EXAMINING EIGHTH GRADE KUWAITI STUDENTS’ RECOGNITION AND INTERPRETATION OF REASONABLE ANSWERS

This research documents Kuwaiti eighth grade students’ performance in recognizing reasonable answers and the strategies they used to determine reasonableness. The results from over 200 eighth grade students show they were generally unable to recognize reasonable answers. Students’ performance was consistently low across all three number domains (whole numbers, fractions, and decimals). There was no significant difference in students’ performance on items that focused on the practicality of the answers or on items that focused on the relationships of numbers and the effect of operations, or on both. Interview data revealed that 35% of the students’ strategies were derived from two criteria for judging answers for reasonableness: the relationships of numbers and the effect of operations, and the practicality of the answers. They used strategies such as estimation, numerical benchmarks, real-world benchmarks, and applied their understanding of the meaning of operations. However, over 60% of the students’ strategies were procedurally driven. That is, they relied on algorithmic techniques such as carrying out paper-and-pencil procedures. Additionally, some of the students’ strategies reflected misunderstandings of how and when to apply certain procedures. Given these findings, mathematics education in Kuwait should shift the emphasis from paper-and-pencil procedures and provide systematic attention to the development of number sense and computational estimation so Kuwaiti students will be more adept at recognizing reasonable answers.     

NUMBER SENSE-BASED STRATEGIES USED BY HIGH-ACHIEVING SIXTH GRADE STUDENTS WHO EXPERIENCED REFORM TEXTBOOKS

The purpose of this study was to explore strategies used by high-achieving 6th grade students in the United Arab Emirates (UAE) to solve basic arithmetic problems involving number sense. The sample for the study consisted of 15 high-achieving boys and 15 high-achieving girls in grade 6 from 2 schools in the Emirate of Abu Dhabi, UAE. Data for the study were collected through individual interviews in which students were presented with 10 basic problems. The results showed that a low percentage of solutions involved aspects of number sense such as appropriate use of benchmarks; using numbers flexibly when mentally computing, estimating, and judging reasonableness of results; understanding relative effect of operations; and decomposing or recomposing numbers to solve problems. It was also found that students were highly dependent on school-taught rules. In many cases, these rules were confused and misused.     

Feeling number: grounding number sense in a sense of quantity

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.     

An investigation of 3rd‐grade Taiwanese students’ performance in number sense

The main purpose of this study was to investigate the number sense performance of 3rd‐graders in Taiwan, and to diagnose areas of weakness or deficiency in number sense development. A total of 808 3rd‐graders participated in this study. The results indicated that these students did not perform well on each of the five number sense components (correct rates approx. 34%), and they appeared worst on the performance of “Judging the reasonableness of computational results”. Boys and girls did not show any appreciable difference in their ability to solve number sense problems. The importance of number sense should be highlighted both by teachers and in textbooks and more time and opportunity provided for students to work on this type of exercise at lower grade levels. This would require that “drill and practice” exercises in mathematics should not indeed be over‐ taught, and the teaching of number sense to children should begin as early as possible.     

Productive failure in mathematical problem solving

This paper reports on a quasi-experimental study comparing a “productive failure” instructional design (Kapur in Cognition and Instruction 26(3):379–424, 2008) with a traditional “lecture and practice” instructional design for a 2-week curricular unit on rate and speed. Seventy-five, 7th-grade mathematics students from a mainstream secondary school in Singapore participated in the study. Students experienced either a traditional lecture and practice teaching cycle or a productive failure cycle, where they solved complex problems in small groups without the provision of any support or scaffolds up until a consolidation lecture by their teacher during the last lesson for the unit. Findings suggest that students from the productive failure condition produced a diversity of linked problem representations and methods for solving the problems but were ultimately unsuccessful in their efforts, be it in groups or individually. Expectedly, they reported low confidence in their solutions. Despite seemingly failing in their collective and individual problem-solving efforts, students from the productive failure condition significantly outperformed their counterparts from the lecture and practice condition on both well-structured and higher-order application problems on the post-tests. After the post-test, they also demonstrated significantly better performance in using structured-response scaffolds to solve problems on relative speed—a higher-level concept not even covered during instruction. Findings and implications of productive failure for instructional design and future research are discussed.     

Sixth-Grade Students’ Views of the Nature of Engineering and Images of Engineers

This study investigated the views of the nature of engineering held by 6th-grade students to provide a baseline upon which activities or curriculum materials might be developed to introduce middle-school students to the work of engineers and the process of engineering design. A phenomenographic framework was used to guide the analysis of data collected from: (1) a series of 20 semi-structured interviews with 6th-grade students, (2) drawings created by these students of “an engineer or engineers at work” that were discussed during the interviews, and (3) field notes collected by the researchers during the interviews. The 6th-grade students tended to believe that engineers were individuals who make or build products, although some students understood the role of engineers in the design or planning of products, and, to a lesser extent in testing products to ensure that they “work” and/or are safe to use. The combination of drawings of “engineers or engineering at work” and individual interviews provided more insight into the students’ views of the nature of engineering than either source of data would have offered on its own. Analysis of the data suggested that the students’ concepts of engineers and engineering were fragile, or unstable, and likely to change within the time frame of the interview.     

GEOMETRIC AND ALGEBRAIC APPROACHES IN THE CONCEPT OF “LIMIT” AND THE IMPACT OF THE “DIDACTIC CONTRACT”

The present study explores students’ abilities in conversions between geometric and algebraic representations, in problem- solving situations involving the concept of “limit” and the interrelation of these abilities with students’ constructed understanding of this concept. An attempt is also made to examine the impact of the “didactic contract” on students’ performance through the processes they employ in tackling specific tasks on the concept of limit. Data were collected from 222 12th-grade high school students in Greece. The results indicated that students who had constructed a conceptual understanding of limit were the ones most probable to accomplish the conversions of limits from the algebraic to the geometric representations and the reverse. The findings revealed the compartmentalized way of students’ thinking in non-routine problems by means of their performance in simpler conversion tasks. Students who did not perform under the conditions of the didactic contract were found to be more consistent in their responses for various conversion tasks and complex problems on limits, compared to students who, as a consequence of the didactic contract, used only algorithmic processes.     

中小学"数学情境与提出问题"教学的实验研究

This research tends to make the experimental study on the mathematics teaching model of “situated creation and problem-based instruction” (SCPBI), namely, the teaching process of “creating situations—posing problems—solving problems—applying mathematics”. It is aimed at changing the situation where students generally lack problem-based learning experience and problem awareness. Result shows that this teaching model plays a vital role in arousing students’ interest in mathematics, improving their ability to pose problems and upgrading their mathematics learning ability as well.      

Number Sense Strategies Used by Pre-Service Teachers in Taiwan

This study examined number sense strategies and misconceptions of 280 Taiwanese pre-service elementary teachers who responded to a series of real-life problems. About one-fifth of the pre-service teachers applied number sense-based strategies (such as using benchmarks appropriately or recognizing the number magnitude) while a majority of pre-service teachers relied on rule-based methods. This finding is consistent with earlier studies in Taiwan that fifth, sixth, and eighth grade students tended to rely heavily on written methods rather than using number sense-based strategies. This study documents that the performance of pre-service elementary teachers on number sense is low. If we want to improve elementary students’ knowledge and use of number sense, then action should be taken to improve the level of their future teachers’ number sense.     

A Teacher’s Mediation of a Thinking-Aloud Discussion in a 6th Grade Mathematics Classroom

This article presents a Vygotsky-inspired analysis of how a teacher mediated a “thinking aloud” whole-group discussion in a 6th grade mathematics classroom. This discussion centered on finding patterns in a triangular array of consecutive numbers as a phase towards building recursive and direct algebraic formulas. By a “thinking aloud” discussion we mean a conversation wherein students exchange and further develop ideas-in-the-making with their peers under the teacher’s guidance. Drawing upon Halliday’s systemic functional linguistics (SFL), we treated the selected discussion as a text. We then analyzed how the teacher mediated the conjoined making of this text so that it served as an interpersonal gateway for students to practice searching for patterns and signifying these patterns in propositional form. This analysis was guided by the following questions: How did the discussion as a text-in-the-making mean what it did? What was the role of the teacher in the conjoined making of this text? Our study illustrates the power of SFL for capturing the inner grammar of instructional conversations thus illuminating the complexities and subtleties of the teacher’s role in mediating semiotic mediation in mathematics classrooms.     

A Quality Math Curriculum in Support of Effective Teaching For Elementary Schools

This paper presents a curriculum, textbook and test result analysis for the new (to California) elementary school “Key Standard” mathematics curriculum, transplanted in 1998 from it's foreign roots in Asia and Europe, locations with far different cultural and economic backgrounds. Based on topic analysis methods developed by Michigan State University, this curriculum is a “quality” curriculum, since it is closely aligned with the curriculum of the six leading TIMSS math countries. Five-year test results are presented for two cohorts totaling over 13,000 students, all from four “early adoption” urban districts where 68% of the students were economically disadvantaged. Included are two cohorts of English learning immigrants totaling over 4,400 students. Performance was found to be statistically superior to similar (control) districts which continued with the old 1991 curriculum and textbooks (0.003 < p < 0.015). The focus of this paper is on the transition from far-below to above average learning performance of these students over the 1998–2002 period.     

Effects of consistency and adequacy of language information on understanding elementary mathematics word problems

Two types of elementary mathematics word problems involving different linguistic structures were devised to examine the understanding and solution of these problems by 91 Grade 3, 4, and 5 children divided into “more able” and “less able” subgroups. One task consisted of 12 consistent and 12 inconsistent language problems on the basic processes of addition, subtraction, multiplication and division. Another task consisted of a total of 36 word problems with 12 items each containing adequate, inadequate, and redundant information, respectively, for problem solution. Subsidiary tasks of general ability, vocabulary, reading comprehension, mathematics concepts, reflection on mathematics learning, and working memory were also administered to provide estimates of the contribution of these “nonmathematics” tasks to the solution of elementary mathematics problems. Analyses of variance and covariance of group data showed significant main effects of grade, consistency, and adequacy of linguistic information in problem solution. Word problems containing inconsistent information were more difficult than those with consistent information. Further, word problems containing inadequate and redundant information were more difficult to classify, and for the children to explain, than those items with just enough information. Interviews with 12 individual children provided further insight into their strategies for problem solutions. Both cognitive and developmental perspectives are important for mathematics learning and teaching for children with or without learning disabilities.     

Mathématiques de la vie quotidienne au Burkina Faso: une analyse de la pratique sociale de comptage et de vente de mangues

Mathematics teaching in Burkina Faso is faced with major challenges (high illiteracy rates, students’ difficulties, and high failure rates in mathematics, which is a central topic in the curriculum). As evidenced in many of these studies, mathematics is reputed to be tough, inaccessible, and far from what students live daily. Students here look as though they are living in two seemingly distant worlds, school and everyday life. In order to better understand these difficulties and to contribute in the long run to a more adapted teaching of mathematics, we tried to document and elicit the “mathematical resources” mobilized in various daily life social practices. In this paper, we focus on one of them, the counting and selling of mangoes by unschooled peasants. An ethnographic approach draws on the observation of the situated activity of counting and selling mangoes (during harvesting) and on “eliciting interviews” of the involved actors. The analysis of results highlights a richness of structuring resources mobilized and distributed through this practice, related to what Lave (1988) call “the experienced lived-in-world” and “constitutive order.” The mathematical resources take the form of “knowledge in action” and “theorems in action” (Vergnaud, Rech Didact Math 10(23):133–170, 1990), embedded in the social, economic, and even cultural structures of actors.     

Sample, Random and Variation: The Vocabulary of Statistical Literacy

This paper considers the development of school students’ ability to define three terms that are fundamental to statistical literacy: sample, random, and variation. A total of 738 students in grades 3, 5, 7, and 9 were asked in a survey to define and give an example for the word “sample.” Of these, 379 students in grades 7 and 9 were also asked about the words “random” and “variation.” Responses were used to describe developmental levels overall and to document differences across grades on the understanding of these terms. Changes in performance were also monitored after lessons on chance and data, emphasizing variation for 335 students. After 2 years, 132 of these students and a further 209 students who were surveyed originally but did not take part in specialized lessons, were surveyed again. The difference after 2 years between the performance of students who experienced the specialized lessons and those who did not was considered, revealing no differences in performance longitudinally. For students in grades 7 and 9, the association of performance on the three terms was explored. Implications for mathematics and literacy educators are discussed.     

The effects of the cooperative learning method supported by multiple intelligence theory on Turkish elementary students’ mathematics achievement

In the present experimental study, the effects of the cooperative learning method supported by multiple intelligence theory (CLMI) on elementary school fourth grade students’ academic achievement and retention towards the mathematics course were investigated. The participants of the study were 150 students who were divided into two experimental (used CLMI) and two control groups (used traditional method). “Mathematics Achievement Test,” “Teele Inventory for Multiple Intelligences” and “Personal Information Form” were used as the measurement instruments of the study. The findings of this research have indicated that CLMI has a more significant effect on academic achievement than the traditional method. Yet, regarding the retention scores, CLMI has not significant effect on retention.     

The Massachusetts math wars

This article recounts the battle in the “math wars” that took place in Massachusetts, United States in 1999–2000 over the scope, content and teaching of the state’s K-12 mathematics curriculum. Harsh controversies arose between the partisans of a “reform-math” movement stressing an undefined “conceptual understanding” and student-created algorithms and those, including the author, advocating an academically stronger mathematics curriculum as well as fluency in students’ computational skills with whole numbers and fractions. While “reform-math” supporters privileged and fought for a radical constructivist view of mathematics learning, the Massachusetts Board of Education decided to implement mathematics standards that linked strong academic content to the development of authentic computational competencies in students. Following the introduction of newly revised mathematic standards in 2000, real progress was reached in terms of student achievement. According to the results of the 2007 tests in reading and in mathematics for Grade 4 and Grade 8, reported by the National Assessment of Educational Progress (NAEP), Massachusetts ranked first nationwide in mathematics and tied for first place in reading, with its students having made significant gains from 2005 to 2007. The article makes a strong case for evidence-based curriculum design and implementation, freed, as much as possible, of mythologies and misconceptions. It explains why it was necessary to reject the theoretical assumptions and pedagogical strategies embedded in the National Council of Teachers of Mathematics’ 1989 and 2000 standards documents. It also highlights the importance of a strong personal life and working “principles” underpinning the mission of curriculum developers: successful reform “strategies” are indeed meaningless in the absence of such durable personal beliefs and values.

Mathematics-for-Teaching: an Ongoing Investigation of the Mathematics that Teachers (Need to) Know

In this article we offer a theoretical discussion of teachers' mathematics-for-teaching, using complexity science as a framework for interpretation. We illustrate the discussion with some teachers' interactions around mathematics that arose in the context of an in-service session. We use the events from that session to illustrate four intertwining aspects of teachers' mathematics-for-teaching. We label these aspects “mathematical objects,” “curriculum structures,” “classroom collectivity,” and “subjective understanding”. Drawing on complexity science, we argue that these phenomena are nested in one another and that they obey similar dynamics, albeit on very different time scales. We conjecture (1) that a particular fluency with these four aspects is important for mathematics teaching and (2) that these aspects might serve as appropriate emphases for courses in mathematics intended for teachers.     

Is there a gap between students’ preference and university presidents’ concern over college ranking indicators?: a case study of “College Navigator in Taiwan”

In order to help students make well-informed choices, reliable college ranking systems with comparable information about higher education institutions worldwide have been welcomed by many students. Because traditional college rankings had many methodological problems, a new type of user-based ranking, called “personalized college ranking” started to develop in many nations in the late 1990s. In 2008, Higher Education Evaluation and Accreditation Council of Taiwan (HEEACT), launched a ranking project called “College Navigator in Taiwan” which developed the first Asian student-based college search engine to provide local and international students with transparent information on Taiwan’s higher education institutions. The main objective of this paper, therefore, is to compare the rational, strategies and pathways for establishing personalized college rankings. In order to analyze the gap between students’ preferences and university presidents’concerns over ranking indicators, HEEACT’s “College Navigator in Taiwan” is adopted as a case study at the end of paper.     

Bridging the macro- and micro-divide: using an activity theory model to capture sociocultural complexity in mathematics teaching and its development

This paper is methodologically based, addressing the study of mathematics teaching by linking micro- and macro-perspectives. Considering teaching as activity, it uses Activity Theory and, in particular, the Expanded Mediational Triangle (EMT) to consider the role of the broader social frame in which classroom teaching is situated. Theoretical and methodological approaches are illustrated through episodes from a study of the mathematics teaching and learning in a Year-10 class in a UK secondary school where students were considered as “lower achievers” in their year group. We show how a number of questions about mathematics teaching and learning emerging from microanalysis were investigated by the use of the EMT. This framework provided a way to address complexity in the activity of teaching and its development based on recognition of central social factors in mathematics teaching–learning.