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Specialised Content Knowledge (SCK) is defined by Ball, Hoover-Thames, and Phelps (2008) as mathematical knowledge essential for effective teaching. It is knowledge of mathematics that is beyond knowledge which would be required outside of teaching; for instance, the capacity to determine what misconception(s) may lie behind an error in calculation. Such knowledge should be core business of teachers of mathematics, and any perceived shortfall in SCK viewed as problematic. The research reported on here is part of a large study about multiplicative thinking involving approximately two thousand children between nine and twelve years of age and their teachers. Data were generated from semi-structured interviews and a written diagnostic assessment quiz. As part of that large project, forty-four Australian and New Zealand primary and middle school teachers were asked to respond to student work related to multiplicative thinking, particularly to concepts of numbers of equal groups and commutativity. Participants' responses reflected confusion about a pivotal piece of SCK, the convention for naming arrays. As well, questionable assumptions about the children's work samples were made. Given that there is not unanimous agreement amongst mathematics educators about naming conventions, these observations may not be surprising. Due to the sample size, broad generalisations cannot be made, but the results suggest that many teachers may have limited SCK with regards to the important mathematical area of Multiplicative Thinking (MT).

Factors and multiples are important aspects of mathematical structure that support the understanding of a range of other ideas including multiplication and division, and later on, factorization. At primary school level, it is important that factors and multiples are taught as a connected enterprise and as vital parts of the multiplicative situation; that is multiplication and division. The primary objective of the study on which this paper is based was to determine the extent of children's understanding of factors and multiples. A written quiz containing questions about factors and multiples and asking for children to explain their responses, was administered. Results suggest that the language involved with factors and multiples may play a role in the extent to which children develop a conceptual understanding of them. Also, most children know some things about factors and multiples but struggled to connect and articulate ideas when factors and multiples were presented in a different context. In conclusion, the inconsistency of participant responses suggests that teaching about factors and multiples needs to emanate from a more conceptual and connected standpoint.

Multiplicative thinking underpins much of the mathematics learned beyond the middle primary years. As such, it needs to be understood conceptually to highlight the connections between its many aspects. This paper focuses on one such connection; that is how the array, place value partitioning and the distributive property of multiplication are related. It is important that students understand how the property informs the written multiplication algorithm. Another component of successful use of the standard multiplication algorithm is extended number facts and the paper also explores students' ability to understand and generate them. One purpose of the study was to investigate the extent to which students used the standard multiplication algorithm and if their use of it is supported by an understanding of the underpinning components of the array, partitioning, the distributive property, and extended number facts. That is, we seek to learn if students have a conceptual understanding of the multiplication algorithm and its underpinning mathematics that would enable them to transfer their knowledge to a range of contexts, or if they have procedurally learned mathematics. In this qualitative study, data were generated from the administration of a Multiplicative Thinking Quiz with a sample of 36 primary aged students. Data were analyzed manually and reported using descriptive statistics. The main implications of the study are that the connections among the multiplicative array, place value partitioning, base ten property of place value, distributive property of multiplication, and extended number facts need to be made explicit for children in terms of how they inform the use of the written algorithm for multiplication.

This small study sought to determine students' knowledge of multiplication and division and whether they are able to use sets of bundling sticks to demonstrate their knowledge. Manipulatives are widely used in primary and some middle school classrooms, and can assist children to connect multiplicative concepts to physical representations. Qualitative data were generated from semi-structured interviews with 32 primary and middle school children aged nine to eleven years. Participants were asked to work out the answer to multiplication and division examples and explain their thinking using bundling sticks. Results suggest that the majority of participant students may have a limited knowledge of aspects of the multiplication process and even less knowledge of the division process. The study also identified that many of the students appeared uncomfortable and/or unfamiliar with using bundling sticks and a number of them had difficulty in using bundling sticks to explain the multiplication and division processes. We conclude that manipulatives such as bundling sticks do not magically lead children to mathematical learning but are sufficiently powerful to warrant teachers familiarising themselves with how manipulatives can be used to develop conceptual understanding.

Multiplicative thinking is widely accepted as a critically important 'big idea' of mathematics that underpins much mathematical understanding beyond the primary years. It is therefore important to ensure not only that children can think multiplicatively, but that they can do so at a conceptual rather than procedural level. This paper reports on a large study of 530 primary school children in Australia, New Zealand and the United Kingdom. The research question was "To what extent do children of 10 and 11 years of age understand what happens to digit values when numbers are multiplied and divided by powers of ten?" A written multiplicative thinking quiz was administered and one section of four questions asked students to multiply and divide two digit whole and decimal numbers by a power of ten and then explain what happened to the numbers. Correct response rates for the four calculations ranged from 38.3% to 61.7%. Response rates for appropriate explanations about what happened to the numbers ranged from 2.6% to 5.5%. Most students who attempted to explain what happened did so at a 'pseudo-procedural' level with responses such as 'a zero is added' or 'we take off the zero'. The students who did explain it conceptually did so in terms of the digits moving a place for each power of ten. The implication is that teaching of multiplication and division needs to be done at a conceptual level, with attention paid to the underlying mathematical structure, rather than relying on 'pseudo-procedures' such as 'adding a zero' which are unsustainable and will likely lead to errors.

Multiplicative thinking has been widely accepted as a critically important "big idea" of mathematics and one which underpins much mathematical understanding beyond the primary years of schooling. It is therefore of importance to consider the capacity of children to think multiplicatively but also to consider the capacity of their teachers to teach multiplicative thinking in a conceptual manner. This article focusses specifically on the conceptual links between the multiplicative array, the notion of numbers of equal groups in the multiplicative situation, factors and multiples, the commutative property of multiplication, and the inverse relationship between multiplication and division. A study involving a large sample of primary school students found that whilst most students demonstrated an understanding of some of the aforementioned elements, hardly any of the students were able to connect the ideas or to explain them in terms of each other. As a consequence of the findings, the impact of teacher knowledge on children's capacity to think multiplicatively was considered.

Multiplicative thinking is a "big idea" of mathematics that underpins much of the mathematics learned beyond the early primary school years. This article reports on a recent study that utilised an interview tool and a written quiz to gather data about children's multiplicative thinking. Our research has so far revealed that many primary aged children have a procedural view of multiplicative thinking which we believe inhibits their progress. There are two aspects to this article. First, we present some aspects of the interview tool and written quiz, along with some of findings, and we consider the implications of those findings. Second, we present a key teaching idea and an associated task that has been developed from our research. The main purpose of the article is to promote the development of conceptual understanding of the multiplicative situation as opposed to the teaching of procedures. In doing so, we encourage the explicit teaching of the many connections within the multiplicative situation and between it and other "big ideas" such as proportional reasoning and algebraic thinking.

Multiplicative thinking is a critical component of mathematics which largely determines the extent to which people develop mathematical understanding beyond middle primary years. We contend that there are several major issues, one being that much teaching about multiplicative ideas is focussed on algorithms and procedures. An associated issue is the extent to which algorithms are taught without the necessary explicit connections to key mathematical ideas. This article explores the extent to which some primary students use the algorithm as a preferred choice of method and whether they can recognise and use alternative ways of calculating answers. We also consider the extent to which the students understand ideas that underpin algorithms. Our findings suggest that most students in the sample are 'prisoners to procedures and processes' irrespective of whether or not they understand the mathematics behind the algorithms. [The title on the pdf is "Algorithms and Multiplicative Thinking: Are Children Prisoners of Process?"]

Evidence suggests that some students have learned procedures with little or no underpinning understanding while others have a much more connected and conceptual levels of understanding. In this article, the work of four primary students is discussed in terms of their contextual understanding of multiplicative concepts. The difference between teaching for understanding and procedural teaching is highlighted. An analogy is drawn with Charles Dickens' character, Mr Thomas Gradgrind, who would have endorsed procedural teaching and abhorred teaching that encouraged understanding. The work of four primary students is discussed in terms of their contextual understanding of multiplicative concepts. The difference between teaching for understanding and procedural teaching is highlighted.

This is the first of two articles on the use of a written multiplication algorithm and the mathematics that underpins it. In this first article, the authors present a brief overview of research by mathematics educators and then provide a small selection of some of the many student work samples they have collected during their research into multiplicative thinking. They contend that many primary-aged children are taught algorithms for multiplication and division without an appropriate understanding of the mathematical structure and concepts that underpin those algorithms.

This article describes some of the essential mathematics that underpins the use of algorithms through a series of learning pathways. To begin, a graphic depicting the mathematical ideas and concepts that underpin the learning of algorithms for multiplication and division is provided. The understanding and use of algorithms is informed by two important ideas--grid multiplication and extended multiplication facts. The graphic combines a number of learning pathways that lead to those two ideas. The article goes on to demonstrate how each element builds part of the underpinning structure needed to understand algorithms and to use them efficiently.

A journey into multiplicative thinking by three teachers in a primary school is reported. A description of how the teachers learned to identify gaps in student knowledge is described along with how the teachers assisted students to connect multiplicative ideas in ways that make sense.

Background: Teacher knowledge continues to be a topic of debate in Australasia and in other parts of the world. There have been many attempts by mathematics educators and researchers to define the knowledge needed by teachers to teach mathematics effectively. A plethora of terms, such as mathematical content knowledge, pedagogical content knowledge, horizon content knowledge and specialised content knowledge, have been used to describe aspects of such knowledge. Purpose: This paper proposes a model for teacher knowledge in mathematics that embraces and develops aspects of earlier models. It focuses on the notions of contingent knowledge and the connectedness of "big ideas" of mathematics to enact what is described as "powerful teaching". It involves the teacher's ability to set up and provoke contingent moments to extend children's mathematical horizons. The model proposed here considers the various cognitive and affective components and domains that teachers may require to enact "powerful teaching". The intention is to validate the proposed model empirically during a future stage of research. Sources of evidence: Contingency is described in Rowland's Knowledge Quartet as the ability to respond to children's questions, misconceptions and actions and to be able to deviate from a teaching plan as needed. The notion of "horizon content knowledge" (Ball et al.) is a key aspect of the proposed model and has provoked a discussion in this article about students' mathematical horizons and what these might comprise. Together with a deep mathematical content knowledge and a sensibility for students and their mathematical horizons, these ideas form the foundations of the proposed model. Main argument: It follows that a deeper level of knowledge might enable a teacher to respond better and to plan and anticipate contingent moments. By taking this further and considering teacher knowledge as "dynamic", this paper suggests that instead of responding to contingent events, "powerful teaching" is about provoking contingent events. This necessarily requires a broad, connected content knowledge based on "big mathematical ideas", a sound knowledge of pedagogies and an understanding of common misconceptions in order to be able to engineer contingent moments. Conclusions: In order to place genuine problem-solving at the heart of learning, this paper argues for the idea of planning for contingent events, provoking them and "setting them up". The proposed model attempts to represent that process. It is anticipated that the new model will become the framework for an empirical research project, as it undergoes a validation process involving a sample of primary teachers.

Abstract The Anthropocene offers an opportunity to (re)imagine and enact futures that care for the geological and ecological times and temporal rhythms of kin, and to expand beyond human politics of time. This article engages posthumanism to disrupt anthropocentrism and attune to times as embedded within relations of being together-with nonhumans. In this methodological article, I consider what it might look like to be together-in-time(s) with nonhuman kin and offer some possible practices for attuning to nonhuman times and the co-created times of leisure. From these temporal attunements emerge creative and affectively disruptive (re)presentations of being together-in-times with rocks, chipmunks, a shaggy mane mushroom, and a log in two protected areas in Ontario, Canada. (Re)imagining leisure in the Anthropocene as an ethic of togetherness, this article seeks to care for affective, relational, and embodied practices of being-with, knowing-with, and writing-with kin and kin times, and for flourishing more-than-human futures. [ABSTRACT FROM AUTHOR]

It is important for students to make informed decisions about computation. This article highlights this importance and develops a framework which may assist teachers to help students to make effective computational choices.

Multiplicative thinking is accepted as a "big idea" of mathematics that underpins important mathematical concepts such as fraction understanding, proportional reasoning, and algebraic thinking. It is characterised by understandings such as the multiplicative relationship between places in the number system, basic and extended number facts, and properties of operations and associated relationships. Using examples from a current Year 6 research project, this article highlights the importance of a combination of conceptual understanding and procedural fluency in developing multiplicative thinkers.

The recent review of Australian Curriculum represents an opportunity to significantly change the teaching of primary mathematics in Australia and perhaps elsewhere. The current curriculum document presents content in a linear and compartmentalized manner that dominates its structure and doesn't explicitly accentuate the links and connections found in the "big ideas" of mathematics. As well, it pays lip service to the "big process ideas"--the Proficiencies--which should be the vehicles for developing and making explicit links between and within the "big content ideas". The nature of key content needs to be reconsidered and organised on the basis of the "big ideas" of mathematics, and to emphasize the links and connections within and between them, as well as between them and real contexts. Such connections should be made explicit for children and hence teachers may need to hold their mathematical knowledge in different ways. The review document acknowledges the issue of the "crowded curriculum" and makes recommendations such as increasing the depth of content whilst reducing its breadth. Exactly how this will be achieved remains to be seen but it is strongly suggested in this position paper that there needs to be a reorganization of content on the basis of "big ideas of mathematics" or at least a substantial acknowledgement that this is important.

The relationships between three critical elements, and the associated mathematical language, to assist students to make the critical transition from additive to multiplicative thinking are examined in this article by Chris Hurst.

Most adults' attitudes to mathematics come from their experiences of mathematics in school when they were children. Children's mathematical worlds are complex places containing both cognitive and affective elements. One cannot ignore the affective domain if one wishes to understand children's mathematical learning. Teacher education students consistently use the following words in describing their memories of mathematics: repetition, monotony, boredom, not understanding, mystery, failure irrelevance, and humiliation. These memories group into feelings relating to the teaching of mathematics and their personal reaction to themselves as mathematics learners. In this article, the authors remind readers of the importance of helping students to develop positive attitudes to mathematics and provide practical suggestions as to how to engage students in a variety of stimulating activities. (Contains 4 figures.)

The plethora of data that confronts teachers on a daily basis requires that they know more than simply being able to calculate. It demands that they understand the context in which the mathematical ideas are embedded and what those ideas are telling them in relation to that context. Also, it necessitates that they are prepared to question how data, and the associated claims, are presented. For primary teachers, the implication is that they must be prepared to help their students to develop these skills. The "Finding the Maths" strategy described in this article has been shown to be successful in helping students achieve this outcome. The aim of using this strategy is to encourage students to develop the three types of thinking referred to in the numeracy framework (fluent operator, mathematical learner, and critical mathematician) and to assume the roles of the "mathematical learner" and "critical mathematician." This strategy was used in tandem with the "Debriefing the Numeracy" strategy that was designed to focus student thinking on the embedded mathematics in a particular context. (Contains 2 figures and 6 student work samples.)

This study contributes to the conceptualization of cultural sensitivity in tourism based on an analysis of tourism and social science literatures associated with Indigenous peoples in Canada. The study used a systematic literature review methodology, which consisted of thematically analysing 56 English language, tourism and social science journal articles. Findings indicate that while cultural sensitivity is rarely explicitly defined, several themes are consistently used to illustrate what cultural sensitivity entails or should entail in tourism and other social science encounters. Themes include: (i) respect, (ii) trust, (iii) ethics, (iv) cultural identity, (v) mutual understanding and cultural exchange, (vi) self-determination, governance, and capacity building, and (vii) unique healing, wellness, and spiritual needs. In addition to describing these interrelated themes, the study proposes a definition of cultural sensitivity in tourism tethered to the contexts of Indigenous peoples in Canada. [ABSTRACT FROM AUTHOR]

The role of alanine and lactate in the regulation of oxalate synthesis was analyzed based on an experimental model utilizing isolated guinea pig peroxisomes and purified lactate dehydrogenase (LDH). Physiological concentrations of glycoxylate and lactate inhibited glycoxylate oxidase activity during the conversion of glycoxylate to oxalate. However, glycoxylate concentrations did not alter LDH-mediated hepatic oxalate synthesis while alanine elicited minor alterations only indicating the role of LDH in catalyzing the conversion of glycoxylate to oxalate.

The urinary excretion of oxalate in guinea pigs increases in the presence of glucagon, a peptide hormone. Although the enzymes involved in the synthesis of oxalate decreases in the presence of glucagon, oxalate synthesis in vitro does not change. This is probably because glucagon decreases the amount of alanine in the liver which increases glyoxylate availability for oxalate synthesis. Glucagon also decreases the lactate and pyruvate in the liver which increases the formation of oxalate from glyoxylate.

In January 2020, with support from the U.S. Department of Homeland Security (DHS), CDC instituted an enhanced entry risk assessment and management (screening) program for air passengers arriving from certain countries with widespread, sustained transmission of SARS-CoV-2, the virus that causes coronavirus disease 2019 (COVID-19). The objectives of the screening program were to reduce the importation of COVID-19 cases into the United States and slow subsequent spread within states. Screening aimed to identify travelers with COVID-19-like illness or who had a known exposure to a person with COVID-19 and separate them from others. Screening also aimed to inform all screened travelers about self-monitoring and other recommendations to prevent disease spread and obtain their contact information to share with public health authorities in destination states. CDC delegated postarrival management of crew members to airline occupational health programs by issuing joint guidance with the Federal Aviation Administration.* During January 17-September 13, 2020, a total of 766,044 travelers were screened, 298 (0.04%) of whom met criteria for public health assessment; 35 (0.005%) were tested for SARS-CoV-2, and nine (0.001%) had a positive test result. CDC shared contact information with states for approximately 68% of screened travelers because of data collection challenges and some states' opting out of receiving data. The low case detection rate of this resource-intensive program highlighted the need for fundamental change in the U.S. border health strategy. Because SARS-CoV-2 infection and transmission can occur in the absence of symptoms and because the symptoms of COVID-19 are nonspecific, symptom-based screening programs are ineffective for case detection. Since the screening program ended on September 14, 2020, efforts to reduce COVID-19 importation have focused on enhancing communications with travelers to promote recommended preventive measures, reinforcing mechanisms to refer overtly ill travelers to CDC, and enhancing public health response capacity at ports of entry. More efficient collection of contact information for international air passengers before arrival and real-time transfer of data to U.S. health departments would facilitate timely postarrival public health management, including contact tracing, when indicated. Incorporating health attestations, predeparture and postarrival testing, and a period of limited movement after higher-risk travel, might reduce risk for transmission during travel and translocation of SARS-CoV-2 between geographic areas and help guide more individualized postarrival recommendations. [ABSTRACT FROM AUTHOR]

Tourism literature on animal ethics and animal welfare has given scarce consideration to how tourists become enrolled into caring, responsible practices towards animals. The objective of this paper is to contextualize a process of moral development – and specifically the emergence of an ethic of care – through the narratives and experiences of captive elephant volunteer tourists in Thailand. Guided by tenets of ecofeminism and a narrative methodology, our study forefronts how relational experiences prompted compassion and empathy as storied by 12 women volunteers. These volunteer tourists described how they shaped their own moral and ethical patterns through practices of witnessing abuse, questioning moral responsibilities, connecting with elephants, and advocating for improved conditions of captive individuals. As storied by the volunteers, processes of witnessing–questioning–connecting–advocating were deeply transformational, and inspired what we interpret as the development of an ethic of care. The research advances understandings of how intentional, relational engagements that prioritize animal wellbeing have the potential to facilitate among tourists processes of becoming care-full. [ABSTRACT FROM AUTHOR]

Predatory publishing has risen with the development of open access publishing. This study examines how many potential predatory journals were used by Brandon University students by analyzing their bibliographies. In total, 245 bibliographies including 2,359 citations were analyzed. Of the 1,485 citations to journals in these citations, five were found to cite journals on Beall's List of Predatory Journals and Publishers. The probable sources of these journals in the students' bibliographies were examined. [ABSTRACT FROM AUTHOR]

• Posthumanism is used to elevate the status of animals in tourism research. • Sentience is a limiting concept for posthuman tourism and animal ethics research. • Agency is a more generative and affirming concept for posthuman approaches. [ABSTRACT FROM AUTHOR]

Multiplicative thinking is a critical component of mathematics which largely determines the extent to which people develop mathematical understanding beyond middle primary years. We contend that there are several major issues, one being that much teaching about multiplicative ideas is focussed on algorithms and procedures. An associated issue is the extent to which algorithms are taught without the necessary explicit connections to key mathematical ideas. This article explores the extent to which some primary students use the algorithm as a preferred choice of method and whether they can recognise and use alternative ways of calculating answers. We also consider the extent to which the students understand ideas that underpin algorithms. Our findings suggest that most students in the sample are 'prisoners to procedures and processes' irrespective of whether or not they understand the mathematics behind the algorithms. [ABSTRACT FROM AUTHOR]

The article discusses the efforts and studies of three primary school teachers within their professional learning community regarding developing multiplicative thinking in students. Topics discussed include the impact of mathematical concepts taught by the primary school, suggestions for teachers on how to develop an action plan, and the development of mental computation strategies.

Background:Teacher knowledge continues to be a topic of debate in Australasia and in other parts of the world. There have been many attempts by mathematics educators and researchers to define the knowledge needed by teachers to teach mathematics effectively. A plethora of terms, such as mathematical content knowledge, pedagogical content knowledge, horizon content knowledge and specialised content knowledge, have been used to describe aspects of such knowledge. Purpose:This paper proposes a model for teacher knowledge in mathematics that embraces and develops aspects of earlier models. It focuses on the notions of contingent knowledge and the connectedness of ‘big ideas’ of mathematics to enact what is described as ‘powerful teaching’. It involves the teacher’s ability to set up and provoke contingent moments to extend children’s mathematical horizons. The model proposed here considers the various cognitive and affective components and domains that teachers may require to enact ‘powerful teaching’. The intention is to validate the proposed model empirically during a future stage of research. Sources of evidence:Contingency is described in Rowland’s Knowledge Quartet as the ability to respond to children’s questions, misconceptions and actions and to be able to deviate from a teaching plan as needed. The notion of ‘horizon content knowledge’ (Ball et al.) is a key aspect of the proposed model and has provoked a discussion in this article about students’ mathematical horizons and what these might comprise. Together with a deep mathematical content knowledge and a sensibility for students and their mathematical horizons, these ideas form the foundations of the proposed model. Main argument:It follows that a deeper level of knowledge might enable a teacher to respond better and to plan and anticipate contingent moments. By taking this further and considering teacher knowledge as ‘dynamic’, this paper suggests that instead of responding to contingent events, ‘powerful teaching’ is about provoking contingent events. This necessarily requires a broad, connected content knowledge based on ‘big mathematical ideas’, a sound knowledge of pedagogies and an understanding of common misconceptions in order to be able to engineer contingent moments. Conclusions:In order to place genuine problem-solving at the heart of learning, this paper argues for the idea of planning for contingent events, provoking them and ‘setting them up’. The proposed model attempts to represent that process. It is anticipated that the new model will become the framework for an empirical research project, as it undergoes a validation process involving a sample of primary teachers. [ABSTRACT FROM PUBLISHER]

The article discusses the significance of a combination of conceptual understanding and procedural fluency in developing multiplicative thinkers in the field of mathematics. Multiplicative thinking underpins important mathematical concepts and is characterized by an ability to recognise and solve a range of mathematical problems.

The recent review of Australian Curriculum represents an opportunity to significantly change the teaching of primary mathematics in Australia and perhaps elsewhere. The current curriculum document presents content in a linear and compartmentalized manner that dominates its structure and doesn't explicitly accentuate the links and connections found in the 'big ideas' of mathematics. As well, it pays lip service to the 'big process ideas'- the Proficiencies - which should be the vehicles for developing and making explicit links between and within the 'big content ideas'. The nature of key content needs to be reconsidered and organised on the basis of the 'big ideas' of mathematics, and to emphasize the links and connections within and between them, as well as between them and real contexts. Such connections should be made explicit for children and hence teachers may need to hold their mathematical knowledge in different ways. The review document acknowledges the issue of the 'crowded curriculum' and makes recommendations such as increasing the depth of content whilst reducing its breadth. Exactly how this will be achieved remains to be seen but it is strongly suggested in this position paper that there needs to be a reorganization of content on the basis of 'big ideas of mathematics' or at least a substantial acknowledgement that this is important. [ABSTRACT FROM AUTHOR]