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This study is the first to examine the associations between the occurrence, frequency, and adaptivity of children's subtraction by addition strategy use (SBA; e.g., 712 - 346 = ?; 346 + 54 = 400, 400 + 300 = 700, 700 + 12 = 712, and 54 + 300 + 12 = 366) and their underlying conceptual knowledge. Specifically, we focused on two rarely studied components of conceptual knowledge: children's knowledge of the addition/subtraction complement principle (i.e., if a + b = c, then c - b = a and c - a = b) and their knowledge of different conceptual subtraction models (i.e., understanding that subtraction can be conceived not only as "taking away" but also as "determining the difference"). SBA occurrence was examined using a variability on demand task, in which children had to use multiple strategies to solve a subtraction. SBA frequency and strategy adaptivity were investigated with a task in which children could freely choose between SBA and direct subtraction (e.g., 712 - 346 = ?; 712 - 300 = 412, 412 - 40 = 372, and 372 - 6 = 366) to solve 15 subtractions. We measured both children's knowledge of the addition/subtraction complement principle, and whether they understood subtraction also as "determining the difference." SBA occurrence and frequency were not related to conceptual knowledge. However, strategy adaptivity was related to children's knowledge of the addition/subtraction complement principle. Our findings highlight the importance of attention to conceptual knowledge when teaching multi-digit subtraction and expand the literature about the relation between procedural and conceptual knowledge.

Most research on children's arithmetic concepts is based on (a) additive concepts and (b) a single concept leading to possible limitations in current understanding about how children's knowledge of arithmetic concepts develops. In this study, both additive and multiplicative versions of six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and associativity) were investigated in Grades 5, 6, and 7. The multiplicative versions of the concepts were more weakly understood. No grade-related differences were found in conceptual knowledge, but older children were more accurate problem solvers. Individual differences were examined through cluster analyses. All children had a solid understanding of identity and negation. Some children had a strong understanding of all the concepts, both additive and multiplicative; some had a good understanding of equivalence or commutativity; and others had a weak understanding of commutativity, equivalence, inversion, and associativity. Associativity was the most difficult concept for all clusters. Grade did not predict cluster membership. Overall, these results demonstrate the breadth of individual variability in conceptual knowledge of arithmetic as well as the complexity in how children's understanding of arithmetic concepts develops., (Copyright © 2017 Elsevier Inc. All rights reserved.)

Students with learning disabilities in mathematics often struggle with the underlying concepts of multidigit addition and subtraction. To help students build a conceptual understanding of these computations, teachers can utilize evidence-based practices such as the concrete-semi-concrete-abstract framework and the use of multiple visual representations. This column presents five key strategies that incorporate evidence-based practices and teach whole number operations that rely on place value understanding.

Background: A major model of the cerebral circuits that underpin arithmetic calculation is the triple-code model of numerical processing. This model proposes that the lateralization of mathematical operations is organized across three circuits: a left-hemispheric dominant verbal code; a bilateral magnitude representation of numbers and a bilateral Arabic number code., New Method: This study simultaneously measured the blood flow of both middle cerebral arteries using functional transcranial Doppler ultrasonography to assess hemispheric specialization during the performance of both language and arithmetic tasks. The propositions of the triple-code model were assessed in a non-clinical adult group by measuring cerebral blood flow during the performance of multiplication and subtraction problems. Participants were 17 adults aged between 18-27 years. We obtained laterality indices for each type of mathematical operation and compared these in participants with left-hemispheric language dominance. It was hypothesized that blood flow would lateralize to the left hemisphere during the performance of multiplication operations, but would not lateralize during the performance of subtraction operations., Results: Hemispheric blood flow was significantly left lateralized during the multiplication task, but was not lateralized during the subtraction task., Comparison With Existing Method(s): Compared to high spatial resolution neuroimaging techniques previously used to measure cerebral lateralization, functional transcranial Doppler ultrasonography is a cost-effective measure that provides a superior temporal representation of arithmetic cognition., Conclusions: These results provide support for the triple-code model of arithmetic processing and offer complementary evidence that multiplication operations are processed differently in the adult brain compared to subtraction operations., (Copyright © 2017 Elsevier B.V. All rights reserved.)

Most research on children's arithmetic concepts is based on one concept at a time, limiting the conclusions that can be made about how children's conceptual knowledge of arithmetic develops. This study examined six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and addition and subtraction associativity) in Grades 3, 4, and 5. Identity (a-0=a) and negation (a-a=0) were well understood, followed by moderate understanding of commutativity (a+b=b+a) and inversion (a+b-b=a), with weak understanding of equivalence (a+b+c=a+[b+c]) and associativity (a+b-c=[b-c]+a). Understanding increased across grade only for commutativity and equivalence. Four clusters were found: The Weak Concept cluster understood only identity and negation; the Two-Term Concept cluster also understood commutativity; the Inversion Concept cluster understood identity, negation, and inversion; and the Strong Concept cluster had the strongest understanding of all of the concepts. Grade 3 students tended to be in the Weak and Inversion Concept clusters, Grade 4 students were equally likely to be in any of the clusters, and Grade 5 students were most likely to be in the Two-Term and Strong Concept clusters. The findings of this study highlight that conclusions about the development of arithmetic concepts are highly dependent on which concepts are being assessed and underscore the need for multiple concepts to be investigated at the same time., (Copyright © 2016 Elsevier Inc. All rights reserved.)

The present study aims to identify factors that may influence the dissociability of number magnitude processing and arithmetic fact retrieval at the behavioural level. To that end, we assessed both subtraction and multiplication performance in a within-subject approach and evaluated the interdependence of unit-decade integration measures on the one hand as well as sex differences in the interdependence of performance measures on the other hand. We found that subtraction items requiring borrowing (e.g. 53-29 = 24, 3 < 9) are more error prone than subtraction items not requiring borrowing (e.g. 59-23 = 34, 9 > 3), thereby demonstrating a borrowing effect, which has been suggested as a measure of unit-decade integration in subtraction. Furthermore, we observed that multiplication items with decade-consistent distractors (e.g. 6 × 4 = 28 instead of 24) are more error prone that multiplication items with decade-inconsistent distractors (e.g. 6 × 4 = 30 instead of 24), thereby demonstrating a decade-consistency effect, which has been suggested as a measure of unit-decade integration in simple multiplication. However, the borrowing effect in subtraction was not correlated with the effect of decade consistency in simple multiplication in either men or women. This indicates that unit-decade integration arises from different systems in subtraction and multiplication. Nevertheless, men outperformed women not only in subtraction, but also in multiplication. Furthermore, subtraction and multiplication performance on correct solution probes were correlated in women, but unrelated in men. Thus, the view of differential systems for number magnitude processing and arithmetic fact retrieval may not be universal across sexes.

This experiment aimed to investigate proto-arithmetic ability in a wild population of New Zealand robins. We investigated numerical competence from the context of computation: behavioural responses to arithmetic operations over small numbers of prey objects (mealworms). Robins' behavioural responses (such as search time) to the simple addition and subtraction problems presented in a Violation of Expectancy (VoE) paradigm were measured. Either a congruent (expected) or incongruent (unexpected) quantity of food items were hidden in a trap door out of view of the subject. Within view of the subject, a quantity of items were added into (and in some cases subtracted from) the apparatus which was either the same as that hidden, or different. Robins were then allowed them to find a quantity that either preserved or violated addition and subtraction outcomes. Robins searched around the apparatus longer when presented with an incongruent scenario violating arithmetic rules, demonstrating potential proto-arithmetic awareness of changes in prey quantity. This article is part of a Special Issue entitled: Cognition in the wild., (Copyright © 2014 Elsevier B.V. All rights reserved.)

A deep understanding of fraction concepts and operations is necessary if pre-service teachers (PSTs) are to present the concepts in multiple forms to learners. Such an understanding needs to be grounded in rich conceptual knowledge. In the present study, we explore the development of this understanding by supporting a cohort of 103 PSTs, who had previously demonstrated poor conceptual understanding of fraction concepts and operations, with a Representational Reasoning in Teaching and Learning (RRTL) approach aimed at strengthening their conceptual knowledge. A comparison of pre- and post-test results indicated that participants showed a significant improvement in shifting the balance of their fraction knowledge to the conceptual end of the procedural-conceptual spectrum. Insights into how this approach assisted in developing PSTs' conceptual understanding were explored through interviews with four participants and an analysis of their pre- and post-test responses. We suggest that the use of teaching strategies such as RRTL are necessary in order to assist PSTs develop strong conceptual knowledge of fractions. [Note: The volume number (1) shown on the PDF is incorrect. The correct citation is v21 n1.]

Spatial features of mathematical equations may influence how people solve and interpret those equations. This study examined whether manipulations of spatial features affected how participants solved and interpreted equations involving the minus sign. Undergraduate participants (N = 91) solved multioperation arithmetic equations involving addition, subtraction, and multiplication (e.g., 25 - 3 + 2 × 5 = &#95;&#95;). We varied the spacing of the final three operands and the position of the first operator relative to the adjacent operands. Participants also generated a story problem to correspond with a given equation. We evaluated the procedures that participants used in solving the equations by analysing both their solutions and their written work. Both close spacing of the final three operands and position of the first operation sign influenced the procedures that participants used. Both of the spatial manipulations also influenced participants' interpretations of the conceptual structure of the equations, as revealed in the story problems that they generated. These results have implications for understanding how people process mathematical symbols and for mathematics education.

The purpose of the study was to determine whether there was any significant difference in speed, accuracy, retention and transferability between the Decomposition (DEC) and the Base-Complement Additions (BCA) algorithms for performing compound subtraction in non-decimal bases. Fifty-nine students with a mean age of approximately 15 years from two Agona Swedru Junior High Schools in the Agona District participated in the study. The study employed the pretest-posttests non-equivalent design. The two schools were randomly assigned to the treatment groups. The data collected on the four achievement tests namely Pretest, Posttest, a Third test and Retention test administered were then studied and analysed by employing the t-test at 0.05 level of significance. From the study it was found out that the mean performance of the BCA group was significantly higher than the DEC group on measures of accuracy. There was no relationship between the BCA and the DEC groups on the measure of speed. The BCA group produced significantly better computational accuracy than the DEC group. The mean score of the BCA group in the retention measures were significantly higher than that of the DEC group. The third test which was the test for understanding and application of the two methods showed that the BCA group performed better than the DEC group though the difference was slightly significant. Based on the findings, it was established that BCA method has substantial gains over the DEC method and as such recommended that the BCA should thus be included in the Junior Hugh mathematics curriculum as alternative method of solving compound subtraction in non-decimal bases.

Algorithms and representations have been an important aspect of the work of mathematics, especially for understanding concepts and communicating ideas about concepts and mathematical relationships. They have played a key role in various mathematics standards documents, including the Common Core State Standards for Mathematics. However, there have been some public misunderstandings about the standards and the role that algorithms and representations have in the teaching and learning of mathematics. In this article, I will first look at how algorithms and representations are discussed in the standards, and then examine and unpack some of the public conceptions around algorithms and representations.

This paper argues for teaching pre-service teachers about remediation strategies for learners who encounter problems in mathematics in the early grades. The premise is that all teachers should be equipped with theory-based practical knowledge to support learning. A few teaching sessions to develop the concepts that underlie the mathematical operations of addition and subtraction are introduced in this paper. An empirically validated, comprehensive model of cumulative arithmetic competence development from the ages of four to eight years forms the basis for the construction of the suggested teaching unit. The model distinguishes five competence levels of arithmetical conceptual development, and proposes that concepts build on one another hierarchically. A 'part plus part is equivalent to whole' model was constructed based on this hierarchical structure and the understanding that the concept of addition is a dynamic process. The teaching examples include exercises for all children, not only ones who struggle. Possibilities for adapting the exercises to the individual development level of slower or faster learners are also included. All exercises are accompanied by a reflection on the procedure and strategies applied in order to support meaningful and sustainable learning and to give student teachers the opportunity to use knowledge of mathematical cognition theory during their pre-service years.

Many sets of state and national mathematics standards have come and gone in the past two decades. The Common Core State Mathematics Standards (CCSMS), which were released in June of 2010, have been adopted by almost all states and will be phased in across the nation in 2014. The main difference between these standards and most of the others is that the CCSMS are mathematically very sound overall. They could serve--at long last--as the foundation for creating proper school mathematics textbooks and dramatically better teacher preparation. In this article, the author gives two examples of the kind of change the CCSMS (if properly implemented) will bring to the mathematics classroom. (Contains 7 footnotes and 10 endnotes.)

I present a viable learning trajectory for prospective elementary teachers' number sense development with a focus on whole-number place value, addition, and subtraction. I document a chronology of classroom mathematical practices in a Number and Operations course. The findings provide insights into prospective elementary teachers' number sense development. These include the role of standard algorithms and their relationship to the evolution of classroom mathematical practices that involve reasoning flexibly about number composition, sums, and differences.

This article contributes to the research literature concerning prospective elementary teachers' mathematical thinking and learning with a focus on flexibility. We present a case study of a prospective elementary teachers' development of flexibility in mental addition and subtraction during a Number and Operations course. Building upon the construct of strategy ranges, we introduce scaffolded strategy ranges, which describe the sets of strategies that people use given the opportunity to solve a task in multiple ways. Like many prospective elementary teachers, Brandy initially appeared inflexible in mental addition and subtraction. In fact, her unscaffolded strategy ranges were limited to just the mental analogs of the standard algorithms. However, Brandy's scaffolded strategy ranges revealed greater potential for flexibility. Furthermore, the way of reasoning that appeared in Brandy's scaffolded strategy ranges (a) influenced her interpretations of nonstandard strategies that she encountered in the Number and Operations course and (b) foreshadowed the direction in which her flexibility would develop.

This article explains how to teach long division using hands-on materials (skittles, base ten blocks, colored counters, numeral cards) and investigates reviewing division word problems with middle school students (N=27) with learning disabilities in mathematics by this approach compared to spending the same amount of time on the typical practices of computerized drill and paper/pencil work. Student performance related to division was assessed by an identical pretest/posttest. Both groups participated in ten thirty-minute lessons during which the experimental group used place value color-coded numeral cards, counters and skittles, and the control group reviewed division with computerized practice and worksheet activities. A significant difference was found between the average gain scores of the two groups, favoring the experimental (17.1% versus 4.5%). The experimental group scored as well or better than the control group on all test sections. (Contains 15 figures and 3 tables.)

Although students' invented strategies typically prove to be meaningful and effective in improving the students' mathematical understanding, much remains unexplored in the current literature. This study examined, through a teaching-scenario task, the nature of 80 preservice teachers' reasoning and responses to students' informal and formal strategies for whole number subtraction. This study also examined challenges reported by preservice teachers attempting to connect students' informal strategies to a traditional method. The broader implications of this study for the international community are discussed, and recommendations for teacher education programs are presented in accordance with the findings of the study.

This study explores the impact that co-designing a virtual manipulative, Fractions Lab, had on teachers' professional development. Tapping into an existing community of practice of mathematics specialist teachers, the study identifies how a cooperative enquiry approach utilising workshops and school-based visits challenged 23 competent primary school teachers' technological, pedagogical and content knowledge of fraction equivalence, addition and subtraction. Verbal and written data from the workshops alongside observations and interviews from the school visits were analysed using the technological, pedagogical and content knowledge (TPACK) framework. The findings show that the assumptions of even experienced teachers were challenged when Fractions Lab was shared as an artefact on which they were asked to co-design and subsequently interact with, using it in subsequent phases of the cooperative inquiry process. Two original aspects of successful co-design of virtual manipulatives through communities of practice are identified and offered to others: (1) careful bootstrapping of the first design iteration that gathers intelligence about the content area and the technological affordances and constraints available; and (2) involvement of highly motivated teachers who perceive themselves as agents of change in the domain area.

Experts think in patterns and structures using the specific "language" of their domains. For mathematicians, these patterns and structures are represented by numbers, symbols and their relationships (Stokes, 2014a). To determine whether elementary students in the United States could learn to think in mathematical patterns to solve addition and subtraction, a pilot curriculum using an Asian model of base-10 counting was introduced in kindergarten and first grade. Children in the pilot program, like their Asian counterparts, did not appear to have any difficulty with the concept of place value. They also did not have significant difficulties with single- and double-digit addition and subtraction as is often the case with American students (Stokes, 2010, 2013, 2014a, 2014b). To continue the pattern-based process in second grade, a Multi-Operation Chart was designed to make the relationships between multiplication, division and fractions clear. Testing at the end of the school year showed that students not only met, but exceeded the Standards set by the Common Core standards for second grade. Medians and modes for correct single- and double-digit multiplication, single-digit division and fractions were 100%. Educational implications are offered and discussed.

There is a call for enabling students to use a range of efficient mental and written strategies when solving addition and subtraction problems. To do so, students should recognise numerical structures and be able to change a problem to an equivalent problem. The purpose of this article is to suggest an activity to facilitate such understanding in an algebraically fruitful way. Methods that make use of the associative rule for students to use a range of efficient mental and written strategies for addition and subtraction problems are examined, with an eye for making the transition to using the associative rule in algebra in later years.

A small number of studies have investigated student understanding of vector addition and subtraction in generic or introductory physics contexts, but in almost all cases the questions posed were in the vector arrow representation. In a series of experiments involving over 1000 students and several semesters, we investigated student understanding of vector addition and subtraction in both the arrow and algebraic notation (using "i," "j," "k") in generic mathematical and physics contexts. First, we replicated a number of previous findings of student difficulties in the arrow format and discovered several additional difficulties, including the finding that different relative arrow orientations can prompt different solution paths and different kinds of mistakes, which suggests that students need to practice with a variety of relative orientations. Most importantly, we found that average performance in the "ijk" format was typically excellent and often much better than performance in the arrow format in either the generic or physics contexts. Further, while we find that the arrow format tends to prompt students to a more physically intuitive solution path, we also find that, when prompted, student solutions in the "ijk" format also display significant physical insights into the problem. We also find a hierarchy in correct answering between the two formats, with correct answering in the "ijk" format being more fundamental than for the arrow format. Overall, the results suggest that many student difficulties with these simple vector problems lie with the arrow representation itself. For instruction, these results imply that introducing the "ijk" notation (or some equivalent) with the arrow notation concurrently may be a very useful way to improve student performance as well as help students to learn physics concepts involving vector addition and subtraction.

Students learning to separate variables in order to solve a differential equation have multiple ways of correctly doing so. The procedures involved in "separation" include "division" or "multiplication" after properly "grouping" terms in an equation, "moving" terms (again, at times grouped) from one location on the page to another, or simply carrying out "separation" as a single act without showing any steps. We describe student use of these procedures in terms of Hammer's resources, showing that each of the previously listed procedures is its own "piece" of a larger problem solving activity. Our data come from group examinations of students separating variables while solving an air resistance problem in an intermediate mechanics class. Through detailed analysis of four groups of students, we motivate that the mathematical procedures are resources and show the issues that students must resolve in order to successfully separate variables. We use this analysis to suggest ways in which new resources (such as "separation") come to be. [This paper is part of the Focused Collection on Upper Division Physics Courses.]

Current curriculum initiatives (e.g., National Governors Association Center for Best Practices and Council of Chief State School Officers 2010) advocate that models be used in the mathematics classroom. However, despite their apparent promise, there comes a point when models break, a point in the mathematical problem space where the model cannot, or arguable should not, be used. In this work, we explore the breaking point of the chip model for integer subtraction and the area model for fraction addition. Breaking is inevitable--either because no one model is robust enough to be applicable in a very large problem space and/or because the modifications required to keep the model functioning weaken or even eliminate its benefits. While models are often intended to serve as visual illustrations or embodiments of a concept, adaptions at the model breaking point can turn model use into nothing more than executing the graphical analogue to a not-well-understood procedure. The act of identifying model breaking points illuminates the affordances and constraints of the model. This provides students a unique opportunity to discriminate across mathematical models to develop a meta-level understanding of the relationship between models and the mathematics those models are intended to support.

This paper reports an intervention phase of a design study aimed to assist second-grade Filipino children in solving addition word problems in English, a language they primarily encounter only in school. With Filipino as the medium of instruction, an out-of-school pedagogical intervention providing linguistic and representational scaffolds was implemented with 17 children. Pre-intervention, children experienced linguistic difficulties and were limited to conceptualising and solving simple additive structures. Post-intervention interviews revealed improved performance and understanding of more complex structures, but only when linguistic difficulties were minimised. The study identified socially and culturally driven barriers to learning: superficial strategies, children's engagement, and learning in an urban poor context.

The present study examined the relations between home numeracy experiences (i.e., parent-child numeracy activities and parents' numeracy expectations) and basic calculation skills (i.e., addition and subtraction) of children with Specific Language Impairment (SLI) and their peers with Normal Language Achievement (NLA), while taking into account their cognitive and linguistic capacities. Fifty children with SLI and 100 children with NLA were tested on cognitive, linguistic, and basic calculation skills, and their parents filled in questionnaires on home numeracy activities and numeracy expectations. The results showed parents of children with SLI report engaging in fewer numeracy-related activities and have lower numeracy expectations for their children than parents of children with NLA. Furthermore, parent-child numeracy activities were more strongly associated with addition and subtraction for children with SLI. It is thus especially important that parents of children with SLI are made aware of their important role in the development of their child's basic calculation skills. (Contains 4 tables.)

PEMDAS is a mnemonic device to memorize the order in which to calculate an expression that contains more than one operation. However, students frequently make calculation errors with expressions, which have either multiplication and division or addition and subtraction next to each other. This article explores the mathematical reasoning of the Order of Operations and the effectiveness of a new approach.

Mathematics continues to be one of the most challenging aspects of the school curriculum for students with learning disabilities. The provision of pedagogically sound foundational computation and word problem instruction in the elementary grades is critical for students to progress through the mathematics curriculum successfully. A multiple-probe-across-participants design was used to investigate the effects of explicit instruction using the concrete-representational-abstract sequence with integrated cognitive strategies to teach five fourth- and fifth-grade students how to subtract when regrouping is required. All five participants benefited from the scripted instructional lessons. Practical implications and suggestions for future research in this area are provided. (Contains 4 figures and 2 tables.)

This study investigated how elementary teachers in a mathematics lesson study made sense of student learning, teaching, and content, as related to using representations in teaching multidigit subtraction, and how changes occurred over time in their talk and practice. The lesson-study process paved a group talk path along which teacher talk shifted from superficial to deeper consideration of student learning. By providing a context in which interactions of diverse ideas drove teacher learning, lesson study facilitated teachers making connections between the craft knowledge of teaching and scholarly knowledge. Individual teacher talk paths varied within the group path, and one teacher's learning path and the interaction of different learning paths is discussed. (Contains 6 figures, 1 table and 8 footnotes.)

In considering mathematics problem solving as a model-eliciting activity (Lesh & Doerr, 2003; Lesh & Harel, 2003; Lesh & Zawojewski, 2008), it is important to know "what" students are modeling for the problems: situations or solutions. This study investigated Grade 3 students' mathematization process by examining how they modeled different types of multi-digit subtraction situation problems. Students' modeling processes differed from one problem type to another due to their prior experiences and the complexity of the problems. This study showed that students make their own distinctions between solution and situation models in their mathematization process. Mathematics curricula and teaching should consider these distinctions to carefully facilitate different model development of and support student understanding of a content topic. (Contains 6 tables and 8 figures.)

This study developed a computer-assisted mathematical problem-solving system in the form of a network instruction website to help low-achieving second- and third-graders in mathematics with word-based addition and subtraction questions in Taiwan. According to Polya's problem-solving model, the system is designed to guide these low-achievers through the parts of the problem-solving process that they often ignore. The situations of verbal questions are visualised to walk the students through the course of thinking so they can solve the question with proper understanding of its meaning. We found that the mathematical problem-solving ability of experiment group students was significantly superior to that of control group students. Most of the participants were able to continue the practice of solving word-based mathematical questions, and their willingness to use the system was high. This indicates that the computer-assisted mathematical problem solving system can serve effectively as a tool for teachers engaged in remedial education. (Contains 7 tables and 8 figures.)

Estimation is more than a skill or an isolated topic. It is a thinking tool that needs to be emphasized during instruction so that students will learn to develop algorithmic procedures and meaning for fraction operations. For students to realize when fractions should be added, subtracted, multiplied, or divided, they need to develop a sense of size for various fraction quantities. They also need number sense, which will enable them to use estimation as a tool with fractions. When wrestling with quantities like three-fifths or nine-tenths, as individual fractions and in relation to each other, students will develop a sense of size and quantity for individual fractions. This experience is necessary if students are to develop meaning for fraction operations. This article explores the ways that estimation is used when students are learning to add, subtract, multiply, and divide fractions. (Contains 1 table and 2 figures.)

This study explores the extent to which a teacher elicited students' mathematical reasoning through the use of challenging tasks and the role her knowledge played in doing so. I characterised the teacher's knowledge in terms of a "local theory" of instruction, a form of pedagogical content knowledge that involves an empirically tested set of conjectures situated within a mathematical domain. Video data were collected and analysed and used to stimulate the teacher's reflection on her enactments of an instructional sequence. The teacher, chosen for how she consistently elicited student reasoning, showed evidence of possessing a local theory in that she articulated the ways student thinking developed over time, the processes by which that thinking developed, and the resources that facilitated the development of student thinking. Her knowledge informed how she revised and enacted challenging tasks in ways that elicited and refined student thinking around integer addition and subtraction. Furthermore, her knowledge and practices emphasised the progressive formalisation of students' ideas as a key learning process. A key implication of this study is that teachers are able to develop robust knowledge from enacting challenging tasks, knowledge that organises how they elicit and refine student reasoning from those tasks. (Contains 3 tables and 1 footnote.)

Cognitive diagnostic models (CDMs) attempt to uncover latent skills or attributes that examinees must possess in order to answer test items correctly. The DINA (deterministic input, noisy "and") model is a popular CDM that has been widely used. It is shown here that a logistic version of the model can easily be fit with standard software for latent class analysis. A partly Bayesian approach to estimation, posterior mode estimation, is used as a simple alternative to a fully Bayesian approach via Markov chain Monte Carlo methods. A latent-class analysis of a widely analyzed data set, the fraction subtraction data of K. K. Tatsuoka, reveals some neglected problems with respect to the classification of examinees; for example, examinees who get all of the items incorrect are classified as having most of the skills. It is also noted that obtaining large estimates of the latent class sizes can indicate misspecification of the Q-matrix, such as the inclusion of an irrelevant skill. It is shown, analytically and via simulations, that the problems are largely associated with the structure of the Q-matrix. (Contains 5 tables and 3 notes.)

Recent findings suggest deficits in coherent motion sensitivity, an index of visual dorsal stream functioning, in children with poor mathematical skills or dyscalculia, a specific learning disability in mathematics. We extended these data using a longitudinal design to unravel whether visual dorsal stream functioning is able to "predict" individual differences in subsequent "specific" mathematical skills, i.e., single-digit subtraction and multiplication. We measured children's sensitivity to coherent motion in kindergarten (mean age: 5 years 8 months) and evaluated their subtraction and multiplication skills in third grade (mean age 8 years 3 months). Findings revealed an association between subtraction but not multiplication performance and coherent motion sensitivity. This association remained significant even when intellectual ability and reading ability were additionally controlled for. Subtractions are typically solved by means of quantity-based procedural strategies, which reliably recruit the intraparietal sulcus. Against the background of a neural overlap between the intraparietal sulcus and visual dorsal stream functioning, we hypothesize that low-level visuospatial mechanisms might set constraints on the development of quantity representations, which are used during calculation, particularly in subtraction.

A dedicated, non-symbolic, system yielding imprecise representations of large quantities (approximate number system, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5-7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the outcome of the transformation at above-chance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithmetic instruction. (Contains 1 table and 4 figures.)

Textbooks, like many other resources teachers have at hand, are meant to be an aid for instruction; however there is little research with textbooks or on their potential to develop metacognitive knowledge. Metacognitive knowledge has received substantial attention in the literature, in particular for its relationship with problem-solving in mathematics. In psychology, metacognition is defined as "knowledge of cognition in general as well as awareness of one's own cognition". A further distinction is made in mathematics education to include individual's monitoring, control, and regulation of cognition while engaged in solving problems. In other words, it corresponds to the knowledge about what needs to be done to set up a problem, what resources to deploy and when, and how to determine that the problem has been successfully solved. In particular, I investigate the extent to which textbook authors make explicit the decision-making process for solving problems, how they indicate that an answer has been found, and that the answer is correct. (Contains 1 figure.)

This article discusses a triangle of numbers that are related to the derangement numbers. These numbers satisfy a Pascal-like recurrence relation with subtraction instead of addition. We describe how they relate to numbers studied by other authors and use them to generalize Euler's famous recurrence relation for the derangement numbers.

The study described herein represents an initial, exploratory attempt to understand what it means to translate between external representation systems. Researchers have traditionally considered translation as an all-or-none activity. We hypothesize that translation is comprised of both knowledge and skill components, and accordingly construe translation as an activity that our framework allows us to define as partial or complete. We examined the effects of part-whole knowledge in change unknown subtraction situations and the structure of three different external representations on first grade students' ability to translate in a partitioning task. Based on three interviews conducted with each student, results from our mixed methods analysis showed that translation skill was affected by what students knew about subtraction and the structure of the ER. Further, whereas high knowledge students demonstrated greater difficulty with translating "to" the ERs, low knowledge students had difficulty translating "from" the problem statement as well as translating "to" the ERs. (Contains 5 tables and 2 figures.)

Educated adults solve simple addition problems primarily by direct memory retrieval, as opposed to by counting or other procedural strategies, but they report using retrieval substantially less often with problems in written-word format (four = eight) compared with digit format (4 = 8). It was hypothesized that retrieval efficiency is relatively low with word operands compared with digits and that this promotes a shift to procedural backup strategies. Consistent with this hypothesis, Experiment 1 demonstrated greater word-format costs on retrieval usage for addition than subtraction, which was due to increased counting for addition but not subtraction. Experiment 2 demonstrated greater word-format costs on retrieval for division than multiplication, which was due to increased use of multiplication-fact reference to solve division problems. Format-related strategy shifts away from retrieval reflected both the efficiency of retrieval for a given operation and the availability of viable alternative strategies. The results demonstrate that calculation processes are not abstracted away from problem surface form. The authors propose that retrieval efficiency for arithmetic connects diverse performance and strategy-related effects across key arithmetic factors, including arithmetic operation, numerical size, and numeral format. (Contains 4 tables and 4 figures.)

This paper studies three models for cognitive diagnosis, each illustrated with an application to fraction subtraction data. The objective of each of these models is to classify examinees according to their mastery of skills assumed to be required for fraction subtraction. We consider the DINA model, the NIDA model, and a new model that extends the DINA model to allow for multiple strategies of problem solving. For each of these models the joint distribution of the indicators of skill mastery is modeled using a single continuous higher-order latent trait, to explain the dependence in the mastery of distinct skills. This approach stems from viewing the skills as the specific states of knowledge required for exam performance, and viewing these skills as arising from a broadly defined latent trait resembling the theta of item response models. We discuss several techniques for comparing models and assessing goodness of fit. We then implement these methods using the fraction subtraction data with the aim of selecting the best of the three models for this application. We employ Markov chain Monte Carlo algorithms to fit the models, and we present simulation results to examine the performance of these algorithms.

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 - 9 yields 12. Here, we investigate whether preschool children's approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding.

A paucity of research has directly compared empirically supported interventions to examine their effectiveness among students with different mathematics fluency skills. The purpose of this study was to compare the effects of two empirically supported interventions and a control condition on the mathematics fluency of 98 second and third grade students. Specifically, effects of explicit timing and cover-copy-compare on subtraction fluency were examined each session, which occurred twice weekly for a total of six weeks. HLM was used to examine progress during the intervention sessions and determine the effect of initial fluency level on intervention effectiveness. Results suggested students' initial level of fluency impacted intervention effectiveness. That is, for students whose fluency levels fell within the frustrational range CCC and control conditions resulted in the best performance over time but for children whose fluency was in the instructional range explicit timing was the most effective treatment.

Three studies compared 98 children with spina bifida myelomeningocele (SBM)--a disorder associated with high rates of math disability and spatial deficits--to 94 typically developing children on multidigit subtraction and cognitive addition tasks. Children with SBM were classified into those with reading decoding and math disability, only math disability, and no reading or math disability. Study 1 showed that visual-spatial errors in multidigit arithmetic were not elevated in children with SBM. In Study 2, deficits in accuracy, speed, and strategy-use in single-digit addition characterized groups with math disability regardless of reading status. Accuracy and speed on single-digit addition was strongly related to performance on multidigit subtraction. A math-level matching design in Study 3 revealed less mastery of math facts by the group with SBM. The results are discussed with reference to cognitive and neuropsychological models of math disability.

The aim of this research is to devise a cognitive tool for meeting the diverse needs of learners for comprehending new procedural knowledge. A model of affordances on teaching fraction equivalence for developing procedural knowledge for adding/subtracting fractions with unlike denominators was derived from the results of a case study of an initial prototype of a graphical partitioning model. Offering affordances in our model makes available profitable spaces for learners to interact in ways that meet their needs. This model of affordances was evaluated by a pre-test-post-test control group design to study the performance of the experimental group in learning with the model. Results of the study indicated that the model afforded learners, with various abilities for learning, knowledge of fraction equivalence. The key for mediating the generation of procedural knowledge for adding/subtracting fractions with unlike denominators in working with our cognitive tool was the concept of fraction equivalence and the capability of computing this.

An alternative to the IQ-achievement discrepancy model for identifying students with learning disabilities is the response to intervention (RTI) model. There is a growing literature base in support of the RTI model, and within the reauthorization of IDEA there is an allowance for an RTI approach whereby RTI "may" be used by school psychologists when evaluating students for special education. The research base on RTI is expanding but there has been little empirical work directed toward understanding RTI as it is implemented by school-based practitioners. The current study examined a simple three-phase RTI model. Universal screening in mathematics was used in Phase I in conjunction with a brief skill/performance deficit assessment to identify students in need of instructional intervention. In Phase II, a classwide intervention was implemented to evaluate the extent to which poor academic performance was a function of lack of instruction. Phase III provided intensive intervention to five students who did not respond adequately to the classwide intervention. Only one student did not respond adequately to individual intervention.

This study argues that maximizing early childhood educators' abilities to create social opportunities for co-construction of knowledge rests on two understudied assumptions, one theoretical and one empirical. Theoretically this study rejects the notion of language as an impartial conveyor of knowledge in favor of one in which math and language interact. This alternative framework is termed Math-Mediated Language (MML) and argues that the perception of common terms that adults possess is an important part of the knowledge that practitioners possess about linking conceptually related linguistic and mathematical knowledge. Empirical findings from a survey recording participants' reactions to seven categories of terms with mathematical meanings and three categories of distracter terms were analyzed. The data indicated that when asked to think about math, practitioners more readily accessed words for operation terms than relational terms. Additionally, participants demonstrated stronger tendencies toward additive terms conveying addition or subtraction concepts over multiplicative ones conveying multiplication or division concepts. The findings point to patterns in the ways that participants view mathematical language demonstrating that language interacts with even simple interpretations of basic mathematical terminology. The implications of this are that practitioners' interpretations of everyday language may influence their ability to see opportunities for teaching mathematical concepts not only in the context of an explicit math lesson but throughout the broader early childhood curriculum.

Tested effects of a structured intervention in teaching children to represent addition and subtraction. Found significant differences between two study groups in terms of affecting the addition and subtraction operations and ability to determine which operation was appropriate. Results suggest children's exploration of real world situations should precede practice in arithmetical symbol manipulation. (Author/SD)