Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Games as an educational resource in the teaching and learning of mathematics: an educational experiment in Portuguese middle schools

This article is based on an experiment using the game ‘Caminhando e Calculando’ (Moving and Calculating) in order to analyse the potential of the game as an educational resource for the teaching and learning of mathematics in Portuguese middle schools, where most students are 10 or 11 years old. Students' data obtained during the games will be used to analyse the different options used for solving the game, identifying its potential and its weaknesses. We start with a theoretical analysis of games as an inherent element of human culture. Combining our innate desire for fun with the different types of teaching and learning styles allows for fun and knowledge to be combined into more efficient and meaningful types of knowledge. Playing games are a primordial aspect of what it means to be a child and they develop within a motivating environment; therefore, not to take advantage of games as a learning resource would be to neglect an important asset. With regard to mathematics, emphasis will be given to the advantages that this teaching and learning tool provides for certain mathematical processes, such as problem-solving.     

A new paradigm for serious games: Transmedia learning for more effective training and education

Serious games present a relatively new approach to training and education for international organizations such as NATO (North Atlantic Treaty Organization), non-governmental organizations (NGOs), the U.S. Department of Defense (DoD) and the U.S. Department of Homeland Security (DHS). Although serious games are often deployed as stand-alone solutions, they can also serve as entry points into a comprehensive training pipeline in which content is delivered via different media to rapidly scale immersive training and education for mass audiences. The present paper introduces a new paradigm for more effective and scalable training and education called transmedia learning. Transmedia learning leverages several new media trends including the peer communications of social media, the scalability of massively openonline course (MOOCs), and the design of transmedia storytelling used by entertainment, advertising, and commercial game industries to sustain audience engagement. Transmedia learning is defined as the scalable system of messages representing a narrative or core experience that unfolds from the use of multiple media, emotionally engaging learners by involving them personally in the story. In the present paper, we introduce the transmedia learning paradigm as offering more effective use of serious games for training and education. This approach is consistent with the goals of international organizations implementing approaches similar to those described by the Army Learning Model (ALM) to deliver training and education to Soldiers across multiple media. We discuss why the human brain is wired for transmedia learning and demonstrate how the Simulation Experience Design Method can be used to create transmedia learning story worlds for serious games. We describe how social media interactions and MOOCs may be used in transmedia learning, and how data mining social media and experience tracking can inform the development of computational learner models for transmedia learning campaigns. Examples of how the U.S. Army has utilized transmedia campaigns for strategic communication and game-based training are provided. Finally, we provide strategies the reader can use today to incorporate transmedia storytelling elements such as Internet, serious games, video, social media, graphic novels, machinima, blogs, and alternate reality gaming into a new paradigm for training and education: transmedia learning.     

The Level of Interpretation of Games

This paper argues that the level of detail at which the effects or experiences of a game are supposed to be interpreted is a useful classificatory criterion, and is a valuable component of a classification scheme for games according to their purpose. Research, teaching and learning games intended to be interpreted at fine levels and coarse levels of detail are discussed and compared. It is noted that coarse-level educational games may have unintentional fine-level effects, and some methods of preventing such effects are suggested. It is also noted that fine-level games require greater accuracy of game models and more realistic player behaviour than coarse-level games. The implications of this for game design are discussed.     

The Role of Teacher Stance When Infusing Inquiry Questioning Into Middle School Science Classrooms

The purpose of this study was to describe how a teacher's stance of wonder, curiosity, and exploration during the use of hypothetical inquiry situations served as a way for the teacher to address her goals for students to be more vocal members of the learning community; to encourage critical and creative thinking in the students; to provide them with meaningful, context-rich opportunities for synthesizing personal past experience with ongoing classroom instruction; to enhance socialization skills in the middle school environment; and to provide an improvisational atmosphere for both learning and teaching.     

Mathematical Level Raising Through Collaborative Investigations with the Computer

Investigations with the computer can have different functions in the mathematical learning process, such as to let students explore a subject domain, to guide the process of reinvention, or to give them the opportunity to apply what they have learned. Which function has most effect on mathematical level raising?We investigated that question in the context of developing learning materials for 16-year-old students in the domain of probability theory,consisting of computer simulations based on a gambling game and investigation tasks about these games. We compared the difference in level raising between three versions of the learning materials: investigations with the computer before, during or after the learning of a mathematical concept. It was shown that there was no significant difference in the final mathematical level that students attained in the three conditions (the product). However, there were differences in the level on which students approached the investigation tasks (the process). Furthermore,we found evidence of new categories in the students' answers, lying between the perceptual and conceptual levels, which may give important insight into the process of level raising. This revised version was published online in July 2006 with corrections to the Cover Date.     

Optimal Client-Server Configuration of Mobile Ad-Hoc Networks

Interactive online gaming is a widely successful application that is becoming more and more popular even on mobile devices. In this context, an emerging scenario is represented by having playersʼ mobile devices directly connected to each other, without resorting to the Internet. To enable this scenario, we leverage on a hybrid architecture that allows client-server games to be played in ad-hoc mode: a number of players take turn in acting also as the server of the game. However, the problem on how to choose the servers so as to maximize the duration of the network still remains an issue. To this aim, we propose an Integer Programming model and a heuristic based on it: preliminary experimental results show the effectiveness and the potential of the approach.     

Online study collaboration to improve teachers’ expertise in instructional design in mathematics

Online study collaboration is a recent professional development approach that goes beyond school and regional boundaries and even helps reach rural schools in China. In this study, we focused on a specific online study collaboration program to examine its potential benefits for improving participating teachers’ expertise in instructional design. Data were collected from the online study collaboration organizers and four main participating schools. The results reveal the program’s well-structured process and organization for planning and conducting the online study collaboration. Participating teachers benefited from their sharing and discussions with experts and other teachers. Their instructional designs show many important changes that are aligned with experts’ comments. Selected teachers’ expertise improvement includes their knowledge about the textbook and content, their perspectives about students’ learning and instruction, and their learning of different instructional approaches to engage students in classroom instruction. The use of online study collaboration for improving teachers’ expertise and the study’s limitations are then discussed.     

Understanding STEM‐focused elementary schools: Case study of Walter Bracken STEAM Academy

Increasingly, STEM focused high schools are used prepare students for college STEM majors and launch them into STEM careers. Yet a new focus on STEM education at the elementary levels suggests that the importance of STEM education is much broader than a preparation for workforce needs in high school or college. This paper describes a case study designed to articulate the mission and design of an effective and nationally recognized STEM‐focused elementary school. As described through the six most impactful components of STEM‐focused elementary school design at Walter Bracken STEAM Academy, the case study emphasizes the school's strong and inclusive school leadership, with staff organized into grade level groups empowered to innovate and honing their teaching practices. External partnerships are leveraged to broaden student learning opportunities. Students at Bracken engage in active learning opportunities and multidisciplinary lessons where STEM is used as a way of thinking and as a way to coherently combine content into active learning opportunities that are engaging for learners. By organizing the structural components of an exemplary STEM‐focused elementary school, we hope to deliver actionable reforms for elementary schools wanting to increase their STEM‐focused offerings.     

Experimental results on ultimatum games with incomplete information

This paper is about experiments on two versions of ultimatum games with incomplete information, called the offer game and the demand game. We apply the strategy method, that is, each subject had to design a complete strategy in advance instead of reacting spontaneously to a situation which occurs in the game. Game theory predicts very similar outcomes for the offer and the demand games. Our experiments, however, show significant differences in behavior between both games. Using the strategy method, allows us to explore the motivations leading to those differences. Since each subject played the same version of the game eight rounds against changing anonymous opponents we can also study subjects' learning behavior. We propose a theory of boundedly rational behavior, called the “anticipation philosophy”, which is well supported by the experimental data.     

Planning District‐Wide Professional Development: Insights Gained From Teachers and Students Regarding Mathematics Teaching in a Large Urban District

This paper describes the mechanism used to gain insights into the state of the art of mathematics instruction in a large urban district in order to design meaningful professional development for the teachers in the district. Surveys of close to 2,000 elementary, middle school, and high school students were collected in order to assess the instructional practices used in mathematics classes across the district. Students were questioned about the frequency of use of various instructional practices that support the meaningful learning of mathematics. These included practices such as problem solving, use of calculators and computers, group work, homework, discussions, and projects, among others. Responses were analyzed and comparisons were drawn between elementary and middle school students' responses and between middle school and high school responses. Finally, fifth‐grade student responses were compared to those of their teachers. Student responses indicated that they had fewer inquiry‐based experiences, fewer student‐to‐student interactions, and fewer opportunities to defend their answers and justify their thinking as they moved from elementary to middle school to high school. In the elementary grades students reported an overemphasis on the use of memorization of facts and procedures and sparse use of calculators. Results were interpreted and specific directions for professional development, as reported in this paper, were drawn from these data. The paper illustrates how student surveys can inform the design of professional development experiences for the teachers in a district.     

Handheld technology for mathematics education: flashback into the future

In the 1990s, handheld technology allowed overcoming infrastructural limitations that had hindered until then the integration of ICT in mathematics education. In this paper, we reflect on this integration of handheld technology from a personal perspective, as well as on the lessons to be learnt from it. The main lesson in our opinion concerns the growing awareness that students’ mathematical thinking is deeply affected by their work with technology in a complex and subtle way. Theories on instrumentation and orchestration make explicit this subtlety and help to design and realise technology-rich mathematics education. As a conclusion, extrapolation of these lessons to a future with mobile multi-functional handheld technology leads to the issues of connectivity and in- and out-of-school collaborative work as major issues for future research.     

不同权力结构对制造商双渠道供应链的影响研究

在一个制造商和一个零售商组成的供应链中, 制造商拥有线下传统渠道及线上直销双渠道。首先根据消费者剩余理论, 构建了制造商双渠道需求函数;接着分别构建了两类Stackelberg博弈及Nash博弈三种权力结构下制造商和零售商的利润模型并对模型进行了求解;最后分析了三种不同权力结构对制造商双渠道供应链的影响。研究发现:制造商线上直销渠道价格不受三种博弈权力结构的影响;线下传统渠道零售价格、需求, 线上直销渠道需求及制造商双渠道供应链总利润不受两类Stackelberg博弈权力结构的影响;线下传统渠道零售价格及线上直销渠道需求在Nash博弈权力结构下最小, 而线下传统渠道需求及供应链总利润在Nash博弈权力结构下最大;制造商批发价格、利润及零售商利润对三种博弈权力结构较敏感, 随供应链成员自身博弈权力地位的下降而逐渐降低。     

How students use the ‘see similar example’ feature in online mathematics homework

Homework is one of students’ opportunities to learn mathematics, but we know little about what students learn from homework. This study employs the instructional triangle and didactic contract to explore how students used the ‘see similar example’ feature in an online homework platform and how that use reflected their learning goals. Findings indicate students used similar examples to troubleshoot, to check if they were on the right track, and to see the form of the answer. Students also sought to unpack the reasoning in solution steps, used solutions as templates for solving their own problems, and sometimes copied answers. One student did a ‘see similar example’ problem for more practice. Students’ goals included completing the homework, maximizing their score, and understanding the content. This research lays groundwork for future work characterizing what students learn from homework and how features that provide students with similar examples help or hinder their learning.     

The impact of the termination rule on cooperation in a prisoner’s dilemma experiment

Cooperation in prisoner’s dilemma games can usually be sustained only if the game has an infinite horizon. We analyze to what extent the theoretically crucial distinction of finite versus infinite-horizon games is reflected in the outcomes of a prisoner’s dilemma experiment. We compare three different experimental termination rules in four treatments: a known finite end, an unknown end, and two variants with a random termination rule (with a high and with a low continuation probability, where cooperation can occur in a subgame-perfect equilibrium only with the high probability). We find that the termination rules do not significantly affect average cooperation rates. Specifically, employing a random termination rule does not cause significantly more cooperation compared to a known finite horizon, and the continuation probability does not significantly affect average cooperation rates either. However, the termination rules may influence cooperation over time and end-game behavior. Further, the (expected) length of the game significantly increases cooperation rates. The results suggest that subjects may need at least some learning opportunities (like repetitions of the supergame) before significant backward induction arguments in finitely repeated game have force.     

Examining the didactic contract when handheld technology is permitted in the mathematics classroom

The use of mathematics analysis software (MAS) including handheld scientific and graphics calculators offers a range of pedagogical opportunities. Its use can support change in the didactic contract. MAS may become an alternative source of authority in the classroom empowering students to explore variation and regularity, manipulate simulations and link representations. Strategic use may support students to direct their own learning and explore mathematics, equipping them to share their findings with the teacher and the class with more confidence. This paper offers a framework for examining the impact of the use of MAS on the didactic contract. Lessons were observed in 12 grade 10 classes, with 12 different teachers new to MAS. MAS technology was used with a variety of didactic contracts, mostly traditional. The framework drew attention to many ways in which the teaching differed. Analysis of the didactic contract must consider both the teaching of mathematics and of technology skills, because these have different characteristics. In all classes, both teachers and students saw the teacher as having a responsibility to teach technology skills. Students saw technology skills as the main point of the lesson, but the teachers saw the lesson as primarily teaching mathematics—one of the mismatches which may need negotiation to adapt didactic contracts to teaching with MAS.     

Totally balanced combinatorial optimization games

Combinatorial optimization games deal with cooperative games for which the value of every subset of players is obtained by solving a combinatorial optimization problem on the resources collectively owned by this subset. A solution of the game is in the core if no subset of players is able to gain advantage by breaking away from this collective decision of all players. The game is totally balanced if and only if the core is non-empty for every induced subgame of it.?We study the total balancedness of several combinatorial optimization games in this paper. For a class of the partition game [5], we have a complete characterization for the total balancedness. For the packing and covering games [3], we completely clarify the relationship between the related primal/dual linear programs for the corresponding games to be totally balanced. Our work opens up the question of fully characterizing the combinatorial structures of totally balanced packing and covering games, for which we present some interesting examples: the totally balanced matching, vertex cover, and minimum coloring games. Received: November 5, 1998 / Accepted: September 8, 1999?Published online February 23, 2000     

Handheld calculators between instrument and document

The new generations of handheld calculators can be considered either as mathematical tools with opportunities for calculation and representation or as a part of the teachers’ and students’ sets of resources. Framed by the Theory of Didactical Situations and the documentational approach, we take advantage of a particular experiment on introducing complex calculators in scientific classes to investigate the position and the role of this handheld technology both for students and teachers. The results show how different functionalities can be shared among teachers and students, but also how other functionalities remain private and may even conflict with the teacher’s intentions.     

An Exploration into How Physical Activity Data-Recording Devices Could be Used in Computer-Supported Data Investigations

There is a great potential opportunity to use portable physical activity monitoring devices as data collection tools for educational purposes. Using one such device, we designed and implemented a weeklong workshop with high school students to test the utility of such technology. During that intervention, students performed data investigations of physical activity that culminated in the design and implementation of their own studies. In this paper, we explore some of the mathematical thinking that took place through a series of vignettes of a pair of students engaged in analyzing some of their own activity data. A personal connection to the data appeared to aid these students in recognizing their own errors, and ultimately helped them move from a point-based analytical approach for making sense of the data to an aggregate one. From our observations of this designed learning experience, we conclude that physical activity data recording devices can afford students the opportunity to reason with personally relevant data in meaningful ways.     

Supporting Middle School Teachers’ Implementation of STEM Design Challenges

We describe and analyze a professional development (PD) model that involved a partnership among science, mathematics and education university faculty, science and mathematics coordinators, and middle school administrators, teachers, and students. The overarching project goal involved the implementation of interdisciplinary STEM Design Challenges (DCs). The PD model targeted: (a) increasing teachers’ content and pedagogical content knowledge in mathematics and science; (b) helping teachers integrate STEM practices into their lessons; and (c) addressing teachers’ beliefs about engaging underperforming students in challenging problems. A unique aspect involved low‐achieving students and their teachers learning alongside each other as they co‐participated in STEM design challenges for one week in the summer. Our analysis focused on what teachers came to value about STEM DCs, and the challenges in and affordances for implementing DCs. Two significant areas of value for the teachers were students’ use of scientific, mathematical, and engineering practices and motivation, engagement, and empowerment by all learners. Challenges associated with pedagogy, curriculum, and the traditional structures of the schools were identified. Finally, there were four key affordances: (a) opportunities to construct a vision of STEM education; (b) motivation to implement DCs; (c) ambitious pedagogical tools; and, (d) ongoing support for planning and implementation. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.