Developing a cognitive model of solid geometry based on Interpretive Structural Modeling method

With the advancement of science and artificial intelligence, education is experiencing significant innovation. The adaptive learning system is emerging as a promising approach to achieving personalized learning. The cognitive model plays a crucial role as the fundamental rationale behind the adaptive learning system. Currently, there is no uniform and highly operational method for constructing cognitive models. This study adopts Interpretive Structural Modeling (ISM) as the foundational approach for constructing a cognitive model of solid geometry. Based on literature and expert opinions, 17 cognitive attributes of high school solid geometry were identified. Subsequently, a questionnaire survey involving 40 experts was conducted to establish the contextual relationships among these attributes. Applying the ISM method resulted in a seven-level model. This model was then revised based on expert opinions to create the final cognitive model, revealing three primary paths within the domain of high school solid geometry. This paper contends that the use of the ISM method for constructing cognitive models is effective and objective. The resulting cognitive model unveils the content structure of high school solid geometry, and provides an innovative perspective on the construction of cognitive models.

A review of learning analytics opportunities and challenges for K-12 education

Use of learning analytics to improve educational outcomes is a growing area of research. While learning analytics research has been more prevalent in higher education contexts, this study presents findings from a qualitative metasynthesis of 47 research publications related to opportunities and challenges relevant to learning analytics design, implementation, and research at the PK-12 level. Our findings indicate that, while many see the educational benefits of learning analytics (e.g., more equitable instruction, individualized learning, enhanced assessment for learning); others remain unconvinced by a lack of evidence of improved outcomes and concerned about persistent challenges and potentially harmful impacts (e.g., infringement on users' privacy, misuse or misinterpretation of data by educators). We conclude by considering implications for mathematics education stemming from our analysis of learning analytics and posing questions to shape future research and key developments in mathematics education as learning analytics become more prevalent.

Analyzing teacher-student interactions through graph theory applied to hyperscanning fNIRS data

Teacher-student relationships have been found consistently important for student school effectiveness in mathematics in the last three decades. Although this observation is generally made from the teacher's perspective, neuroscience can provide new insights by establishing the neurobiological underpinning of social interactions. This paper further develops this line of research by utilizing graph theory to represent interactions between teachers and students at the neural level. Through hyperscanning with functional near-infrared spectroscopy (fNIRS), we collected data from the prefrontal cortex and the temporoparietal junction of 24 dyads composed of a teacher and a student. Each dyad used a board game to perform a programming logic class that consisted of three steps: independent activities (control), presentation of concepts, and interactive exercises. Graph theory provides results regarding the strength of teacher-student interaction and the main channels involved in these interactions. We combined graph modularity and bootstrap to measure pair coactivation, thus establishing the strength of teacher-student interaction. Also, graph centrality detects the main brain channels involved during this interaction. In general, the teacher's most relevant nodes rely on the regions related to language and number processing, spatial cognition, and attention. Also, the students' most relevant nodes rely on the regions related to task management.

Exploring the potential of eye tracking on personalized learning and real-time feedback in modern education

Eye tracking is one of the techniques used to investigate cognitive mechanisms involved in the school context, such as joint attention and visual perception. Eye tracker has portability, straightforward application, cost-effectiveness, and infant-friendly neuroimaging measures of cognitive processes such as attention, engagement, and learning. Furthermore, the ongoing software enhancements coupled with the implementation of artificial intelligence algorithms have improved the precision of collecting eye movement data and simplified the calibration process. These characteristics make it plausible to consider eye-tracking technology a promising tool to assist the teaching-learning process in school routines. However, eye tracking needs to be explored more as an educational instrument for real-time classroom activities and teachers' feedback. This perspective article briefly presents the fundamentals of the eye-tracking technique and four illustrative examples of employing this method in everyday school life. The first application shows how eye tracker information may contribute to teacher assessment of students' computational thinking in coding classes. In the second and third illustrations, we discuss the additional information provided by the eye-tracker to the teacher assessing the student's strategies to solve fraction problems and chart interpretation. The last illustration demonstrates the potential of eye tracking to provide Real-time feedback on learning difficulties/disabilities. Thus, we highlight the potential of the eye tracker as a complementary tool to promote personalized education and discuss future perspectives. In conclusion, we suggest that an eye-tracking system could be helpful by providing real-time student gaze leading to immediate teacher interventions and metacognition strategies.

Decoding of spatial proportions using somatosensory feedback in sighted and visually impaired children

BACKGROUND AND PURPOSE

Humans can naturally operate with ratios of continuous magnitudes (proportions). We asked if sighted children (S) and visually impaired children (VI) can discriminate proportions via somatosensory feedback.

PROCEDURES

Children formed a proportion by tracing a pair of straight lines with their finger, and compared this proportion with a second proportion resulting from the tracing of another pair of lines.

MAIN FINDINGS

Performance was 68% in S, thus significantly lower (p < 0.001) compared to VI (75%). Tracing velocity (p < 0.01) and trial-to-trial variability of tracing velocity (p < 0.05) was higher in S compared to VI.

CONCLUSIONS

Operating with proportions solely from somatosensory feedback is possible, thus tracing lines might support learning in mathematics education. Kinematic variables point to the reason for the difference between S and VI, in that higher trial-to-trial variability in velocity in S leads to biased estimation of absolute line lengths.

Teachers' instructional responses to the COVID-19 pandemic

In response to the COVID-19 pandemic, schools transitioned to Emergency Remote Teaching (ERT). In May 2020, as part of an existing study of flipped Algebra instruction, we interviewed eleven Missouri teachers to understand how their instruction changed as they moved to ERT. Drawing on practical rationality, we found the pandemic led to a breach of norms (assigning grades, synchronous meeting times, delivering new content, and students' completion of work). Findings highlight the interconnected nature of norms and reveal differences in teachers' responses to the breach of norms. We found administrative policies, particularly around grading, significantly impact teachers' decisions during ERT.

Subtraction by addition in young multi-digit subtraction learners: A choice/no-choice study

When solving subtraction problems such as 83-46, children use the direct subtraction (DS) strategy (e.g., 83 - 40 = 43, 43 - 6 = 37) or the subtraction by addition (SBA) strategy (e.g., 46 + 4 = 50, 50 + 30 = 80, 80 + 3 = 83, so the answer is 4 + 30 + 3 = 37). This study is the first to use the choice/no-choice method to examine DS and SBA use in third-graders (8- and 9-year-olds) with varying mathematical achievement levels. All children (N = 66) solved a series of small difference subtractions (e.g., 72 - 64) and large difference subtractions (e.g., 94 - 8) in one choice condition (choice between DS and SBA) and two no-choice conditions (obligatory use of either DS or SBA). Results showed that, although only the DS strategy was taught to these children, in the choice condition nearly half the children already made use of the SBA strategy and that SBA was used on one in five subtractions. Whereas DS was the fastest strategy on large difference items, interestingly, children achieved a similar level of accuracy with the SBA strategy compared with DS. Finally, 1 in 5 children made problem-based adaptive strategy choices, and children were generally adaptive to their individual strategy speed. This study clearly demonstrates the nascent use of SBA in third-graders and provides evidence for the plea to introduce SBA as an alternative to DS already in the lower grades of elementary school.

Online physically active academic lessons in COVID-19 times: A pilot study

Schools play an important role in promoting physical activity among students. This paper studies the perception of educators, students, and parents about the use of online physically active academic lessons during COVID-19 in the north of Chile. Starting the first week of November 2020, and for a period of five weeks, 323 students, alongside 11 educators, practiced a geometry-based dance routine online. The qualitative analysis results reveal a positive perception of the experience and an increase in physical activity without reducing the amount of time spent on academic activities. There were also improvements in learning, social relationships, and enjoyment.

Do You Need the Machine? Tipping in Canada Is Unconscious (Part II)

While recovering from a major personal tipping point (see Part I), I was still able to keep on the lookout for Canadian mathematics education matters. After all, if Canadian mathematics education matters, Canadian mathematics education matters. In doing so, I ran into a number of other financial problems. Everywhere I turned was a financial problem: from tipping in the sharing (or platform) economy; to spending your way to savings with credit cards; the proliferation of sportsbooks and online casinos; trying to reconcile the Canadian cost of living with the seemingly high accepted standard level of consumption; and the outrageous fee to take $20 out of my very own bank account. Each taken on their own, I clearly have some financial problems. Taken together, I contend that 'Egan's Financial Problems', albeit unconventional, could, one day, be the impetus for financial education and literacy leaving math class and becoming a class of its own in Canadian schools. Until then, I guess we go with the School of Hard Knocks for our financial education.

The Effects of Mobile Technology on Learning Performance and Motivation in Mathematics Education

Due to rapid developments, mobile technologies started to play an essential role in designing seamless learning environments. Due to the availability of mobile technologies, students can access learning materials without being bound by time and place. On the grounds that these applications allow information exchange, time and space limitations such as classrooms or school bells have been eliminated. Therefore, this study aims to assess mobile-assisted seamless learning environments' effects on students' success and motivation in the secondary school 7th grade mathematics class algebra unit and student opinions about the application. The research is designed using the descriptive pattern of mixed-method research. The sample of the study is 73 middle school students (30 male and 43 female) in Turkey. Augmented Reality (AR) applications developed in teaching algebra to support individual learning and to utilize mobile technologies, WhatsApp groups were created. Algebra Achievement Test (AAT), Mathematics Motivation Scale (MMS), and semi-structured interview forms were used as data collection tools in the research. The results of the study showed that there were statistically significant differences in favor of the experiment group in AAT and MMS scores. However, no significant difference was found between the groups in intrinsic goal orientation and test anxiety scores, which are motivation sub-dimensions. The findings obtained from AAT, MMS, and the students' opinions showed that mobile technology applications used in out-of-school learning environments positively affect the learning process.

Mentoring Undergraduate Research in Mathematical Modeling

In writing about undergraduate research in mathematical modeling, I draw on my 31 years as a mathematics professor at the University of Nebraska-Lincoln, where I mentored students in honors' theses, REU groups, and research done in a classroom setting, as well as my prior experience. I share my views on the differences between research at the undergraduate and professional levels, offer advice for undergraduate mentoring, provide suggestions for a variety of ways that students can disseminate their research, offer some thoughts on mathematical modeling and how to explain it to undergraduates, and discuss the challenges involved in broadening research participation to include early career students and mid-tier students and how to deliver a research experience in a classroom setting. While different situations pose different challenges, different problems require different approaches, and different experiences lead to different conclusions, it is my hope that my experiences will be of broad value to a wide audience.

Financial Numeracy in Mathematics Education: Research and Practice

Financial Numeracy in Mathematics Education: Research and Practice is a book that brings unique and innovative ideas for using financial numeracy in learning mathematics in the classroom. The purpose of this book is to describe the different ways to integrate financial numeracy into mathematics classrooms. Empirical and conceptual studies related to the application of financial numeracy in learning are successfully discussed in this volume. This is perhaps the first book to comprehensively cover the theory and practice of financial numeracy in mathematics instruction. This book is ideal for instructors, lecturers, researchers, stakeholders, and anybody else interested in financial numeracy.

Welcome to the era of vague news: a study of the demands of statistical and mathematical products in the COVID-19 pandemic media

In this article, we report on a typology of the demands of statistical and mathematical products (StaMPs) embedded in media items related to the COVID-19 (coronavirus) pandemic. The typology emerged from a content analysis of a large purposive sample of diverse media items selected from digital news sources based in four countries. The findings encompass nine categories of StaMPs: (1) descriptive quantitative information, (2) models, predictions, causality and risk, (3) representations and displays, (4) data quality and strength of evidence, (5) demographics and comparative thinking, (6) heterogeneity and contextual factors, (7) literacy and language demands, (8) multiple information sources, and (9) critical demands. We illustrate these categories via selected media items, substantiate them through relevant research literature, and point to categories that encompass new or enhanced types of demands. Our findings offer insights into the rich set of capabilities that citizens (including both young people and adults) must possess in order to engage these mass media demands, critically analyze statistical and mathematical information in the media, evaluate the meaning and credibility of news reports, understand public policies, and make evidenced-informed judgments. Our conclusions point to the need to revise current curricular frameworks and conceptual models (e.g., regarding statistical and probability literacy, adult numeracy), to better incorporate notions such as blended knowledge, vagueness, risk, strength of evidence, and criticality. Furthermore, more attention is needed to the literacy and language demands of media items involving statistical and mathematical information. Implications for further research and educational practice are discussed.

Do You Need 'The Machine'? Tipping in Canada Is Unconscious (Part I)

Constantly on the lookout for Canadian mathematics education matters, I recently experienced a major personal tipping point. The juxtaposition of two different customer service situations was simply too much for me to handle. Now through the looking glass, it was abundantly clear that tipping in Canada is unconscious, and the evidence was everywhere. The current state of financial literacy education in Canadian schools, the opportunity that COVID-19 has provided for us to renew Canada's implied gratuity guidelines, and an investigation into pre- and post-tax bill totals all supported my assertion that the tipping culture in Canada is a habit in many senses of the word. A look back at how tipping in Canada has evolved from parting with a few coins every once and a while, and a look at the evolution of the point of sale terminal, which I refer to as 'The Machine', helped me realize that I am unable to move on and start looking for other Canadian mathematics education matters just yet. After all, if Canadian mathematics education matters, then Canadian mathematics education matters. As such, Part II of this article follows in the next issue. Stay tuned.

Examining How Emergency Remote Teaching Influenced Mathematics Teaching

The COVID-19 pandemic forced teachers worldwide to shift to emergency remote teaching (i.e., virtual teaching). As teachers return to their classrooms for in-person teaching, there is a need to examine how remote teaching influences teachers' instruction. This study examined teachers' use of digital technologies and specific mathematics activities both during remote teaching and during in-person teaching after returning to their classrooms. The study also examined how teacher participants reported how the pandemic influenced their mathematics teaching. Data analysis indicated statistically significant differences in the frequency of use of all digital technologies except for mathematics games, meaning that mathematics games are used now as much during in-person teaching as remote teaching. Teacher participants also reported that the largest influences of the pandemic and remote teaching have had on their in-person mathematics teaching was the use of general, non-mathematics specific technologies to support organization, the use of hands-on or virtual manipulatives, and the benefit of formative assessment. Implications for future research include the need to examine teachers' use of digital technologies and mathematics activities more closely during in-person teaching and leverage interviews as a possible way to more closely study teachers' experiences.

Future teachers' perception of the usefulness of SketchUp for understanding the space and geometry domain

This article reflects the opinion of future Early Childhood Education teachers at the Universidad de Cádiz on the usefulness and degree of satisfaction of SketchUp, a 3D modelling software programme, after they participated in a workshop for didactic-mathematical training. They had to use the software to design and model their ideal nursery school in 3D, supported by clearly stated and well-defined educational pillars. This study aims to ascertain the students' perceptions of the use of this resource with the intention of assessing its suitability to offer more appropriate initial training regarding mathematics education. It seeks to make the most of using the software programme and minimise the obstacles encountered. Opinions were collected from a sample of 203 students who responded to two questionnaires designed ad hoc. The results are organised around a SWOT analysis and show a satisfactory global evaluation.

Health and pathology: a brief history of the biopolitics of US mathematics education

Concerns about health and disease have long pervaded mathematics education research, yet their implications have been underappreciated. This article focuses on three contemporary relationships amplified by the COVID-19 pandemic: (1) school mathematics and national health, (2) mathematics educators' roles in distinguishing the health needs of students, and (3) mathematics instruction as either enhancing or threatening students' mental health and social adjustment. We argue that these concerns are foundational preoccupations of mathematics education research that have persistently shaped debates over who should learn mathematics, how, and to what ends. Our study examines histories of school mathematics and health discourses to explore how particular notions of health entered US mathematics education during the 19th and early twentieth centuries in ways that resonate with recent research trends and responses to COVID-19. We especially attend to how health/pathology distinctions reconfigured hierarchies of nationality, sex, race, and dis/ability within exclusionary, segregated, colonial, and tracked mathematics instruction. By mapping some of the shifting contours of health and pathology over time, we emphasize the potential dangers of the pandemic reanimating long-circulating dividing practices, such as in emerging trends comparing national metrics of well-being, responding to perceived trauma with differentiated instruction, and seeking to calibrate healthy mathematics identities in marginalized groups.

Distance mathematics teaching in Flanders, Germany, and the Netherlands during COVID-19 lockdown

The COVID-19 pandemic has confronted mathematics teachers with the challenge of developing alternative teaching practices-in many cases at a distance through digital technology-because schools were closed. To investigate what distance practices in secondary mathematics education have emerged and how teachers experienced them, we set out online questionnaires in Flanders-the Dutch-speaking part of Belgium-, Germany, and the Netherlands. The questionnaire focused on teaching practices, teacher beliefs, didactics, and assessment. Data consisted of completed questionnaires by 1719 mathematics teachers. Results show that the use of video conferencing tools increased massively, while the use of mathematics-specific tools that teachers used before the lockdown reduced substantially. Further findings are that teachers' confidence in using digital technologies increased remarkably during the lockdown and that their experiences and beliefs only marginally impacted their distance learning practices. Also, we observed some differences between the three countries that might be explained by differences in educational policies and in technological facilities and support. For future research, it would be relevant to investigate long-term changes in teachers' practices, as well as students' views and experiences related to the teacher's practices.

Learning number patterns through computational thinking activities: A Rasch model analysis

Despite the increasing presence of computational thinking (CT) in the mathematics context, the connection between CT and mathematics in a practical classroom context is an important area for further research. This study intends to investigate the impact of CT activities in the topic of number patterns on the learning performance of secondary students in Singapore. The Rasch model analysis was employed to assess differences of ability between students from the experimental group and control group. 106 Secondary One students (age 13 years old) from a secondary school in Singapore took part in this study. A quasi-experimental non-equivalent groups design was utilized where 70 students were assigned into the experimental group, and 36 students were assigned into the control group. The experimental group was given intervention with CT-infused activities both on- and off-computer, while the control group received no such intervention. Both groups were administered the pretest before the intervention and the posttest after the intervention. The data gathered were analyzed using the partial credit version of the Rasch model. Analysis of pretest and posttest results revealed that the performance of the experimental group was similar to the control group. The findings did not support the hypothesis that integrating CT in lessons can result in improved mathematics learning. However, the drastic improvement was observed in individual students from the experimental group, while there is no obvious or extreme improvement for the students from the control group. This study provides some new empirical evidence and practical contributions to the infusion of CT practices in the mathematics classroom.

Navigating (and Disrupting) the Digital Divide: Urban Teachers' Perspectives on Secondary Mathematics Instruction During COVID-19

This study examines the perspectives and lived experiences of 10 urban secondary mathematics teachers from two epicenters of COVID-19 in the United States regarding their transition to digital learning during the 2019–2020 academic year. We use case study methodology with phenomenological interviews to gather insights into the teachers’ efforts to modify their mathematics instruction and curriculum while navigating observed digital inequities and new digital tools for mathematics teaching. We also report on the teachers’ targeted attempts to bridge home and school while problematizing the threatened humanistic aspect of remote teaching and learning. These frontline experiences recognize technology-associated systemic inequities in marginalized, urban communities and the need to strategize ways to implement equity-oriented technology integration that benefits all learners, especially urban youth. By critically examining digital education in the urban context, crucial conversations can transpire that critique (and disrupt) the digital divide in mathematics education and open doors for other stakeholders to broadly discuss the logistics and implications of digital education to enhance new ways of teaching and learning.

Students' experiences with remote learning during the COVID-19 school closure: implications for mathematics education

This paper reports the findings of a descriptive survey research that explored secondary school students' experiences with mathematics remote learning during the Corona Virus Disease 2019 (COVID-19) school closure. The study involved 367 students of ages 13 to 21 selected from six secondary schools in Kitwe district of Zambia using the cluster random sampling method. Using a mixed-methods research approach, quantitative and qualitative data were merged to provide a comprehensive analysis of the main findings in the context of the existing literature, the government's response to COVID-19 school closure, and the challenges associated with remote learning during that time. Research findings show that more than 56% of the respondents did not have sufficient access to Information and Communication Technologies (ICT), electricity, and internet services. Most of these respondents also held a belief that mathematics is a subject that is best learned with face-to-face interactions between the teacher and students, and among students. These results suggest a need for the education systems in Zambia and other similar contexts to put up infrastructure that supports the blended and online learning models during and after the COVID-19 pandemic.

Implementing GeoGebra integrated with multi-teaching approaches guided by the APOS theory to enhance students' conceptual understanding of limit in Ethiopian Universities

The notion of limit is one of the fundamental concepts which underpins advanced calculus of one or more variables in the field of analysis. However, understanding the concept of limit has been an impenetrable problem for many students in Ethiopian Universities. Only very few literatures were documented focusing on overcoming the difficulty of learning the concept of limit. For this reason, the overarching aim of the present study is to enhance students' conceptual understanding of limit by empowering their visualization skills using GeoGebra integrated with multi-teaching approaches. The study employed mixed methods experimental (intervention) design within an APOS paradigm. Both qualitative and quantitative data were collected. Qualitative data was collected using students' reflections and interviews, whereas quantitative data was collected through pretest and posttest using diagnostic tests. The results of the qualitative data analysis revealed that the learning milieu created a positive impact on students' understanding of the concept of limit. Additionally, students provided coherent and viable reasons while making mental constructions and their coordination in the learning process based on the genetic decomposition grounded in APOS theory. Furthermore, the results of the quantitative (posttest) data analysis proved that students' mean scores on conceptual understanding of limit in the experimental group was significantly better than those in the control group. Thus, it could be possible to conclude that students’ conceptual understanding of limit is improved using GeoGebra integrated with multi-teaching approaches within an APOS paradigm. The findings open a great opportunity to suggest technology integrated mathematics curriculums for the teaching and learning of mathematics.

The Mandate of Scholarly Mathematics Education Research: Moving Ourselves Forward

The 20th anniversary of this journal finds us at a crossroads in many ways. Mathematics education itself has always existed at an unclear intersection of mathematics and education. As well, in mathematics education, the poles of reform and traditional learning can pull the community apart. And now, in the world, we find ourselves at a crossroads in terms of priorities related to the future of the planet. In this article, I reflect on past work, and theorise about the future role of mathematics education research, given these new realities, in moving forward.

COVID-19 and the use of digital technology in mathematics education

Once the COVID-19 crisis is over, will everything" return to normal" or will we instead witness an ongoing boom in online learning? A time of crisis is an opportunity for all education systems to look to the future; there is enormous potential for digital technology in mathematics education, regardless of the impact of COVID-19. In this paper, the researcher focuses on answering two research questions: (1) Is COVID-19 the gateway for digital learning in mathematics education? (2) What type of digital technology is being used in mathematics education during the COVID-19 pandemic? The study also provided a discussion on the implications that such digital technologies could have on research into the field of mathematics education and practice in addition to suggestions for future research directions on this topic. Interviews were chosen as techniques for the purpose of this research, which were undertaken with hundred and twenty mathematics teachers from different secondary schools in the Kingdom of Saudi Arabia. The researcher found that 98% of participants believed that COVID-19 is the gateway for digital learning in mathematics education. In addition, 97% claimed that the use of online education by schools had expanded greatly following the coronavirus outbreak. This has resulted in various forms of software being used to facilitate communicate between teachers and students included mobile technologies, touchscreens and pen tablets, digital library and designing learning objects in mathematics education, Massive Open Online Courses (MOOCs) in mathematics, and computer algebra systems (CAS) such as Mathematical, Maple, MuPAD, MathCAD, Derive and Maxima.

Professional Growth and Identity Development of STEM Teacher Educators in a Community of Practice

Quality STEM teacher education is predicated on teacher educators who are well-equipped to design learning experiences, provide feedback, guide the development of teachers across their career span, and conduct rigorous research to advance education theory and praxis. While numerous models and approaches to professional development for teachers exist, few parallels can be drawn between the professional development of teachers and teacher educators (Loughran, 2014). To support the multi-faceted identity (trans)formation of STEM teacher educators, self-directed learning opportunities can help bridge knowledge and practice, enhance productive collaboration, and support efforts to negotiate multiple and conflicting agendas (Goodwin & Kosnik, 2013). The purpose of this empirical study was to explore the identity (trans)formation of teacher educators participating in a long-term interdisciplinary STEM-based Community of Practice (CoP; Wenger, 1998), which began in 2012. An analysis of our experiences through the figured worlds lens informs how a CoP can impact curricular approaches and teacher PD, imploring members to move through their comfort zones into innovative spaces. We conclude with suggestions for our STEM teacher educator colleagues who seek opportunities to challenge their own positions and best support preservice and in-service STEM teachers in a way that allows them to model for their students the value of community.

The Case for Algebraic Biology: from Research to Education

Though it goes without saying that linear algebra is fundamental to mathematical biology, polynomial algebra is less visible. In this article, we will give a brief tour of four diverse biological problems where multivariate polynomials play a central role-a subfield that is sometimes called algebraic biology. Namely, these topics include biochemical reaction networks, Boolean models of gene regulatory networks, algebraic statistics and genomics, and place fields in neuroscience. After that, we will summarize the history of discrete and algebraic structures in mathematical biology, from their early appearances in the late 1960s to the current day. Finally, we will discuss the role of algebraic biology in the modern classroom and curriculum, including resources in the literature and relevant software. Our goal is to make this article widely accessible, reaching the mathematical biologist who knows no algebra, the algebraist who knows no biology, and especially the interested student who is curious about the synergy between these two seemingly unrelated fields.

The Disappearing "Advantage of Abstract Examples in Learning Math"

When teaching a novel mathematical concept, should we present learners with abstract or concrete examples? In this experiment, we conduct a critical replication and extension of a well-known study that argued for the general advantage of abstract examples (Kaminski, Sloutsky, & Heckler, 2008a). We demonstrate that theoretically motivated yet minor modifications of the learning design put this argument in question. A key finding from this study is that participants who trained with improved concrete examples performed as well as, or better than, participants who trained with abstract examples. We argue that the previously reported "advantage of abstract examples" manifested not because abstract examples are advantageous in general, but because the concrete condition employed suboptimal examples.

Elementary students' perceptions of 3Dmetric: A cross-sectional study

Rapid changes in the 21st century demand the use of technology in learning geometry in elementary schools. One such technology is augmented reality (AR). 3Dmetric (3D and Geometric) is a geometry learning medium on AR-based 3D space material. Students' perceptions, which refer to their interpretation, are a key factor in studying the changes in their interpretations of a particular phenomenon. The purpose of the current study was to investigate the perceptions of elementary school students after using 3Dmetric to learn geometric shapes. The differences and the relationship between the students’ level of perception and level of spatial ability were also investigated. This study applied a cross-sectional approach with quantitative and qualitative designs. A total of 36 students in one elementary school in Indonesia participated in this study. The instruments used were the Perception Scale for Using 3Dmetric in Geometry Teaching, Spatial Ability Scale, and In-Depth Interview Form. Results showed that the positive perception of elementary school students regarding the use of 3Dmetric does not depend on the level of their spatial ability. Moreover, the difference in their perceptions is not caused by the level of their spatial ability. The positive findings in this cross-sectional study can contribute to the success of AR-based learning and teaching in the 21st century, especially with regard to learning materials for 3D geometry. They can also lead to the formation of the spatial abilities and improvement in the academic performance of elementary school students.

"It gives you that sense of hope": An exploration of technology use to mediate student engagement with mathematics

Despite the predicted need for a more mathematically capable workforce, the proportion of students undertaking advanced mathematics courses in Australia and other comparable countries has stagnated or fallen, in part due to a lack of student engagement with mathematics in school. In society in general, technology use is commonplace, leading some educators to speculate that technology use for the teaching and learning of mathematics can improve student engagement. In this paper, using multiple case studies, we examine how teachers (n = 10), recognised by their peers as exemplary users of technology, take advantage of technological affordances to optimise student engagement with mathematics. Data was collected from three participant groups: Teachers, Leaders (n = 10), and student focus groups (n = 6). We examine both student and teacher perspectives, through the lens of the Framework for Engagement with Mathematics (FEM), to tease out the ways in which exemplary teachers use technology to enhance pedagogical relationships with students and their pedagogical repertoires. We find that the teachers and students reported evidence of all elements of the FEM, but to differing degrees. In particular, we identified that teachers used technological tools to enhance teacher awareness of individual student learning needs and to promote student-centred pedagogies leading to greater student engagement with mathematics. We contend that a greater awareness of the nuanced pedagogical affordances of a range of technological tools could lead teachers toward practices that enhance student engagement with mathematics, leading to an increase in students wishing to extend their mathematical knowledge beyond the compulsory school years.

Numeracy, adult education, and vulnerable adults: a critical view of a neglected field

This survey paper examines selected issues related to the intersection of three broad scholarly areas: numeracy, adult education, and vulnerability. Numeracy encompasses the ways in which people cope with the mathematical, quantitative, and statistical demands of adult life, and is viewed as an important outcome of schooling and as a foundational skill for all adults. The focus on vulnerability stems from the realization that concerns of policy makers and educators alike often center on populations seen as vulnerable. The paper is organized in five sections. After a brief introduction, Section 2 examines adult numeracy, focusing on five numeracy domains (health, financial, digital, civic, and workplace numeracy), literacy-numeracy linkages, functional and critical aspects of numeracy, and the centrality of numeracy practices, and notes sources of vulnerability for each of these. Section 3 sketches formal, non-formal and informal contexts in which adults learn or develop their numeracy, and examines factors that may be potential sources of vulnerability, including systemic factors and dispositional and affect factors. Section 4 reflects more broadly on the concept of vulnerability, introduces selected aspects of the papers published in this issue of ZDM Mathematics Education, and points to findings regarding adult learners who may be deemed vulnerable. The closing section summarizes conclusions and research directions regarding the intersection of the three core domains. Overall, the paper points to emerging research needs and educational challenges that are relevant to scholars, practitioners, and policy makers interested in developing the numeracy of adults as well as in the mathematics education of younger learners.

Eye-tracking data and mathematical tasks with focus on mathematical reasoning

This data article contains eye-tracking data (i.e., dwell time and fixations), Z-transformed cognitive data (i.e., Raven's Advanced Progressive Matrices and Operation span), and practice and test scores from a study in mathematics education. This data is provided in a supplementary file. The method section describes the mathematics tasks used in the study. These mathematics tasks are of two kinds, with and without solution templates, to induce different types of mathematical reasoning.

An outlook on self-assessment of homework assignments in higher mathematics education

BACKGROUND

We discuss first experiences with a new variant of self-assessment in higher mathematics education. In our setting, the students of the course have to mark a part of their homework assignments themselves and they receive the corresponding credit without that any later changes are carried out by the teacher. In this way, we seek to correct the imbalance between student-centered learning arrangements and assessment concepts that keep the privilege to grade (or mark) completely with the teacher.

RESULTS

We present results in the form of student feedback from a course on functional analysis for third- and fourth-year students. Moreover, we analyze marking results from two courses on real analysis. Here, we compare tasks marked by the teacher and tasks marked by the students.

CONCLUSIONS

Our experiments indicate that students can benefit from self-assessment tasks. The success depends, however, on many different factors. Promising for self-assessment seem to be small learning groups and tasks in which a priori weaker students can catch up with stronger students by increasing their practising time.

SKOPE-IT (Shareable Knowledge Objects as Portable Intelligent Tutors): overlaying natural language tutoring on an adaptive learning system for mathematics

BACKGROUND

This study investigated learning outcomes and user perceptions from interactions with a hybrid intelligent tutoring system created by combining the AutoTutor conversational tutoring system with the Assessment and Learning in Knowledge Spaces (ALEKS) adaptive learning system for mathematics. This hybrid intelligent tutoring system (ITS) uses a service-oriented architecture to combine these two web-based systems. Self-explanation tutoring dialogs were used to talk students through step-by-step worked examples to algebra problems. These worked examples presented an isomorphic problem to the preceding algebra problem that the student could not solve in the adaptive learning system.

RESULTS

Due to crossover issues between conditions, experimental versus control condition assignment did not show significant differences in learning gains. However, strong dose-dependent learning gains were observed that could not be otherwise explained by either initial mastery or time-on-task. User perceptions of the dialog-based tutoring were mixed, and survey results indicate that this may be due to the pacing of dialog-based tutoring using voice, students judging the agents based on their own performance (i.e., the quality of their answers to agent questions), and the students' expectations about mathematics pedagogy (i.e., expecting to solving problems rather than talking about concepts). Across all users, learning was most strongly influenced by time spent studying, which correlated with students' self-reported tendencies toward effort avoidance, effective study habits, and beliefs about their ability to improve in mathematics with effort.

CONCLUSIONS

Integrating multiple adaptive tutoring systems with complementary strengths shows some potential to improve learning. However, managing learner expectations during transitions between systems remains an open research area. Finally, while personalized adaptation can improve learning efficiency, effort and time-on-task for learning remains a dominant factor that must be considered by interventions.

Preparing facilitators to use and adapt mathematics professional development materials productively

BACKGROUND

Determining whether a professional development program can be enacted with integrity in different settings and by different facilitators is critical to understanding efficacy. In this paper, we describe the two-stage preparation process of a facilitator as she prepared to use and adapt the highly specified Learning and Teaching Geometry video-based professional development materials with fidelity. The latter stage of the preparation process involved a rehearsal, during which the research team used two instruments to measure fidelity.

METHODS

Two existing instruments were used to explore fidelity through different lenses, including timing and modification of activities and learning goals.

RESULTS

Results from both fidelity instruments indicate that the facilitator used the materials as intended by the developers. However, these instruments did not capture important information regarding modifications the facilitator made, including timing and content-focused adaptations.

CONCLUSIONS

Suggestions are made with respect to measuring fidelity, preparing facilitators, and supporting productive adaptations.

The Use of Concrete Experiences in Early Childhood Mathematics Instruction

Addressed are four key issues regarding concrete instruction: What is concrete? What is a worthwhile concrete experience? How can concrete experiences be used effectively in early childhood mathematics instruction? Is there evidence such experiences work? I argue that concrete experiences are those that build on what is familiar to a child and can involve objects, verbal analogies, or virtual images. The use of manipulatives or computer games, for instance, does not in itself guarantee an educational experience. Such experiences are worthwhile if they target and further learning (e.g., help children extend their informal knowledge or use their informal knowledge to understand and learn formal knowledge). A crucial guideline for the effective use of concrete experience is Dewey's principle of interaction-external factors (e.g., instructional activities) need to mesh with internal factors (readiness, interest). Cognitive views of concrete materials, such as the cognitive alignment perspective and dual-representation hypothesis, provide useful guidance about external factors but do not adequately take into account internal factors and their interaction with external factors. Research on the effectiveness of concrete experience is inconclusive because it frequently overlooks internal factors.

Notice, Explore, and Talk About Mathematics: Making a Positive Difference for Preschool Children, Families, and Educators in Australian Communities That Experience Multiple Disadvantages

Let's Count is a preschool mathematics intervention implemented by The Smith Family from 2012 to the present in "disadvantaged" communities across Australia. It is based on current mathematics and early childhood education research and aligns with the Australian Early Years Learning Framework. Let's Count has been shown to be effective in enhancing mathematics learning and dispositions of young children, early childhood educators, and families through a longitudinal evaluation undertaken from 2012 to 2015. In this chapter, the authors explore the development, implementation, and evaluation of Let's Count and highlight the importance of adults noticing, exploring, and talking about children's mathematics. The findings from the longitudinal evaluation of Let's Count suggest that when adults notice children's mathematics, then children's learning thrives, and the positive dispositions and confidence of parents and educators increase. Let's Count has made a positive difference for many children and adults across Australia.

Coaching in Early Mathematics

Falling scores in math have prompted a renewed interest in math instruction at early ages. By their own admission, early childhood educators are generally underprepared and not always comfortable teaching math. Professional development (PD) in early mathematics is widely considered a main way to increase teachers' skills and efficacy (e.g., Guskey, 2000; Hyson & Woods, 2014; Munby, Russell, & Martin, 2001; Piasta, Logan, Pelatti, Capps, & Petrill, 2015; Richardson & Placier, 2001; Sarama, Clements, Wolfe, & Spitler, 2016; Sarama & DiBiase, 2004; Zaslow, 2014). However, it has been documented that stand-alone PD is not as effective in changing practice (e.g., Biancarosa & Bryk, 2011; Garet et al., 2008; Guskey, 2000; Hyson & Woods, 2014; Institute of Medicine and National Research Council, 2015; Joyce & Showers, 2002; Zaslow, 2014). Site-embedded ongoing support in the form of coaching or mentoring has been shown to be critical for successful implementation (Neuman & Cunningham, 2009; Powell, Diamond, Burchinal, & Koehler, 2010). In this chapter, we describe coaching models and abstract characteristics of effective coaching from the research. With this background, we provide an in-depth view of the coaching aspect of two large empirical studies in early mathematics. We introduce the theoretical framework from which the coaching models for these projects were developed and describe the research on which they were based. We then summarize how the planned models were instantiated and challenges to their implementation within each project. In the final section, we summarize what we have learned and described implications and challenges for the field.

Integrating STEM in elementary classrooms using model-eliciting activities: responsive professional development for mathematics coaches and teachers

BACKGROUND

This research highlights a school-university collaboration to pilot a professional development framework for integrating STEM in K-6 mathematics classrooms in a mid-Atlantic suburban school division. Because mathematics within STEM integration is often characterized as the calculations or the data representations in science classrooms, technology labs, or outside-of-school programs, developing a reasonable and realistic conceptualization of STEM integration for mathematics teachers and coaches may be especially challenging. Using design-based implementation research, university facilitators worked with eight mathematics teachers and coaches to construct an accessible vision of STEM integration built upon the design features of model-eliciting activities (MEAs). The research team strategized a flexible and fluid professional development that would (1) situate participants' breadth of experiences on a STEM curriculum integration continuum; (2) elicit a new vision of STEM integration through open-ended mathematics problems with client-driven, real-life contexts; and (3) focus on making mathematics content explicit.

RESULTS

Qualitative analysis of participant discussions and written reflections from a four-day summer institute indicates that the daily tailoring of the professional development design supported an evolving participant envisioning of STEM integration. Opportunities to engage with MEAs as learners, contrast MEAs with problem-based learning and draw from MEA design features to modify existing curricular tasks allowed participants to think more broadly about mathematics content within STEM integration. Participants communicated a readiness to use MEAs as a vehicle for K-6 STEM integration which maintains an important grounding in the teaching realities of grade-level standards and standardized test preparation. They also acknowledged the need for ongoing support as they considered the challenges of curricular pacing and administrative expectation.

CONCLUSIONS

The researchers continued to support the school division during monthly academic-year professional development sessions as the teachers and coaches created and enacted prototype lessons. Their shared investment in building STEM integration capacity with a specific focus on mathematics content can offer a model for STEM integration using MEAs that challenges one-size-fits-all professional development, encourages STEM instructional leadership, and promotes mathematical readiness for STEM citizenship and careers.

Support of mathematical thinking through embodied cognition: Nondigital and digital approaches

Research on mathematics education has shown that learners’ actions can influence how they think and vice versa. Much of this work has been rooted in the use of manipulatives, gestures, and body movements. Our article dissects the mechanisms that underscore the impact of embodied activities and applies this lens to explore how to harness the affordances of new technology to enhance mathematical thinking. This is especially crucial given the increasing accessibility of technology—such as digital touch devices, 3D printers, and location sensors—for constructing embodied experiences. Providing guidance for incorporating those tools, we focus on the role that embodied cognition can play in communicating mathematical concepts as well as in allowing learners to experiment and evolve their ideas. To inspire future integration of theory in the development of technologically enhanced embodied mathematics experiences, we provide examples of how this can be done. Finally, we outline future directions in the areas of design, implementation, and assessment of embodied learning of mathematics.

Grounded and embodied mathematical cognition: Promoting mathematical insight and proof using action and language

We develop a theory of grounded and embodied mathematical cognition (GEMC) that draws on action-cognition transduction for advancing understanding of how the body can support mathematical reasoning. GEMC proposes that participants' actions serve as inputs capable of driving the cognition-action system toward associated cognitive states. This occurs through a process of transduction that promotes valuable mathematical insights by eliciting dynamic depictive gestures that enact spatio-temporal properties of mathematical entities. Our focus here is on pre-college geometry proof production. GEMC suggests that action alone can foster insight but is insufficient for valid proof production if action is not coordinated with language systems for propositionalizing general properties of objects and space. GEMC guides the design of a video game-based learning environment intended to promote students' mathematical insights and informal proofs by eliciting dynamic gestures through in-game directed actions. GEMC generates several hypotheses that contribute to theories of embodied cognition and to the design of science, technology, engineering, and mathematics (STEM) education interventions. Pilot study results with a prototype video game tentatively support theory-based predictions regarding the role of dynamic gestures for fostering insight and proof-with-insight, and for the role of action coupled with language to promote proof-with-insight. But the pilot yields mixed results for deriving in-game interventions intended to elicit dynamic gesture production. Although our central purpose is an explication of GEMC theory and the role of action-cognition transduction, the theory-based video game design reveals the potential of GEMC to improve STEM education, and highlights the complex challenges of connecting embodiment research to education practices and learning environment design.

The DREME Network: Research and Interventions in Early Childhood Mathematics

The DREME Network was created to advance the field of early mathematics research and improves the opportunities to develop math competencies offered to children birth through age 8 years, with an emphasis on the preschool years. All four main Network projects will have implications for interventions. Section 1 introduces the Network and its four projects. The remainder of the chapter focuses on one of these four projects, Making More of Math (MMM), in depth. MMM is directly developing an intervention for children, based on selecting high-quality instructional activities culled from the burgeoning curriculum resources. We first report a review of 457 activities from 6 research-based curricula, which describes the number of activities by content focus, type (nature), and setting of each activity. Given the interest in higher-order thinking skills and self-regulation, we then identified activities that had the potential to, develop both mathematics and executive function (EF) proficiencies. We rated these, selecting the top 10 for extensive coding by mathematics content and EF processes addressed. We find a wide divergence across curricula in all these categories and provide comprehensive reports for those interested in selecting, using, or developing early mathematics curricula.

Subtypes and comorbidity in mathematical learning disabilities: Multidimensional study of verbal and visual memory processes is key to understanding

A large body of research suggests that mathematical learning disability (MLD) is related to working memory impairment. Here, I organize part of this literature through a meta-analysis of 36 studies with 665 MLD and 1049 control participants. I demonstrate that one subtype of MLD is associated with reading problems and weak verbal short-term and working memory. Another subtype of MLD does not have associated reading problems and is linked to weak visuospatial short-term and working memory. In order to better understand MLD we need to precisely define potentially modality-specific memory subprocesses and supporting executive functions, relevant for mathematical learning. This can be achieved by taking a multidimensional parametric approach systematically probing an extended network of cognitive functions. Rather than creating arbitrary subgroups and/or focus on a single factor, highly powered studies need to position individuals in a multidimensional parametric space. This will allow us to understand the multidimensional structure of cognitive functions and their relationship to mathematical performance.

Non-formal mechanisms in mathematical cognitive development: The case of arithmetic

The idea that cognitive development involves a shift towards abstraction has a long history in psychology. One incarnation of this idea holds that development in the domain of mathematics involves a shift from non-formal mechanisms to formal rules and axioms. Contrary to this view, the present study provides evidence that reliance on non-formal mechanisms may actually increase with age. Participants - Dutch primary school children - evaluated three-term arithmetic expressions in which violation of formally correct order of evaluation led to errors, termed foil errors. Participants solved the problems as part of their regular mathematics practice through an online study platform, and data were collected from over 50,000 children representing approximately 10% of all primary schools in the Netherlands, suggesting that the results have high external validity. Foil errors were more common for problems in which formally lower-priority sub-expressions were spaced close together, and also for problems in which such sub-expressions were relatively easy to calculate. We interpret these effects as resulting from reliance on two non-formal mechanisms, perceptual grouping and opportunistic selection, to determine order of evaluation. Critically, these effects reliably increased with participants' grade level, suggesting that these mechanisms are not phased out but actually become more important over development, even when they cause systematic violations of formal rules. This conclusion presents a challenge for the shift towards abstraction view as a description of cognitive development in arithmetic. Implications of this result for educational practice are discussed.

Productive failure in learning math

When learning a new math concept, should learners be first taught the concept and its associated procedures and then solve problems, or solve problems first even if it leads to failure and then be taught the concept and the procedures? Two randomized-controlled studies found that both methods lead to high levels of procedural knowledge. However, students who engaged in problem solving before being taught demonstrated significantly greater conceptual understanding and ability to transfer to novel problems than those who were taught first. The second study further showed that when given an opportunity to learn from the failed problem-solving attempts of their peers, students outperformed those who were taught first, but not those who engaged in problem solving first. Process findings showed that the number of student-generated solutions significantly predicted learning outcomes. These results challenge the conventional practice of direct instruction to teach new math concepts and procedures, and propose the possibility of learning from one's own failed problem-solving attempts or those of others before receiving instruction as alternatives for better math learning.