This purpose of this study was to examine the reliability and validity of a curriculum-based measure of word problem solving (CBM-WPS) as an indicator of performance and progress in a sample of 136 ...third-grade students at risk for mathematics difficulties (MDs) instructed in a standards-based mathematics curriculum. Students completed the CBM-WPS measure every 2 weeks across 12 school weeks. Results indicated that the CBM-WPS measure was reliable and significantly correlated with measures of arithmetic WPS, number combinations fluency, and a standardized test of mathematics achievement. Results of growth modeling indicated that students showed significant growth on the CBM-WPS measure, with an average increase of 0.33 problems correct per week. Additional analyses revealed that students identified as high at-risk demonstrated similar growth as students identified as low at-risk. Furthermore, the CBM-WPS growth slopes were a significant predictor of students' spring performance on a standardized test of mathematics achievement, demonstrating their predictive validity. Implications for practice and future research for assessing mathematics skill development are discussed.
This study investigated the role of strategy instruction and working memory capacity (WMC) on word problem solving accuracy in children with (n = 100) and without (n = 92) math difficulties (MD). ...Within classrooms, children in Grades 2 and 3 were randomly assigned to one of four treatment conditions: verbal-only strategies (e.g., underlining question sentence), verbal + visual strategies, visual-only strategies (e.g., correctly placing numbers in diagrams), or untreated control. Strategy interventions included 20 sessions in both Year 1 and Year 2. The intent-to-treat as well as the “as-treated” analyses showed that treatment effects were significantly moderated by WMC. In general, treatment outcomes were higher when WMC was set to a high rather than low level. When set to a relatively high WMC level, children with MD performed significantly better under visual-only strategy conditions and children without MD performed better under verbal + visual conditions when compared to control conditions.
Children's number sense in kindergarten was used to predict their calculation fluency in second grade (N = 198). Using block entry regression, usual predictors of age, reading, memory, and verbal and ...spatial cognition were entered in the first block and number sense measures were added in the second block. Number sense measures contributed a significant amount of variance over and above the more general predictors (26%—42%). Uniquely predictive subareas were active memory for numbers, number knowledge, and number combinations, with number combinations standing out as the strongest single predictor. Number sense screening in kindergarten, using “at-risk” versus “not-at-risk” criteria, successfully ruled out 84% of the children who did not go on to have calculation fluency difficulties and positively identified 52% of the children who later showed fluency difficulties. The relation of early number skills to later calculation fluency has important implications for math screening and intervention.
In many discussions of the ways in which abstraction is applied in computer science (CS), researchers and advocates of CS education argue that CS students should be taught to consciously and ...explicitly move among levels of abstraction (Armoni
Journal of Computers in Mathematics and Science Teaching
,
32
(3), 265–284,
2013
; Kramer
Communications of the ACM
,
50
(4), 37–42,
2007
; Wing
Communications of the ACM
,
49
(3), 33–35,
2006
). In this paper, we describe one way that attention to levels of abstraction could also support learning in mathematics. Specifically, we propose a framework for using abstraction in elementary mathematics based on Armoni’s (
2013
) framework for teaching computational abstraction. We propose that such a framework could address an enduring challenge in mathematics for helping elementary students solve word problems with attention to context. In a discussion of implications, we propose that future research using the framework for instruction and teacher education could also explore ways that attention to levels of abstraction in elementary school mathematics may support later learning of mathematics and computer science.
Various solving strategies in probability of events could complicate the students in choosing the best strategy in problem solutions. This research was carried out to identify the favourite solving ...strategies used by college students to solve probability and event problems. As a preliminary of a case study, only three students were selected as participants. In this study, the participants have learnt the probability topic in secondary schools as they must apply the knowledge in college level. Think aloud analysis was used to evaluate the influence thought and behaviour of the participants when solving the task. An in-depth semi-structured interview was conducted together with the think-aloud analysis to gain rich and thick descriptions from the participants. Protocols were coded and analyzed to determine the strategies used and the reason of choosing those strategies to solve the problems. Results indicated that participants loved to use tree diagram as solving strategies in solving probability problems without knowing the accuracy of its effect. Therefore, educators can plan strategies, set problems, or prepare activities for students to understand the probabilistic situation and probabilistic reasoning to solve problems. Otherwise the students are merely copying procedures presented in class.
Implementing an integrated sequence of concrete-representational-abstract depictions of mathematics concepts (CRA-I) can improve the mathematics achievement of students with disabilities, and ...explicit instructional strategies involving problem-solving heuristics and student verbalizations can help facilitate students’ conceptual understanding of mathematics. Combining CRA-I and explicit instructional strategies may increase students’ conceptual understanding and ability to express mathematical reasoning through writing. This study included three ninth-grade students with disabilities, and employed a multiple-probe design across-participants to investigate a functional relation between an explicit instructional strategy within a CRA-I framework and high school students’ with disabilities proficiency in solving rate of change problems. Results showed that all three students improved their mathematics scores (combined Tau-U effect size = 0.77, p < .001) and maintained improvements during a 1- to 7-week post-instruction phase. Implications for research and practice related to mathematics instruction and intervention specifically for students with learning disabilities are discussed.
In the literature, problem-posing abilities are reported to be an important aspect/indicator of creativity in mathematics. The importance of problem-posing activities in mathematics is emphasized in ...educational documents in many countries, including the USA and China. This study was aimed at exploring high school students' creativity in mathematics by analyzing their problem-posing abilities in geometric scenarios. The participants in this study were from one location in the USA and two locations in China. All participants were enrolled in advanced mathematical courses in the local high school. Differences in the problems posed by the three groups are discussed in terms of quality (novelty/elaboration) as well as quantity (fluency). The analysis of the data indicated that even mathematically advanced high school students had trouble posing good quality and/or novel mathematical problems. We discuss our findings in terms of the culture and curricula of the respective school systems and suggest implications for future directions in problem-posing research within mathematics education.
The aim of this paper is to investigate the pupils' word problem solving strategies in lower secondary school education. Students prior experiences with solving word problems by arithmetic methods ...can create serious difficulties in the transition from arithmetic to algebra. The arithmetical methods are mainly based on manipulation with numbers. When pupils are faced with the methods of algebra they often have difficulty in formulating algebraic equations to represent the information given in word problems. Their troubles are manifested in the meaning they give to the unknown, their interpretation what an equation is, and the methods they choose to set up and solve equations. Therefore they mainly use arithmetical and numerical checking methods to solve word problems. In this situation it is necessary to introduce alternative methods which make the transition from arithmetic to algebra more smooth. In the following we will give a detailed presentation of the false position method. In our opinion this method is useful in the lower secondary school educational processes, especially to reduce the great number of random trial-and-error problem solving attempts among the lower secondary school pupils. We will also show the results of some problem solving activities among grade 6-8 pupils. We analysed their problem solving strategies and we compared our findings with the results of other research works.
The present study attempted to explore the relationship between computational estimation and problem solving in a sample of Year 6 children and adults (N=72). For this purpose, participants were ...presented with two tasks which asked them to estimate the computational result in eight arithmetic operations (Computational Estimation Task) as well as to provide solution to eight mathematical word problems, all coming from four different areas of mathematics (Problem Solving Task). The analysis of the results showed that adult participants accomplished more successful estimations compared to children, mostly with the use of rounding and algorithm strategies in both age groups. Age differences were also found in the Problem Solving Task with geometry and arithmetic problems being the most difficult for children and adults, respectively. Last, a significant positive correlation was revealed between estimation computational ability and problem solving ability: participants who had great success in computational estimation tended to indicate high success rates in problem solving. Educational implications are further discussed in terms of the role of computational estimation and problem solving in mathematics learning.
Differential bundle functioning (DBF) analyses were conducted to determine whether seventh and eighth grade second language learners (SLLs) had lower probabilities of answering bundles of math word ...problems correctly that had heavy language demands, when compared to non-SLLs of equal math proficiency. Math word problems on each of four test forms (two at Grade 7 and two at Grade 8) were bundled together if they used the passive voice, conditional clause, relative clause, or a combination of any two. The results showed that the average total scores for SLLs was significantly lower than that for non-SLLs on each test form. However, only two bundles (passive voice and conditional clause at Grade 7 on Form 2) indicated statistically significant DBF against SLLs in favor of matched non-SLLs. An additional step was taken to determine whether the two bundles that showed statistically significant DBF against SLLs had biased the mean total scores for this group.The Walker, Zhang, Banks, and Cappaert (2012) procedure established for this purpose showed that this was not the case. Implications of the results are provided as well as suggestions for future research.
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