A LONGITUDINAL STUDY OF THE DEVELOPMENT OF MATHEMATICS STRATEGIES AND UNDERLYING COUNTING SCHEMES

The purpose of this study was to determine whether patterns of strategy use in second, third and fourth grade children’s arithmetic supported Steffe’s model of numerical development. In addition to student-generated strategies, we looked at commonly taught algorithms not considered in Steffe’s model to determine whether these algorithms reflected underlying schemes. 206 children were assessed on their strategy use while solving arithmetic problems in the second grade, the third grade and again in the fourth grade. Factor analyses indicated some support for Steffe’s model, but there were exceptions. Not all children progressed in their number knowledge and strategies as a function of schooling and schooling appeared to affect the types of strategies children use independent of underlying schemes.     

Grade level, study time, and grade retention and their effects on motivation, self-regulated learning strategies, and mathematics achievement: a structural equation model

The present study complements previous research findings with new data to improve our understanding of the relationship between motivational variables and academic performance in math mediated by self-regulated learning (SRL). A structural equation model with predictor (i.e., grade retention, grade level, and study time), process (i.e., perceived usefulness of SRL strategies, self-efficacy for the use of SRL strategies, and reported use of SRL), and product variables (i.e., academic achievement in mathematics) is proposed. The model was analyzed in two samples of data (calibration and validation samples). The first sample served to fit and respecify the model, and the second one was used to analyze the consistency of the findings of the first sample. A sample of 756 middle school Portuguese students participated in the current study. The results indicate that SRL is positively and significantly related to academic achievement and that the latter is, in turn, powerfully determined by perceived usefulness and self-efficacy, although students’ reported use of SRL strategies decreases from 7th to 9th grades. Self-efficacy and perceived usefulness of SRL strategies were also found to decrease as grade retention increased. These results are discussed with regard to the relevance of self-efficacy and perceived usefulness of SRL strategies in increasing academic achievement.     

To what extent does children’s spelling improve as a result of learning words with the look,say, cover,write, check,fix strategy compared with phonological spelling strategies?

Abstract

The ubiquitous weekly spelling test assumes that words are best learned by memorisation and testing but is this the best way? This study compared two well-known approaches to spelling instruction, the rule based and visual memory approaches. A group of 55 seven-year-olds in two Year 3 classrooms was taught spelling in small groups for three lessons a week, 20-min per lesson, over ten weeks. In the first intervention, students learned statistically likely spelling strategies for vowel sounds, syllable breaking strategies, and the doubling rule. In the second intervention, students used a look, say, cover, write, check, fix strategy, listed words in alphabetical order, and wrote them in sentences. The control group completed non-spelling activities. Results showed that although both intervention groups learned to spell taught words better than the control group, the rule-based approach had greater transfer to spelling of new words for both proficient and less proficient spellers.     

Spelling Remediation: a systematic long‐term approach to teaching spelling to an aphasic boy

TJ, a 10‐years, 7‐months‐old spelling‐disabled boy suffering from aphasia, was exposed to a spelling remediation programme introduced in the autumn term of Grade 4 and concluded in the spring term of Grade 6. A systematic strategy instructional approach was used to teach the boy 65 phonetically irregular words. The remedial spelling programme was successful in developing appropriate spelling strategies which were applied to the processing of these words and maintained at follow‐up at 2 months. Generalisation of programme effects was shown on standardised spelling tests administered during training. However, the programme did not seem to be very successful in lessening the disparity between the boy's spelling performance and average spelling performance for his grade. It was concluded that spelling‐disabled students suffering from neurological impairment probably need more cumulative training than other spelling‐disabled students. The potential role of a verbally orientated strategy approach in spelling remediation was also discussed in relation to a visually orientated approach.     

How children choose among serial recall strategies

A theoretical and methodological approach originally developed to investigate preschoolers' arithmetic strategies was extended to the study of 5-8-year-olds' serial recall strategies. Immediately retrospective verbal reports, together with observation of overt behavior, indicated a number of parallels between strategy choices in the 2 areas. In serial recall, as in arithmetic, most children used multiple approaches. Even among the youngest children, the majority sometimes used the most effective strategy, repeated rehearsal. Changes in strategy choices were adaptive in the sense of children rehearsing more often when not rehearsing on the previous trial had led to incorrect recall. Developmental changes occurred in how often children rehearsed, in how accurately they executed each strategy, and in the type of rehearsal that they used most frequently. Only small changes were apparent, however, in whether the 5-8-year-olds rehearsed at all.     

Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education

Some years ago, Hatano differentiated between routine and adaptive expertise and made a strong plea for the development and implementation of learning environments that aim at the latter type of expertise and not just the former. In this contribution we reflect on one aspect of adaptivity, namely the adaptive use of solution strategies in elementary school arithmetic. In the first part of this article we give some conceptual and methodological reflections on the adaptivity issue. More specifically, we critically review definitions and operationalisations of strategy adaptivity that only take into account task and subject characteristics and we argue for a concept and an approach that also involve the sociocultural context. The second part comprises some educational considerations with respect to the questions why, when, for whom, and how to strive for adaptive expertise in elementary mathematics education.     

The relevance of the Good Strategy User Model to the Teaching of Mathematics

This article briefly reviews the strategic, metacognitive, and knowledge components of good strategy use. Five principles of teaching that follow from this good strategy user model are illustrated by examples from arithmetic instruction. Practitioners are advised: (a) to teach strategies; (b) to teach knowledge about when, where, and how to use strategies; (c) to teach general knowledge about factors; that promote strategy functioning; (d) to teach relevant nonstrategic knowledge; and (e) to have students practice components of good strategy use and the coordination of components. The good strategy user model for math instruction is compared to Polya's conception of how to teach problem solving.     

Writing fluency and quality in kindergarten and first grade: the role of attention,reading, transcription,and oral language

In the present study, we examined the influence of kindergarten component skills on writing outcomes, both concurrently and longitudinally to first grade. Using data from 265 students, we investigated a model of writing development including attention regulation along with students’ reading, spelling, handwriting fluency, and oral language component skills. Results from structural equation modeling demonstrated that a model including attention was better fitting than a model with only language and literacy factors. Attention, a higher-order literacy factor related to reading and spelling proficiency, and automaticity in letter-writing were uniquely and positively related to compositional fluency in kindergarten. Attention and higher-order literacy factor were predictive of both composition quality and fluency in first grade, while oral language showed unique relations with first grade writing quality. Implications for writing development and instruction are discussed.     

Student and school SES,gender, strategy use,and achievement

A multilevel mediated regression model was fit to Programme for International Student Assessment achievement, strategy use, gender, and family‐ and school‐level socioeconomic status (SES). Two metacognitive strategies (i.e., understanding and summarizing) and one learning strategy (i.e., control strategies) were found to relate significantly and positively to achievement. These strategies were used more by females and students attending higher SES schools. In contrast, males and students attending lower SES schools tended to use a greater number of learning strategies that did not relate to achievement, including memorization and elaboration. In addition, the strategies that did not relate to achievement were used more frequently by students from higher SES families. The findings suggest that schools, as opposed to families, may be the primary vehicle for developing effective strategy use practices for students and thus, targeted interventions may be particularly useful for male students attending low SES schools.     

Mathematical tasks designed to foster a reconceptualized view of early arithmetic

We discuss mathematical tasks used in a first mathematics content course for elementary teachers at our university to foster a deep conceptual understanding of early arithmetic, including basic concepts of number, number relationships and strategies, and coordinating units of different rank. Our approach is to immerse our students in a base 8 world for up to six weeks. A key aspect is that we develop base 8 vocabulary. We use base 8 analogs of instructional sequences developed in classroom teaching experiments in the elementary grades that have been proven successful to promote deep conceptual understandings of basic arithmetic and place-value numeration in young children. As a result, our students have unique opportunities to develop a reconceptualized view of early arithmetic and learn how it can be advanced.     

Are first graders' arithmetic skills related to the quality of mathematics textbooks? A study on students’ use of arithmetic principles

A major goal of the first years of schooling is students' development from using counting strategies to using calculation strategies – or even recall – to solve addition or subtraction problems. It requires the perception and usage of arithmetic principles. Although the continued use of counting strategies is problematic for further learning progress, they are frequently used by many students beyond Grade 1. While learning resources possibly play a role in students' development in this field, our knowledge about their impact is limited. In this study we investigated the presented learning opportunities of four different textbooks regarding arithmetic principles in Grade 1, and subsequently the relation to first graders' strategy use by analyzing a one-year dataset of 1614 students from 86 classes. The analyses show differences in the textbooks' quality concerning arithmetic principles as well as a considerable connection to students' strategy use at the end of Grade 1.     

The spelling strategies of francophone dyslexic students

The development of spelling skill is a very difficult task for students with dyslexia. Spelling in French involves the consideration of various types of knowledge, procedures and strategies. This study aims to describe the spelling strategies of 32 dyslexic students (DYS) aged from 8 to 12 years and to establish links between spelling strategies and spelling skill. Students had to spell 24 dictated words and provide comments on the strategy employed for each word. The performances of DYS were compared to 25 children of the same chronological age (CA) and of 24 children of the same reading age (RA). The results show that phonological strategies are the most commonly used by all groups of participants. If no particular strategy is related to the spelling skill of DYS, visuo-orthographic strategy generally accounts for the spelling skill results of CA and RA.     

A cross-sectional study of mathematics achievement, estimation skills, and academic self-perception in students of varying ability

This study investigated students' mathematics achievement, estimation ability, use of estimation strategies, and academic self-perception. Students with learning disabilities (LD), average achievers, and intellectually gifted students (N = 135) in fourth, sixth, and eighth grade participated in the study. They were assessed to determine their mathematics achievement, ability to estimate discrete quantities, knowledge and use of estimation strategies, and perception of academic competence. The results indicated that the students with LD performed significantly lower than their peers on the math achievement measures, as expected, but viewed themselves to be as academically competent as the average achievers did. Students with LD and average achievers scored significantly lower than gifted students on all estimation measures, but they differed significantly from one another only on the estimation strategy use measure. Interestingly, even gifted students did not seem to have a well-developed understanding of estimation and, like the other students, did poorly on the first estimation measure. The accuracy of their estimates seemed to improve, however, when students were asked open-ended questions about the strategies they used to arrive at their estimates. Although students with LD did not differ from average achievers in their estimation accuracy, they used significantly fewer effective estimation strategies. Implications for instruction are discussed.     

Solution strategies and adaptivity in multidigit division in a choice/no-choice experiment: Student and instructional factors

Adaptive expertise in choosing when to apply which solution strategy is a central element of current day mathematics, but may not be attainable for all students in all mathematics domains. In the domain of multidigit division, the adaptivity of choices between mental and written strategies appears to be problematic. These solution strategies were investigated with a sample of 162 sixth graders in a choice/no-choice experiment. Children chose freely when to apply which strategy in the choice condition, but not in the no-choice conditions for mental and written calculation, so strategy performance could be assessed unbiasedly. Mental strategies were found to be less accurate but faster than written ones, and lower ability students made counter-adaptive choices between the two strategies. No teacher effects on strategy use were found. Implications for research on individual differences in adaptivity and the feasibility of adaptive expertise for lower ability students are discussed.     

Response Strategies of Culturally Deprived Children

The response strategies of culturally deprived children as a function of race, sex, and grade level were investigated. A 2x2x3 balanced factorial design was used with twenty children at each grade level (kindergarten, first grade, second grade) . A test situation allowed children to make choices for each of five pairs of stimuli in a tray before receiving reinforcement for the “correct” choice. Chi-square analysis indicated that eleven of the twelve groups were using strategies in their choices at a rate greater than would be expected by chance alone. The groups differed by grade level in the number of times specific strategies were selected, although total number of strategies used did not differ across grade level, race, and sex. Constant strategies were used more often by kindergarten children while alternating strategies were utilized more often by second grade children.     

STRATEGIES FOR SOLVING THREE FRACTION-RELATED WORD PROBLEMS ON SPEED: A COMPARATIVE STUDY BETWEEN CHINESE AND SINGAPOREAN STUDENTS

This paper presents the results obtained from a study comparing the strategies used by 1,070 Chinese students and 1,002 Singaporean students from primary grade 6 to secondary year 2 in solving three fraction-related problems. It is part of the author's Ph.D. study, which involves more word problems on speed. The Chinese students performed better than the Singaporean students on two of the three problems, while the Singaporean students performed better than the Chinese students on the other. The strategy analyses reveal that the Chinese students used the traditional methods like arithmetic and algebraic strategies more frequently than the Singaporean students, whereas the Singaporean students used the model and unitary methods more frequently than the Chinese students. Implications for the teaching and learning of word problems on speed, as well as problem solving, are also provided.     

Using a Think-Aloud with Diverse Students: Three Primary Grade Students Experience Chrysanthemum

Many struggling readers, students with English as a second language, and children with disabilities do not engage in the strategies that good readers use when reading for understanding. Reading comprehension depends upon the students’ ability to successfully use strategies to monitor and control their own comprehension. Teachers need to help students develop skills that will aid in reading comprehension. The think-aloud is one strategy that can help struggling readers improve fluency and comprehension. One of the most important components to determine the success of the think-aloud is the teacher’s ability to model and facilitate the think-aloud procedure. The teacher needs to have a basic understanding of what is meant to be accomplished using this method. This article profiles how three teachers model and facilitate a think-aloud with three struggling readers.     

Reliability and validity of rubrics for assessment through writing

This experimental project investigated the reliability and validity of rubrics in assessment of students’ written responses to a social science “writing prompt”. The participants were asked to grade one of the two samples of writing assuming it was written by a graduate student. In fact both samples were prepared by the authors. The first sample was well written in terms of sentence structure, spelling, grammar, and punctuation; however, the author did not fully answer the question. The second sample fully answered each part of the question, but included multiple errors in structure, spelling, grammar and punctuation. In the first experiment, the first sample was assessed by participants once without a rubric and once with a rubric. In the second experiment, the second sample was assessed by participants once without a rubric and once with a rubric. The results showed that raters were significantly influenced by mechanical characteristics of students’ writing rather than the content even when they used a rubric. Study results also indicated that using rubrics may not improve the reliability or validity of assessment if raters are not well trained on how to design and employ them effectively.     

Spelling without phonology: A study of deaf and hearing children

To what extent does phonology play a role in spelling English words? The written responses of deaf students and groups of hearing children to five tasks were subjected to quantitative and qualitative analyses. The first three tasks were used to see if deaf students utilized phonology when they generated their own words and to compare their spelling performance with that of hearing subjects. The fourth and fifth tasks were designed to compare the spelling performance of deaf and hearing subjects when they were required to reproduce visually presented common words. Results showed that deaf students, who were chronologically much older, were not better spellers than hearing children from the fifth grade. Analysis of data revealed little evidence that the deaf students involved in the present study utilize phonology in spelling. Nor did word-specific visual memory for entire words appears to play a role in spelling by deaf students. Rote visual memory for letter patterns and sequences of letters within words, however, appears to play a role in the spelling by deaf students. It is concluded that sensitivity to the stochastic-dependent probabilities of letter sequences may aid spelling up to certain point but phonology is essential for spelling words whose structure is morphophonemically complex.

     

Strategies and shoes: Can we ever have enough? Teaching and using reading comprehension strategies in general and vocational programmes

This qualitative study examines a reading comprehension strategy use in English as a second language in upper-secondary schools one year after teachers participated in a professional development course. The data comprises observations, teacher narratives, and student interviews. A key finding is that the teachers used a repertoire of strategies from the course, suggesting an impact of the course. A second finding was a clear difference between how students in general and vocational programmes used the strategies taught: the vocational students used them in ways that indicated their relevance to them as learners, while the students in general programmes did not. The findings also suggest how and why students employ strategies. The article discusses implications for strategy instruction and how to enhance the reading proficiency of adolescent readers.