Content of Vol. 27 No. 3 (2020)

Part 1 ESCO 2018: Special Issue on Computers and ICT in Mathematics Education

Revisiting Elementary Arithmetic Through the Development of Iterative Sentences: Some Brief Reflections about the Ordering of the Corresponding Teaching Activities

By Eugenio Roanes-Lozano and Pavel Solin

This experience has taken place at the School of Education of the Universidad Complutense de Madrid.  The goals of the subjects considered are to show how the computer can help in the teaching-learning process in Primary and/or Secondary Education and to clarify and strengthen the mathematical concepts of our students.  Mastering a specific computational language or package is not a goal.  We’ll focus here on a very specific problem, discussed and precised together with the second author: implementing arithmetic operators as nontrivial examples of iterative sentences (not exactly using Peano’s arithmetic, but a similar constructive approach).  We’ll analyse along the article why we believe the best ordering for developing the corresponding algorithm and implementing the code for the operators: power (with positive integer exponent), multiplication (by a positive integer) and addition (of a positive integer), is the reverse ordering in which they are introduced in elementary mathematics or introduced in a an axiomatic.

 

Research for Curricular Innovation in Action in the Mathematics Classroom

By José Luis Galán-García, José Luis González-Marí, Yolanda Padilla-Domínguez and Pedro Rodríguez-Cielos

It is difficult for research in Mathematics Education to have any real impact on teaching practice in the classroom.  Conscious of this, we have developed a research model for curricular innovation, involving the teachers responsible for its implementation, improving their training, and providing scientific information whilst maintaining the integrity of the phenomena analyzed “where they occur and exactly as they occur”.  The model is based on an analysis of the principal existing trends, and is tested in the development of Mathematical subjects at distinct educational levels, chief among them university courses in Engineering, where mixed research methods are used to analyse the effect that carrying out commands in a Computer Algebra System (CAS) has on the performance and attitude of students taking various Mathematics courses.  Each teacher leads the study in his/her group and collaborates with the rest of the groups as an observer and assistant.  Teaching protocols are employed in experimental and control groups; observation protocols, testing objectives, and individual questionnaires and interviews are also employed.  The complete analysis provides data that confirms the fitness of the specific innovations applied (with a notable improvement of more than one point over 10 with α = 0, 05), which enables a direct and solidly-based change to the programmes and the development of their corresponding curriculums.

 

Using Computer Programming as an Effective Complement to Mathematics Education: Experimenting with the Standards for Mathematics Practice in a Multidisciplinary Environment for Teaching and Learning with Technology in the 21^st Century

By Pavel Solin and Eugenio Roanes-Lozano

Many mathematics educators are not aware of a strong connection that exists between the education of computer programming and mathematics.  The reason may be that they have not been exposed to computer programming. This connection is worth exploring, given the current trends of automation and Industry 4.0.  Therefore, in this paper we take a closer look at the Common Core’s eight Mathematical Practice Standards.  We show how each one of them can be reinforced through computer programming.  The following discussion is virtually independent of the choice of a concrete programming language.  Therefore, in the interest of simplicity, we will use a well-known educational programming language named Karel the Robot based on (Pattis, 1995) which is freely available online in NCLab (NCLab, 2019).  The visual character of this language will allow us to provide more illustrative examples than would be possible with a standard programming language such as Java, C++ or Python.

 

Part 2

Developing Preschoolers’ Combinatorial Thinking with the Help of ICT: The Case of Arrangements

By Konstantina Frantzeskaki, Sonia Kafoussi and George Fessakis

In recent years, the learning and teaching of combinatorics presents particular educational research interest from the primary up to higher education levels.  The combinatorial problems constitute a valuable opportunity for mathematical exploration, as combinatorics is a branch of mathematics with many applications, providing a complex network of connections with many areas of mathematics.  The studies which examine the development of combinatorial thinking to the preschoolers are limited.  The purpose of this study is to investigate the effect of a microworld in the development of combinatorial thinking of kindergarten children.  Specifically, the research concerns the production of arrangements two by two, three by two and four by two, from sets of discrete objects in the context of a digital microworld and embedded in a game and narrative context.  The research findings show that the designed microworld comprises a developmentally appropriate learning environment for the meaningful learning and the development of combinatorial thinking by the preschoolers.  The participants’ interaction with the microworld showed that the children can understand the concept of arrangement in a simulated concrete situation and that they can produce possible arrangements with various ability levels.  A strong relation was revealed between the ability level of arrangements production of the children and their ability to pay attention to and interpret the feedback available in the microworld.

 

Exploring Students’ Online Homework Completion Behaviors

By Mustafa Demir and Joannis Souldatos

With the advent of information technology, various online homework systems have often been utilized to enhance students’ knowledge and skills. Several studies analyzed the effects of web-based homework systems on students’ learning through comparing them with paper-pencil homework. However, few studies explored students’ homework completion behaviors (e.g., starting early or late to solve homework problems) in online homework settings. To fill the gap, this study examined students’ use of the WeBWorK, an online homework system, in three undergraduate mathematics courses and the impact of their usage of the system on their final exam and overall homework grades. The findings revealed that students who made more attempts on solving homework problems displayed considerably higher performance on homework and final exam than their peers having less attempts. Although students starting early to answer homework questions obtained higher homework scores than the late starters, they were unable to reflect their homework performance on the final exam.