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Children's knowledge of the ordinal relations among number symbols is related to their mathematical learning. Ordinal knowledge has been measured using judgment (i.e., decide whether a sequence of three digits is in order) and ordering tasks (i.e., order three digits from smallest to largest). However, the question remains whether performance on these two ordinal tasks tap into similar cognitive processes. Canadian children (N = 87; Age M = 8.7 years, Grade 3) completed symbolic number tasks (i.e., number comparison, ordering, and order judgment) and measures of arithmetic fluency (i.e., addition and subtraction) and working memory (i.e., digit span backward). For both ordinal tasks, there was a reverse distance effect for ordered sequences such that children responded faster to adjacent than to non-adjacent sequences (e.g., 2 3 4 vs. 4 7 9) and a canonical distance effect for unordered sequences such that children responded faster to non-adjacent than to adjacent sequences (e.g., 4 2 3 vs. 4 9 7). Working memory and number comparison each predicted unique variance in the ordinal measures (ordering, order judgment, and a latent ordinal factor based on the two measures). Furthermore, ordinal skills superseded the role of number comparison as the key predictor of arithmetic, controlling for children's gender and working memory skills. In summary, although both ordering and order judgment tasks index ordinal knowledge, a latent factor that excludes task-specific error may be a better index than either task separately.

We investigated the effect of conceptual transparency in the physical structure of manipulatives on place-value understanding in typically developing children and those at risk for mathematics learning disabilities. Second graders were randomly assigned to one of three manipulatives conditions: (a) attachable beads that did not make the denominations or ones in the denominations transparent, (b) pipe cleaners that made only the denominations transparent, and (c) string beads that made both the denominations and the ones in the denominations transparent. Participants used the manipulatives to represent double- and triple-digit numerals. Statistical analyses indicated that the transparency of the denominations, but not the transparency of the ones in the denominations, is responsible for children's number representation and place-value understanding. Descriptive analyses of their responses revealed that the at-risk children were at a greater disadvantage than their typically developing peers with the attachable beads, failing to use place-value concepts to interpret their representations.

The hundreds chart is a valuable tool for teachers in elementary mathematics classrooms. The present study investigated the effects of spatial configurations of virtual hundreds charts on children's knowledge of base-ten concepts. Kindergarten and first-grade students were assigned to three instructional conditions in which they worked with a researcher to solve arithmetic word problems. The charts used in the conditions differed by the spatial configuration of the numbers: (a) top down, with numbers increasing from top to bottom; (b) bottom up, with increases from bottom to top; and (c) explicit bottom-up, with a bottom-up chart and an explicit visual cue that prompted a comparison between the "up-is-more" structure to a cylinder filling with water. Participants were more likely to count by tens than by ones to solve problems when the numbers in the chart increased from top to bottom than when they increased from bottom to top. No effects of spatial configuration on place-value knowledge was observed. One account of the findings is that participants were more accustomed to the movements in the top-down chart because they matched the direction in which children learn to read. The role of spatial affordances of instructional materials in children's mathematics learning is discussed.

Canadian students enrolled in either French-immersion or English-instruction programs were followed from Grades 2 to 3 (M[subscript age] = 7.8 years to 8.9 years; N = 244; 55% girls). In each grade, students completed two mathematical tasks that required oral language processing (i.e., word-problem solving and number transcoding from dictation) and two that did not (i.e., arithmetic fluency and number line estimation). Students in both English-instruction (n = 92) and French-immersion programs (n = 152) completed tasks in English. Students in French-immersion programs also completed word-problem solving and transcoding tasks in French. The models were framed within the Pathways to Mathematics model, with a focus on the linguistic pathways for students in English-instruction and French-immersion programs. For tasks with oral language processing, performance in Grade 3 was predicted by students' English receptive vocabulary for both English-instruction and French-immersion students, even when French-immersion students were tested in French, controlling for performance in Grade 2. In contrast, for tasks without oral language processing, receptive vocabulary in either English or French did not predict performance in Grade 3, controlling for performance in Grade 2. These results have implications for teaching mathematics within the context of immersion education.

Language skills play an important role in mathematics development. Students (7 to 10 years of age) learning school mathematics either in the same language used at home (first-language learners; n = 103) or in a different language (second-language learners; n = 57) participated in the study. Relations among cognitive skills (i.e., receptive vocabulary, working memory, quantitative skills), domain-specific language skills (i.e., mathematical vocabulary, mathematical orthography), word-problem solving, arithmetic fluency, and word reading were investigated. Second-language learners had lower scores on measures with strong language components (i.e., receptive vocabulary, subitizing, and word-problem solving) than first-language learners, whereas they performed equally well on other tasks. Mathematical vocabulary and receptive vocabulary contributed to word-problem solving success for first-language learners, whereas only receptive vocabulary in the language of instruction related to mathematical outcomes for second-language learners. Mathematical vocabulary was related to arithmetic fluency for both groups, but mathematical orthography was not. For both groups, students' word reading was predicted by receptive vocabulary but not by quantitative skills, highlighting the domain-specific nature of these skills. These findings have implications for supporting mathematical learning in second-language students.

Math anxiety is a common correlate of math performance. However, the causal direction of this relation is unclear.Research with young students is limited but critical for determining patterns of development. Students (N = 147) completedmath measures (i.e., number comparison, arithmetic fluency, and math problem solving) and math anxiety assessmentstwice, first in grade 2 (Mage = 7 years:10 months) and then a year later in grade 3. Correlational analysis revealed thatmath anxiety is related only to arithmetic fluency, related to other types of math performance. Cross-lagged analyses wereconducted to evaluate causal relations between math anxiety and arithmetic fluency. These analyses showed that arithmeticfluency in grade 2 predicted change in math anxiety from grade 2 to grade 3, however, math anxiety in grade 2 did notpredict the change in arithmetic fluency from grade 2 to grade 3. These results suggest that math anxiety may be the resultof poor math performance.

In the present research, we provide empirical evidence for the process of symbolic integration of number associations, focusing on the development of simple addition (e.g., 5 + 3 = 8), subtraction (e.g., 5 - 3 = 2), and multiplication (e.g., 5 × 3 = 15). Canadian children were assessed twice, in Grade 2 and Grade 3 (N = 244; 55% girls). All families were English-speaking, and parent education levels ranged from high school to postgraduate, with a median of community college. In Grade 2, children completed general cognitive tasks (i.e., receptive vocabulary, working memory, nonverbal reasoning, and inhibitory control). In both grades, children completed single-digit addition and complementary subtraction problems. In Grade 3, they completed single-digit multiplication problems and measures of applied mathematics, specifically, word-problem solving, algebra, and measurement. We found that addition and subtraction were reciprocally related (controlling for cognitive skills). Subtraction fluency predicted multiplication in Grade 3, whereas addition fluency did not. In Grade 3, both subtraction and multiplication fluency were predictors of applied mathematics, with multiplication partially mediating the relation between subtraction and applied mathematics performance. These findings support the view that learning arithmetic associations is a hierarchical process. As students practice each new skill, individual differences reflect the integration of the novel component into the developing associative network.

Children enrolled in language-immersion programmes may be required to learn math in the immersion language. Fol-lowing the framework of the Pathways Model (LeFevre et al., 2010; Sowinski et al., 2014), the goal of the present studywas to understand how instructional language supports math learning by comparing patterns of performance of immersionand non-immersion students. Participants included 182 grade 2 students (Mean age= 7.8 years): 108 students were en-rolled in French immersion programs and were learning math in French (their second language) and 74 were enrolled innon-immersion programs and were learning math in English (their home language). Participants were tested on a numberof general cognitive measures as well as math specific outcome measures. Results show that overall, across both immer-sion and non-immersion students, linguistic, quantitative and working memory components contributed to math problemsolving. However, within the linguistic component there were differences between the direct and indirect pathways.

Little is known about whether manipulatives can support children's inferences of the mathematical structure of word problems. The objective was to test the effects of using manipulatives during problem solving on students' understanding of the additive relationships in word problems. Third and fifth graders (N = 45) solved one-step addition and subtraction word problems that described either a mathematical action or relation. Children in each grade were randomly assigned to either a manipulatives or a paper-and-pencil condition. Problem structure understanding was assessed by the degree to which students' strategies and verbal justifications reflected the quantitative relationships described in the problem text. Performance of the fifth graders was not impacted by the use of manipulatives, but in the third grade, strategy performance with manipulatives was superior to performance without manipulatives on Relation problems. The degree to which the third graders' justifications were aligned with corresponding problem structure was positively impacted by manipulatives regardless of problem type.

Solving word problems is challenging for many children, but particularly for those with language difficulties. The objective was to examine the nature of the challenges experienced by children with language difficulties as they solved word problems in the context of a developmental‐trajectory instructional sequence. We recruited 45 third graders with (n = 17) and without (n = 28) language difficulties from public and private schools and one speech‐language therapy clinic. They solved word problems with additive change and compare structures and language that was either consistent or inconsistent with the structure. The instruction was based on successive simplifications to the problems whenever the child was unable to apply a structurally appropriate strategy: removing irrelevant literal information from the problem, simplifying the text syntactically, and removing irrelevant numerical information. The performance of children with language difficulties was lower than that of the typically developing children and the simplifications did not support performance to the same extent. The presence of irrelevant numerical information was challenging, especially for children with language difficulties. Removing irrelevant information from the problem was followed by stronger performance on change problems and those with consistent language, revealing that the instructional supports were not sufficient for compare and inconsistent problems. [ABSTRACT FROM AUTHOR]

The objective of the present study was to examine the relationship between mathematics anxiety and the procedural and conceptual knowledge of fractions in prospective teachers. Thirty-two preservice teachers enrolled in an elementary mathematics methods course were administered the Revised Mathematics Anxiety Rating Scale (RMARS, Baloglu, 2002). Procedural and conceptual knowledge of fractions was assessed with a validated paper and pencil test (Saxe, Gearhart, & Nasir, 2001) and with four additional problems created by the researchers. Results indicated that as mathematics anxiety scores increased, scores on the validated measure of procedural knowledge of fractions decreased. The same relationship was found between the RMARS and the validated measure of conceptual knowledge. The findings provide some insight on the cognitive and pedagogical factors associated with mathematics anxiety in the preservice teacher population, underscoring the importance of facilitating their proficiency in both mathematical procedures and concepts. (Contains 4 tables and 1 figure.)

The effects of prior encodings of manipulatives (red and blue plastic chips) on children's ability to use them as representations of quantity were tested. First graders (N = 73) were assigned to four conditions in which the encoding of plastic chips was experimentally manipulated. All children then participated in an addition activity that relied on the chips' quantitative representations. Children who were given quantitative encodings were better able to use the chips with meaning relative to those who played with them during the encoding phase. Children who perceived the chips as toys after encoding were similarly hindered on their ability to use them as quantities. The findings suggest that prior encodings are predictive of children's quantitative representations of manipulatives.

Transcoding is the process of translating between spoken and written numbers, and it is correlated with other mathematical skills. In the present study, we investigated the link between French number writing of 49 students in the third grade (aged 7–9 years) and their language skills. Transcoding in French is of particular interest because the spoken number language system does not completely correspond to that of the written digits (e.g., quatre-vingt-dix [four-twenty-ten] and 90). We hypothesised that the complex linguistic structure of spoken numbers in French would be challenging for students who are learning to transcode. First and second French-language learners' accuracy and errors were recorded during a writing task of 3- to 7-digit numbers. Children also completed linguistic tests (e.g., receptive vocabulary, receptive syntax). Results showed that first- and second-language learners did not differ in their transcoding accuracy. Number size, decade complexity of stimulus number words in French (i.e., numbers containing a complex decade, operationalized as a number between soixante-dix, 70, and quatre-vingt-dix-neuf, 99), and receptive vocabulary predicted children's French transcoding skills. Students were more likely to produce errors (e.g., 68 or 6018 for 78) when they transcoded complex decade numbers compared with simple decade numbers. When an error was made on the complex decade portion of a number, it was likely a lexical error. In conclusion, third graders, both first- and second-language learners, found complex decade numbers challenging and their performance was related to their general vocabulary skills. [ABSTRACT FROM AUTHOR]

This paper describes the implementation and quasiexperimental evaluation of a three-week instructional project designed in accordance with theories and assumptions of constructivism and socially situated cognition. Our goal was to develop students' ability to reason about real-life problems, where "good reasoning" was conceptualized in terms of a normative thinking model derived from cognitive research in decision making, probabilistic reasoning, and argumentation. In the spring of 1994, students in two middle school classrooms worked in teams that collected evidence, constructed arguments, and prepared presentations while engaged in activities that culminated in a mock legislative hearing. Through instruction and mentoring, students were encouraged to use statistics and probability as tools for reasoning. The effectiveness of the program was evaluated by comparing the written arguments of students from the two treatment classrooms with those of students from eight comparison classrooms. Students' arguments were scored in terms of how well they captured essential features of model reasoning and avoided particular thinking fallacies. That the reasoning abilities of students developed through social negotiation and shared problem solving was su

It is not surprising, as research has shown, that fractions are one of the most difficult of the elementary school math topics to teach and learn in ways that are meaningful. The authors reference a work by James Hiebert, "Mathematical, Cognitive, and Instructional Analyses of Decimal Fractions" (1992), that mathematical concepts should be linked to symbols and rules during teaching to foster students' authentic, personally meaningful understanding. More specifically, Dr. Hiebert suggested three precise sites, (symbol interpretation site, procedural execution site, and solution evaluation site) or points in real time, during problem solving where such links could be especially useful to learning. His sites theory aligns nicely with best teaching practices in mathematics for pictures and manipulatives can help children learn the meaning of the mathematical symbols they use. They can also help children think about problems and reason about ways to solve them. However, Hiebert's theory goes beyond simply presenting students with pictures and objects to understand symbols and think about problems, but also helps children "reflect" on what they did with their pictures or concrete objects. This article presents teaching methods designed to take the learner through the entire problem-solving sequence, beginning with the interpretation of mathematical symbols (site 1), followed by the use of the procedures (site 2), and through to the evaluation of the solution (site 3) and helps teachers make explicit links between concepts and procedures at each of these three points while students are working on problems in class. The authors conclude that this method has proven to be effective in that students showed a more meaningful understanding of fractions after the unit, and they propose that this method can be used with most, if not all, mathematical topics, and likely at any grade level.

Background: Although making explicit links between procedures and concepts during instruction in mathematics is important, it is still unclear the precise moments during instruction when such links are best made. Aims: The objective was to test the effectiveness of a 3-week classroom intervention on the fractions knowledge of grade 5/6 students. The instruction was based on a theory that specifies three sites during the learning process where concepts and symbols can be connected (Hiebert, 1984): symbol interpretation, procedural execution, and solution evaluation. Sample: Seventy students from one grade 5/6 split and two grade 6 classrooms in two public elementary schools participated. Method: The students were randomly assigned to treatment and control. The treatment (Sites group) received instruction that incorporated specific connections between fractions concepts and procedures at each of the three sites specified by the Sites theory. Before and after the intervention, the students' knowledge of concepts and procedures was assessed, and a random subsample of 30 students from both conditions were individually interviewed to measure their ability to make specific connections between concepts and symbols at each of the three sites. Results: While all students gained procedural skill (p less than 0.001), the students in the Sites condition acquired significantly more knowledge of concepts than the control group (p less than 0.01) and were also better able to connect fractions symbols to conceptual referents (p less than 0.025). Conclusions: The current study contributes to the literature because it describes when it might be important to link concepts and procedures during fractions instruction. (Contains 4 figures, 2 tables, and 4 footnotes.)

This study evaluated the effects of Adapted Reciprocal Teaching (ART) on the receptive and expressive flight-word vocabulary of 1st-grade students. During ART, classroom interactions produced narrative contexts within which students assumed responsibility for applying new flight words in personally meaningful ways. Students in the control group also received interactive storybook instruction, but classroom interactions were led primarily by the teacher and focused only on the meanings of unfamiliar flight words. Students were assessed using the Receptive Flight Word Vocabulary Test (RFVT) and the Expressive Flight Word Vocabulary Test (EFVT). The data demonstrated that after the instructional intervention, the students in the ART group acquired significantly more target words (as measured by performance on the RFVT and the EFVT) than students in the control group. Results are interpreted in light of generative learning theory, and practical implications for introducing vocabulary in the early school years are addressed.

Children’s knowledge of the ordinal relations among number symbols is related to their mathematical learning. Ordinal knowledge has been measured using judgment (i.e., decide whether a sequence of three digits is in order) and ordering tasks (i.e., order three digits from smallest to largest). However, the question remains whether performance on these two ordinal tasks tap into similar cognitive processes. Canadian children (N = 87; Age M = 8.7 years, Grade 3) completed symbolic number tasks (i.e., number comparison, ordering, and order judgment) and measures of arithmetic fluency (i.e., addition and subtraction) and working memory (i.e., digit span backward). For both ordinal tasks, there was a reverse distance effect for ordered sequences such that children responded faster to adjacent than to non-adjacent sequences (e.g., 2 3 4 vs. 4 7 9) and a canonical distance effect for unordered sequences such that children responded faster to non-adjacent than to adjacent sequences (e.g., 4 2 3 vs. 4 9 7). Working memory and number comparison each predicted unique variance in the ordinal measures (ordering, order judgment, and a latent ordinal factor based on the two measures). Furthermore, ordinal skills superseded the role of number comparison as the key predictor of arithmetic, controlling for children’s gender and working memory skills. In summary, although both ordering and order judgment tasks index ordinal knowledge, a latent factor that excludes task-specific error may be a better index than either task separately. [ABSTRACT FROM AUTHOR]

This paper presents some results of research aimed at contributing to the development of a professional knowledge base for teachers of elementary mathematics methods courses, called here "teacher educators." We propose that a useful unit of analysis for this knowledge could be the tasks in which teacher-educators engage pre-service teachers, and that the tasks could be indexed by a system of nested categories of actions required in the tasks and the analytic tools for carrying them out. The paper develops the seed of such an indexing framework, based on analyses of tasks in two methods courses. We argue that this seed is robust enough to be expandable, in the sense that categories could be added to it without destroying the existing ones. An expanded index, accompanied by a bank of tasks, would facilitate the development of a unified discourse to bring together, represent, and communicate, the vast experience of teacher educators.

In this study, we implemented one-on-one fractions instruction to eight preservice teachers. The intervention, which was based on the principle of Progressive Formalization (Freudenthal, 1983), was centered on problem solving and on progressively formalizing the participants' intuitive knowledge of fractions. The objectives of the study were to examine the potential effects of the intervention and to uncover specific difficulties experienced by the preservice teachers during instruction. Results revealed improvement on one measure of conceptual knowledge, but not on a transfer task, which required the teachers to generate word problems for number sentences involving fractions. In addition, the qualitative analysis of the videotaped instructional sessions revealed a number of cognitive obstacles encountered by the participants as they attempted to construct meaningful solutions and represent those solutions symbolically. Based on the findings, specific suggestions for modifying the intervention are provided for mathematics teacher educators. (Contains 6 tables and 6 figures.)

This study extended the work of S. Siddiqui, R. F. West, and K. E. Stanovich (1998), who studied the link between general print exposure and syllogistic reasoning. It was hypothesized that exposure to certain text structures that contain well-delineated logical forms, such as popularized scientific texts, would be a better predictor of deductive reasoning skill than general print exposure, which is not sensitive to the quality of an individual's reading activity. Furthermore, it was predicted that the ability to generate explanatory bridging inferences while reading would also be predictive of syllogistic reasoning. Undergraduate students (N = 112) were tested for vocabulary, nonverbal cognitive ability, exposure to general print, exposure to popularized scientific literature, and the ability to comprehend texts distinguished by the number of inferences that must be generated to support comprehension. Hierarchical multiple regression analyses showed that a combined measure of exposure to general and scientific literature was a significant predictor of syllogistic reasoning ability. Additionally, the ability to comprehend high-inference-load texts was related to solving syllogisms that were inconsistent with world knowledge, indicating an overlap in deductive reasoning skill and text comprehension processes.

The objective of this study was to examine the nature of preservice teachers' evaluations of elementary mathematics problems using the Mathematical Tasks Framework (MTF), a model designed to discriminate among tasks according to their cognitive complexity. We also tested the relationship between mathematics content knowledge and problem length on the preservice teachers' evaluations. Twenty-six undergraduate students enrolled in an elementary mathematics methods course at a large urban university were introduced to the MTF and cognitive complexity during a class lecture and were subsequently required to sort 32 mathematics problems according to the framework. Results demonstrated that overall, the preservice teachers had more difficulty accurately classifying problems considered to represent high levels of cognitive complexity compared to problems that were less complex. Those with strong mathematics content knowledge, as measured by a standardized test, were able to sort the problems more accurately than those with weaker content knowledge. Two open-ended items assessing content knowledge were not related to sorting performance. Finally, the preservice teachers were influenced by the surface characteristic of task length; the data indicated that the teachers tended to label short problems as less cognitively demanding and long problems as more so. Implications for preservice professional development include an increased emphasis on mathematics content knowledge as well as expert modeling of the identification of deep conceptual principles at the heart of the mathematics curriculum.

The authors implemented a cognitive apprenticeship learning community (Collins, Brown, & Newman, 1989) in a class of preservice teachers at the University of Missouri - Columbia to enhance their argumentation and critical thinking skills about complex, educational problems. A detailed rubric based on the literature in argumentation and scientific reasoning was developed to evaluate the students' performance before and after the intervention. The rubric was designed to measure students' (a) conceptions and use of evidence, (b) notions about research and its applicability in evaluating complex social problems, and (c) ability to consider alternative perspectives. This article describes the intervention, the theoretical underpinnings of the rubric, and the use of the rubric in measuring the development of critical thinking in a group of preservice teachers. Strengths and weaknesses of the rubric are outlined and avenues for future research are discussed.

This case study addressed the meanings two preservice teachers ascribed to four expert strategies used in reciprocal teaching (questioning, summarizing, clarifying, and predicting) and four learning principles upon which reciprocal teaching is built (cognitive apprenticeship, scaffolding, zone of proximal development, and proleptic teaching). Results highlight conceptual growth and misinterpretations of the procedures and learning principles as the student teachers developed their understanding across training sessions. (SM)