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Connecting All Applications to Mathematics and Technology



ATCM 2022, December 9-12, Prague, hosted by Czech University of Life Sciences Prague (CZU)


1.     Abstracts for Invited Papers

2.     Abstracts for Full Papers

3.     Abstracts for Presentations with Abstract Only

4.     Abstracts for Hands-on workshops

Link to the ATCM 2022 program with Zoom ID and Passcode (For registered participants)

Abstracts for Invited Papers

Abstract for 21945

Olympic geometry problems: human vs. machine

Authors: Belen Ariño-Morera, Tomas Recio, Piedad Tolmos

Affiliations: Universidad Antonio de Nebrija, Madrid, Universidad Rey Juan Carlos, Madrid

We discuss, through a collection of examples, the behavior of GeoGebra Discovery automated reasoning tools concerning the solving of some difficult elementary geometry problems, such as those proposed in mathematics contests, or in entrance exams for talented students’ training courses (e.g. ESTALMAT ), or in selection processes for becoming a mathematics teacher for the public administration (Spanish: “oposiciones”). Considering just statements coming from these sources, and such that GeoGebra Discovery is currently programmed to deal with (e.g., not involving inequalities in the hypotheses), we observe, on the one hand, the good behavior of GeoGebra in many instances and, in the other, the remarkable coincidence on the perception of the difficulty, for both human and machine solvers (see, for example, [1]). It is surprising to verify, once more, that there is some hidden coincidence between the difficulty of solving the given problems, for humans and for machines, even considering only those cases that machines can address without restrictions. See [2], where it is remarked how Computer Algebra troubles for deriving conclusions from equalities through rewrite rules, coincide with-- and may actually help understanding-- students’ difficulties with such tasks, proposed in the well-known in the 70’s, Chelsea College Concepts in Secondary Mathematics and Science (CSMS) Project. We think that both facts (the use, in relevant contexts, of test items for humans that can be solved by a machine; the ability to detect the idea of “difficulty” through the behavior of a machine) should be further explored and discussed, regarding their potential consequences in mathematics education. In this respect we would like to call the attention of the community of educators about the need to understand and to compare the notion of “difficult” geometric problem, both for humans and for GeoGebra automated reasoning tools, as it could

a) help improving the performance of GeoGebra, by advertising the user beforehand that a sought query is of high difficulty and that might expect a delayed or no answer, or by selecting, in the Discover command output, to exhibit only interesting theorems, avoiding obvious, trivial ones, etc. Likewise, it could

b) help improving and adapting GeoGebra’s reasoning tools to the needs of students with high capacities. Finally, as the most direct consequence, a deeper consideration of the concept of difficulty for geometric problems, regarding the machine vs. human behavior, could contribute to

c) foster the potential role of GeoGebra concerning geometric problems posed in mathematical contests, such as the mathematical Olympiads, etc.

[1] Ariño-Morera, B. (2022): GeoGebra Discovery en la (European Girls’ Mathematical Olympiad) EGMO 2022. Actas de las Jornadas de Aprendizaje y Enseñanza de las Matemáticas, (JAEM 2022). Valencia.

[2] Recio, T. (1998): Didactical relevance of meaningless mathematics. International Journal of Computer Algebra in Mathematics Education (IJCAME), vol.5, no. 1, 15-27.


Abstract for 21946

Spreadsheets: A Technological Nexus between Big Ideas in Mathematics and Computational Thinking in Mathematics Classrooms

Authors: Weng Kin Ho, Jonaki Ghosh, Berinderjeet Kaur, Chee Kit Looi

Affiliations: Nanyang Technological University, Department of Elementary Education Lady Shri Ram College for Women University of Delhi Delhi, India, National Institute of Education Mathematics and Mathematics Education Nanyang Technological University Singapore

Recent years in Mathematics Education research saw the emergence of ‘Big Ideas in Mathematics’ and ‘Computational Thinking’ as two areas of intense research. This paper highlights the technological and pedagogical affordances of electronic spreadsheets, which make them a convenient and practical technological nexus between ‘Big Ideas in Mathematics’ and ‘Computational Thinking’ in a Mathematics classroom. To demonstrate how such a connection is realized in authentic classroom settings, some instructional exemplars that have been designed and deployed in Asian (senior) high schools are provided here. Our present approach for generating such mathematics lessons may be considered as one of the ways for mathematics teachers to transform their understanding of big ideas of mathematics into tasks and lessons in the classroom.


Abstract for 21948

Ordering Question with Clue in Moodle

Authors: Kosaku Nagasaka, Takahiro Nakahara

Affiliations: Kobe University, Sangensha LLC

The ordering question is one of the third-party question types for the quiz activity in Moodle. It displays several draggable items in a random order, that may include a couple of mathematical expressions rendered by MathJax or KaTeX. The students are required to rearrange/drag them in the correct order specified in the question text. This question type is very useful to evaluate or drill the both of procedural and conceptual knowledge in mathematics if the questions in use are well considered and structured. In this paper, we propose a new feature for the ordering question type, that allows some items to be fixed as clues (static/non-draggable items). This feature helps the students to understand how to solve the problem and force the students to think deeply the hidden relationship among the given items. We also show our actual ordering questions with clues in use, that are pre-class learning activities as one of self-assessment tasks in linear algebra lectures.


Abstract for 21949

Ellipsoid is Tangent to its Locus under a Linear Transformation, Isometries and Sheared Maps

Authors: Wei-Chi Yang

Affiliations: Radford University

In ([10]) and ([9]), we describe an antipodal linear transformation L on an ellipsoid S; and see S is inside and tangent to L(S): In this paper, we discuss how two geometry figures are congruent and are related by an isometry through this linear transformation L. We describe how a locus ellipsoid L(S) can be written as a standard form of XMX^t = 1, and how the rotated ellipsoid Sr stays tangent to XMX^t = 1: Next, we also explore how the ellipsoid, XMX^t = 1, can be tilted so its minor and medium axes form a circle. We first note that the minor and mean axis of XMX^t = 1 span a plane P that intersects the ellipsoid in the smallest possible ellipse. We rotate this plane by keeping the mean axis fixed, and tilting the minor axis towards the major axis. At some unique point, one obtains a plane P that intersects the ellipsoid in a round circle. We shall explore finding such sheared map T. This paper will benefit those students who have backgrounds in Linear Algebra and Multivariable Calculus. In particular, we need the eigendecomposition for the ellipsoid of L(S).


Abstract for 21950

Shift and Vigenère Ciphers with Maplets

Authors: Rick Klima, Neil Sigmon

Affiliations: Appalachian State University, Radford University

Cryptology, the art and science of communicating in secret, provides an excellent tool for illustrating practical uses of mathematics. Topics from number theory, linear and abstract algebra, probability and statistics, and other areas all appear prominently throughout cryptographic methods and their cryptanalysis. Although historical, or "classical," ciphers are no longer widely used in our modern digital society, they can still be used to directly connect mathematics and cryptology, with advanced technology significantly enhancing how classical ciphers can be implemented and broken. As examples of this, in this paper we describe and illustrate the implementation and cryptanalysis of shift and Vigenère ciphers. To help with this, technology involving Maplets will be used.


Abstract for 21951

Computational thinking in mathematical modelling: an investigative study

Authors: Keng Cheng Ang, Liang Soon Tan

Affiliations: Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, Academy of Singapore Teachers Ministry of Education Singapore

It has been suggested that computational thinking, based on fundamental concepts of computing science, provides a useful approach to everyday problem solving. It has also been seen that computational approaches can play a major role in the work of professionals in various fields, and it is therefore pertinent that computational thinking be given some attention in schools to prepare students for the future workforce. One way to do so is to expose students to modelling challenges, and through these, provide opportunities for students to learn, practise and refine their skills and competencies in both computational thinking and practical problem solving. In this paper, we describe the interaction and interplay between computational thinking and mathematical modelling through students’ experiences in an international mathematical modelling contest. Students’ ability to apply computational thinking in the contest was inferred and investigated via case studies. Data sources in the form of report artefacts, videos, interviews and judges’ comments form the basis of the case studies. The investigation reveals that the constructs of computational thinking such as pattern recognition, abstraction, decomposition and algorithm creation play a critical role in the successful completion of the students’ modelling tasks.


Abstract for 21952

Real Quantifier Elimination in the Classroom

Authors: Tomás Recio, Zoltan Kovacs

Affiliations: The Private University College of Education of the Diocese of Linz, Institute of Initial Teacher Training, Nebrija University, Madrid, Spain

We present the experimental command RealQuantifierElimination in GeoGebra Discovery. The command provides quantifier elimination over the reals. We describe how this new command can be used in certain classroom situations.

In our examples we focus on mathematical logic and elementary calculus (in particular, on definitions of basic notions and proving inequalities). Finally, we conclude a potential impact of this new command in the educational world.


Abstract for 21955

Riemann Sphere and Complex Plane Transformations with Sphere Movements (Software Aspect)


Affiliations: Holon Institute of Technology

There are various ways to visualize complex functions, but most of them provide only passive show, even with support for manually controlled and/or parameterized animations.

The Riemann sphere offers a very different instrument for modeling complex plane transformations, often allowing them to be defined as the result of a transformation of the sphere itself.

In this paper, we present the use of the Riemann sphere model in discovering the features of complex functions and the construction of these functions as results of sphere movements with the help of the author’s non-commercial software VisuMatica.


Abstract for 21958


Authors: CELINA A. A. P. ABAR

Affiliations: Pontifical Catholic University of Sao Paulo

This paper aims to present the contribution of GeoGebra in research carried out in Brazil. Dynamic geometry systems (DGS) have evolved significantly and the contribution to research and education is noticeable in the papers presented here. The easy access and dynamism offered by GeoGebra make it possible to reduce the gaps between what is taught at school and the context of the students, in addition to contributing greatly to the training of teachers. Between 2009 and 2022, researchers identified 140 studies aimed at teacher training throughout Brazil and, in another study, on papers published in a specific journal on GeoGebra, it appears that all of them focus on didactic issues, concerning about the analysis of the effects on the teaching and learning process of students and analysis of the types of thinking/reasoning developed in the proposals presented. In the research, we identify strategies that enable teacher training that reflects positively on schools and indicates ways to improve teaching practice. Hence the importance of some theoretical contributions presented and considered in the studies, which allow validation of the results obtained. It appears that using GeoGebra is an alternative in the search for ways to build situations that help overcome teaching and learning difficulties, not only in Mathematics, but also in other contexts of science. The available and constantly evolving tools, as in the experimental proposal of GeoGebra Discovery, make it go beyond the context of mathematics to other sciences, revealing multidisciplinary aspects.


Abstract for 21963

Deployment of Mathematical Resources to a Philippine High School through a Community LTE Network

Authors: Ma. Louise Antonette De Las Penas, Maria Alva Q. Aberin, Agnes D. Garciano, Juan Carlo F. Mallari, Jumela F. Sarmiento, Mark Anthony C. Tolentino, Debbie Marie B. Verzosa

Affiliations: Ateneo de Manila University, University of Southern Mindanao

In the Philippines, one challenge that continues to be faced by the Department of Education in bringing educational content in a blended learning modality is the lack of internet access of the learners. This paper discusses the distribution, through a community LTE network, of mathematical resources for Grades 7 to 10 to teachers and students of a particular high school in the Philippines. It also gives details on particular technological tools (mathematical applications) that were created to help the mathematical learning of students in a remote setting.


Abstract for 21964

Sphere and Spheric Geometry: The Power of Visualization and Investigation of Dynamic Geometry

Authors: Jean-Jacques Dahan

Affiliations: IRES of Toulouse

This article aims to provide a dynamic approach to the basic notions of spherical geometry. We will use the power of visualization of Cabri 3D to make more comprehensible the detail of the proofs establishing the formulas giving the area and the volume of a sphere. Experimenting with real spheres is a challenge already overcome with half hollow and transparent plastic balls on which you can stretch strings and write with markers. The use of Cabri 3D greatly enriches such a teaching of spherical geometry in facilitating the 3D representation of the objects of this geometry as well as their manipulation. We will show how. All the results presented here are known, but it is the way in which they are approached and detailed that constitutes the originality of this article.


Abstract for 21966

Exploration of parameterized families of surfaces: envelopes, offsets and canal surfaces

Authors: Noah (Thierry) Dana-Picard

Affiliations: Jerusalem College of Technology

The study of envelopes and offsets requires a strong dialog between algebraic computations and graphic representations. Both are available using a Computer Algebra System, and sometimes accompanied by a Dynamic Geometry System. The two kinds of software have different affordances, whence different characteristics of the animations. Visualization raises specific problems. We illustrate this with examples of 1-parameter and 2-parameter families of surfaces in 3D space, in particular canal surfaces and pipe surfaces. In certain cases, the envelopes have non-isolated singularities; we study them switching between parametric and implicit presentations. Finally, networking between them and between visualization and symbolic computations is discussed.


Abstract for 21968

Link to an expanded abstract

7-Circle Waltz

Authors: Jen-chung Chuan

Affiliations: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300

Evelyn, Money-Coutts, and Tyrrell first published the Seven Circle Theorem in 1974. It shows that there are many beautiful relationships involving only lines and circles on a plane still waiting to be discovered. The earliest study of polyhedra dates back to Apollonius, Pappus and Archimedes some 23 centuries ago. The author first read about the “circumscriptible tetrahedron”. The term is used to define a tetrahedron whose six edges (not produced) are tangent to the same sphere. Our project here has constructed, with the dynamic geometry software Cabri 3D, a set of 27 mathematically curious animations of the 7 circles formed with the faces of a circumscriptible heptahedron.

Each animation offers a concrete example in the discipline of:

1) Graph Theory: the contact graph formed with the 7 circles in each animation remains the same.

2) Topology: The 27 heptahedra accompanying the animations are topologically distinct.

3) Geometry: For each animation there exists a pair of circles having the same axis throughout the animation.

4) Combinatorics: There could be more than one pair of circles for the same circmscriptible heptahedron.

5) Connections between 2D and 3D: Each “snapshot” of the animation can be transformed under the stereographic projection into a Sangaku figure of 7 circles among which two are concentric.

6) Prism: The term “porism” is obscure. Michel Floréal Chasles showed his appreciation of the interest in the Porisms from the point of view of modern geometry. With Cabri 3D this work leads a new perspective of a generalization of Stein''s Prorism.

Since each animation is based on a simple harmonic motion, we simply call it a “7-Circle Waltz”.


Abstract for 21969

Augmented intelligence with GeoGebra and Maple involvement

Authors: M. Pilar Vélez, Tomás Recio

Affiliations: Universidad Antonio de Nebrija, Escuela Polit�cnica Superior Universidad Antonio de Nebrija Madrid, Spain

We reflect on the necessary interaction between human and machine in the mathematical activity of our days. The emergence of computers, and more recently everything related to artificial intelligence, has shaped a new way of doing and learning mathematics. The role of mathematics in this scenario is twofold. On the one hand, mathematics is the hidden support of artificial intelligence and, therefore, of the digitization of everything; on the other hand, personal computers provide digital tools that perform incredible calculations (including most of the tasks required in the current mathematics curriculum) or facilitate the drawing of dynamic graphs that help visualize mathematical objects, and even perform mathematical demonstrations. Clough’s conjecture example will illustrate these items from our experience.


Abstract for 21973

Exploring some elementary results of Ramanujan with modern tools

Authors: Alasdair McAndrew

Affiliations: Victoria University, Melbourne Australia

The Indian mathematician Srinivasa Ramanujan (1887--1920) was one of the most extraordinary mathematicians in history. Almost entirely self-taught, he produced results of such originality that nobody understood him until he gained the confidence of the English mathematician Godfrey Harold Hardy. As one of the most expert mathematicians of his time, even Hardy was forced to admit that some of Ramanujan''s results ``defeated me completely; I had never seen anything in the least like them before''''~\cite{hard37}. Hardy and Ramanujan worked together for five years (during which time Ramanujan was admitted to both the Royal Society and to a Fellowship of Trinity College, Cambridge), after which Ramanujan, by then a very sick man, travelled home to India where he died aged only 32. His results have kept scores of mathematicians busy for the last century---indeed some have devoted their lives to his work---and his influence is if anything even greater now. Many of his results are deep and difficult, and quite beyond an article such as this. But there are also some lovely gems which are more easily grasped; and of varying difficulties to prove. In this paper we explore a few of his simpler results with the aid of modern technology.


Abstract for 21977

Mathematics and Technology: Does it work?

Authors: Vanda Santos

Affiliations: Research Centre on Didactics and Technology in the Education of Trainers, University of Aveiro, Portugal, Centre for Informatics and Systems of the University of Coimbra

The objective of this paper is to show that technologies have a great impact on the teaching and learning of mathematics. Technologies can be programs, applications, or even automatic deduction systems that can be used to teach one or several mathematical concepts. The use of learning environments with or without tutors, which integrate a dynamic geometry system or another, can be used in the teaching and learning of mathematics. Active methodologies that involve students in the learning process have as their fundamental objective to develop mathematical skills that allow them to know how to solve problems. Some tasks that were developed will be presented, either with the use of programs or teaching and learning environments, online or offline. The results have been favorable in the teaching and learning of mathematics, with the use of technological tools and involving active methodologies


Abstract for 21982

Using virtual reality to teach linear algebra with a focus on affine geometry

Authors: José L. Rodríguez

Affiliations: University of Almería

We present an innovative experience of the use of virtual reality in a first undergraduate course of mathematics at a Spanish university carried out during the last 3 academic years. This complements the theoretical and practical classes of the subject of linear algebra with notions of affine geometry that will be used in later courses. Mathematics, with the use of technological tools and involving active methodologies.


Abstract for 21987

Understanding Geometric Pattern and its Geometry, (part 9) - On Walking Pentagons and Isfahani Inflation

Authors: Miroslaw Majewski

Affiliations: New York Institute of Technology, Abu Dhabi Campus

We introduce Isfahani inflation, a simple inflation technique for decagonal tessellations. We discuss the creation of geometric patterns using Isfahani inflation. Using examples from Bukhara and Iran, we show how one can reconstruct them with this inflation and produce numerous variants. Some extensions of Isfahani inflation will also be mentioned.


Abstract for 21989

A Computer Algebra Approach to Billiards

Authors: Setsuo Takato, Jose A Vallejo

Affiliations: Universidad Autonoma de San Luis Potosi, Toho University

Billiards are very interesting examples of mathematical objects where geometry, analysis and physics merge. From a dynamical systems perspective, they offer examples of integrable and chaotic behavior, so they are quite appropriate to illustrate the main features of these. In the talk, we will review the basics of billiards and we will see how to study them with the aid of the free packages Maxima and KeTCindy.


Abstract for 21991

A Topological View of Curves and Surfaces Inspired by 2D and 3D Locus Problems

Authors: Wei-Chi Yang, Guillermo Dávila

Affiliations: Radford University, Universidad de Sonora

Motivated by some locus problems discussed in [8], and further in [9] and [10] for 2D and 3D cases, we explore scenarios in which if two topological manifolds are topologically equivalent with technological tools. We derive some interesting facts about topological equivalence that could be used in the classroom.



Abstract for 21995

Is it so time consuming to start using a new piece of mathematical software?

Authors: Eugenio Roanes-Lozano, Carolina Fernández-Salinero

Affiliations: Universidad Complutense de Madrid

In this paper we summarize an experience carried out with pre-service teachers, students of the Master's Degree in Secondary Teacher Training (MFPES) of a public university, about their ability to self-learn the use of the computer algebra system for smartphones “Maple Calculator”. It is focused on determining if they were able to find in a short session the main features of the new software and the advantages and disadvantages of its use compared to the use of “Maple” on a computer (note that even if the students handled the latter, it does not resemble the smartphone software). This research has a qualitative approach, of an ethnographic nature. The responses to the questionnaire have shown great maturity, highlighting the expected points and pointing out others.


Abstract for 21996

Upskills Teachers’ Competencies in Constructing Sangaku Puzzles with the Geometer’s Sketchpad

Authors: Krongthong Khairiree

Affiliations: International College, Suan Sunandha Rajabhat University Bangkok Thailand

The purpose of this study was to describe how to increase mathematics teachers’ knowledge and skills in using the Geometer’s Sketchpad to construct Sangaku puzzles. Survey research was conducted in Bangkok, Thailand in the year 2021. A total of 35 mathematics teachers participated in this study. Based on the research findings, the teachers revealed that because of the Covid-19 pandemic, they must teach mathematics through an online platform until the year 2022. More than 50% of the teachers wanted to use the Geometer’s Sketchpad to design Sangaku puzzles as e-learning materials that made students’ learning challenging and fun. The Sangaku puzzle was not easy to construct with a compass and a ruler. The Geometer’s Sketchpad enhanced teachers’ competencies to construct Sangaku puzzles easily, correctly, and precisely.


Abstract for 30001

Visualizing the Derivative Relationship Between “Volume” and “Surface Area”

Author: Douglas B. Meade

Affiliations: Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA

Many students note the obvious derivative relationship between the “volume” and “surface area” of a sphere or circle, but that it fails for a cube or square. In this presentation we explore this topic both analytically and visually. Of particular interest whether this is relationship is just a curiosity or whether it can be observed in other regions and in other dimensions.


Abstract for 30006

“What the majority of mathematics teachers are missing.”

Author: Douglas Butler

Affiliation: iCT Training Centre, UK

The latest research suggests that only 20% of mathematics teachers are making any use of online resources in their teaching. As a result, the majority are therefore missing out on the many possibilities to enhance their students’ understanding through visualisation and interaction. In this talk Douglas will run through a number of possibilities using items selected from his wide-ranging TSM RESOURCES website, covering topics from junior levels to advance higher, and he will include the use of the latest web-version of Autograph.


Abstracts for Regular Full Papers

Abstract for 21941

Research on Sangaku and the use of ICT.

Authors: Hideyo MAKISHITA

Affiliations: Shibaura Institute of Technology

The author proposes geometry teaching materials based on Sangaku, the mathematical culture of the Edo period, together with utilization of information and communications technology (ICT). As described in this paper, the author proposes the use of computer algebra systems (CAS) and dynamic geometry software (DGS) as ICT supporting mathematics education in junior and senior high schools, and reports the use of CAS for algebraic equations expressed by algebraic equations and the use of DGS for drawing diagrams of Sangaku problems.


Abstract for 21942

Computations of statistical power in R

Authors: Sharad Silwal

Affiliations: Radford University

Hypothesis testing is one of the foundational topics in statistics and a must teach in every introductory statistics course. The entire theory of hypothesis testing rides on the concept of power. Yet, its computation is generally perceived to be daunting and is avoided by most introductory courses in statistics. For instance, it is not covered in AP Statistics curriculum. Understanding of power, on the other hand, is reinforced when you know how to compute power against an alternate hypothesis. This paper presents an exposition of computations and illustrations of the critical value and statistical power for a t-test and its nonparametric analog in R.


Abstract for 21959

Combining Brute Force and IT to Solve Difficult Problems

Authors: Jozef Hvorecký, Lilla Korenova, Tomáš Barot

Affiliations: University of Ostrava, Ostrava, Czech Republic, Institute of Technology and Economics, České Budějovice, Czech Republicc,, Faculty of Pedagogy, University of Ostrava, Ostrava, Czech Republic

The brute force method is a typical approach to solving unknown or non-standard problems: without having a more appropriate tool, one tries to test all relevant candidates whether they are or are not the result using the trial-and- error approach. During the process, one may look for a clue or a hypothesis leading to more effective solution and/or to its generalization. The brute force allows learners to solve tasks which are for them otherwise unsolvable because they require other – more sophisticated, beyond curricula – methods. In the paper, we demonstrate two principal steps of the method: the generation of the space of relevant candidates and their testing. Depending on the space size and complexity of calculations, testing may last long. We accelerate is using spreadsheet calculations.

The problem-solving method is explained using examples. First, we find out a brute force solution and then discuss it. Depending on the type of problem, the discussion hypothesizes “theory-based” solutions, generalizes the findings and outlines similar problems which can be solved using brute force.


Abstract for 21961

Different types of test questions and examining systems to assess students'' knowledge in Applied mathematics for Informatics

Authors: Helena Brozova, Jan Rydval, Milan Jelínek

Affiliations: Czech University of Life Sciences, Faculty of Economics and Management, Dep. of Systems Engineering, Faculty of Economics and Management, Czech University of Life Sciences Prague Czech Republic, Faculty of Economics and Management, Czech University of Life Sciences Prague Czech Republic

Appropriately chosen teaching and testing methods enable students to adequately understand the taught matter through direct contact teaching in classrooms and through distance e-learning and self-study at home as well. When teaching the subject Applied mathematics for Informatics, students must not only be able to recognize which mathematical model to apply to solve a given problem, but also to analyze and interpret their obtained results. In this paper, we demonstrate our approach of teaching and testing Applied mathematics for Informatics using e-learning system Moodle for students of the Bachelor's degree at Czech University of Life Sciences. Within the framework of Applied mathematics for Informatics teaching, students are provided with practical seminars in addition to classical lectures. In these seminars’ students solve specific problem situations from practice, through which they learn to recognize which mathematical models to apply to solve the problem situation and students are encouraged to work independently and think critically. Students demonstrate this knowledge in the final exams, which are designed as "open book" standardized examination system, and contained a practical management story in which students as managers are forced to solve many practical problems corresponding to different mathematical models. Students can use their notes, books and all resources available in Moodle. The method of teaching and testing knowledge is chosen to increase the students'' ability to understand the problems taught in Applied mathematics for Informatics, increases the ability to recognize and apply the appropriate mathematical model to solve a given practical problem and to analyze and interpret the results obtained.


Abstract for 21962

Applying Flipped Classroom to Increase Students’ Achievement and Investigating students’ Satisfaction in Learning Mathematics

Authors: Supotch Chaiyasang, Asst.Prof. Dr. Supotch Chaiyasang

Affiliations: Suan Sunandha Rajabhat University, Bangkok, Thailand

The objectives of this classroom action research were to increase students’ mathematical achievement and to survey students’ satisfaction in learning by using flipped classroom. The participants were 32 grade 11 students who enrolled in the second semester of the academic year 2019 at a high school. The topic used in this study was Vectors in Three Dimensions. The instruments were 7 lesson plans, achievement test and satisfaction survey. Learning management by using flipped classroom comprised 3 steps: 1) before class, students studied online learning through video clips, handouts and quizzes, 2) during class, teacher reviewed key concepts and students discussed the contents that they had studied from home, solved harder problems, and got individual help from teacher and 3) additional skills or extended knowledge. Data were collected from pretest, posttest, and satisfaction survey. Data were analyzed by using effectiveness index, mean, mode, and standard deviation. The results showed that: 1) the effectiveness index of the flipped classroom was 0.8 which revealed that students’ achievement was increased 0.8 from the beginning and 2) students’ satisfactions in three categories: students’ understanding category, learning activities category, and learning atmosphere category by using flipped classroom were at satisfied, satisfied, and very satisfied, respectively.


Abstract for 21965

Carral Geometric Proof of a Steiner Ellipse Property Attempt of Generalisation

Authors: Jean-Jacques Dahan

Affiliations: IRES of Toulouse

In this paper, we will first prove using the solution proposed by Michel Carral, that the ellipse inscribed in a triangle of maximum area is its Steiner ellipse. This proof is based on a purely geometric reasoning which greatly simplifies the first complete proof proposed by Minda and Phelp ([5]), in the sense that it uses more basic tools and knowledge. Recall that in a previous paper ([6]), an approach with dynamic geometry had already allowed a very simplified approach even if a step was incomplete insofar as it was only justified experimentally. As I said above, this problem was solved geometrically by my colleague Michel Carral to whom I had submitted it. He has included it as an exercise in the area geometry book he is finalizing. I thought that his solution deserved better than an exercise in his geometry book and I decided with his agreement to write it trying to respect the spirit of the author but detailing it enough to be understood by all. I will then propose an attempt at generalization to ellipses inscribed in polygons. In passing I will propose a still purely geometric proof of the fact that polygons of minimum area having a given inscribed circle are regular polygons. Eventually a construction under the New Cabri, using macros, illustrates the sequence of polygons circumscribed to a given ellipse approaching the polygon of minimum area circumscribed to this ellipse.


Abstract for 21967

Multiple choice questions using STACK with partial score and feedback

Authors: Kentaro Yoshitomi

Affiliations: Osaka Metropolitan University

Online testing systems in mathematics education are useful tools for both teachers and students. STACK is a system that is widely used in Europe and other parts of the world, and it is a question plugin for Moodle, a Learning Management System (LMS). On the other hand, the use of multiple-choice questions in mathematics has been limited due to the randomization problem. The use of multiple-choice questions with STACK solves this problem, but the method described in the help document does not support partial score and feedback, which are important features of a learning tool. In this paper, we introduce a method for providing partial score and feedback by devising a coding method that utilizes the framework of STACK's multiple-choice questions, and present a template. In addition, we will present some examples in which multiple-choice questions, together with incorrect answer patterns, are considered to be important educational resources.


Abstract for 21970

Understanding the Problem Structure in Computational Thinking in Mathematics Classrooms

Authors: Masanori Fukui, Yuji Sasaki

Affiliations: Center for University Education, Tokushima University, Graduate School of Media and Governance, Keio University

Understanding the structure of problems is essential not only in mathematics education but also in all subjects. Particularly in structured problems such as those in mathematics, it is crucial to understand the problem structure before solving it to improve the solution method. It has been reported that problem making, or problem-posing, is effective in promoting creativity and understanding problem structures. However, it is not always easy for majority of the students to create problems, and it is not clear whether creating problems is connected to problem solving. Thus far, other scholars have proposed �modified problem-posing� as a method to enable many students to solve problems. However, it is unclear whether this method improves problem-solving ability in mathematics. This study proposes a methodology to enhance problem-solving skills by focusing on the relationship between modified problem-posing and computational thinking.


Abstract for 21971

Engaging Learners through Data: Senso Eskwela Pilipinas

Authors: Mark Anthony Tolentino, Ma. Louise Antonette N. De Las Pe�as, Maria Alva Q. Aberin, Agnes D. Garciano, Mark L. Loyola, Jumela F. Sarmiento, Juan Carlo F. Mallari, Debbie Marie B. Verzosa

Affiliations: Ateneo de Manila University, University of Southern Mindanao

The use of real data in teaching and learning statistics has been recommended in the literature. This paper talks about the development and implementation of the Senso Eskwela Pilipinas, the first database in the Philippines that builds and provides real data with the intention of making the study of statistics engaging, understandable, relatable, and relevant for Filipino learners.


Abstract for 21972

Develop Computational Thinking in Portuguese Mathematics Curricula with Collatz Conjecture

Authors: José Manuel Dos Santos, Alexandre Emanuel Batista Trocado, Zsolt Lavicza

Affiliations: Centre for Research and Innovation in Education (inED), Center for Research and Development in Mathematics and Applications (CIDMA), School of Education - Johannes Kepler University, Linz, Austria

Recently several countries have adopted as a strategy the introduction of Computational Thinking (CT) in the curricula of compulsory education. In Europe this movement has been driven by decisions of the institutions of the European Union, arguing that the technological capacity of citizens must go beyond their use, and it is necessary to promote the development of CT. In this context, some European countries have included the development of CT as a capacity to develop in mathematics. Portugal was one of the countries that adopted this strategy, which is object of public debate. With this work we intend to analyze some of the potentialities of the introduction of CT as an ability to develop in the mathematics curriculum, namely those related to the type of mathematical activity that can be developed in the classroom. The Collatz Conjecture (CC) will be the context for us to show how we can work with a open mathematical problem, simultaneously developing Mathematical Thinking and different concepts and procedures of CT. The CC approach, as a classroom task, may illustrate how a contemporary subject of mathematics can help itself from computing, highlighting the use of various technological tools or programming languages depending on the age student group.


Abstract for 21976

Link to extended abstract

The Enjoy Origami on Mathematics and Science Education in STEAM

Authors: Minoru Itoh

Affiliations: Tokyo University of Science

Abstract: The origin of Japanese origami culture came from the Ancient China as the Buddhist tradition (around the 6th to 7th centuries). Especially, the paper on which the Buddhist scriptures are written is the root of thin and durable Japanese paper. Washi, called very traditional Japanese paper, is still used in various places even in modern Japanese society. Origami paper is very popular for young children in primary school in Japan. Therefore, origami is great material for mathematics education. In Japanese mathematics education, using origami as a teaching material that is very familiar to children stimulates not only children�s intellectual academic ability but also emotional aspects, and is improving the educational effect. For example, in Euclid geometry, angle trisection and doubling problems are famous as impossible to draw on mathematics, but it is possible by using origami. Similarly, the problem of drawing with a ruler and compass can be easily drawn by children using geometric PC software; GeoGebra, Grapes, Cinderella, etc. By using origami and geometric software in combination, children can expect more effective mathematics education than ever before. On the STEAM, why don�t you go on a fun geometry learning journey with your children and math teachers using origami?


Abstract for 21979

Limit calculation outside the domain of definition of real functions using computer algebra systems: an educational panoramic view

Authors: Enrique Ferres-López, Eugenio Roanes-Lozano, Angélica Martínez-Zarzuelo

Affiliations: Complutense University of Madrid, Centro Universitario de Tecnología y Arte Digital

We recently realized that one of the best-known pieces of mathematical software (GeoGebra)
evaluated the limits of certain functions at a point outside their domain of definition
in different ways according to the so called “View” used (GeoGebra is a very powerful dynamic geometry system with algebraic capabilities). The examples used would classically be classified as “removable discontinuities” or “jump discontinuities”. In a previous paper we described this fact and compared the output of GeoGebra with the output of the computer algebra system Maple. In this new paper we have checked the output of some of the best-known computer algebra systems in more examples, providing a panoramic view of the situation. Clearly, the behavior of the pieces of software depend on the decisions made during the design step, and condition their applications on mathematics teaching.


Abstract for 21984

Scissors, Cardboard and GeoGebra: Technology as instrument, not only as artefact

Authors: Mathias Tejera, Franco Mariani, Zsolt Lavicza

Affiliations: Johannes Kepler University, CICATA-IPN

This concept paper presents an example of technology integration by modifying an optimisation problem as an initial activity of the calculus course in the last year of high school. This proposal articulates three ideas; the importance of the instruments used for the mathematical activity, the concrete, pictorial and abstract model, and the notion of instrumental genesis. Technological tools appear in this proposal initially as an artefact, but this design generates that they become an instrument for learning progressively. Experimentally noted that the activity generates a high level of commitment, causing students to put in play and refine their ideas about working with functions to model reality and make better decisions. This model for technology integration into the mathematics classroom by modifying existing tasks could promote teachers to integrate technology in an easy and more meaningful way.


Abstract for 21985

How to prepare a digital geometric model which is enclosed by an assembly of surfaces for 3D printing

Authors: Petra Surynková

Affiliations: Department of Mathematics Education, Faculty of Mathematics and Physics, Charles University

In this article we address the process of manufacturing of 3D geometric models on 3D printers for the educational purposes. The manufacturing of the geometric model includes the design, 3D computer modeling with Constructive Solid Geometry (CSG), 3D computer modeling of parametric surfaces based on differential geometry, 3D scanning of real objects, and the process of fabrication itself. Students can join the whole process of designing the models for 3D printers or its parts and the printed models can be included in teaching and learning geometry at every stage of education (at the university and the secondary school in our case). We present the possibilities how to model geometric objects in 3D computer modeling software; in addition to commercial ones in free GeoGebra dynamic system too. We present particular constructions of selected geometric models whose boundary consists of parametric surfaces. All examples of printed geometric models presented in this article are intended to be used in mathematics instructions at the secondary school and in the undergraduate courses on descriptive geometry at Charles University (mandatory courses for secondary pre-service mathematics teachers who study teaching mathematics and descriptive geometry). All phases of designing the models for 3D printing together with physical 3D printed models bring a new light in teaching and learning geometry. It engages students in real-world problem solving and promotes students� knowledge in geometry while introducing them into 3D computer modeling and 3D printing technologies. 3D virtual models and 3D printed models themselves can serve as the educational manipulative aids.


Abstracts for Abstract Only

Abstract for 21937

Multiple-Choice questions using STACK

Authors: Kentaro Yoshitomi

Affiliations: Osaka Metropolitan University

We have developed Mathematica-programs to generate XML files of abundant multiple-choice questions (MCQ) for using on Moodle with random question function. However, the problem is the correctness of the questions. In some cases. there are bugs which are almost impossible to be fixed since scale of the number of questions. On the other hand, STACK can provide MCQ, but does not have feedback messages, which are quite important for the students'' self-study. We have developed STACK question template, which can provide partially scoring and appropriate feedbacks.


Abstract for 21938

Review of stochastic volatility option pricing models

Authors: Abby Tan

Affiliations: Universiti Brunei Darussalam

The aim of this work is to review extension to the Black-Scholes pricing model. More specifically, we look at various models where the constant volatility assumption is relaxed. Perhaps the simplest model is the one where volatility function follows a deterministic process. A natural extension of this model is to consider a diffusion process for volatility process. We consider a range of diffusion processes with varying ‘memory’ parameter. For such cases, there are no closed form solutions for option prices. One non-conventional option pricing model is to allow risk to exist and price the risk into the option pricing model. The risk manifested itself as pricing bands or confidence interval around classical Black-Scholes prices. An interesting feature of this risk is that it is independent of detailed statistical characteristics of the volatility process. To take into account memory effects, we can define specific ‘moments’ exhibited by the volatility process which will result in different pricing bands. These pricing bands give good approximation to market price. Interestingly, the volatility ‘smile’ curve can be recovered.


Abstract for 21940

Presentation: The Survey Toolkit Curriculum for Researching Information, Survey Development, and Data Analysis Using TinkerPlots�

Author: Thomas Walsh

Affiliations: Ames Community Schools (retired), Iowa State University

The Survey Toolkit project and use of the curriculum with upper elementary (middle school) students is presented. The curriculum is discussed as a field-tested program found to be effective in guiding students choosing research questions, writing a research report using a paragraph cluster information strategy, developing unbiased survey questions using reliable sampling, analyzing survey data with TinkerPlots®, and sharing results. The development and understanding of data analysis is most important for students living in an era of misinformation and unreliable findings from surveys used to influence the choices for decision-making.

The PowerPoint presentation will focus on these topics about The Survey Toolkit:

• Integration of the curriculum with other disciplines including language arts studies.

• The design model and development of the curriculum for collecting and organizing information for testing and analysis.

• Application of the design model following a student project on Alternative Energy from choosing a research question (goal), writing a report using an inquiry-based composing process to support the development of survey questions, developing the survey questions, hypothesis testing of survey questions, choosing a sample to administer the survey, analyzing survey data with TinkerPlots®, and sharing results on a poster board.

• Examples of other student projects showing various research topics, questions, and data plots on Indian culture, color affects on mood, interest in Rome, usefulness of video cameras, and understanding earthquakes are displayed.

The presentation focuses on the author’s implementation of The Survey Toolkit using TinkerPlots® with student project examples provided following a lesson plan sequence in the text and integrated with supplemental activities provided in The Survey Toolkit Resource Manual. Following the PowerPoint, a demonstration of the TinkerPlots® software will be presented. Access to The Survey Toolkit, TinkerPlots® and resource manual are available from the author’s site at Journal publication links discussing use of the curriculum and teaching strategies along with a literature review on teaching statistics with students is provided. The author is currently conducting a literature review focused on the use of TinkerPlots® by elementary and middle school students in the schools. The Survey Toolkit Resource Manual is available as a pdf download at the web site. A reference business card will be provided, with a QR code, for access to the site.

The need for further research to evaluate the effectiveness of the curriculum materials, student learning, alternative teaching strategies, use of TinkerPlots®, and staff development is needed. Students show different levels of understanding of statistical data and reasoning, which needs teacher scaffolding and support to evaluate graphs. This author has found supporting students learning through mediated teacher intervention using TinkerPlots® has been effective to support creation and understanding of data set graphs for reporting findings and making inferences. Evaluation of The Survey Toolkit curriculum will be needed to improve students’ ability to think statistically, since data analysis and statistical reasoning are becoming part of the mainstream school curriculum in many countries.


Abstract for 21957


Authors: CELINA A. A. P. ABAR, Alexandre Matias Russo, Tomas Recio

Affiliations: Pontifical Catholic University of Sao Paulo, Pontifical Catholic University of São Paulo Brazil, Universidad Antonio de Nebrija Spain

This contribution is about some work that is part of a doctoral dissertation, in progress, in the context of geometry education. More precisely, it deals with the study of plane geometry properties through an experimental version of GeoGebra called Discovery, which has some Automated Reasoning Tools (ART). The objective is to identify if there are contributions to the geometry learning process of students in the 9th year of Elementary School and if the tools available in Discovery allow the development of geometric thinking, through constructions and conjectures passing through the levels proposed by Van Hiele. In our presentation we will describe the development of the exploration of an activity on a circumference, and the corresponding reflections presented by a pair of students working on this activity, all this in the framework of the Design Research methodology. Students were asked to address this activity through GeoGebra Discovery. It can be concluded that the Relation and Discover tools from GeoGebra Discovery helped in the process of exploring, verifying, and validating the students'' conjectures, pointing out relevant contributions to the teaching and learning of plane geometry properties.


Abstract for 21960

Exploring the Nature of Online Discussion Forums in Mathematics

Authors: Pragashni Padayachee

Affiliations: University of Cape Town

Mathematics plays a fundamental role in engineering studies. Students need to successfully navigate learning in mathematics to apply mathematical knowledge and skills to other engineering courses. However, mathematics poses a challenge for students, delaying graduation and contributing to increasing student attrition rates. Online discussion forums provide an added pedagogical opportunity to encourage student engagement with mathematics concepts in an online mathematics community of learning. Careful consideration should be taken in designing this course activity so that it encourages, facilitates and reflects a deep level of student learning. An expectation exists in this anytime available and open to all asynchronous activity for students to take responsibility for their own learning and to share knowledge with their peers. Commonly used in other disciplines, not much is written about discussion forums in the often traditionally taught undergraduate mathematics courses. This research sought to understand students'' approaches to learning in mathematics discussion forums. students taking Vector Calculus during 2020 in an engineering support program at the University of Cape Town were the focus of this research. This qualitative research study rooted in constructivist theories of learning employed a conceptual framework to explore student’s discussion forum participation. The three-pronged framework focused on content, interaction and objective measures. Quantitative and qualitative data are used to evaluate the nature of the discussion forums based on the 11 categories set out by this framework. Key findings indicate a positive correlation between student engagement in discussion forums and student learning and achievement in the vector calculus course.


Abstract for 21998

Learning mathematics through 3D printing in Secondary Education: a case study in Spain

Authors: Miguel Angel Fuertes-Prieto, Bárbara María Alonso-Ruano, María Álvarez-Díez, Laura Delgado-Martín

Affiliations: Universidad de Salamanca, Colegio Sagrado Corazón – Salamanca

The progressive affordability of printers and printing material has made 3D printing go from being a technology used only in professional and higher education environments to being a teaching resource available in many primary and secondary schools [1]. The introduction of 3D printing technology in the field of education has been considered by some researchers as a great advance compared to current traditional education [4], having a positive effect on the general performance of students in their technical and mathematical skills [3]. Learning environments centered in the students and with integration of technology produce students who are better able to think critically, solve problems, collaborate with others and engage deeply in the learning process. When teachers know how to effectively use technology features, they can better address the different cognitive strengths and needs of the different students [3].

In this presentation, a case study where 3D printing has been used in Secondary Education in a Spanish school will be described, in relation with the main ways in which 3D printing is being used in the educational system, according to Ford and Minshall [2].

It has been a cross-curricular project carried out mainly in the areas of mathematics and technology, but also with integration of other areas. Secondary students have learnt about 3D printing, it has been used as support technology during teaching, not only to produce artefacts that aid learning but also to support outreach activities and even community services.

In the technology area, students learnt all the steps of the 3D printing process, from the design to the making. That knowledge was applied in the mathematics area to calculate areas and volumes of simple and composite shapes, experimentally verify the results of mathematical calculations, or carry out experimental verifications of mathematical statements such as the Pythagorean Theorem.

Preliminary results show that the learning process of most of the students has been improved when using 3D printing and almost all the students gave very positive feedback of it.


[1] Canessa, E., Fonda, C. and Zennaro, M. (2013). Low cost 3D printing for science, education and sustainable development. Trieste: ICTP—The Abdus Salam International Centre for Theoretical Physics.

[2] Ford, S. and Minshall, T. (2019). Where and how 3D printing is used in teaching and education. Additive Manufacturing, 25 131-150.

[3] Kwon, H. (2017). Effects of 3D printing and design software on students’ interests, motivation, mathematical and technical skills. Journal of STEM Education, 18(4).

[4] Sun, Y. and Li, Q. (2017). The application of 3D printing in mathematics education. In 2017 12th international conference on computer science and education (ICCSE) (pp. 47-50). IEEE.


Abstract for 21999

Computational Thinking in the Spanish Secondary School Curriculum of Mathematics. Meaning and implications.

Authors: Belen Palop, Juan José Santaengracia, Luis José Rodríguez-Muñiz

Affiliations: Universidad de Valladolid, Universidad de Oviedo (Spain)

The digital transformation of our society has opened a new gap between those who can understand and control the technology that surrounds us and those who don’t. For this transformation to happen, we need on the one hand, the digital competence that needs to be learned and taught, especially since 2020, when 99% of Spanish households with dependent children have smartphones and access to the internet Instituto Nacional de estadística (INE, 2021). On the other hand, we have to achieve an even more ambitious goal, which has been included by the Spanish Ministry of Education in the new K-12 curricula [LOMLOE,22]: the development of children’s Computational Thinking.

We can define Computational Thinking (CT) as the way of reasoning that allows people to tackle a problem using input data with the aim of having a computer solve it [Diaz&Palop]. This definition is based on three main computing aspects: Algorithms, Data and Problems, and includes all dimensions of CT (Abstraction, Pattern Recognition, Decomposition, Implementation, Debugging, Parallelization, Simulation, Data collection, Data analysis and Data representation, Modellization and Generalization).

The broad meaning of the term that some authors simply define as “thinking as a Computer Scientist”, makes it a big challenge to include CT in K-12 curricula. Many questions on how this should be done are still unanswered, from the best learning paths to aspects such as when is the child’s mind ready to apprehend certain abstract concepts. To name an example that has received special attention in teaching: when are children’s minds ready to open the black-boxes of Machine Learning and fully understand the Computational Thinking behind the cat-picture vs. dog-picture classifiers?

With all these aspects in mind, in this work we analyze the new Spanish Compulsory Secondary School Curriculum regarding the actual meaning of each of the 30 appearances of the concept. Among those, one appears under the generic term of Digital Competence; 3 in the Biology and Geology course description; 14 in the Technology courses; 1 in Basic Vocational Training and 10 under the description of the Mathematics courses. Our research questions are: (1) What components among Algorithms, Data and Problems is the Mathematics curriculum proposing to work on? (2) When explicit descriptions are given, which of the cited 12 dimensions are addressed? and (3) Can these components/dimensions be added or infused in the traditional Mathematics curriculum?

Diaz, I. & Palop, B. A holistic approach to Computational Thinking in K-12 Education [Unpublished manuscript]. U. Oviedo, U. Valladolid.

INE. (2021). Hogares que tienen acceso a Internet y hogares que tienen ordenador. Porcentaje de menores usuarios de TIC. Madrid: Instituto Nacional de Estadística.

Ministerio de Educación y Formación Profesional [MEFP]. (2022). Real Decreto 1105/2014, de 26 de diciembre, por el que se establece el currículo básico de la Educación Secundaria Obligatoria y del Bachillerato.. Boletín Oficial del Estado, 03/01/2022.


Abstract for 22000

A didactic process for learning the logical conditional and its correct application through the use of computer tools with prospective Primary teachers

Authors: Laura Sánchez-Pascuala, Eloísa Montero

Affiliations: Centro Universitario Escuni, Universidad Complutense de Madrid

The objective of this communication is to present a possible didactic process to be followed in order to facilitate the understanding and use of the conditional and its operation by prospective Primary teachers (PPT).

The optional subject "ICT for Mathematics in Primary" of the Primary Education Degree seeks to develop competences such as: i) Discovering the possibilities of various ICT tools for the teaching of Mathematics in Primary, and ii) Knowing how to translate to mathematical language and know how to deal with ICT tools some problems of daily life. In this communication we will focus on two of its objectives: i) Discover the possibilities of the three large groups of software for Mathematics Education: Logo/Scratch, Dynamic Geometry and Computer Algebra; and ii) Begin to program in a computational language.

At Escuni University Center for Education (Madrid, Spain) we have planned the aforementioned subject with the aim that PPT can develop these skills and objectives, among others, so that they learn how to design teaching-learning experiences for their future Primary students through the learning of three software pieces: Scratch, GeoGebra and Excel. The use of conditionals is among the basic learnings to be developed in this subject, in addition to the use of these software, so that its assimilation is not a goal itself, but a knowledge required to complete different projects associated with the PPT’s professional development. The goal of this communication is to introduce a didactic process to facilitate the understanding of the logical value of a conditional expression and its operation, in order to implement it in a practical way in daily life and to develop the computational thinking skills.

The associated computational language comprehension process consists of different levels, beginning with a written natural language (or close to it) (Scratch) to end with an abstract computational language (Excel), going through an intermediate stage (GeoGebra). Our goal is to plan the learning of computer tools based on this progression in the use of computational languages.

For PPT located at the first level we recommend the use of Scratch. The fact that this software has a visual interface allows learners to focus on understanding the conditional itself and how to use it empirically, with a language close to written language.

At the second level, the use of the GeoGebra program is appropriate because this software’s interface remains essentially visual. Carrying out self-correcting exercises is a good context to motivate PPT. Some of the examples in which this concept appears are: when we give a condition for the appearance of an object or when we describe the functionality of a created button.

For PPT of the last stage the use of Excel is possible. In this program we find a very basic level of the use of the conditional, such as the conditional formatting of cells, and a wide range of formulas that can help us work on this concept, from the simplest ("IF", "IF.ERROR") to other more complex uses such as the concatenation of conditionals.


Abstract for 22001

Math trails in initial teacher training with MathCityMap and Augmented Reality

Authors: Alvaro Nolla de Celis, Angélica Benito, Ariadna Gomezescobar, Elena Sánchez, Carlos Ajenjo

Affiliations: Universidad Autónoma de Madrid, Universidad a Distancia de Madrid, IES Ángel Corella

We present an ongoing project in teacher training at Universidad Autónoma de Madrid (UAM) which incorporates math trails with mobile technologies. In particular, it includes the use of MathCityMap and the Augmented Reality (AR) capabilities of GeoGebra in math trails and proposes the creation of math trails by the students as a rich activity in initial teacher training.

Designing a math trail is a creative activity that starts from the consideration of the surroundings as an educational space in itself. It is presented as an open challenge and becomes a mathematical learning process which requires the creation of contextualized tasks, allowing a shift from usual academic settings to real-life mathematical situations. The inclusion of math trails in initial teacher training is presented as an opportunity to develop problem-solving and problem-posing skills in prospective teachers, and to provide them with a creative and collaborative mathematical experience that can incorporate in their professional future.

In addition, the use of mobile technology has been successfully implemented in several outdoor learning proposals. These tools give more flexibility and autonomy during the activities, provide teachers with the possibility to monitor and interact with their students and may include gamification features. Learning how to use and experiment with these tools give prospective teachers the confidence and opportunity to include them in their own math trails.

We will describe the work carried out by the project since the academic year 2019-2020 with students of mathematics subjects of Early Childhood, Primary and Secondary Education Degrees and Masters at UAM. The project followed two steps: (i) Students experience a math trail using MathCityMap and GeoGebra AR with pre-design routes, and (ii) students in groups create math trails aimed for their future pupils. The routes were mostly designed at various locations around the city (streets, parks, museums), highlighting the connections of mathematics with architecture, history, nature, arts… which shows the STEAM nature of the project. For some groups of students, the trails were designed around the UAM Campus, and it was possible to carry out an evaluation session with a peer (group) review process.

The project has had a very positive evaluations by the students. They mostly found the inclusion of MathCityMap and GeoGebra engaging and motivating, and the creation process of math trails as meaningful.

The most experimental part of the project was the inclusion of AR features in one of the pre-designed routes, which consisted of a MathCityMap trail augmented with GeoGebra 3D applets which were linked at every task. This augmented math trail contains various uses of AR in this outdoor setting, and the ones that required more interaction with a GeoGebra 3D model to solve the task were considered more engaging by the students. The AR experience with future Secondary School math teachers showed good responses in participation and motivation, as well as noticing room for improvements.


Abstracts for Hands-on Workshops

Abstract for 21943

Workshop: Exploring Computer Science with Lynx to Learn Geometry and Logo Programming Code

Authors: Thomas Walsh

Affiliations: Ames Community Schools (retired), Iowa State University

Research on Logo programming contributing to student learning has appeared in the literature during the last four decades. Empirical and meta-analysis studies support of teaching Logo coding in developing student cognitive problem-solving skills has been documented using teacher guided instruction. Utilizing guided instruction with teacher-mediated scaffolding Exploring Computer Science with MicroworldsEX (Walsh 2013-2017) has been found as an effective curriculum in preparing the author’s elementary and middle school students to apply Logo code language to create geometric graphics, animation, and gaming projects. The instructional curriculum e-book updated to a cloud-based platform Exploring Computer Science with LYNX (Walsh 2020) is anticipated to provide continued support to students and teachers in learning Logo coding. Michael Quinn is the lead designer of LYNX and president of Logo Computer Systems Incorporated (LCSI). Quinn reports about 99% of the users of LYNX are students and teachers in grades 4 to 9 including use reported by graduate students for their program of study. LYNX is part of the Canada CanCode Program started in 2017 to support digital skill development to Canadian youth (K-12) and providing teachers with professional development.

The e-book is published by LCSI and posted at on the cloud. Visiting the website the user can get started exploring LYNX on their computerat the Home Page by:

• Viewing the Learner Mode and Advanced folders showing math projects.

While opening the projects, by selecting the EDIT button, grey words in the

Procedures Pane explain the commands.

• Selecting the Help Section documents and videos are available along with

downloads for the Getting Started Guide and Activity Cards. In this section

Exploring Computer Science with LYNX is available under Resource


Returning to the home page at select the yellow link CREATE A LYNX PROJECT when ready to code exploring and creating explore and create your own project.

The workshop will introduce LYNX coding to participants with no previous experience using Logo or other coding languages. A PowerPoint presentation with provide an overview of the use and need to teach coding in the schools along with a brief background about the development and features of Logo. Research support in the literature for teaching Logo is presented along with a sample of the author’s student projects. On the participant’s computer (Mac or Windows) an introduction to coding using LYNX will be provided experimenting with turtle commands for creating graphics. Time will then be provided for applying the commands to create a Logo program.

Experience teaching Logo on changing platforms has supported the author at the elementary and middle school level for over 30 years. The classroom field experience along with research on Logo programming, based on dissertation study, has contributed to development of the e-book curriculum. More research will be needed to study teacher scaffolding and mediation skills to support learning Logo using the LYNX platform along with transfer to other domains including programming environments like Python or JavaScript. Future employment of computer-programming jobs will be best for applicants with experience in a variety of programming languages and newest programming tools (Bureau of Labor Statistics, 2021).


Abstract for 22002

Developing Mathematical and Computational Thinking through Spreadsheets Authors: Jonaki Ghosh, Weng Kin Ho

Affiliations: Lady Shri Ram College, Delhi University, National Institute of Education, Mathematics and Mathematics Education, Nanyang Technological University, Singapore

We assume Excel is installed in participants' computer prior to the workshop. The workshop will illustrate the possible linkages between the educational goals of the ‘Big ideas in mathematics’ and ‘computational thinking’ through the use of spreadsheets. Electronic spreadsheets, if used appropriately, can serve as a nexus between these two domains in mathematics education. Mathematics teachers need to able to convey the ‘Big ideas’ to their students by engaging them in meaningful problem solving and explorations. Spreadsheets, equipped with advanced numerical and graphing capabilities can be great enablers in this regard and can be used as primary vehicles for exploration in many mathematical tasks.

Technological and pedagogical affordances, offered by spreadsheets in exploring mathematical concepts, which in turn foster computational thinking, will be the focus of the workshop. The easy accessibility and powerful features of spreadsheets make them conducive to designing inquiry-based tasks. In fact, spreadsheets are easily accessible to school students, as they are easy to use and do not require an extensive knowledge of coding. The workshop will emphasize the synergistic relationship between mathematical and computational thinking whereby participants will be given the opportunity to visualize and analyze problems using MS Excel. The problems will be drawn from a range of topics, which illustrate the ‘Big ideas’, such as exploring numerical patterns, graphing and analyzing data, simulating experiments in probability and working with matrices.


Abstract for 30002

Introduction to Sage: Part 1

Author: Alasdair McAndrew

Affiliation: Victoria University, Melbourne, Victoria, Australia

We assume participants installed Sage in their computers prior to the workshop. Sage ( is an open-source computer algebra system which has been in continuous development since 2004.  The original aim was to collect the best of open source mathematical software together under the one interface; this being the programming language Python.  Currently Sage consists of nearly 100 different packages covering almost all branches of mathematics, as well as many tens of thousands of lines of code specifically written for it, many contributed by the leading researchers in their fields.  It also contains interfaces to commercial software systems.  It remains fully open source.

The first workshop will introduce Sage briefly, provide a whistle-stop tour of its major components, including its documentation and its interfaces, and look at some undergraduate-level material: algebra, calculus, linear algebra, geometry.


Abstract for 30003

Introduction to Sage: Part II

Author: Alasdair McAndrew

Affiliation: Victoria University, Melbourne, Victoria, Australia

We assume participants installed Sage in their computers prior to the workshop. The second workshop will delve a bit deeper, looking at some of Sage's more advanced capabilities, such as animated graphics, Gröbner bases and geometry theorem proving, number theory, optimization, and proposals from the first workshop. No particular expertise will be expected, and the only thing that participants need to bring is enthusiasm. The aim of these workshops is to provide the participants with enough information to be able to determine whether Sage might be useful for their own work, and how they can teach themselves further. These two Sage workshops will be delivered using Zoom, as the presenter is unable to be physically present at the conference.

Presenter: Alasdair McAndrew is a long time user of Sage (more or less since its inception) and is a developer - he created the initial version of the Derangements package: see He has written about Sage extensively in articles and on his blog. He has also written a book "Introduction to Cryptography with Open-Source Software", based around Sage. See for details.


Abstract for 30004

GeoGebra in Action: Dynamic Geometry in 2D and 3D

Author: Petra Surynková

Affiliation: Charles University, Faculty of Mathematics and Physics, Department of Mathematics Education, Praha, Czechia

We need everybody to install GeoGebra 5 or 6, the GeoGebra 6 is prefered, before joining this workshop. (See We will introduce the GeoGebra 6 dynamic system and the bases of using the software. We will work with dynamic features of GeoGebra, 2D and 3D Graphics View, and Computer Algebra System. In the first part of the workshop, we will solve geometric constructions in the Euclidean plane, for instance circumscribed and inscribed circles of a triangle or common tangent lines to two circles. We will demonstrate known theorems and statements in the planar geometry, such as Pythagorean and Thales's theorems. For the pedagogical purposes, we can show the differences in using defined GeoGebra’s functions and constructions which simulate the hands-on drawing using only the ruler and the compass. In the second part of the workshop, we will focus on analytic and differential geometry in the two- and three-dimensional space. We will combine defined tools for geometry and computer algebra system to solve equations such as constructions of conic sections, the intersection of a plane and a line in the three-dimensional space, or the intersection of surfaces.