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In school, at least in the US, we were taught to multiply by hand according to a standard algorithm. Most people find that algorithm difficult to use, and many children fail to learn it. We propose a new way to make sense of this difficulty: to treat explicit computation as perceptually supported physical and mental action. Based on recent work in neuroscience, we trace the flow of arithmetic information to emphasize demands on visual working memory and attention. We predict that algorithms that make moderate demands on memory and attention will work better than others that make stronger demands. We suggest that the judicious use of spatial schemas can reduce such cognitive demands. Experimental evidence from children in an inner-city school supports this claim. Our work suggests new ways to think about instruction. The goal should be to minimize demands that present obstacles and maximize instead what human eyes, bodies, and brains do well. (Contains 7 figures and 2 tables.)

Background: National ranking from the triennial Programme of International Student Assessment (PISA) often serves as a barometer of national performance and human capital. Though excessive student- and school-level covariates (n > 700) may prove intractable for traditional least-squares estimate procedures, shrinkage methods may be more suitable for subset selection., Aims: With a focus on the United States, this paper proposes sparse regression for PISA 2012 to discover salient student- and school-level predictor variables for mathematical literacy achievement., Sample: The sparse regression analysis was conducted on 10 top-tiered OECD countries/economies, Canada, and the United States in mathematical literacy on the 2012 PISA. Two- and three-level hierarchical regression analyses were performed on Canadian and US students (N = 26,522) along with five of the ten top-tiered countries/economies (N = 58,385)., Methods: Using the 'least absolute shrinkage and selection operator' (LASSO) technique, the study (1) identified salient predictor variables of mathematical literacy performance for the top-tiered countries/economies, Canada, and the United States and (2) used these salient variables to perform two- and three-level hierarchical regression on data from Canada and the United States along with five top-tiered countries/economies. Weights and replicates were used to account for complex sample design. A weighted, two-level confirmatory factor analysis was performed to identify latent constructs. Missing data were handled through multiple imputation., Results: Separate two-level hierarchical models accounted for 32-35% student-level and 58-70% school-level variance in Canada and the United States, respectively; three-level models accounted for 33% of level-one variance, 62-65% level-two variance, and 13-44% of level-three variance for the US/Canada and US/Canada/top-tiered students, respectively. Following top-tiered countries/economies, Canadian students had high levels of self-efficacy, were more likely to encounter advanced concepts in class, were less activity/small group-centred, and were more likely to consider truancy a learning hindrance. Factor analyses revealed a positive relation with rigour and class organization (teacher-centred) for top-tiered countries and Canada, though not for the United States. For all countries, there was a strong relation between rigour and self-beliefs., Conclusion: Compared to top performers, a less rigorous curriculum, coupled with class and school factors, may explain lag in US performance., (© 2018 The British Psychological Society.)

A "crisis account" of shortages of well-qualified scientists, engineers, mathematicians and technologists has shaped education policy in the UK and the USA for decades. The apparent poor quality of school science education along with insufficient numbers of well-qualified teachers have been linked to skills shortages by government and other agencies since at least the time of the Second World War. There is, however, an alternative account that challenges the received view of a skills deficit and questions the evidence for sustained and long-term shortages across the sector. This paper provides an historical account of some of the main events that have characterised debates over the supply and demand of science and engineering professionals in the UK and the USA and the implications that this has had for science education policy. Starting from the end of the Second World War, the paper looks at the key challenges to the evidence that underpins the shortage debate and considers the consequences that more than seven decades of crisis accounts have had on the recruitment and retention of highly skilled scientists and engineers. The paper shows that while the shortage debate has a long history, it is one that is characterised by poor quality data as well as methodological and conceptual challenges. It argues that there is no consensus view about the existence of a skills deficit and that while there may have been short-lived shortfalls in specialist areas, there is little evidence in support of widespread and far reaching shortages as the rhetoric often claims.

Over the last 25 years, reforms in undergraduate biology education have transformed the way biology is taught at many institutions of higher education. This has been fueled in part by a burgeoning discipline-based education research community, which has advocated for evidence-based instructional practices based on findings from research. This perspective will review some of the changes to undergraduate biology education that have gained or are currently gaining momentum, becoming increasingly common in undergraduate biology classrooms. However, there are still areas in need of improvement. Although more underrepresented minority students are enrolling in and graduating from biology programs than in the past, there is a need to understand the experiences and broaden participation of other underserved groups in biology and ensure biology classroom learning environments are inclusive. Additionally, although understanding biology relies on understanding concepts from the physical sciences and mathematics, students still rarely connect the concepts they learn from other STEM disciplines to biology. Integrating concepts and practices across the STEM disciplines will be critical for biology graduates as they tackle the biological problems of the twenty-first century.

The article presents three activities that promote probabilistic thinking with middle school students in the U.S. Activity 1 uses experimental probability as students play the game of rock, paper, and scissors. In activity 2, a table and a tree diagram help students understand the theoretical probability behind the game. Activity 3 allows three people to play the game. The results of these activities can give teachers valuable insight into adapting these activities to fit their classroom needs.

Math education in the United States remains resistant to systemic change, and our country pays the price. Stemhagen's article "Democracy and School Math" further confirms this trend. Despite repeated calls for reform, decades of research on how people learn, millions of dollars invested in teacher professional development, and years of politicized debate, the math wars rage on--between those who believe students have the capacity to construct their own mathematical ideas and others who insist mastery of the traditional canon must come first. Meanwhile, algebra failure among secondary students remains rampant and elementary education majors report the greatest rates of math anxiety of any college major. Adults and children alike joke about being terrible at math, seemingly unaware of the extent to which this innumeracy serves as a barrier to full participation in democracy as well as to the realization of their individual goals, hopes, and dreams. In the math education community itself, there is little discussion of the unique role mathematics can play in preparing students for democracy. In this short paper, I offer a more detailed conceptualization of democratic mathematics education and discuss the role of constructivism in bringing these ideas to fruition. I suggest that a shift in the power dynamic that characterizes most mathematics classrooms will be a key component in moving beyond the gridlock.

Calls to advance students' ability to engage in mathematical reasoning practices including conjecturing, generalising and justifying (CGJ) place significant new demands on teachers. This case study examines how Mathematics Studio provided opportunities for a team of U.S. middle school teachers to learn about these practices and ways to promote them in the classroom. Findings demonstrate how CGJ readings and focused discussions, coupled with repeated cycles of collaborative lesson planning, observation and debrief, supported the development of teacher knowledge, professional community, and teaching resources. In addition, this paper explores the role school leadership played in facilitating Math Studio to ensure these learning opportunities were realised. Documenting how Math Studio features and participants contributed to teachers' ability to implement CGJ focused lessons not only provides insights into the difficulties teachers have shifting instruction, but also adds to our understanding of school-embedded professional development more generally.

The authors comment on the article by Morris and Hiebert in three ways. First, they add thoughts about why improvement efforts often focus on teachers, rather than teaching. Second, they offer evidence from U.S. lesson study research that focus on teaching can improve both students' learning and teachers' learning. Finally, they suggest that the instructional products and common assessments advocated by Hiebert and Morris are not sufficient, and that they need to be accompanied by practice-based, collegial learning in which teachers build shared knowledge and commitments for the hard work of improvement. Their research indicates that lesson study focuses on teaching, but improves teachers as well, increasing mathematical knowledge and beliefs that support instructional improvement, as well as improving student learning. (Contains 2 figures.)

The claim that faculty conduct research is one of the main justifications for the modern university. Supposedly, academe carries out important, cutting-edge inquiries in which society has an interest. In fact, states this author, research at American universities is becoming narrow and artificial, out of touch with social realities, and of interest mainly to other academics. This development the author calls scholasticism, after the medieval philosophers whom one remembers as being similarly ingrown. In recent decades, conservative critics of academe have focused on political correctness--the tendency of academics to privilege liberal perspectives to the exclusion of conservative ones. But scholasticism may pose a greater threat to the heart of university life, which is the intelligent construction of arguments. Rather than address important issues thoughtfully, today's academics are becoming pedants, mostly writing only for other specialists. Scholasticism is squeezing the intellectual life out of academe, and driving it toward other venues, including government and the blogosphere. In this article, the author describes the rise of scholasticism and asks about causes and cures: Why has obscurantism mushroomed in recent decades, and what can be done about it? He relies mostly on the trends he has seen in his own department and at New York University over more than thirty years. (Contains 35 footnotes.)

This synthesis paper explores current leadership training in science, technology, engineering and mathematics (STEM) education in Bulgaria. The analysis begins with discussion of global factors influencing the implementation of leadership training in STEM education in general and then presents information about the current status of leadership training in Bulgaria with emphases on the country's economics, politics and geographical location as specific factors influencing leadership education. A short background of Bulgaria is presented with regard to population, gross domestic product, educational system, engineering force and possible need for leaders in industry in Bulgaria and the European Union. The paper provides an overall view about the current status of leadership training in all Bulgarian universities offering STEM education and concentrates specifically on two major universities by examining their currently offered programmes. As part of the discussion, similar training elements in other European countries and the USA are presented. (Contains 1 table.)

Definitions play an essential role in mathematics. As such, mathematics teachers and students need to flexibly and productively interact with mathematical definitions in the classroom. However, there has been little research about mathematics teachers' understanding of definitions. At an even more basic level, there is little clarity about what teachers must know about mathematical definitions in order to support the development of mathematically proficient students. This paper reports on a qualitative study of pre-service secondary mathematics teachers choosing, using, evaluating, and interpreting definitions. In an undergraduate capstone course for mathematics majors, these future teachers were assigned three tasks which required them to (1) choose and apply definitions of functions, (2) evaluate the equivalence of definitions of functions, and (3) interpret and critique a secondary school textbook's definition of a specific type of function. Their performances indicated that many of these pre-service mathematics teachers had deficiencies reasoning with and about mathematical definitions. The implications of these deficiencies are discussed and suggestions for teacher educators are proposed. (Contains 3 tables and 1 figure.)

STEM (science, technology, engineering and mathematics) has been a powerful engine of prosperity in the US since World War II. Currently, American students' performances and enthusiasm in STEM education are inadequate for the US to maintain its leadership in STEM professions unless the government takes more actions to motivate a new generation of US students towards STEM careers. Despite of coherent actions taken by the government and various institutions, the US cannot ensure the production of a sufficient number of experts in STEM fields to meet its national and global needs. The current situation is that the US is largely dependent on the foreign-born STEM workforce. This paper starts with a deeper look at the participation rate of American students in STEM careers and the basis of career choices by the US students. The discussion is driven by barriers and misconceptions about STEM education. It concludes with recommendations for how to motivate more US students to pursue STEM careers. (Contains 1 figure.)

Despite being the inventing country of the Internet, the US is not satisfied with its current state in Internet speed and broadband adoption. Although, more students, teachers and researchers in the US use Internet than any other country in the world, it is not satisfactory for the US educators and legislators to maintain US competitiveness in the achievement of science, mathematics and technology. Another alarming situation is that the mean scores in science and mathematics of US, students are lower than expected. Even with coherent action taken by the government and various institutions, the US cannot produce a sufficient number of experts in science, mathematics and technology fields to meet national and global needs. These situations are not satisfactory for educators and legislators to reach US education goals. To improve this situation, President Obama's government has taken several action plans. This paper presents a closer look at US science, mathematics and technology education as well as the President's plan to improve the situation. Conclusions are made regarding whether the US plan is too ambitious as well as whether the vision is comprehensive enough but still possible to execute. (Contains 1 table.)

This paper draws from a broad study that explored mathematics teachers’ engagement with textbooks within the Lesson Study context. In the current paper, we report on teachers’ use of mathematics textbook activities to inform their instructional decisions during collaborative lesson planning of numeric and geometric patterns. The study took place within the Lesson Study (LS) context – a teacher development practice that emphasises, inter alia, collaborative lesson planning. Data was collected by observing teachers during collaborative lesson planning and through interviews. Data analysis was informed by the Mathematics Knowledge for Teaching (MKT) framework and Modes of teacher engagement with the textbook. Findings suggest that teachers’ instructional decisions are mainly stimulated by the textbooks, teachers use textbook activities with fidelity, and they seldom adapt the activities drawn from the textbooks. However, where adaptation occurs teachers do it superficially. We contend that the capacity to conduct in-depth interrogation of the text is a derivative of mathematics subject matter knowledge (SMK) as well as pedagogical content knowledge (PCK). Therefore, the widespread adoption of text without interrogation could be indicative of the lack of requisite content knowledge and pedagogical skills. [ABSTRACT FROM AUTHOR]

This paper reports upon particular aspects of a study carried out by Delice in 2003, the main aim of which was to compare the performance of students in the 16-19 age group from Turkey and England on trigonometry of "A-level standard" and then to compare the curriculum and assessment provision in each country to seek possible explanations for differences in performance. Overall, the Turkish students did much better with the algebraic, manipulative aspects of trigonometry, the English students did better with the application of trigonometry to practical situations. The comparison of curriculum and assessment provision in the light of these results raises several issues about the current A-level mathematics syllabus. It may be that there is a tendency to think that mathematics is very much the same in any country, however this research supports previous work [Kawanaka, T., Stigler, J.W. and Hiebart, J. (1999) Studying Mathematics Classrooms in Germany, Japan and the United States: Lessons from the TIMSS Video Study] where mathematics was indeed found to be taught and learnt differently and the roles of teachers and students were different across three countries (Germany, Japan and the USA). The paper begins by briefly describing the research and presenting some of the results. There then follows a discussion of these results within the context of curriculum and assessment provision within each country. Finally, issues for the current A-level syllabus arising out the results are presented.

Computerized adaptive testing (CAT) has unsurpassable advantages over traditional testing. It has become the mainstream in large scale examinations in modern society. This paper gives a brief introduction to CAT including differences between traditional testing and CAT, the principles of CAT, psychometric theory and computer algorithms of CAT, the advantages and cautions of CAT. In the end, the development of CAT in China is reviewed. (Contains 1 figure and 1 table.)

The ten scholarly papers in Sex, Drugs and Body Counts explore the generation and propagation of numbers that drive policy decisions in the U.S. government regarding human trafficking, drug trade, and armed conflict (including the war on terror). Each of these papers, written by different authors, provides an illuminating insight into how some of the numbers we hear or read in the news are derived. While the general message of the papers tends to be depressing, namely that the numbers driving U.S. policy vary from being slightly suspect to clearly fabricated, the book does provide positive examples of how accurate numbers can be obtained and how the numbers that are being used can be interpreted. As a teaching resource, the book provides instructors an opportunity to deepen their understanding of how quantitative data are used in U.S. policy, allowing them to explore these issues in class. Individual papers from the book could be used in a general education course (either in mathematics or in a field related to the topics) as a way to introduce students to reading quantitatively dense material. In a course more focused on the topics of the book and with a more quantitatively literate audience, the entire book could probably be assigned as reading. [ABSTRACT FROM AUTHOR]

Introduction: The authors examined a tablet computer application (iPad app) for its effectiveness in helping students studying pre-algebra to solve mathematical word problems. Methods: Forty-three visually impaired students (that is, those who are blind or have low vision) completed eight alternating mathematics units presented using their traditional literacy medium or an iPad app. Twenty percent of the mathematics problems included graphics such as maps, line graphs, and bar graphs. During each session, teachers of visually impaired students rated the amount of support they provided for students and the student motivation. Results: Students answered more mathematics problems correctly when using the iPad app and, overall, teachers reported that their students were more motivated with the app than with their traditional literacy medium. Students often used the hints provided in the app when they did not solve a problem correctly the first time. Discussion: Visually impaired students and their teachers found the app and graphics to be easy to use and motivating. The built-in Scratch pad was used by almost all students who were print users. Implications for practitioners: The study results contribute to the growing body of knowledge about the potential value of tablet-type devices for use by visually impaired students. [ABSTRACT FROM AUTHOR]

This paper seeks to identify significant trends in mathematics curricula and teaching approaches in two education systems: the United States (a highly decentralised education system) and England (a highly centralised education system), with focus on 16-to-19-year-olds. The paper adopts a two-fold perspective: an historical overview, and comparison of the areas of convergence and divergence across both education systems. The trajectory of mathematical development is expressed through timelines of core concepts and ideas which chronicle the sequence of events and philosophies that have shaped the development of mathematics teaching and learning. By tracing the trajectory of mathematics through history, the paper provides a greater awareness of how different factors influence how mathematics is taught across two disparate educational jurisdictions. The paper affords opportunities to reflect on and draw conclusions about what constitutes meaningful mathematics teaching and curriculum approaches for 21st century learner. [ABSTRACT FROM AUTHOR]

The author notes that the sciences are now something one must latch onto early and successfully as a teenager, and endure against all odds. This pivotal point requires maturity and even myopia--delayed gratification and voluntary dorkiness--traits not common in the young. As a nation, Americans are losing their hegemony in the sciences. In just the past decade, China quintupled its number of doctorates in engineering and the sciences, Korea and Taiwan doubled theirs, but the U.S. stayed stable. As individuals, Americans operate more and more contraptions, but understand them less and less. While there is no single solution to the problems of the STEM pipeline--where scientific interest is squelched over time and newcomers are barred from entry--the author offers a few thoughts.

The importance of students' problem-posing abilities in mathematics has been emphasized in the K-12 curricula in the USA and China. There are claims that problem-posing activities are helpful in developing creative approaches to mathematics. At the same time, there are also claims that students' mathematical content knowledge could be highly related to creativity in mathematics, too. This paper reports on a study that investigated USA and Chinese high school students' mathematical content knowledge, their abilities in mathematical problem posing, and the relationships between students' mathematical content knowledge and their problem-posing abilities in mathematics.

The purpose of this paper is to show the similarities as well as the differences of fundamental fraction knowledge owned by preservice elementary teachers from the United States (N = 89) and Taiwan (N = 85). To this end, we examined and compared their performance on an instrument including 15 multiple-choice test items. The items were categorized into four different types of fundamental fraction constructs, including part-whole relationship, quotient, equivalence, and meanings of operations. Each item was embedded in the area, linear, or set model except for the items constructed out of the meaning of operations. Several items were featured with a pictorial illustration. Quantitative analysis showed that U.S. preservice teachers were significantly outperformed by their Taiwanese counterparts overall. The difference between the two groups was statistically significant on 12 of 15 items. Findings suggest that preservice elementary teachers from both countries need to be better prepared in their understanding of the meaning of fraction multiplication or division operations. Findings also suggest that U.S. preservice elementary teachers need to be more knowledgeable in dealing with fraction problems embedded in a linear model. Further research is suggested to study the issues raised from the findings. (Contains 9 tables.)

Peer review is the gold standard for scientific communication, but its ability to guarantee the quality of published research remains difficult to verify. Recent modeling studies suggest that peer review is sensitive to reviewer misbehavior, and it has been claimed that referees who sabotage work they perceive as competition may severely undermine the quality of publications. Here we examine which aspects of suboptimal reviewing practices most strongly impact quality, and test different mitigating strategies that editors may employ to counter them. We find that the biggest hazard to the quality of published literature is not selfish rejection of high-quality manuscripts but indifferent acceptance of low-quality ones. Bypassing or blacklisting bad reviewers and consulting additional reviewers to settle disagreements can reduce but not eliminate the impact. The other editorial strategies we tested do not significantly improve quality, but pairing manuscripts to reviewers unlikely to selfishly reject them and allowing revision of rejected manuscripts minimize rejection of above-average manuscripts. In its current form, peer review offers few incentives for impartial reviewing efforts. Editors can help, but structural changes are more likely to have a stronger impact. [ABSTRACT FROM AUTHOR]

Background: Teachers often rely on the use of open‐ended questions to assess students' conceptual understanding of assigned content. Particularly in the context of mathematics; teachers use these types of questions to gain insight into the processes and strategies adopted by students in solving mathematical problems beyond what is possible through more close‐ended problem types. While these types of problems are valuable to teachers, the variation in student responses to these questions makes it difficult, and time‐consuming, to evaluate and provide directed feedback. It is a well‐studied concept that feedback, both in terms of a numeric score but more importantly in the form of teacher‐authored comments, can help guide students as to how to improve, leading to increased learning. It is for this reason that teachers need better support not only for assessing students' work but also in providing meaningful and directed feedback to students. Objectives: In this paper, we seek to develop, evaluate, and examine machine learning models that support automated open response assessment and feedback. Methods: We build upon the prior research in the automatic assessment of student responses to open‐ended problems and introduce a novel approach that leverages student log data combined with machine learning and natural language processing methods. Utilizing sentence‐level semantic representations of student responses to open‐ended questions, we propose a collaborative filtering‐based approach to both predict student scores as well as recommend appropriate feedback messages for teachers to send to their students. Results and Conclusion: We find that our method outperforms previously published benchmarks across three different metrics for the task of predicting student performance. Through an error analysis, we identify several areas where future works may be able to improve upon our approach. Lay Description: What is already known about this topic: Open‐ended questions are used by teachers in the domain of mathematics to assess their students understanding but automated support for these types of questions are limited in online learning platforms. Recent advancements in areas of machine learning and natural language processing have led to promising results in a range of domains and applications. What this paper adds: Emulating how teachers identify similar student answers can be used to build tools that support teachers in assessing and providing feedback to student open‐ended work. Implications for practice: Developing better automated supports can increase the amount of direct feedback students receive to guide their learning. [ABSTRACT FROM AUTHOR]

This paper introduces a bibliometric, citation network-based method for assessing the social validation of novel research, and applies this method to the development of high-throughput toxicology research at the US Environmental Protection Agency. Social validation refers to the acceptance of novel research methods by a relevant scientific community; it is formally independent of the technical validation of methods, and is frequently studied in history, philosophy, and social studies of science using qualitative methods. The quantitative methods introduced here find that high-throughput toxicology methods are spread throughout a large and well-connected research community, which suggests high social validation. Further assessment of social validation involving mixed qualitative and quantitative methods are discussed in the conclusion. [ABSTRACT FROM AUTHOR]

With constantly changing political landscapes affecting the ability of college students with minoritized identities of sexuality and/or gender (MIoSG; Vaccaro et al., 2015) to thrive on campus, higher educators need to understand student reported stressors to design more inclusive learning environments. Building from minority stress theory and using data from a grounded theory study with 56 collegiate STEM students with MIoSG, this paper documents stressors that students reported as contributing to diminished wellbeing. We used constant comparative grounded theory analysis to identify stressors shared by all participants, which included lack of belonging and invisibility in competitive STEM cultures as well as exclusionary interactions with STEM peers. Recommendations include the design and delivery of holistic education and support services on campus. [ABSTRACT FROM AUTHOR]

Background: Valuable safeguards against fast‐spreading conjectures about learning in times of fear and uncertainty are evidence‐based approaches to the assessment of the impact of sudden and unforeseen disruptions on learning practices. The present research focused on physics learning in such times because conceptual and computational literacy in physics is critical to the development of a scientifically and technologically literate society. Objectives: The present research aimed (a) to offer an objective assessment of whether performance differences in a physics course of the general education curriculum existed between the face‐to‐face medium (familiar mode of instruction) and the online medium (unfamiliar mode of instruction) for both male and female students, and then (b) to develop a response to the evidence collected to ensure a quality education for all parties involved. The research intended to fill two critical gaps in the extant literature: mixed findings concerning students' performance in the face‐to‐face and online mediums as well as scarce coverage of specific domains of knowledge that are critical to STEM learners. Methods: Students' performance was examined as a function of the type of assessment (formative and summative), instructional mode (online and face‐to‐face), and gender. An understudied student population of STEM students of Middle Eastern descent without prior formal exposure to online instruction was targeted. Results and Conclusions: In both formative and summative assessments, male students performed better online than face‐to‐face, whereas the performance of female students was either higher online or equivalent between instructional mediums. The evidence collected suggested that consideration be given to remedies that foster academic success in the face‐to‐face instructional medium, particularly for male students. Takeaways: An evidence‐based approach to learning dismantled emotion‐driven expectations regarding the impact of the online medium on physics learning, and encouraged new perspectives about instruction. Lay Description: What is already known?: Ongoing debates on the impact of the pandemic on students' learning shape decisions regarding future reliance on the online medium for instruction. What this paper adds: The extant literature is mixed concerning performance in the face‐to‐face and online mediums, including particular domains of knowledge critical to STEM learning.This study focused on physics learning in an understudied population of STEM students who were unaccustomed to online instruction.Performance was higher online than face‐to‐face, but gender differences emerged. Implications for practice: Lessons learned focused on the flexibility of online learning and on the need to develop materials to enhance learning in males enrolled in face‐to‐face or hybrid/blended classes. [ABSTRACT FROM AUTHOR]

Background: Providing students with worked out problem solutions is a beneficial instructional technique in STEM disciplines, and studying examples that have been worked out incorrectly may be especially helpful for reducing misconceptions in students with low prior content knowledge. However, past results are inconclusive and the effects of incorrect worked examples alone or in combination with correct examples remains unclear. Objectives: We aim to address whether studying incorrect examples alone or in combination with correct examples can support the reduction of students' fraction misconceptions, operationalized as errors made with high confidence. Methods: After incorrectly solving a sampling problem, 130 students in 4th through 11th grade in the U.S. were randomly assigned to a condition in an online problem set focused on fraction equivalence. Students studied either single‐type worked examples (i.e., correct or incorrect; n = 49) or combination‐type worked examples (correct and incorrect; n = 41) or engaged in a problem‐solving control (n = 50). Results: Studying a combination of correct and incorrect worked examples was as effective as the problem‐solving control with feedback at improving fraction equivalence knowledge and reducing the rate of high‐confidence errors. Students in both the combination condition and the problem‐solving with feedback condition outperformed those who studied either correct or incorrect worked examples alone. Conclusions: Results support the inclusion of a combination of correct and incorrect worked examples when teaching students with low prior content knowledge. Studying a combination of example types within an online tutor helps to reduce misconceptions about fractions, a topic students commonly struggle with. A problem‐solving task with corrective feedback worked equally well. Lay Description: What is already known about this topic: Studying worked examples of mathematical solutions improves student learning.Incorrect examples, either alone or in combination with correct examples, are particularly effective for improving learning for students with low prior knowledge in the target content.Comparing correct and incorrect examples is also effective at improving problem‐solving but this effect may be specific to those with high prior knowledge.Still unknown are the impacts of these different combinations of examples on misconceptions in particular. What this paper adds: We show that studying a combination of correct and incorrect examples reduces math misconceptions.Problem‐solving with corrective feedback within an online tutor also has this effect. Implications for practice and/or policy: Using a combination of example types will help reduce students' misconceptions about fractions. [ABSTRACT FROM AUTHOR]

Background: The sudden growth in online instruction due to COVID‐19 restrictions has given renewed urgency to questions about remote learning that have remained unresolved. Web‐based assessment software provides instructors an array of options for varying testing parameters, but the pedagogical impacts of some of these variations has yet to be examined. Objectives: The current study examines whether question order affects student performance on online college math assessments. Drawing on the literature of testing and math anxiety, we hypothesized that difficult questions near the beginning of an assessment would have a destructive effect on student confidence, which would in turn have a deleterious effect on their performance. Methods: We employed an observational 'big data' methodology, analysing 23,468 final exams completed by students in 10 different courses over eight semesters at a Math Emporium in a large technical university in the eastern United States. Students were freshmen and sophomores enrolled in non‐engineering math courses. We regressed the final score on the difficulty level of the first and second questions, controlling for several other factors. Results and Conclusions: We found that several factors—day of the week, amount of time before the deadline, number of minutes spent on the exam—have more of an impact on score than question order. This pattern was consistent across sexes. Takeaways: Our findings contradict some previous studies, which have found that difficult early questions degrade student performance, and that this affect is more pronounced in females. This work enriches our understanding of how students respond to online assessment. Lay Description: What is already known about this topic: Previous studies have had mixed results about the effect of question difficulty ordering on student performance.Furthermore, few studies have examined this issue for computer assessments. What this paper adds: We employ a novel methodology with a uniquely large data set.Our findings indicate that difficulty ordering has no effect on student performance. Implications for practice: Math instructors can use the random ordering feature of computer assessment generating tools because question order does not affect performance. [ABSTRACT FROM AUTHOR]

Mathematics Learning Support (MLS) has been widely available in Higher Education Institutions in the United States (US) for decades. However, until recently, there has been little research that considered the extent of provision. In this paper, we present the results of a survey of MLS with responses from 268 institutions across the US, which indicate that such support is well established across responding institutions. To place these results in context, we compare them to other recent international surveys of provision. We discuss what these findings mean and the collaborative opportunities that are available for US and international practitioners and researchers in this area. [ABSTRACT FROM AUTHOR]

In a recent paper Engle et al. (Commun Math Phys 354:231–246, 2017) showed that there is a unique state on the reduced holonomy–flux ∗ -algebra of homogeneous isotropic loop quantum cosmology, that is invariant under residual diffeomorphims. This result has been claimed to be true both for the Ashtekar–Bojowald–Lewandowski framework and for that introduced by the present author. Unfortunately, the uniqueness proof relies on an incorrect argument which spoils the second case. In our short note, we are going to patch this issue, this way keeping the nice uniqueness result in both cases. Moreover, we will even extend the underlying operator algebraic statements as this might help later for studying higher-dimensional models. [ABSTRACT FROM AUTHOR]