila University 
In the Philippines, the performance task is one of the major summative assessments in the K to 12 curriculum. This paper discusses how performance tasks may utilize mathematical apps within the context of blended learning. Guidelines on designing performance tasks, as well as the GRASPS framework, are discussed. Performance tasks that cut across various grade levels and strands of mathematics are presented. These involve divisibility (Numbers), integer and polynomial operations (Algebra), triangle centers (Geometry), and statistics (Statistics). The performance tasks described in this paper can provide an initial idea for the design of other summative assessments and contribute to the literature on the use of technological tools in assessment and evaluation.
Abstract for 21917
The loci of virtual points constructed with elementary models of planetary orbits
Author: 
Noah (Thierry) DanaPicard 
Affiliation: 
Jerusalem College of Technology 
The orbits of the planets around the Sun are ellipses, close to circles. A simple model is given by concentric circles centered at the Sun. The radius of the orbit and the linear velocity of the planet may be found in numerical form in the literature. The motion of a planet on its orbit is then described by a trigonometric parametrization. Using such parametrizations for two planets, various mobile "virtual points" such as their midpoint or their center of gravity, are defined and their trajectories are studied. A good opportunity to experience different kinds of animations with software, depending on the affordances of the package. By that way, classical plane curves appear in a dynamical way. It appears quickly that the level of accuracy of the involved numerical data influences strongly the shape of the constructed curves. Despite the fact that loci and envelopes are often related, and automated commands exist for both in the same packages, their respective appearances may be different, the one very rich and the other much apparently less. The plane curves which appear as loci of some of the virtual points have been known for a long time, but separately. We have here an opportunity to see them as members of a larger family. A byproduct of the activities is an incitement to study classical plane curves and their properties, in a dynamical framework and in relation with real world. Despite the physical nonaccuracy of the models, this is a typical STEAM approach.
Abstract for 21920
The Potential of Outdoor Mathematics in a Digital Contextg
Authors: 
Matthias Ludwig, Simone Jablonski 
Affiliations: 
GoetheUniversity, GoetheUniversity Frankfurt Germany 
In this paper, we present the MathCityMap system which aims at bringing the idea of mathematics trails into an educational context. To facilitate the preparation and conduct from both, teacher’s and students’ perspective, the system provides two components: a web portal to create outdoor tasks and a smartphone app to guide students to discover mathematics outdoors. In this paper, we focus on the benefits of the math trail idea for educational purposes and, in particular, its enrichment through digital tools in the context of mobile learning. By taking learning theories and empirical findings into consideration, the potentials of the system are analyzed and evaluated. Hereby, the focus is on different aspects of math trails and, in particular, math trails with MathCityMap that differ from “classic” mathematics teaching, i.e. the autonomous learning, the potential for modelling tasks, the use of mobile learning, the role of embodiment and their empirical impact on students’ performance and motivation.
Abstract for 21921
Teaching the Concept of Integral and Related Contents using Software
Authors: 
Vladimir Nodelman 
Affiliation: 
Holon Institute of Technology 
The usual practice of studying the concept of Integral, if it uses any visual images, then they are reduced to considering the Riemann sums.
In the meantime, the contents of this subject are rich in opportunities to facilitate the perception and assimilation of educational material even in the case of rather delicate questions of the course, with the active use of readymade and new dynamic models created by students.
The lecture discusses the main ideas of the course and provides computer models, the use of which allows students to better understand the studied educational material.
Abstract for 21922
Archimedes in the 21st Century: Part I
Author: 
Douglas Meade 
Affiliation: 
Department of Mathematics, University of South Carolina 
“Archimedes: What Did He Do Besides Cry Eureka” is a gem of a book written by Sherman Stein in 1999 with the goal to “make what I view as Archimedes’ most mathematically sophisticated significant discoveries accessible to … anyone who recognizes the equation of a parabola”. In this talk I will begin to supplement Prof. Stein’s efficient exposition with appropriate use of modern dynamic geometry and computer algebra tools. Emphasis will be on the effective use of these tools to deepen understanding of the underlying mathematics.
Results will include as many of the following topics as time allows (others will be the topic of subsequent talks): centers of gravity, area, surface area, and volume of circles, spirals, spheres, paraboloids, and floating bodies.
While we will be utilizing 21st century technologies, the underlying ideas will be as faithful as possible to Archimedes’ methods in the thirdcentury BC – more than 2200 years ago.
Abstract for 21923
Mathematical Knowledge for Teaching of Calculus: An Exploratory Study of Mathematics Teachers'' Knowledge in the Indian Context
Author: 
Jonaki Ghosh 
Affiliation: 
Lady Shri Ram College, University of Delhi, India 
Calculus is perhaps the hardest topic to teach at the senior secondary school level. It forms a significant part of the mathematics syllabus of upper secondary school in most countries and is often considered to be the gatekeeper for accesss to higher mathematics. Needless to say, teaching of calculus at school is fraught with many challenges. Students encounter a set of procedures to be memorised and applied to solving problems of a routine nature. A significant amount of time is spent on developing procedural skills for evaluating limits, derivatives and integrals while very little emphasis is given to visualising these concepts. Much of the practice of teaching calculus and the way students develop the fundamental concepts is dependent on the teachers’ understanding of how these concepts should be taught to students. Thus teacher knowledge with regard to calculus plays a critical role in shaping students’ understanding of the concepts and is an important area of research.
This talk will present a study conducted in the Indian context, which focuses on teachers’ conceptual as well as pedagogical knowledge with regard to teaching of calculus. It delves into how they interpret and adapt knowledge presented in textbooks for teaching purposes, how they sequence their teaching material and the nature of explanations and examples they provide. It also aims to understand how teachers assess the cognitive demands of certain tasks, how they identify students’ errors and rectify their misconceptions.
The theoretical framework, which forms the backbone of this study, is the Mathematical Knowledge for Teaching (MKT) framework proposed by Ball. D. L. et. al. [1] . It refers to fundamental knowledge categories required by mathematics teachers for the professional work of teaching. The MKT framework distinguishes between two subdomains of knowledge  Subject Matter Knowledge (SMK) and Pedagogical Content Knowledge (PCK). SMK and PCK are further subdivided to six subdomains. This study focuses on two subdomains specialized content knowledge (SCK) which is a part of SMK and knowledge of content and students (KCS), which belongs to the domain of PCK, specifically in relation to teaching of calculus at the level of grades 11 and 12. SCK is the knowledge of mathematics exclusively needed for the teaching profession [2], which includes “building and examining alternative representations, providing representations and understanding unconventional students method” [3]. KCS can be defined as the content knowledge of how students think about, know, or learn a particular content as well as their conceptions and misconceptions.
The study aimed to address the following research questions
1. What is the nature of conceptual knowledge of secondary school mathematics teachers related to the fundamental concepts of limits, continuity and derivatives? What metaphors, examples and visual representations do they use when teaching these concepts to students?
2. How flexible is teachers’ knowledge related to calculus teaching? Which modes of representation  graphical, numerical or symbolic do they use?
3. What are teachers’ understandings of students’ notions and abilities related to the learning of calculus? How do they assess the cognitive demands of various tasks? What is the nature of remedial measures used by teachers to scaffold students’ conceptual development?
There are 25 senior secondary mathematics teachers from a diverse range of schools of New Delhi, which follow the curriculum proposed by the Central Board of Secondary Education (CBSE) [4] participated in this study. Based on the MKT framework, 16 items were designed to elicit Specialised Content Knowledge (SCK) and Knowledge of Content and Students (KCS) of the teachers related to the fundamental concepts of limits, continuity and derivatives. The items comprised multiple choice questions (MCQ) as well as open response questions (ORQ). Teachers’ responses to these items were analysed and some interesting facts were revealed with regard to their understanding of these fundamental concepts, their pedagogical approaches and their thinking related to students’ conceptions or misconceptions. An analysis of the SCK items revealed that majority of teachers preferred procedural methods to illustrate concepts. The examples chosen by them were mostly algebraic and rarely graphical. They did not seem to be flexibile in using different modes of representation while explaining concepts and their explanations were restricted to those available in the textbook. They experienced discomfort when attempting items which required graphical reasoning. Teachers however responded more confidently to the KCS items. They demonstrated firm ideas about students’ understanding of calculus concepts and could state clearly what students find difficult and were able to suggest remedial measures to help students overcome their problems.
The findings of this study seem to suggest that teachers would benefit greatly from professional development programs designed to build upon their existing conceptual knowledge through graphing software. This led the author to develop modules in GeoGebra which emphasised a graphical understanding of the concepts of limits, derivatives, antiderivatives and the Fundamental Theorem of Calculus. Feedback taken from participating teachers at the end of the professional development workshop convinced the author of the need for enhancing teachers’ Mathematical knowledge for Teaching (MKT) especially with regard to calculus. While many MKT studies have been conducted in topics of elementary mathematics (Number and operations, fractions, algebra and geometry), similar studies addressing Calculus are scarce. This was the primary motivation for conducting the study.
The talk will highlight different aspects of teachers’ knowledge with regard to teaching of calculus and the critical role that digital tools can play in enhancing their Mathematical knowledge for Teaching.
References
[1] Ball, D. L., Hill, C. H. & Schilling, S. H. (2008). Unpacking pedagogical content knowledge, conceptualizing and measuring teachers’ topics specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.
[2] Ball, D. L., Hill, C. H. & Bass, H. (2005). Knowing mathematics for teaching. Who knows mathematics well enough to teach third grade and how can we decide? American Educator, 329(1), 1417, 2022, 4346.
[3] Hill, H.C., Schilling, S.G., & Ball, D.L. (2004) Developing measures of teachers’ mathematics knowledge for teaching. Elementary School Journal 105, 1130.
[4] Central Board for Secondary Education. (n.d.). Course structure of mathematics (class XI and XII). Retrieved from http://cbseportal.com/exam/Syllabus/cbse11th12th2011mathematics.
Abstract for 21924
If it does not move . .
Author: 
Douglas Butler 
Affiliation: 
iCT Training Centre (Oundle, UK), AutographMaths 
This talk will discuss the benefits of learning through the movement of mathematical objects on the screen using dynamic software. There will be references to the main players Geogebra, Desmos, and Autograph, all of whom allow users to move objects. However far too many students and teachers use these programs simply to find answers. Hence the title “If it does not move . . .” there is little learning. On the other hand, if you can move objects around, and, importantly, take the opportunity to correctly predict the outcome, then the learning can be profound.
This session will explore the effectiveness of this approach in the teaching of the school mathematics curriculum: elementary geometry, the structure of graphs, concepts in calculus, vectors, complex numbers, and 3D objects.
Email: debutler@argonet.co.uk
Web: www.tsmresources.com
Twitter: @douglasbuter1
Abstract for 21927
A Generalization of the ZeroProbability Theorem
Author: 
Miodrag Lovric 
Affiliation: 
Radford University, Virginia, USA 
In this paper, a generalized form of the ZeroProbability Theorem (initially called a paradox) is established. Originally, it was proved for the standard hypothesized form that contains a single number. The new version demonstrates that when testing a mean of the normal population using a point null hypothesis which is formulated as a set of all countably infinite algebraic numbers, the probability of such a hypothesis is zero within the set of all real numbers. This result shows that the pointnull hypothesis paradigm is based on a flimsy foundation and points to one of the most important root causes of the current reproducibility crisis in science. We explore implications of this theorem on the Fisherian significance testing, NeymanPearson hypothesis testing, and Bayesian testing based upon the Bayes factor. We recommend that pointnull hypotheses of a normal mean should be immediately abandoned. For decades, their testing has produced countless criticisms and recently even methodological crises in many fields of science and has done serious damage to the image of statistics and statisticians. The new model should be based on the negligible null hypotheses accompanied by the practically relevant alternative hypotheses. We regard this simply achieved modification as a new paradigm in the Kuhn''s sense. Enough arguments are provided to confirm that this will not only eradicate most of the objections against frequentist testing and breathe a new life into them, but also considerably reconcile communication in inference between frequentist and Bayesian approaches. In conclusion, no frequentist nor Bayesian should test pointnull hypotheses for continuous parameters.
Abstract for 21932
On curves with circles as their isoptics
Author: 
Witold Mozgawa 
Affiliation: 
Maria CurieSklodowska University, Poland 
The main part of my lecture is a joint work with prof. W. Cieślak. Let us fix α ∈ (0,π). Let Cα be the locus of vertices of a fixed angle π−α formed by two tangent lines of a closed convex plane curve C of class C1. The curve Cα will be called an αisoptic of C, see [1][5]. In the talk we describe the family of all closed convex plane curves of class C1 which have circles as their isoptics and explain why these curves form a generalization of ovals of constant width, [3, 7]. In the first part of the talk we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the talk contains characterization of curves which have circles as their isoptics and we describe the form of support functions of these curves. In this framework, at the end, we present a generalization of the theorem characterizing ovals of constant width proved by Mellish in 1931, [6, 7].
References
[1] K. Benko, W. Cieślak, S. Góźdź, W. Mozgawa, On isoptic curves, An. Stiint. Univ. ”Al. I. Cuza”, 36, 4754 (1990).
[2] W. Cieślak, A. Miernowski, W. Mozgawa, Isoptics of a strictly convex curve, Global differential geometry and global analysis, Proc. Conf., Berlin/Ger. 1990, Lect. Notes Math. 1481, 2835 (1991).
[3] W. Cieślak, W. Mozgawa, On curves with circles as their isoptics. Aequat. Math. (2021), https://doi.org/10.1007/s00010021008284
[4] Th. DanaPicard, A. Naiman, W. Mozgawa, W. Cieślak, Exploring the isoptics of Fermat curves in the affine plane using DGS and CAS, Math.Comput.Sci., 14, 2020, 4567 (2020).
[5] Th. DanaPicard, W. Mozgawa, Automated exploration of inner isoptics of an ellipse, J. Geom. 111, No. 2, Paper No. 34, 10 p. (2020).
[6] A. P. Mellish, Notes on differential geometry. Ann. Math. (2) 32, 181190 (1931).
[7] W. Mozgawa, Mellish theorem for generalized constant width curves, Aequationes Math. 89, No. 4, 10951105 (2015).
Link to the abstract in PDF .
Abstract for 32101
An interactive visual introduction to curvature
flows
Author: 
Matthias Kawski 
Affiliation: 
School of Mathematical and Statistical Sciences, Arizona State University, USA 
After a short motivation we introduce several different curvature flows: A naive flow
on the curvature under the heat equation, the curveshortening flow, the mean curvature
flow for imbedded surfaces, and the Ricci flow on surfaces of revolution and for abstract
2manifolds. The main focus is on interactive visualizations using animations of curves,
surfaces, and in the case of the Ricci also using flow of a metric field similar to Tissot’s
indicatrix. Our code is written in the computer algebra system MAPLE and thus it is very
easy to vary the examples and experiment with different initial shapes. Link to the abstract in PDF .
Abstract for 32102
ShannonNyquist Sampling with a Student Exploration Application
Author: 
William C Bauldry 
Affiliation: 
Appalachian State University, USA 
The ShannonNyquist theorem provides a minimum number of sample points required
to collect to completely determine an analog signal. First, we discuss the history
following the developments of Nyquist, Shannon, and Whitakker on analyzing analog
functions. Second, we outline digitizing a signal and processing the data along
with aliasing and other anomalies. We finish by presenting a student application for
exploring ShannonNyquist Sampling using Maple™.
Abstract for 32103
PseudoRiemann's quartics
in Finsler's geometry
Author: 
Yaakov Itin 
Affiliation: 
Jerusalem College of Technology
Jeruslaem, Israel 
An extension of Riemmann's geometry into a direction dependent geometric structure is usu
ally described by Finsler's geometry. Historically, this construction was motivated by the
wellknown Riemann's quartic length element example. Quite surprisingly, the same quar
tic expression emerges in solidstate electrodynamics as a basic dispersion relationcovariant
Fresnel equation. Consequently, Riemann's quartic length expression can be interpreted as a
mathematical model of a wellestablished physics phenomena. In this paper, we present various
examples of Riemann's quartic that demonstrate that Finsler's geometry is too restrictive even
in the case of a positive definite Euclidean signature space. In the case of the spaces endowed
with an indefinite (Minkowski) signature, there are much more singular hypersurfaces where
the strong axioms of Finsler's geometry are broken down. We propose a weaker definition of
Finsler's structure that is required to be satisfied only on open subsets of the tangent bundle.
We exhibit the characteristic singular hypersurfaces related to Riemann's quartic and briely
discuss their possible physical interpretation. Link to the abstract in PDF .
Abstracts for Full Papers
Abstract for 21873
A Tragedy at the National Gallery of Art: Problembased learning approach to online teaching of mathematics.
Author: 
Irina Lyublinskaya 
Affiliation: 
Teachers College, Columbia University 
This paper examines an experience of mathematics preservice teachers of using problembased learning (PBL) approach in an online course about teaching applied mathematics. The study results showed that PBL approach to teaching mathematics supports collaborative/cooperative work and increases PSTs’ engagement in and out of class. The structure of the course that involved PBL experiences from students’ perspective followed by analysis of PBL approach from teacher’s perspective provided PSTs with sufficient background to develop their own PBLbased instructional units. The study results also showed that while PSTs’ teams were successful in developing curriculumbased problematic situations with realworld connections, they were challenged to justify the use of technology in these experiences. PSTs also had difficulty with development of authentic assessments within PBL approach.
Abstract for 21879
Effectiveness of ProblemBased Learning Strategy in Improving Academic Performance of Grade 10 Students in Mathematics
Author: 
Royet Cayabyab 
Affiliations: 
Calbeg National High School, SDO1 Pangasinan, Perpetual Help College of Pangasinan, Montemayor St. Malasiqui, Pangasinan

Evidence of poor performance in mathematics by secondary school students highlights the fact that the most desired technological, scientific and business application for mathematics cannot be sustained. This makes it paramount to seek a strategy for teaching mathematics that aims at improving its understanding and performance by students practically. ProblemBased Learning as a method of teaching may be used to accomplish the instructional roles of learning basic facts, concepts, and procedures, as well as goals for problemsolving. We include problemsolving in school mathematics because it can stimulate the interest and enthusiasm of the students. The use of ProblemBased Learning in the mathematics classroom may enhance the quality of mathematics teaching and learn in the Philippines. The purpose of this study is to shed light on this issue. Thus, the aim of the study reported here was to ascertain the extent to which the ProblemBased Learning (PBL) strategy would affect the achievement and retention of students when teaching Mathematics concepts in the Philippine secondary schools. The subject of this research was the twenty Grade 10 students of Calbeg National High School who were diagnosed to have difficulty in solving probability problems during the third quarterly examination of this School Year 20202021.All the needed data for the study were drawn from the questionnaires taken from the Learner’s Material of Grade 10 and other reference books in Mathematics. Frequency and ttest were used to treat the academic performance of the Grade 10 students in the pretest and posttest.
Abstract for 21882
Curved Patterns in the Graphs of PPTs
Author: 
James Parks 
Affiliation: 
SUNY POTSDAM 
In a graph of all Primitive Pythagorean Triples (PPTs), with legs up to length 10,000, conic sectionlike patterns can be observed. We show that they are indeed parabolic curves, which follow in a natural way from the mathematics of the subject matter.
This work lives at the intersection of Arithmetic, H.S. Algebra, and Analytical Geometry. It is easily accessible by students. Computer programs for Dynamic Geometry (Sketchpad, GeoGebra) and Mathematica or Maple are used to build the graphs.
Abstract for 21884
Pomodoro Technique: A Grade 11 STEM Student''s Partner to Achieve Productivity in Mathematics
Authors: 
Jamill Gaspar, Janna Casao 
Affiliation: 
Highway Hills Integrated School 
The digital world of the 21st century is a highly competitive era, where you have to be able to do all your works with maximum efficiency and it’s no different for students, especially now during this period where Mathematics classes are held online, it’s always a makeitwork situation. The one hindrance that has been holding everyone back for a long time is the lack of productivity. The researchers found a possible solution to this phenomenon which is the Pomodoro Technique by Francesco Cirillo that manages and balances the amount of time a person spends on working and taking breaks. This is a qualitative analysis of the Pomodoro Technique’s impact on the productivity in mathematics of selected 11thgrade students that currently taking the Science, Technology, Engineering, and Mathematics strand in Highway Hills Integrated School. A survey with a premade video explanation of the technique is handed out to 30 participants. To elucidate outcomes, the researchers looked for the weighted mean of the responses. The results showed that the method had the ability to refine the learner’s timemanagement skills, focus and concentration, and capacity to manage distractions. The researchers concluded that the Pomodoro Technique is indeed a remarkable method to employ for achieving productivity in the field of mathematics
Abstract for 21888
Using “Turtle Geometry” in the XXIst Century for teaching mathematics? Classic ideas and new software and hardware
Author: Eugenio RoanesLozano 

Affiliation: Universidad Complutense de Madrid 

In “Turtle geometry” there is a graphic cursor (the “turtle”) that obeys elementary orders related to the position and orientation of the turtle itself: move forward, move backward, turn right and turn left. That is, graphics are not based on a classic Cartesian reference system. Turtle Geometry is best known for its incorporation into Logo language. As monitors had no graphical capabilities in the 1960s, the movements ordered to the turtle from the first Logo versions were carried out by electromechanical devices connected by cable to the computers. After a great success in the ''80s and'' 90s Logo has fallen into disuse, but there are also very modern visual computer languages based on the use of “blocks” for programming such as Scratch and Snap! Moreover, nowadays there are affordable programmable robots that use Turtle Geometry. Summarizing, wonderful powerful software and hardware that use Turtle Geometry, appropriate for teaching mathematics are available in 2021.
Abstract for 21892
On the analysis of singularity structure in learning
Authors: 
Tomohiro Washino, Tadashi Takahashi 
Affiliations: 
Konan University, Graduate School of Natural Science, Konan University 
The existence of singularities often affects the learning dynamics in neural network
and caused plateau phenomena. Using Mathematica, we observed near singular regions by
examining the evolution of the parameter and the dynamics of learning on the training loss
surface. Our result is to investigate that the type of dynamics of learning changes when
the overlap and the elimination singularity is approached from a distance by changing the
initial values of the statistical model, and to clarify the plateau phenomenon observed near
singular regions.
Abstract for 21893
On a Chasles construction of Cartesian ovals
Author: 
Jiří Blažek 
Affiliation: 
University of South Bohemia 
This article deals with Cartesian ovals and one of the many methods of their construction
 the socalled Chasles construction. The paper first shows an alternative justification
of why points constructed by the Chasles construction satisfy foci definition of Cartesian
oval, and  as a part of this justification  shows why the oval has three foci and how to
construct the third focus by Euclidean means (assuming the oval is defined by two foci).
Finally, the special case in which one of the circles in the construction is replaced by a
straight line is discussed. It is shown synthetically that in such a case the construction
renders a conic, and moreover, it can be shown that the focal definition of a conic and the
definition using a directrix line are equivalent (the proof is only hinted).
Abstract for 21896
The Efficacy of an Integrative Performance Output in Senior High Subjects among STEM Learners
Author: 
Ryan Jay Verbo 
Affiliations: 
Mataas Na Paaralang Neptali A. Gonzales, Schools Division Office Mandaluyong, Jose Rizal University 
The Department of Education (DepEd) issued DepEd Order No. 012, s. 2020 titled Adoption of the Basic Education Learning Continuity Plan (BELCP) for School Year SY 2020 2021 in light of the COVID19 Pandemic. Thus, the department issues the Interim Policy Guidelines for Assessment and Grading considering the Basic Education Learning Continuity Plan to provide guidance on the assessment of student learning and on the grading scheme to be adopted this school year. In compliance to this order, teachers are compelled to collaboratively design and implement performance tasks that integrate two or more competencies within or across subject areas called “Integrative Performance Output”.
Correlational research method was used in order to determine the significant relationship of the performances of the learners among subjects utilizing integrative performance task/output. Using Ftest value of 4.81 the claim was rejected and concluded that there is a significant difference among the students’ performances in three (3) subjects; Practical Research 1, Statistics and Probability and Reading and Writing using an integrative performance task.
Likewise, using Pearson’s r of 0.89 and 0.82, it can be deduced that there is a high correlation between the students’ performance in Practical Research 1 and Reading and Writing and Statistics and Probability, respectively through the use of Integrative Performance Task/Output.
This implies that the utilization of an integrative assessment affects the students’ performance in making connections among concepts and experience so that information and skills can be applied to novel and complex challenges.
Abstract for 21905
Challenges of Mathematics Teachers in The New Normal
Author: 
Kenwyn Chiu 
Affiliation: 
Jose Rizal University 
This study examined the challenges among teachers in the conduct of online teaching of the Mathematics Faculty of the Jose Rizal University. Using descriptivecorrelational study, the research was able to investigate the teaching competence of the Mathematics teachers during the pandemic and the adjustment done in their teaching style online. It revealed that a nonsignificant relationship existed between the variables when the respondents were grouped according to their age, number of seminars attended about EdTech, and years in teaching mathematics to their adjustment in preparing for online teaching. It also found out that there was no significant relationship between the teaching mode and instructional materials used by JRU Mathematics Faculty in online teaching in terms of internet accessibility, technical expertise and teaching competence.
The chisquare test of the data revealed a nonsignificant relationship between the age, number of seminars attended about EdTech and Years in Teaching mathematics as correlated to the adjustment of JRU Mathematics Faculty in preparing for online teaching. This means that the independent variable mentioned above was not associated with the preparation of JRU Mathematics Faculty in teaching online. The Fisher exact test of the data revealed a nonsignificant relationship between the teaching mode and instructional materials used by JRU Mathematics Faculty as correlated to the internet accessibility, technical expertise and teaching competence. This means that the dependent variable mentioned above is always present for both online and traditional class. From the data collected, the research was able to develop an action plan and strategy map that will enhance or improve the skills of JRU Mathematics Faculty in teaching online.
Abstract for 21906
Integer Partitions using Generating functions and
their Applications in Graph Theory
Authors: 
Faqir Bhatti, Ms. Iqra Zaman, Ms. Tabasum NAZ 
Affiliations: 
Riphah Institute of Computing and Applied Sciences, Dept of Mathematics, Riphah Institute of Computing and Applied Sciences, Lahore Pakistan 
Integer partitions is one of the main subjects in Mathematics and which is taught at various
levels in colleges and Universities. The subject was introduced by Leibniz and Euler. The chief aim
of this paper is to give the ideas about Euler generating function which is the main tool to learn
various properties of partitions of integers along with bijective function. Bijection is used to find
out various identities and it always helps to prove results. It is argued that Mathematica software
is important to learn and teach the subject. We give some applications of integer partitions in
graphs theory.
Abstract for 21907
Learning congruence through ornaments and tilings
Author: 
Šárka Voráčová 
Affiliation: 
Czech Technical University in Prague, Faculty of Transportation 
To investigate movement in the plane and congruent transformations, instead of traditional logic development, we can conveniently use intuitive involvement in the geometric ideas of ornaments with the dynamic geometry software GeoGebra and 3D software Tinkercad. In this paper, we will explore selected examples of ornaments and segmented geometric patterns. We bring up some ideas about what a teacher of mathematic can find in geometric patterns and how he can fit them into his teaching curriculum. In this paper, we present a few proposals of methodologies for teaching symmetry in the plane with digital technologies, manipulative activities, and controlled exploration. We also present the results of the research of this method.
Abstract for 21909
Socscistatistics Integration on Test and Evaluation in Statistics (SITES): Mastering Statistics Using Free Online Resource
Author: 
REYNALDO JR COLLADO 
Affiliation: 
Highway Hills Integrated School, Department of Education Mandaluyong City 
Statistics is kernel part of Mathematics designed to ease future research endeavors of students taking it. With its mastery, learners would be able to easily justify their research results in a manner that is not challenging to them at all. Traditionally, they are taught in a way that exhausting long computations take place so that each test statistic will be determined. Careful solving is necessary for them to achieve the correct values which will then affect the overall interpretation of the results. With such profound tasks, they tend to have very little confidence in achieving their tasks. This study is about shifting the mindset of learners in mastering statistics using free website called socscistatistics. This website can be easily accessed in any search engines and functions like Statistical Package for the Social Sciences (SPSS) but is more friendly to learners and startup researchers. It employed the quasiexperimental design. To determine the overall performance of students in statistics, a pretest was conducted. Scores obtained are subjected to 3 levels namely Poor Level (0% to 70%), Good Level (71% to 85%) and Satisfactory Level (86% to 100%). This test was done after the full discussions of statistical tests such as mean, median, mode, variance standard deviations, correlation analysis, relationship tests and difference tests. Results of the tests were used as a basis in implementing the intervention called SITES or Socscistatistics Integration on Tests and Evaluation in Statistics. A posttest was then given after 2 months of utilizing the mentioned website. Results indicated a higher performance in posttest. Results were tested at 0.05 alpha level of significance using a ttest which indicated that there was a significant difference between the means of pretest and posttest scores of the respondents.
Abstract for 21910
Problem based Learning approach to train preservice teachers in mathematics education: an experience
Authors: 
Piedad Tolmos, Ana I. CidCid, Rocío GuedeCid, Herman Martín 
Affiliations: 
Universidad Rey Juan Carlos, Universidad Rey Juan Carlos, Universidad Rey Juan Carlos, Colegio Montpellier de Madrid 
Societies around the world have focused its attention in motivating scientific vocations in children. STEM education is a priority in many Governments education policy. Actions should begin by improving preservice teachers training, including the acquisition of skills in teaching methodologies and ICT tools, which will contribute to increase the motivation of students in STEM subjects. Among the most used methodologies in teaching STEM subjects, is Problem Based Learning, because of its interdisciplinary vision and scientific working method based on resolution of real situations. Aimed to improve STEM training in preservice teachers, we present an experience in which students of the Master in training teachers on Secondary Education of a Spanish University, design a problem within PBL methodology, which will be solved by real students of a Madrid school. Considering the importance of improving preservice teachers training in ITC tools, we include as mandatory the presence of GeoGebra in the design of the problems. The use of GeoGebra and ITC tools in the problem resolution process and the results presentation, complete PBL methodology as motivating elements for the students, both of school and of the University. The methodology of learning by doing we applied, led the students to know PBL methodology and GeoGebra, increasing their skills to apply them in the future. The experience is evaluated in terms of analysis of the preservice teachers designed problems, and of the results of the indepth interviews conducted with several master’s students, and with the teacher responsible for applying the problems in the school.
Abstract for 21911
Research on Development of TeachingMaterials Put Emphasize Retroductive Inference to Develop Statistical Thinking in Mathematics Education
Author: 
Shinichi KATO 
Affiliation: 
Akita University 
The purpose of this paper is to develop teachingmaterials for 6th grade elementary school students in Japan put emphasize retroductive inference to develop their statistical thinking and discuss it. Recently, MEXT (Ministry of Education, Culture, Sports, Science and Technology) makes “Programming Education” compulsory at elementary schools in Japan. The purpose of “Programming Education” in Japan is to develop their programing thinking based on computational thinking and enhance their academic ability in teaching and learning of a particular subject, not to learn cording. However, surprising few studies have so far been made at developing their programing thinking based on computational thinking and enhancing their academic ability. “Retroduction” is effective to promote computational thinking. Also, development of mathematical thinking and computational thinking are interacted by one another. Therefore, we developed teachingmaterial so that they retroductively infer. Specifically, a context and situation setting so that they retroductively infer is what they consider why coins thrown in coin slot for the automatic vending machine in Japan are returned. Results of discussion of developed teachingmaterial, it seems reasonable to suppose that this teachingmaterial so that they retroductively infer makes it possible to promote their computational thinking and develop their statistical thinking.
Abstract for 21913
Construction and exploration of particular curves and evolutes of theirs using GeoGebra dynamic software
Author: 
Petra Surynková 
Affiliations: 
Department of Mathematics Education, Faculty of Mathematics and Physics, Charles University 
The article deals with the synthetic construction of specific planar curves which are defined by geometric motions and with the modeling of these constructions in GeoGebra dynamic system. We focus on the constructions of trajectories, envelopes, and centers of curvature and osculating circles of specific curves. We present a particular construction of all centers of curvature of these curves, i.e. the construction of the evolute of a curve. Our aim is to investigate the synthetic constructions without the use of coordinates and formulate the proofs of them. We newly model selected types of curves in Geo\Gebra dynamic system which is an additional output to our theoretical conclusions and also a significant teaching aid at the same time. The constructions are meant to be dynamic and the curves are formed gradually as the traces of points or curves. All examples presented in this article are intended to be used in the undergraduate courses on kinematic geometry (mandatory courses for preservice mathematics teachers who study teaching mathematics and descriptive geometry, i.e. the prospective secondary school teachers). The synthetic constructions together with their proofs demonstrated in GeoGebra dynamic system bring a new light into this area. It allows students to imagine the main idea of the proofs of the constructions and to investigate the properties of the curves more easily based on the pure geometry and visual aspects. Dynamic constructions, i.e. the possibility of changing positions of points and curves play the significant role here. GeoGebra offers also algebraic expressions which represent another tool for students how they can study and manipulate with those curves.
Abstract for 21914
Geometry of planetary motion
Author: 
Roman Hašek 
Affiliation: 
University of South Bohemia 
This contribution is inspired by the lecture "The Motion of Planets around the Sun"
given by Richard Feynman, American physicist and Nobel Laureate, in 1964, namely by
steps of Feynman's geometrical proof of the law of ellipses, which was based on Isaac
Newton's approach to the problem. The contribution focuses on selected passages of this
proof, interprets them using the program of dynamic mathematics GeoGebra and oers
their use in the form of activities that can be implemented at dierent levels of mathematics
curriculum of lower and upper secondary school. The activities presented in the paper
are suitable for the implementation of the STEM approach to mathematics education as
they combine the topics of the mathematics and physics curriculum, in addition against
the backdrop of the captivating story of discovering the essence of the functioning of the
universe.
Abstract for 21918
Delineation of the zone of influence of pumping wells using CAS and DGS
Authors: 
David G. Zeitoun, Thierry DanaPicard 
Affiliations: 
Orot College of Education , Elkana Israel, Jerusalem College of Technology Jerusalem Israel 
Groundwater pollution is a general concern in countries using pumping wells for water consumption. Also, the determination of protected zone around pumping wells is a practical concern. The delineation of this zone is frequently obtained using numerical models such as MODFLOW and or FEFLOW. In this contribution we derived the equation of the plane curve of the protected zone using a steady state solution of the groundwater flow equations and the theory of envelopes. Cassini ovals appear in some particular cases, providing new applications of these plane curves. A Dynamic Geometry System, such as GeoGebra and/or Desmos, is used to explore the protected zone. A Computer Algebra System may be used for the computations, in particular to characterize the delineation curves for the zones of influence for wells.
Abstract for 21928
Computation of voting power indices using polynomial rings and ideals
Author: 
Alasdair McAndrew 
Affiliation: 
Victoria University, Melbourne Australia 
In many jurisdictions globally, voting is done not by individuals, but by blocks of voters. Examples are the American Electoral Collage, the nternational Monetary Fund, the European Parliament, and the houses of parliament or of congress in many national legislatures. Voting is thus done by blocks: state, country, or political party; each member of which casts the same vote. It is tempting to assume that a block with the most votes is the most powerful, and has the greatest chance of influencing the outcome of the final ballot. As with so much else in voting theory, this is quite incorrect, and it is often the case that a minor block can have an influence entirely out of keeping with its size. There are various different methods of determining the relative power of each block, and allocating a numerical measure of power; these values are called ``power indices''''. Some of these methods can be computationally intensive, and it took several years after the initial definition to compute power indices for the American Electoral Collage, with its 50 states. In this paper we explore a unified method for computing different power indices using the theory of polynomial rings. This allows not only a relatively simple computation, but one which can be adjusted to consider ``coalitions'''' (two or more blocks which band together to increase their power), and ``quarreling parties'''', where two blocks refuse to agree on any vote.
Abstracts for Papers with Abstract Only
Abstract for 21862
IDENTITY JOURNEY MAPS: A GEOSPATIAL ROAD TRIP OF MATH CONTENT IDENTITY
Author: 
Dennis DeBay 
Affiliations: 
University of Colorado  Denver: School of Education and Human Development, NCTM, CCTM, AERA 
The goal of the Identity Journey Map project (IJM) is to explore the impact of student teachers creating IJMs is to allow for the opportunity to reflect and set the stage for all the (possibly different) ways student teachers will experience their mathematics content identity in the classroom. Through geospatial storytelling software (StoryMaps  https://storymaps.arcgis.com/en/), students will communicate their journey and experiences to this point that have brought them to be connected math educators in the hopes of exploring ways to connect their student’s content identities in math.
As a teacher educator, many math classrooms I observe maintain to incorporate a onesizefitsall approach to supporting students through their learning. There is no wonder why many students are turned off by math and begin to believe, “I’m just not a math person.” They lack an identity or connection to the content that many of our teachers possess  a love for mathematics. We are referring to this as their mathematical content identity. This lack of identity can affect performance in classrooms and lead to a sinking feeling that they do not belong in that math class (Huvard, H., et al., 2020). A growing body of research has revealed the diversity of student strengths and challenges and a wide variety of pathways there are to effective learning (VincentRuz, P., & Schunn, C. D., 2018). How can educators transform their approach and teach math in a way that effectively addresses learner variability? Productive struggle focuses on the math journey instead of simply the destination. It recognizes that there are multiple ways to tackle a problem and views mistakes as opportunities for learning rather than failures. It is this journey that is the focus of this proposal. We are proposing to explore Identity Journey Maps (IJMs). In this project we have students build webmaps using the geospatial technology StoryMaps to explore their own mathematical identities and consider ways to use these journals as entry points to engage students in believing they too have an identity as a mathematician.
Identities are developed and shaped by different experiences influencing how we perceive and relate to the world around us. Identity theory states that our identities are filled with meaning based on how we perceive our roles as an individual and within the larger cultural groups we are part of within society (Burke & Stets, 2009). People hold multiple identities, and each identity is attributed a specific set of meanings. This paper extends this line of inquiry to investigate how participation in the creation of these geospatial stories impacts student teacher’s mathematical identities and in turn their perceptions of their student’s identities as mathematicians.
The aim of this study is to explore how the IJMs impacted their perceptions of their content identity and how this connects to their student’s identities in math. Using case study methodology, we will study graduatelevel courses at our institution to provide an indepth examination of students’ sense of content identity. We use identity theory to explore the nature of content identity development in preservice teachers who engaged in creating identity journey maps at a university in the western USA. Constructs of content identity development are seen as critical outcomes of experiential math inreach and outreach programs.
References:
Burke, P. J., & Stets, J. E. (2009). Identity theory. Oxford University Press.
Huvard, H., Talbot, R. M., Mason, H., Thompson, A. N., Ferrara, M., & Wee, B. (2020). Science identity and metacognitive development in undergraduate mentorteachers. International Journal of STEM Education, 7(1), 117.
VincentRuz, P., & Schunn, C. D. (2018). The nature of science identity and its role as the driver of student choices. International journal of STEM education, 5(1), 11
Abstract for 21863
UNBOXING PEDAGOGY: ENGAGING ONLINE MATH EDUCATION STUDENTS THROUGH THE MAIL
Author: 
Dennis DeBay 
Affiliations: 
University of Colorado  Denver: School of Education and Human Development, NCTM, CCTM, AERA 
Introduction
In a set of 2 preservice education courses, we have explored how play and wonder connected to math learning can impact connectedness and community in online learning environments. The goal of the Unboxing Pedagogy Project (UPP) is to examine the impact of mailing collaborative, tangible projects to students learning in online spaces on the student’s sense of community.
The accepted assumption todate has been that math courses are solely for learning and the role of educators to equip students with the knowledge and skills they need to be successful in the future. However, what we have learned since the move to distance learning, is that maybe this entire assumption is incomplete. In 2018, 34.7% of postsecondary students were enrolled in an online course (U.S. Department of Education, 2019). With another year of overall decreased postsecondary enrollment in 2019 (National Student Clearinghouse, 2019), institutions of higher education (IHEs) need to consider how to effectively and comprehensively meet the unique needs of online learners. The idea of moving to more of a social connection is described by Liu, Magjuka, Bonk, and Lee (2007), “…a deep trust or spirit of community lies in the commitment and mutual engagement of community members”. While there are some studies centering on the importance of creating a sense of community and belonging for learners in IHEs, the literature is still lacking (Tovar & Simon, 2010).
We have been experimenting with sending packages in the mail to students who are engaged in online courses for students to feel part of a larger community. We will examine the UPP using the Community of Inquiry instrument (Arbaugh et al., 2008) and a case study methodology, we will explore students’ sense of connectedness and belonging in these online courses  through the mail.
References
Arbaugh, J.B., ClevelandInnes, M., Diaz, S.R., Garrison, D.R., Ice, P., Richardson, & Swan, K.P. (2008). Developing a community of inquiry instrument: Testing a measure of the Community of Inquiry framework using a multiinstitutional sample. The Internet and Higher Education, 11(34), 133136.
Liu, X., Magjuka, R.J., Bonk, C.J., & Lee, S. (2007). Does sense of community matter? An examination of participants’ perceptions of building learning communities in online courses. The Quarterly Review of Distance Education, 8(1), 924.
National Student Clearinghouse Research Center (2019). Current term estimates: Spring 2019. Retrieved from: https://nscresearchcenter.org/wpcontent/uploads/CurrentTermEnrollmentReportSpring2019.pdf
Thomas, L., Herbert, J., & Teras, M. (2014). A sense of belonging to enhance participation, success and retention in online programs. The International Journal of the First Year in Higher Education, 5(2), 6980.
Abstract for 21866
The Let''s Track Math Methodology
Author: 
Tamar Apel Campo 
Affiliation: 
CimeH Institute 
Modern qualitative research in education and social sciences is using observation, closed or semistructured interviews and local experiments in the classroom to understand the teachinglearning situation. These methods have a limiting property with boundaries that close on the educational scenario producing a twodimensional methodology of inquiry. Tables, drawings, sketches and recordings try to provide a wide picture of interactions and situations in the maths classroom. Moving out of these boundaries the Let''s Track Math Methodology presents a wide possibility of collecting information that remains invisible using the abovementioned techniques. The combination of complex system analysis, hologrammatic procedures and dynamic assessment, joined with ancient tracking skills of our ancestors has enabled the emergence of a dynamic helicoidal model of research suitable for education in general and mathematics in particular.
The Let’s Track Math methodology is celebrating more than ten years of implementation in the educational mathematics environment starting in Israel from where the methodology has moved to Latin America and India corroborating the earlier findings. These point to a clear explanation of the relation between teaching and learning in the mathematics class.
The results of the long term research of more than eight years steered the formulation of the Relational Theory of TeachingLearning of Mathematics, characterized by complex system analysis, fuzzy sets theory and ethical Levinasian foundations. The model presenting the methodology rests on a strong analytical platform including movement and differentiation of sectors of the research activity. The characteristics of the different sectors complement each other coherently, irradiating the same configuration on the final theory.
The LTM methodology provides researchers with the equipment required for scalable research assuring validity from the first step of the clear sequence that underlies its structure. The whole mechanism, examples, guidelines and manuals for its implementation will be presented supporting the results outcoming from its use.
Abstract for 21867
Challenges for a new Math curriculum in the context of STEAMEducation
Author: 
Belen Palop 
Affiliation: 
Universidad de Valladolid 
Research in Mathematics Education has proven that students obtain better results when taught by mathematical experts who, at the same time, have a strong pedagogical foundation. On the other hand, educators worldwide are showing how STEAMEducation provides an integrated approach to all subjects in the curriculum with a holistic perspective that motivates and encourages students to connect their learning in broader ways than when each subject is presented in an isolated way. The specialized deepfirst approach, where students incorporate math facts and procedures during most of their academic time, collides therefore with the STEAM broadfirst approach, where mathematical competence is intertwined with its connections to the real world, with technology and, in its broader sense, with all academical knowledge.
In this round table we will discuss this questions with experts from Math preservice teacher training and STEAM practitioners from highschool. They will be asked to present their views, concerns, approaches and proposals for the coexistence of a deep mathematical education and a broad contextualized learning.
Abstract for 21870
Some of my favourite problems to use when teaching with technology
Author: 
Rob Corless 
Affiliation: 
Western University, University of Waterloo 
We all have our favourite problems to teach with: mine tend to have surprising elements in them. My favourite integrals to use in Calculus class include one which comes from parameterizing an example taken from p. 25 of a paper by Kahan, namely ``Handheld Calculator Evaluates Integrals'''' (link given below; Kahan uses n=64), which is perfectly doable with partial fractions for all positive integer values of the parameter n. Yet for large n the answer obtained becomes unwieldyor does it? With a little more inspection, we can turn Kahan''s point, namely that numerics are better, on its head. Continuing in this vein, my favourite differential equations include one where we can solve everything completely, including the numerics, and really understand what is going onand it''s a lot more than is normally taught. Similarly, my favourite matrix is only 2 by 2, yet quite informative (there''s a famous 3 by 3 example that I like, too). The pedagogical points of all of these examples would be entirely missed by using hand computation alone. The main point of this talk is the pedagogical value of surprise; some of these examples startle the students. I don''t think you will be startled, but you might be amused.
Abstract for 21929
A Processdriven Evaluation Method for a New Online Course, Introductory Mathematics for Artificial Intelligence
Author: 
SangGu Lee 
Affiliation: 
Sungkyunkwan University 
It is essential for students living in the future society to understand artificial intelligence (AI) and machine learning. Accordingly, in this study, a semesterlong “Introductory Mathematics for Artificial Intelligence” course was developed and provided to undergraduate students for various majors to navigate their career opportunities in various AI. We wrote a textbook and provided an online mathematics laboratory that integrated new and effective technology as much as possible for this new online course. In addition, we developed a processdriven evaluation method based on the Problem/ProjectBased Learning report, and it showed a significant contribution to students’ performance in the course. In this talk, we present the newly written textbook for one semester, Introductory Mathematics for Artificial Intelligence, and a new laboratory and evaluation method in our online class. We also share a record of communications with our students and some meaningful insights from the data analysis generated from the class.
Abstract for 21930
Exploring the capability of Scientific Calculator Apps in Solving Trigonometry Problems
Author: 
Maria Isabel Lucas 
Affiliation: 
Pamantasan ng Lungsod ng Maynila, Mathematics Teachers of the Philippines  Tertiary Level 
Today’s learners who belong to the postmillennial generation or Gen Z greatly depend on technology in their learning needs. Dubbed as digital natives, they have a strong preference for ICT devices, especially smartphones, which are already part of everyday life. Teachers who are digital immigrants can take advantage of this by integrating technology as a teaching and learning tool to meet their learning needs and interests. Free apps are available for both Android and iOS operating systems, such as the scientific calculator app. This study verifies whether the use of mobile applications in the calculation of trigonometric problems is a good alternative to the commonly used handheld calculators by using a quasiexperiment between 50 students at a local university in the Philippines or not. The results of the paired sample ttest showed that the average score of the students on their summative test was significantly higher than their score using the handheld calculator. The researcher recommends that teachers allow their students to use mobile computer apps to solve math exercises and tests. Besides saving costs, it will also bring more convenience to students as they don't need to carry two devices in their briefcase or pocket. Its ergonomics are also more favorable as the size of the smartphone corresponds to the user's manual dexterity. With their daily use of gadgets, digital natives have developed an attachment and familiarity in them. The automation they have over time facilitates touchscreen navigation allowing for faster calculations than the usual handheld commercial calculator.
Abstract for 21934
Unconscious gender bias effects when teaching problemsolving strategies
Authors: 
Belen Palop, Juan José Santaengracia de Pedro 
Affiliations: 
Universidad de Valladolid 
Problemsolving is one of the hardest topics to teach in Mathematics. Spanish tradition for school children is known as “dataoperationssolution” (DOS, from now on). Singapore math is starting to penetrate Spanish classrooms promoting a less abstract method, the bar method model [1] (BMM from now on), which is a pictorial diagram that allows the organisation of the abstract information and helps to understand and analyse the problem itself.
Statistically significant differences in achievement between genders have been systematically found by TIMSS (4th graders) and PISA (9th graders). There is a clear need for a careful analysis of our own classroom practices in order to avoid those that can inadvertently enlarge these differences. We already know that students can improve their problemsolving skills when introduced to the BMM and our main goal in this study is to focus on the gender differences if any. Our hypothesis is that girls may benefit more from the BMM than boys, since it gives them an additional tool to rely on for problemsolving before the abstract formulation [2].
The 71 (neurotypical) fifth graders from a school in Segovia (Spain) are assigned to 4 classrooms, each one led by a different math teacher. Three of the rooms (22 boys, 33 girls) have been assigned as the control group, and the remaining one (9 boys, 7 girls) acted as our convenience experimental group.
All children worked for one month on 53 word problems. The control group used DOS while the experimental group was introduced to BMM. Two similar subsets of questions, released from TIMSS (2011), were used as pretest and posttest. The results were analysed according to Content domains (Numbers, Geometry & Measurement and Data) and Cognitive domains (Knowing, Applying, and Reasoning).
• Chisquare tests show no significant differences in any component depending on the teaching strategy for boys, while girls in the experimental group improve their results in the cognitive domain Applying (p=0.003) and in the content domain of Geometry (p=0.023) while the Numbers domain doesn’t reach significance (p=0.072).
• Independently of the teaching method, boys significantly improve their results after the intervention compared to girls with respect to their reasoning skills (p=0.007) and nearlysignificantly in the most worked domain of Numbers (p=0.061).
This smallscale study suggests that girls improved less than boys, independently of the approach, and can benefit more from BMM than DOS without hindering boy’s performance.
References
[1] Hyde, Janet Shibley. 2005. «The Gender Similarities Hypothesis.» American Psychologist 60 (6): 58192. https://doi.org/10.1037/0003066X.60.6.581.
[2] Yeap, V. H. (2011). Bar modelling: A problemsolving tool: From research to practice: An effective Singapore math strategy. Marshall Cavendish Education.
Abstract for 21935
Is innovative virtual classroom simulation better than traditional roleplay simulation? A comparative study
Author: 
YungChi Lin 
Affiliation: 
National Tsing Hua University, Taiwan 
The use of virtual classroom simulations is widely seen as the wave of the future in teacher education. It is expected to provide better opportunities for preservice teachers to practise their teaching and develop teaching competence than traditional roleplay simulations do. However, few studies have actually compared their effectiveness. To help fill that gap, the present mixedmethods research provides a headtohead comparison of the effectiveness of virtual and roleplay classroom simulations in two otherwise similar mathematics teaching methods courses. The participants were 62 preservice teachers, of whom 27 were assigned to the virtual group and the remaining 35 to the roleplay group. A demo clip of the virtual simulation environment, CartoonClass, can be seen at https://youtu.be/s4mF5vvZfL8. Immediately after the simulation, the members of both groups completed an online perception questionnaire, and answered two openended questions, one regarding their perceptions of the simulation, and the other about their teacher noticing skills.
The results of a MannWhitney U test showed no significant intergroup difference in perceptions of whichever classroom simulation the participants had experienced. Our qualitative results likewise showed that both groups had broadly positive perceptions of using the classroom simulation (positive comments vs negative comments: virtual: 54.5%>45.5%, roleplay: 57.7%>42.3%). However, as compared to the roleplay group, the virtual group perceived their simulation as more authentic and enjoyable. On the other hand, the roleplay group exhibited slightly better noticing skills than the virtual group, as measured by the former’s significantly higher total number of comments, and the significantly lower number of surface comments in their responses to the openended noticing question. We, therefore, recommend complementary use of both these types of simulation in teachereducation courses.
Abstract for 32104
New features in GeoGebra
Author: 
Buzek Štěpán Bc. 
Affiliation: 
Faculty of Education, University of South Bohemia, Czech Republic 
There are few interesting features appeared in the program GeoGebra during last year. In this talk, I will introduce some of new features.
In particular, we shall explore the use of commands such as Locus, LocusEquation or Envelope and their respective applications in a classroom.
I will present various examples in geometry by incorporating the use of the commands above by searching for a locus, locus equation or envelope.
Some examples will be focused on mathematical proofs and the way how GeoGebra makes this task easier.
Abstract for 32105
Structural constructions in virtual reality by using Neotrie VR
Author: 
José L. Rodríguez 
Affiliation: 
Departamento de Matemáticas, CITE III, Universidad de Almería, La Cañada de San Urbano, Almería, Spain 
We present some new tools and options in the Neotrie VR software, that will allow highschool students to create
and understand the structure of some geometric figures in space. The tools have been designed to be intuitive, but at the same time to allow students
to understand about the structure in the construction process. These include symmetric figures, fractals, or domes made from a basic polyhedron.
This is a first step towards being able to design and program more sophisticated tools for more complex geometric constructions in virtual reality.
