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Studies show that gamification enhances the mathematics learning experience by increasing student engagement. Gamification can be successfully used in higher education contexts, primarily when it comes to distance learning. It can compensate for the lack of interaction and create a student-centered, customizable, and autonomy-promoting environment. The use of adaptive learning and interactive feedback can be seen as strategies to improve student access and success in open online courses. In particular, a mathematical modelling module for undergraduates developed inside the project start@unito uses such strategies. Moreover, it is taught entirely in English. The Mathematical Modelling course inside the Erasmus+ Project SMART represents another source for research. The purpose of the chapter is to discuss the possibility of combining adaptive learning with gamification techniques to enhance and facilitate the English-mediated learning process while teaching of mathematics, adding the linguistic and STEM education challenge.

Adaptive teaching is defined as applying different teaching strategies to different groups of students so that the natural diversity prevalent in the classroom does not prevent each student from achieving success. Some of the strategies for adaptive teaching are formative assessment, feedback, and self-regulated learning. For the application of these strategies, technology plays a fundamental role. Our research group has developed and tested a model for automatic formative assessment and interactive feedback for STEM using an Automatic Assessment System to allow adaptive teaching. This article presents the design, the implementation, and the analysis of adaptive teaching activities carried out by secondary school teachers of STEM disciplines during a training course. Teachers designed the transformation of questions created for standardized assessment into formative assessment questions, reflecting on how to adapt the requests to students’ different needs and how to create guided learning paths. In this paper, all the steps of the training activity carried out will be analyzed using some representative examples: the design of the questions with an analysis of the adaptive teaching strategies adopted; the reflections of the teachers at the end of the course; carrying out activities with students and analyzing their responses; the reflections of the teachers at the end of the activity.

The health emergency due to COVID-19 has highlighted the need of new and flexible digital methodologies for learning and teaching Mathematics, which can support the individual student’s needs and help shape education. In this paper, we propose to use Automatic Formative Assessment (AFA) activities designed according to an innovative model, framed on the theories of formative assessment and feedback. The model includes: availability of the activities and multiple attempts; algorithmic questions; open mathematical answers; contextualized tasks; immediate and interactive feedback. We analyzed an experience using AFA in a blended module in Mathematics for 96 first-year students in Biotechnology. We collected data from the platform on grades and attempts of AFA activities and cross-checked them with the final exam grades. The results show that the feedback gained from AFA helped students improve their performance and supported them in the exam preparation. In light of these results, we can conclude that similar activities can be an effective solution to support students’ self-study during and after the pandemic.

Because of the Covid-19 pandemic, the need to promote digital education has been felt even more, bringing the DELTA Research Group to reflect on how digital education could optimally facilitate the creation of an educating community able to promote and support educational success exclusively through distance learning. In order to answer, in this paper the experience of the 2019/2020 edition of “Scuola dei Compiti” (a recovery project for secondary schools of the poorest suburbs of Turin) is analysed in the perspective of the priority strategies illustrated in the Digital Education Action Plan for the period 2021-2027. Results will show that this was a forerunner experience perfectly in line with the principles and directions outlined in the document. Moreover, thanks to digital education, even a peculiar edition of the project such as this one has been a success: in fact, it has allowed students to improve from a cognitive and metacognitive point of view, even during a time that was very difficult for some families.

Nowadays, the students are digital natives and they live immersed in technology, but their use is mostly superficial; therefore it is important to educate them to use the technological tools that they have available for educational purposes already since the primary school, both at school and in extra-curricular contexts. Moreover, the digital competence has been included by the European Union among the eight key competences for lifelong learning. This demonstrates that educational institutions should play a fundamental role in this field since the early school years. In this paper, we present the design and the implementation of a vertical path for all five years of primary school within a Digital Learning Environment in the context of Space and Shapes. The goal of the path is to promote the skills development in the Space and Shapes area in a vertical perspective through the methodologies of problem solving and automatic formative assessment. The path was implemented on a Digital Learning Environment, integrated with an Advanced Computing Environment and an Automatic Assessment System suitable for STEM. The path consists in five sections from grade 1 to 5; they are dedicated respectively to plane shapes, solid shapes, perimeter, area, and volume. Each section includes a contextualized problem and one or more questions for the formative assessment have been designed. For the planning of the vertical path, reference was made especially to the Italian National Indications and the "INVALSI" (the Italian institute which annually administers standardized learning tests to evaluate the school system) test archive was consulted. Initially, this paper shows the theoretical framework to which reference was made during the design of the vertical path. In particular, we present the theory of the development of geometric thought conceived by Dina and Pierre Van Hiele between 1957 and 1986 and the methodologies of problem solving and automatic formative assessment. Then, the design of the path in a vertical perspective is presented, and the resource used during the design phase are shown. Finally, the implementation of a section of the path is shown as a representative example and the expected results of a possible experimentation, are discussed.

The transition between different levels of education is delicate, especially when mathematics is concerned; could a digital learning environment and new technologies help and support students in it? In this paper, we discuss the efficiency of a technology-enhanced learning environment for facilitating the transition between lower and upper secondary school. This Online Path regards the review and the enhancement of the main mathematical topics that are fundamental for beginning the upper secondary school. This educational path has been implemented in a Virtual Learning Environment (VLE) integrated with an Advanced Computing Environment (ACE) and with an Automatic Assessment System (AAS); in this way, it is always available and accessible and the educational materials in it are interactive and adaptive. Moreover, the system allows the adoption of a model of formative automatic assessment and interactive feedback that has already been proven to be effective when self-paced learning is concerned. Students could consult the material and self-assess their abilities through those summer holidays between the lower and the upper secondary school but the course is always available during the whole year allowing a perpetual support in the main mathematical concepts studied in class. So far, a total of 500 students have been involved, around two hundred students during the summer of 2018 and three hundred during the summer of 2019. Data analysis shows that students, who have been more involved in this online path, have also achieved a better performance in mathematics at the beginning of the scholastic year in the upper secondary school, proving that methodologies and technologies adopted were successful. For the summer of 2020, we are planning to enlarge the project with two more online paths: one for Latin and one for English language.

Problem solving and computational thinking are the key competences that all individuals need for professional fulfillment, personal development, active citizenship, social inclusion and employment. In mathematics, during contextualized problem solving using Maple, the differences between these two skills become thinner. A very important feature of Maple for problem solving is the programming of animated graphs: an animation obtained by generalizing a static graph, choosing the parameter to be varied and its interval of variation. The first objective of this research is to analyze the computational thinking processes behind the creation of animated graphs for the resolution of a contextualized problem. To this end, we selected and analyzed some resolutions of problems carried out by fourth-grade students of upper secondary schools in Italy (grade 12). The paper shows some examples in which different processes of computational thinking have emerged, which reflect resolutive strategies and different generalization processes. From the analysis it emerged that all the processes underlying the mental strategies of the computational thought useful for solving problems are activated in the creation of animated graphs. In the second part of the article we discuss examples of animations created during training activities with secondary school teachers, and how animations can support the learning of scientific concepts. It is very important to train the teachers in this regard, both to understand the processes that the students would activate during the creation of animated graphics and to enrich the theoretical or practical explanations with animated representations.

In the transition from lower to upper secondary education, Italian students are expected to have achieved a level of competence which allows them to use knowledge and abilities to model and to understand scientific and technical disciplines. National standardized tests show that especially students who attend technical high schools often have gaps or misunderstandings in their basic knowledge, which may hinder them in learning scientific technical disciplines, which are at the core of their curriculum. In this paper we start from items designed for summative assessment which highlight the main difficulties that students face with Mathematics at this stage, and discuss how it is possible to adapt them to automatic formative assessment through a process of expansion and digitalization, with the aim of helping students fill the gaps and develop mathematical competences. The activities are part of an online course, called “MATE-BOOSTER”, conceived to strengthen mathematical skills of students attending the first year of a technical upper secondary school. In this paper the process of design of MATE-BOOSTER, rooted on constructivist assumptions, is outlined; the design of tasks for automatic formative assessment is discussed in details, and some examples of online activities are analyzed in light of a theoretical framework.

Engagement is one of the most powerful driving forces that moves learners forward in a learning experience. The effects of engagement are related to satisfaction, learning results, and reduction of drop-out rates. School engagement can be positively influenced by problem solving and technology-enhanced learning activities. This paper discusses how online interactive activities can promote engagement in Mathematics at school level. After discussing a theoretical framework for defining and measuring engagement, the potentialities of activities with automatic formative assessment used in a blended environment are illustrated. The activities have been experimented with 299 students of grade 8 in Italy; their effects on student engagement are evaluated through questionnaires administered before and at the end of the project. The activities resulted particularly effective with regard to the increase of the engagement level of students who showed a low interest in Mathematics at the beginning of the project.

In the transition from lower to upper secondary education, Italian students are expected to have achieved a level of competence which allows them to use knowledge and abilities to model and to understand scientific and technical disciplines. Gaps or misunderstandings in basic knowledge can hinder the effort of students who attend technical high schools, where the core subjects are based on Mathematics. This paper deals with the design of a project conceived to strengthen mathematical competences of students attending the first year of a technical upper secondary school through an online course named “MATE-BOOSTER”. The online activities on the web-based platform have been developed using didactic methodologies founded on constructivist assumptions, as problem posing and problem solving, collaborative learning, learning by doing, automatic and adaptive formative assessment. In this work the process of design of MATE-BOOSTER is shown, the methodologies chosen are discussed, and the online activities are analysed from a constructivist perspective.

Automatic Assessment Systems empowered by mathematical engines allow the development of online assignments for Mathematics, which goes beyond multiple-choice modality. Automatically assessed assignments, used with formative purposes, can support teaching and learning from several perspectives, such as conceptual and procedural understanding, metacognition, enactment of adaptive strategies, and teachers’ management of the class. This paper reports on an experimentation where automatic assessment has been used in a blended modality according to a model of formative assessment and interactive feedback to enhance learning. The experiment involved a total number of 546 students of 8th grade in the town of Turin (Italy). The use of the automatic assessment is shown and exemplified. Data from learning tests, questionnaire and platform usage are analyzed and used to show the effectiveness of the interactive materials for enhancing mathematical understanding and self-assessment skills. Moreover, a profile of the students who did not use the online opportunities, defined as “reluctant users”, is drawn and discussed.

In an Italian and European context, one of the fundamental skills in Mathematics is the ability to solve problems in everyday situations, often linked to everyday life. For this reason, the problem posing and solving methodology plays a fundamental role in the process of teaching and learning Mathematics. This paper presents the results of the immersive experience "Mathematical Exploration with Problem Posing and Solving", included in the teacher training activities proposed by the national PP&S - Problem Posing & Solving - Project of the Italian Ministry of Education, which aims at enhancing the teaching and learning of Mathematics by using new methodologies and technologies. In particular, the focus will be on the work and considerations of the 50 teachers who took part in the project, from both primary and secondary school. They were guided through the individual step-by-step creation of a contextualized problem, following a process guided through stimulus-based questions. This immersive experience brought about the production of valid problems and was full of very stimulating teachers' considerations on the various phases of the problem posing and solving.

Problem solving, i.e. the ability to model concrete situations in mathematical terms and to solve the problems connected to them, plays an important role in teaching and learning Mathematics to and by students at every level of education and is one of the basic skills for the training of citizens. By learning problem solving in Mathematics, students acquire ways of thinking, creativity, curiosity, collaborative skills and confidence in unfamiliar situations, from which they will benefit also outside the Mathematics classroom. The use of digital technologies helps in both representing and exploring mathematical tasks. Technologies also provide a range of tools and opportunities to structure and support learning environments in which students continually interact to build knowledge and solve problems. The national PP&S - Problem Posing & Solving - Project of the Italian Ministry of Education aims to enhance the teaching and learning of Mathematics and Informatics using new methodologies and technologies. One of the methodologies is problem solving through the use of an Advanced Computer Environment (ACE). An ACE is a system that allows one to perform numerical and symbolic calculation, make graphical representations in 2 and 3 dimensions and create mathematical simulations through interactive components. Moreover, it is able to support students in reasoning processes, in the formulation of exit strategies and in the generalization and re-adaptation of the solution in different contexts. The goal of this paper is to discuss the results of an immersive workshop, entitled "Mathematical Exploration with Problem Posing and Solving", involving 50 secondary school Maths teachers, who were invited to reflect on the role of an ACE in learning and teaching Mathematics through problem posing and solving. During the workshop were presented the problem solving methodology used in the PP&S project and an example of a contextualized problem solved with the use of an ACE. The results emerged from the analysis of the questionnaires (initial and final) compiled during the workshop and a study of the works delivered by the teachers, who were asked to design a problem, its resolution and generalization through the use of an ACE. They are useful to understand teachers' reflections on the methodologies proposed and on the impact of new technologies on teaching.

Enhancing language skills is a duty for many institutions, schools and universities; Information and Communications Technology is nowadays supporting language teaching, heading also to the integration of foreigners. According to previous studies, the use of a Technology Enhanced Learning Environment can foster language competences through online tests. The goal of this paper is to show and discuss different typologies of automatically evaluated questions useful to learn languages and designed with an Automatic Assessment System, already successfully used for teaching STEM disciplines: Science, Technology, Engineering and Mathematics. This work has been carried out thanks to the collaboration between the Department of Mathematics and the Department of Foreign Languages. Five different typologies of questions are analysed: "Combining sentence elements" asks to combine given elements to form at least three correct and meaningful sentences; "Find the mistake" asks whether a sentence is grammatically correct and to explain why; "The Target puzzle" asks to join words together using precise links; "Scrambled text" asks the student the correct sequence of sentences in a scrambled text; "The Encrypted Crossword puzzle" is a crossword puzzle without definition clues. Linguists consider these question types effective to develop language competences and have already used them profitably in traditional teaching and language learning. To engage the learner and increase motivation, we will develop an implementation where the questions are self-evaluated, always available, with immediate and interactive feedback. These tools could be used in online language courses and in all those teaching activities that are carried out remotely by the student alone. All questions are outlined with the explanation of the grading code and a triplet of descriptors; they describe the Performance, Requirements and Objectives of each question and allow further studies on automatic detection of the relatedness between different language learning objects, in order to construct adaptive language tests.

This paper presents a model for school remedial, focusing on improving the digital materials sharing process for the diversification of tutoring strategies. The model involves the characterization of materials for automatic assessment shared within a community of tutors. The characterization expects materials to be linked with natural language descriptors explicating their intended instructional objectives. The possibility of implementing a recommendation system on the basis of natural language processing techniques is discussed taking in consideration an analysis of the application of the model within a local-scale project. Clustering techniques searching for materials that have the same educational purposes but involve the activation of different cognitive processes are proposed, in order to continuously extend the database of shared materials in favour of the effectiveness of ongoing tutoring actions. The results collected from questionnaires submitted to students, tutors, and teachers involved in the project are shown, and clustering data are discussed highlighting the feasibility of the application of the model.

Supporting students in the transition from secondary school to university is an important action to be taken in order to increase the number of people with tertiary education degrees; it has positive consequences on the citizens’ working perspective and on the national economic growth. To this purpose, in the academic year 2014/2015 the University of Turin started the Project Orient@mente, a platform of self-paced open online courses aimed to guide students in the choice of a scientific university program. This paper focuses on the design of the Realignment Course in Mathematics, provided by the platform and aimed at the self-paced review and integration of knowledge that students are required to acquire during secondary school. Materials are created and structured according to the User Experience Design for Learning, with the aim of providing a valuable learning experience to students. Contents are delivered through an accessible learning management system integrated with an advanced computing environment and an automatic assessment system. Data from the usage of the activities and from the results of the online tests are analyzed in order to validate the design choices and the coherent structure of the course.

Formative assessment is a practice aimed to enhance teaching and learning and to develop self-regulation. Among the strategies for activating formative assessment, the provision of feedback is one of the most studied: good feedback helps close the gap between actual and desired performance and it promotes self-regulation. Automatic Assessment Systems can improve the practice of formative assessment, especially for the immediacy of feedback and the flexible solutions for questioning. In this context, the Department of Mathematics of the University of Turin has developed a model for automatic formative assessment using an automatic assessment system suitable for STEM disciplines. The key features of the model, grounded on the models from the literature, are: availability, algorithmic questions, open answers, immediate feedback, interactive feedback, and contextualization. These are detailed and discussed through evidence gained from three experimentations, involving the total number of 553 students. In particular, it emerges that the use of interactive feedback ensures that students process the information from the feedback and use it to improve their performance, thus solving the major problem raised in the literature that feedback are not useful when students do not read them.

Teachers need, in a lifelong learning dimension, occasions that provide them with opportunities to reflect on the activities done by themselves, to learn how to use the newest and most advanced technologies in their teaching, to exchange ideas, to confront themselves on the updated didactic methodologies. Only in this way can they enhance their professional competences and keep the motivation for their job high. Online contexts seem to be the most appropriate option for in-service teacher training because teachers work fulltime in school and, having personal and family responsibilities, it is difficult to find time for scheduled meetings. In this paper we will discuss online asynchronous collaboration within the virtual community of the Italian teachers of Mathematics of secondary schools who participate to the national Problem Posing and Solving (PP&S) Project of the Italian Ministry of Education. One of the main goals of PP&S, which started in 2012, is to enhance teaching and learning Mathematics and Informatics using new methodologies and technologies. The community lives on an e-learning Moodle platform integrated with several tools. Using a multi-method approach, we analyse the online forum discussions among teachers and tutors collected in the last three years; in particular, we explore which kind of activities they suggest to adopt in their real-world classes and the nature of the interactive processes activated in the community. This collaboration among teachers is a peer support and facilitates the process of transformation of practices and of construction of new professional knowledge and competences. Some implications for the training of teachers in pre-service are discussed.

This paper shows a model of teacher training developed by the XXX of the ZZZ, aimed at introducing teachers to the use of innovative methodologies for learning Mathematics and for developing disciplinary and cross-cutting competences. The learning methodologies proposed are mainly based on Problem Posing and Problem Solving, the use of an Advanced Computing Environment, of a Virtual Learning Environment and of an Automated Assessment System. The training model, designed in blended modality, mainly relies on the creation of an online community of practice, where teachers, supported by tutors, collaborate in the creation of interactive learning materials for their classes. They acquire competences not only in the use of learning technologies, but also on sharing and collaborating in virtual environments; they learn how to develop self-tailored didactic methodologies. The key strengths of this model are highlighted and the results, achieved after the experimentation in several projects, are discussed, showing the effectiveness of the model.

[EN] The paper shows how problem solving competences can be developed by solving contextualized problems using an Advanced Computing Environment (ACE). An ACE is a computer system which enables its user to perform numeric and symbolic computations, graphical representations in two and three dimensions, insert embedded components and create interactive worksheet, all in the same user-friendly environment. An ACE allows students to approach a problematic situation in the way that most suits their thinking, to use different types of representations according to the chosen strategy and to display the whole reasoning together with verbal explanation in the same page: in other words, they can fulfill all the processes that problem solving involves. This paper analyzes a problem solving activity with an ACE proposed by the XXX of the ZZZ, and clarifies, also through examples, how the use of the ACE makes it possible to solve real and relevant problems, facilitates the comprehension of the situation and of the Mathematics laying behind and enhance critical thinking., The authors desire to thank the Fondazione CRT of Turin, in particular dott.ssa Daniela Tornielli for the support given for the Project Digital Mate Training and for her remarkable dedication and passion.