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In a 1975 essay, Martin Gardner presented a trick for multiplying two large numbers, when one of the numbers was given by the operator. The explanation of trick involved factoring a number that was one more than a power of ten, thereby obtaining one of the factors being multiplied. In this paper, we explain how to choose one of the factors without needing the factorization step. [ABSTRACT FROM AUTHOR]

The study of the transition from the numerical system of writing to the decimal system of writing, including the quest to date the origin of zero as a place numeral, has actively engaged the attention of modern researchers and scholars alike. The subject is in fact so vast, gripping, as well as baffling that it will most likely continue to keep them engaged for quite a while to come. There is no need to reiterate the fact that India is very rich in epigraphical wealth. Hundreds and thousands of inscriptions have been reported from across the length and breadth of the country, written in various languages and in varied scripts, and following different dating systems. It is because of this abundance and the sheer complexity of these inscriptions, coming from different parts of India and belonging to different dynasties, that it has become an arduous task to pinpoint when exactly the numerical system of writing gave way to the decimal system in India, and when zero first made its appearance in Indian inscriptions. Further, the chronological sequencing of the changes in the system of writing -- from numerical to decimal -- that had taken place over time is also difficult to establish. In this chapter, while keeping the vastness of the subject in mind, an attempt has been made to re-examine some of the early dated inscriptions that have come to light in the last few years, both as part of the reports from archaeological excavations and from epigraphical explorations. Efforts have also been made to connect these new findings with the hypotheses of and the views expressed by earlier scholars. The goal of this exercise was to ascertain and confirm the presence and usage of decimal systems of writing, and also the use of zero as a placeholder, in South India. Thus, the primary task undertaken has been to review some of the already known dated inscriptions from Andhra Pradesh, Karnataka, Kerala, and Tamil Nadu. [ABSTRACT FROM AUTHOR]

Since recent datings of the Bakhshali manuscript confirm our understanding that the zero is indeed an Indian creation, what is it in Indian culture that could have influenced an invention of this kind? From data relating to the Bakhshali manuscript, it is possible to recognize the influence of Buddhism and, from this, to try to see how this movement could have played a role in the theory leading to the inception of the placeholder zero. Moreover, with Brahmagupta this first form and use of zero was developed to the point of becoming a number, and thus adopting an operative role. While the writer of the Brāhmasphuṭasiddhānta was not a Buddhist, it is still possible to show in that document how Buddhist thinking, particularly its speculative developments in the meaning of śūnyatā as described by the great writers of Mahāyāna Buddhism, may provide a favorable setting for theories about the numerical concept of zero, even to the point where it is perceived as a condition of its 'thinkability' or conceptualization. [ABSTRACT FROM AUTHOR]

Who invented zero? Interestingly, there is no one-line answer. And the acceptability of any attempted answer generally depends on the perception of the seeker, as the connotation of the word 'zero' can be perceived from several different but interrelated perspectives. Many ancient civilizations, including India, had their own version of zero or a zero-like concept or symbol as a representative of 'nothingness'; some as a philosophical conundrum, elsewhere some others even in a practical sense, such as a filler or a gap on their counting board; but except in India, none of these early and somewhat hesitant initial concepts did ultimately mature to its true mathematical potential. Today it is generally accepted worldwide that this peerless concept of a decimal place value system of enumeration in tandem with the true zero of our present-day mathematics, evolved in ancient India. Initially philosophically nurtured and analyzed during the early periods of oral tradition in India by several different schools of thought, eventually at some point of time the concept reflected in the Sanskrit word śūnya was transformed into a numeral for mathematical expression of 'nothing'. Perhaps at a later period, it bloomed into its full potential as a number, kha, an integer on which mathematical operations can be performed. Indeed, this unique feature makes kha, the Indian zero, the true progenitor of our modern mathematical zero. However, the exact time frame of this gradual evolution is still hotly debated, a recent controversy in this direction being the outcome of attempted radiocarbon dating of the famous Bakhshali manuscript by the Bodleian Library of Oxford. Going in the other direction, some scholars suggest that a trace of this concept, if not in total operational perspective, might have had a Greek origin that traveled to India during the Greek invasion of the northern part of the country in the pre-Mauryan period. A relatively recent third view professes the Chinese origin of the concept of zero as a placeholder, which might have traveled with the traders from China to the far eastern parts of Asia, to places like Cambodia, then under Buddhist influence that spread from mainland India, where it got the shape of a 'bold dot', the earliest known written form of zero. Scholars belonging to this school of thought want to credit mainland India only for 'garlanding' this concept of zero toward its modern shape, sometime around the eighth or ninth century CE. Embracing all these paradigms of claims and counterclaims about the origin of zero made till date, this article is a nutshell version of an odyssey, an unparalleled journey from a concept to a number in its own right, perhaps the most significant creation of the human mind ever. [ABSTRACT FROM AUTHOR]

Many cultures contributed to the invention of zero, but how to apportion credit? This article explores a new way to do so: imagine royalties were due for every usage of zero, and then figure out how these should be divided among present-day countries. We report on what such a divided allocation might look like, and give reactions, both positive and negative, to it. We also explore how such ideas might be relevant in terms of potential exploitation of the Indian cyber market by foreign corporations. [ABSTRACT FROM AUTHOR]

The term 'Jaina mathematics' represents the mathematical concepts and ideas available in Jaina canonical and non-canonical literature. The literature related to the Karaṇānuyoga section of the Digambara Jaina tradition and the Gaṇītānuyoga section of the Śvetāmbara Jaina tradition have much content to hold mathematicians' interest. The classification of numbers in the three groups Saṃkhyāta (countable), Asaṃkhyāta (uncountable) and Ananta (infinite) can be found in ancient Jaina literature. The definition of Utkrṛsṣṭa saṃkhyāta (maximum countable) can be given only with the help of zero. Without the knowledge of numbers and zero it is not possible to define Uṭkrṛsṣta saṃkhyāta and other related quantities, such as Acalātma and Śīrsṣaprahelikā. But we find the use of zero in ancient mathematical texts such as the Bakhshali manuscript, texts of the Digambara Jaina tradition, such as SṢaṭakhaṇdāgama, Pancāstikāya, Mahābandha, Lokavibhāga, Tiloyapaṇṇattī, and the Dhavalā Commentary etc., and texts of the Śvetāmbara Jaina tradition such as Sthānāṃga sūtra, Jambūdvīpaprajñapti, Anuyogadvāra sūtra, Āvaśyaka Niryukti, Br1;ahatksṛetra Samāsa, etc., in the mathematical sense in many places. All these references are explored and analyzed in this chapter to give a comprehensive picture to the academic world. The study indicates that the use of zero and the decimal place value system was popular in the Jaina tradition during the first and second centuries CE. Keywords: SṢaṭkhaṇdāgama, Anuyogadvāra sūtra, Tiloyapaṇṇattī, Lokavībhāga, Bakhshālī manuscript,aṃkhyāta, Acalātma, Śūnya [ABSTRACT FROM AUTHOR]

If mathematics is the language the universe was written in and mathematics is discovered rather than invented, then Brahmagupta's 628 CE Sanskrit text first defines a zero as old as the universe. Brahmagupta's definition of zero as a sum of equal and opposite negative and positive quantities is the first scientific definition of zero found to be consistent with laws of motion and particle physics. However, having originated in the East, the full power of India's symmetric and scientific zero failed to migrate West via the medieval Arabic world to Renaissance Europe. Thus, only a trivial mathematical concept of zero emerged, representing either nothing on its own or an arithmetical placeholder used alongside other numbers. Zero as an arbitrary midpoint for measurement purposes within a single quantity is similarly trivial, whether it be centuries BCE or CE, or a measure above or below a surveyor's foundation line as used for construction in ancient Egypt. Thus, the oldest original extant definition of zero compatible with laws of physics describing our universe originated with Brahmagupta in India 628 CE. Yet long lost in transmission and translation, the sad aftermath in classrooms today is a zero in which negatives to its left are arbitrarily less than positives to its right. [ABSTRACT FROM AUTHOR]

In China, a decimal system of numeration is reflected in the number of oracle bones scripts between the fourteenth and eleventh centuries BCE. The place value system is clearly shown in the numbers on coinage from the eighth century BCE to the third century BCE. The decimal place value system was completely established in counting rods numeration, a description of the use of which is recorded in Mathematical Canon of Sun Zi of the late third century and the beginning of the fourth century; the earliest written counting rods numbers exist in a Dunhuang manuscript from around the tenth century. The procedures of multiplication, division or extraction of roots show that precise vacant positions in numbers are emphasized and strictly respected by the ancient practitioners. In thirteenth century mathematical works, the written symbol ... is systematically used as a placeholder in numbers. The 'procedure of the positive and the negative' is usually regarded as a general rule of addition and subtraction between positive, negative numbers and zero, but this procedure should be interpreted in the context of the mathematical method fangcheng. The general rules concerning the process of elimination numbers in which zero is involved are clearly expressed and adopted, even though there is no explicit declaration of the existence of the numeral zero. [ABSTRACT FROM AUTHOR]

The aim of this study was to investigate second-grade elementary school students' performance related to their counting skills, place value understanding, and addition operation in natural numbers. A total of 205 second-grade elementary school students from Turkey participated in this study. The data were collected through a 'Personal Information Form,' 'Counting Skills Test (CST),' 'Place Value Understanding Test (PVUT),' and 'Addition Operations Test (AOT).' The data were analyzed using independent samples t-test, correlation coefficient, and regression analysis. It was indicated through the results that the second-grade students demonstrated their highest performance in counting skills and lowest performance in understanding the concept of place value. No significant difference was determined between the female and male students' performance as it related to CST, PVUT, and AOT. There were moderate positive relationships between the students' performance related to counting skills, place value understanding, and addition operation in natural numbers. It was also shown in the results of the regression analyses that the addition operation, place value understanding, and counting skills performance in natural numbers were all important variables predicting students' mathematics achievement and explained 61% of the total variance. [ABSTRACT FROM AUTHOR]

Children often learn abstract mathematics concepts with concrete manipulatives. The current study compared different ways of using specific manipulatives – base-ten blocks – to support children's place value knowledge. Children (N = 112, M age = 6.88 years) engaged in place value learning activities in one of four randomly assigned conditions in a one-on-one tutoring setting: Concrete Only, Two-Step Fading, Three-Step Fading, and Fading-with-Comparison. Performance on a posttest measure was higher in the Two-Step and Three-Step Fading conditions relative to the Fading-with-Comparison condition, suggesting potential benefits of gradual, sequential transitions from concrete objects to written numerals. Children in the Concrete Only condition also exhibited high place value knowledge on the posttest. Finally, across conditions, children who exhibited knowledge of the connections between the base-ten-blocks and written number symbols had higher posttest and transfer test scores relative to children who did not exhibit knowledge of these connections. [ABSTRACT FROM AUTHOR]

Examining how informal knowledge systems change after formal instruction is imperative to understanding learning processes and conceptual development and to implementing effective educational practices. We used network analyses to determine how the organization of informal knowledge about multidigit numbers in kindergartners (N = 279; mean age = 5.76 years, SD = 0.55; 135 females) supports and is transformed by a year of in-school formal instruction. The results show that in kindergarten, piecemeal knowledge about the surface properties of reading and writing multidigit numbers and the use of base-10 units to determine large quantities are strongly associated with each other and connected in a stringlike manner to other emerging skills. After a year of instruction, each skill becomes connected to the "hub" abilities of reading and writing multidigit numbers, which also become strongly connected to more advanced knowledge of base-10 principles. These findings provide new insights into how partial knowledge provides the backbone on which explicit principles are learned. [ABSTRACT FROM AUTHOR]

Place value concepts were measured longitudinally from kindergarten (2017) to first grade (2018) in a diverse sample (n = 279; Mage = 5.76 years, SD = 0.55; 135 females; 41% Black, 38% White, 8% Asian, 12% Latino). Children completed three syntactic tasks that required an explicit understanding of base‐10 symbols and three approximate tasks that could be completed without this explicit understanding. Approximate performance was significantly better in both age groups. A factor analysis confirmed that syntactic and approximate tasks tapped separate latent variables in kindergarten, but not in first grade. Path analyses indicated that only kindergarten approximate performance predicted overall first‐grade place value understanding. These findings suggest that explicit understanding of base‐10 principles develops from implicit, partial knowledge of multidigit numbers. [ABSTRACT FROM AUTHOR]

A part of an investigation regarding the position value and its didactics is presented. A study was designed to explore the didactic potential of a relation-structure approach to positional value. The research was made by means of the application of a course. This enabled the student to attend the relations between components of positional value structure. The didactic application was made in two groups of 3rd. of elementary school. In one of them the location of the units of each order of magnitude in the space arranged in the work sheet with adjacent fields was used. In the other group the representation of the units in the abacus was used. The research purpose is to present effects of the didactic course respect. To the meaning that the students give to the relations attended to each order of magnitude. This is necessary for students to evoke and understand these relationships. [ABSTRACT FROM AUTHOR]

Background: While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding. Aims: The current study aimed at examining whether children's non‐symbolic ratio processing and their understanding of place‐value structure of whole numbers play a role in their understanding of fractions and decimals and whether their roles are different for fractions versus decimal understanding. Sample: A sample of 124 fourth graders was tested. Methods: Participants were tested on their symbolic rational number processing, non‐symbolic ratio processing, place‐value understanding of whole numbers, mathematics achievement, as well as a series of domain‐general and domain‐specific cognitive skills related to symbolic rational number processing. Results: The findings suggest that, while the understanding of place value of whole numbers significantly predicted the understanding of both fractions and decimals, non‐symbolic ratio processing specifically predicted the understanding of fractions, but not decimals. Conclusions: The findings highlight the roles of place‐value understanding and non‐symbolic ratio processing in the acquisition of symbolic rational numbers. [ABSTRACT FROM AUTHOR]

Highlights • We evaluated the carry effect in children with and without math difficulties. • Children with math difficulties took longer and made more errors in general. • Children with math difficulties showed a more pronounced carry effect. • This indicates an impairment in place-value understanding. • Potential differences in strategy use or working memory are discussed. Abstract Recent research has provided initial evidence that children with math difficulties (MD) experience problems in processing place-value information in basic numerical tasks. However, it remains unclear whether these problems generalize to basic arithmetic operations. For instance, multi-digit addition problems with carryover specifically require the computation of place-value information. Yet little is known about the carry effect in children with MD. Therefore, the current study investigated whether problems in processing place-value information among third-grade children with MD (n = 29 9-year-olds) compared with an age-matched control group (n = 50) generalize to two-digit addition. The results indicate an increased carry effect for response latencies and error rates in children with MD. These findings suggest that deficits in processing place-value information among children with MD generalize to place-value computations in multi-digit arithmetic. Potential contributions of strategy use and working memory for difficulties in processing place-value information are discussed. [ABSTRACT FROM AUTHOR]

This paper explores the link between the quality of the built environment and its value, in health, social, economic and environmental terms. This is theorized as 'place value' which, alongside 'place quality', is conceptualized as existing within a virtuous loop in which quality dictates value and value defines quality. To test this, a systematic review brought together wide-ranging international research evidence. The work confirmed a range of definitive associations between the quality of place and its place derived value. It also makes a clear link back from the evidence on place value to the sorts of qualities that enhance or detract from that value. These, in turn, define the constituent elements of place quality. [ABSTRACT FROM AUTHOR]

The article discusses the significance of creating the notion of place value as a rate of ten. The author examines the role of the language of value, problem types of multiplication in many steps, and addition apart from measurement division with ten as an organizing unit. According to author, understanding place value results in understanding multiplication.

This paper looks at spatial and social discontinuities in burial practice and in the intensity with which burial places became contested spaces in Bronze Age Cyprus. Early and Middle Bronze Age cemeteries are typically viewed as communal landscapes associated with ancestral claims to place and community, a process that involved varied levels of social negotiation and posturing linked at some sites with ritual and material culture elaboration and the likely use of some tombs as mortuary shrines. In the Late Bronze Age there was a shift, in some settlements, to intramural burial. This 'privatisation' of mortuary space has been linked with the rise of more rigid political and economic structures and the development of an 'urban mindset'. The public/private dichotomy across the Middle to Late Bronze Age transition, however, appears to have been overstated. While earlier, relatively fluid associations did give way to more structured relationships, likely reducing the need for community-wide strategies of affiliation, this did not happen evenly across the island and at many settlements it did not happen at all. The boundary between intraand extramural burial, furthermore, was never absolute and the inclusive egalitarian village cemetery may never have been a reality, with the dead perhaps always viewed as belonging to lineage groups rather than to a wider ancestral community. Variations in the place value of burial are visible throughout the Bronze Age and the apparent shift to intramural burial is likely to have been determined by historically contingent circumstances rather than changes in ideology or emerging notions of privacy or hierarchy. [ABSTRACT FROM AUTHOR]

The article discusses the development and beginning implementation of sequence of concepts, with the intention of supporting teachers to support students to understand place value. Topics discussed include used concept mapping to describe place value as a complex system; the teaching of number concepts requires the student learning from examples and activities provided by the teacher; and the development of students' conceptual understanding in mathematics by the use of hands-on materials.

The article describes an activity program for grades 2 to 5 mathematics class which asks students to work with groups and use base-ten blocks to come up with different ways to represent their number and understand place value in the base-ten number system.

A robust understanding of place value is essential. Using a problem-based approach set within meaningful contexts, students’ attention may be drawn to the multiplicative structure of place value. By using quotitive division problems through a concrete–representational– abstract lesson structure, this study showed a powerful strengthening of Year 3 students’ conceptual understanding of place value. [ABSTRACT FROM AUTHOR]

Using a game-based context and concentrating explicitly on language, students in the early years are able to make sense about place value amid the vagaries of the English language in naming numbers. This conceptual approach to understanding place value allows students to further develop number strategies beyond counting by ones. [ABSTRACT FROM AUTHOR]

Concrete objects used to illustrate mathematical ideas are commonly known as manipulatives. Manipulatives are ubiquitous in North American elementary classrooms in the early years, and although they can be beneficial, they do not guarantee learning. In the present study, the authors examined two factors hypothesized to impact second-graders' learning of place value and regrouping with manipulatives: (a) the sequencing of concrete (base-ten blocks) and abstract (written symbols) representations of the standard addition algorithm; and (b) the level of instructional guidance on the structural relations between the representations. Results from a classroom experiment with second-grade students (N = 87) indicated that place value knowledge increased from pre-test to post-test when the base-ten blocks were presented before the symbols, but only when no instructional guidance was offered. When guidance was given, only students in the symbols-first condition improved their place value knowledge. Students who received instruction increased their understanding of regrouping, irrespective of representational sequence. No effects were found for iterative sequencing of concrete and abstract representations. Practical implications for teaching mathematics with manipulatives are considered. [ABSTRACT FROM AUTHOR]

Two experiments examined whether concrete models support place value learning. In Experiment 1 (N = 149), 7-year-olds were trained with either a) symbols alone or b) symbols and base-10 blocks. Children in both groups showed significant growth overall, but there were specific effects favoring one training type over another. Symbols-only training led to higher scores on a number line estimation task and was particularly effective among high-ability students, whereas blocks training led to better understanding of base-10 structure and was particularly effective among low-ability learners. In Experiment 2 (N = 68), Montessori students, for whom concrete models play a major role in mathematics instruction, also demonstrated better understanding of base-10 structure than did their matched peers enrolled in mainstream elementary schools. [ABSTRACT FROM AUTHOR]

A new property of the Symmetric group, Sn, arises when each element is assigned a unique place value, which enables the ordering of the elements (numerically). It is shown that the di erences between successive elements of this ordered, place-value assigned symmetric group Sn, give rise to a palindromic sequence (Pk)β1≤k≤n!-1. We define the family of palindromic sequences, associated to Sn. Sieving out a given number at a time in the maximal palindromic sequence of the group, Sn, of length n!-1, results in a hierarchy of palindromic sequences, ending with a single element, which will be the central element of (Pk)β1≤k≤. Consequences of the concept of place value ordering of the elements of Sn, are presented in this article. [ABSTRACT FROM AUTHOR]

A new property of the Symmetric group, Sn, arises when each element is assigned a unique place value, which enables the ordering of the elements (numerically). It is shown that the di erences between successive elements of this ordered, place-value assigned symmetric group Sn, give rise to a palindromic sequence (Pk)β1≤k≤n!-1. We define the family of palindromic sequences, associated to Sn. Sieving out a given number at a time in the maximal palindromic sequence of the group, Sn, of length n!-1, results in a hierarchy of palindromic sequences, ending with a single element, which will be the central element of (Pk)β1≤k≤. Consequences of the concept of place value ordering of the elements of Sn, are presented in this article. [ABSTRACT FROM AUTHOR]

Using a dog's paw as a basis for numerical representation, sixth grade students explored how to count and regroup using the dog's four digital pads. Teachers can connect these base-4 explorations to the conceptual meaning of place value and regrouping using base-10. [ABSTRACT FROM AUTHOR]

In multi‐digit numbers, the value of each digit is determined by its position within the digit string. Children's understanding of this place‐value structure constitutes a building block for later arithmetic skills. We investigated whether a number line estimation task can provide an assessment of place‐value understanding in first grade. We hypothesized that estimating the position of two‐digit numbers requires place‐value understanding. Therefore, we fitted a linear function to children's estimates of two‐digit numbers and considered the resulting slope as a measure of children's place‐value understanding. We observed a significant correlation between this slope and children's performance in a transcoding task known to require place‐value understanding. Additionally, the slope for two‐digit numbers assessed at the beginning of grade 1 predicted children's arithmetic performance at the end of grade 1. These results indicate that the number line estimation task may indeed constitute a valid measure for first‐graders' place‐value understanding. Moreover, these findings are hard to reconcile with the view that number line estimation directly assesses a spatial representation of numbers. Instead, our results suggest that numerical processes involved in performing the task (such as place‐value understanding) may drive the association between number line estimation and arithmetic performance. [ABSTRACT FROM AUTHOR]

The recent determination of the local value of the Hubble constant by Riess et al., 2016 (hereafter R16) is now 3.3 sigma higher than the value derived from the most recent CMB anisotropy data provided by the Planck satellite in a ΛCDM model. Here we perform a combined analysis of the Planck and R16 results in an extended parameter space, varying simultaneously 12 cosmological parameters instead of the usual 6. We find that a phantom-like dark energy component, with effective equation of state w = − 1.29 − 0.12 + 0.15 at 68 % c.l. can solve the current tension between the Planck dataset and the R16 prior in an extended ΛCDM scenario. On the other hand, the neutrino effective number is fully compatible with standard expectations. This result is confirmed when including cosmic shear data from the CFHTLenS survey and CMB lensing constraints from Planck. However, when BAO measurements are included we find that some of the tension with R16 remains, as also is the case when we include the supernova type Ia luminosity distances from the JLA catalog. [ABSTRACT FROM AUTHOR]

The objective of this study is to determine the depth characteristic of a heat transfer coefficient in the wellbore heat exchanger with the use of the original calculation procedure called an internal functions method. During the geothermal heat recovery, the substances change their pressure and temperature together with current (local) depth of the wellbore. Together with a change of these thermophysical parameters, there are also changes in the substance thermophysical properties, i.e.: specific heat, heat conductivity, density and viscosity. These properties decide on Prandtl Pr and Reynolds Re criterion numbers. These numbers, in connection with Nusselt number, enable to determine the local values of the heat transfer coefficient. The values of thermophysical properties and a heat transfer coefficient, assumed in the calculation models of the geothermal heat recovery, usually have a decisive influence on the reliability of the obtained results. In the considered case, the value of the heat transfer coefficient has changed – from the depth of 500 to the depth of 2100 m – by about 70%. Therefore assuming the constant value of this coefficient in the simulation models of the geothermal heat recovery is a significant simplification that may considerably affect the calculated effectiveness of the geothermal systems. [ABSTRACT FROM AUTHOR]

The purpose of this longitudinal study is to investigate whether an early childhood intervention using an explicit and transparent number naming system will have a lasting benefit to English and Spanish speaking children in their mathematics achievement related to number sense by accelerating their acquisition of concepts of numeracy and place value recognition. Statistical analysis of the results of the Stanford Achievement Test administered to school children in grades 1 through 5 at a school in Texas (USA) over a 6-year period of time showed consistent differences between those children who had participated in the intervention program by learning to count using explicit numbers names in Pre-Kindergarten and those who had not participated in the program. The advantages of this study were most apparent in children in the fourth grade, 6 years after their Pre-Kindergarten experience with explicit number names, particularly with those who had 2 years of experience with explicit number names. [ABSTRACT FROM AUTHOR]

Considering eye-fixation behavior is standard in reading research to investigate underlying cognitive processes. However, in numerical cognition research eye-tracking is used less often and less systematically. Nevertheless, we identified over 40 studies on this topic from the last 40 years with an increase of eye-tracking studies on numerical cognition during the last decade. Here, we review and discuss these empirical studies to evaluate the added value of eye-tracking for the investigation of number processing. Our literature review revealed that the way eye-fixation behavior is considered in numerical cognition research ranges from investigating basic perceptual aspects of processing non-symbolic and symbolic numbers, over assessing the common representational space of numbers and space, to evaluating the influence of characteristics of the base-10 place-value structure of Arabic numbers and executive control on number processing. Apart from basic results such as reading times of numbers increasing with their magnitude, studies revealed that number processing can influence domain-general processes such as attention shifting-but also the other way round. Domain-general processes such as cognitive control were found to affect number processing. In summary, eye-fixation behavior allows for new insights into both domain-specific and domain-general processes involved in number processing. Based thereon, a processing model of the temporal dynamics of numerical cognition is postulated, which distinguishes an early stage of stimulus-driven bottom-up processing from later more top-down controlled stages. Furthermore, perspectives for eye-tracking research in numerical cognition are discussed to emphasize the potential of this methodology for advancing our understanding of numerical cognition. [ABSTRACT FROM AUTHOR]

The article focuses on progression of place value in mathematics under the Common Core State Standards for Mathematics (CCSSM) across elementary schools. Topics discussed include decomposition of numbers by kindergarten students, comparison of whole numbers with decimals by students of fifth grade and suggestions for teachers who implement mathematics instruction like the use of electronic resources and arrangement of teachers into teams.

The article focuses on a conversation between a teacher educator and preservice teachers that helps the latter remember the importance of unitizing when teaching place value to students. It mentions a discussion about the numbers two, 20 and 200, the tendency for teachers to overlook the value of the zero in teaching students the concrete representations of whole numbers and the need to allow students to unitize before teaching them base-ten manipulatives.