Observing mathematical fluency through students’ oral responses

‘Procedural’ fluency in mathematics is often judged solely on numerical representations. ‘Mathematical’ fluency incorporates explaining and justifying as well as producing correct numerical solutions. To observe mathematical fluency, representations additional to a student’s numerical work should be considered. This paper presents analysis of students’ oral responses. Findings suggested oral responses are important vantage points from which to view fluency – particularly characteristics harder to notice through numerical work such as reasoning. Students’ oral responses were particularly important when students’ written (language) responses were absent/inconsistent. Findings also revealed the importance of everyday language alongside technical terms for observing reasoning as a fluency characteristic. Students used high modality verbs and language features, such as connectives, to explain concepts and justify their thinking. The results of this study purport that to gain a fuller picture of students’ fluency, specifically their explanations or reasoning, students’ oral responses should be analyzed, not simply numerical work.     

Relationships between students’ learning and their participation during enactment of middle school algebra tasks

This study examines a sequence of four middle school algebra tasks through their enactment in three teachers’ classrooms. The analysis centers on the cognitive demand—the kinds of thinking processes entailed in solving the task—and the participatory demand—the kinds of verbal contributions expected of students—of the task as written in the instructional materials, as set up by the three teachers, and as discussed by the teachers and their students. Relationships between the nature of the task enactments and students’ performance on a pre- and post-test are explored. Findings include the fact that the enacted tasks differed from the written tasks with regard to both the cognitive demand and the participatory demand, which related to students’ lack of success on the post-test. Specifically, cognitive demand declined in the enacted curriculum at different points for different classes, and the participatory demand during enactment tended to involve isolated mathematical terms rather than students verbally expressing mathematical relations.     

Černý’s conjecture and group representation theory

Let us say that a Cayley graph Γ of a group G of order n is a Černy Cayley graph if every synchronizing automaton containing Γ as a subgraph with the same vertex set admits a synchronizing word of length at most (n−1)². In this paper we use the representation theory of groups over the rational numbers to obtain a number of new infinite families of Černy Cayley graphs.     

Mathematics teachers’ enactment of cognitively demanding tasks and students’ perception of racial differences in opportunity

This study examined whether secondary students in an urban school district perceived racial differences in opportunity to be successful in mathematics, whether those perceptions differed between students of color and white students, and the relation of those perceptions to teachers’ choice and implementation of mathematical tasks. The results of multi-level regression models based on student survey and teacher observation data revealed two primary findings: (a) students of color were more likely to perceive opportunity differences than were white students; and (b) this difference was greater in classrooms in which teachers attempted to use cognitively demanding tasks but allowed the cognitive demand to decline during the lesson. Implications for both future research and mathematics teacher education are discussed.     

Engineering students’ conceptions of the derivative and some implications for their mathematical education

This paper explores Mechanical Engineering students’ conceptions of and preferences for conceptions of the derivative, and their views on mathematics. Data comes from pre-, post- and delayed post-tests, a preference test, interviews with students and an analysis of calculus courses. Data from Mathematics students is used to make comparisons with Mechanical Engineering students. The results show that Mechanical Engineering students’ conceptions of and preferences for the derivative develop in the direction of the rate of change aspects while those of Mathematics students develop in the direction of tangent aspects, and that Mechanical Engineering students view mathematics as a tool and want the application aspects in their course. Students’ developing conceptions, preferences and views with regard to teaching and departmental affiliation are considered and educational implications are suggested for the mathematical education of engineering students.     

Researching young students’ mathematical world views

As an alternative to questionnaires suitable for young students, pictures, texts and interviews are used as data sources for studying mathematical world views of fifth and sixth graders in a several-step design. The project was developed in three successive studies. In the first study, the approach of using pictures, texts and interviews for researching young students’ mathematical world views was investigated. Object of the second study was the development of an interrater-method for determining mathematical world views which delivered a satisfactory degree of reliability. The empirical results in the second study indicated as well that quite often mathematics courses were dominated by a view on mathematics emphasizing numbers or calculations. An analysis of students’ utterances suggests that some young students might have mixed world views. This motivates a modified rating approach in a third study in which raters can give weights to several world views. The procedure indicates that various mixed forms of the world views can be observed. This brings up the question as to whether this phenomenon is due to the methodology or whether it describes the formation of mathematical world views at that age.     

An in-service primary teacher’s implementation of mathematical tasks: the case of length measurement and perimeter instruction

The purpose of this paper is to examine the cognitive demand levels of tasks used by an in-service primary teacher during length measurement and perimeter instruction and to examine a possible link between these tasks and the teacher’s mathematical knowledge in teaching. For this purpose, a case study approach was used and the data was drawn from classroom observations, semi-structured interviews, and field notes. Specific tasks from length measurement and perimeter instruction were presented and analyzed according to the Mathematical Tasks Framework. Then, how these tasks gave information about the teacher’s mathematical knowledge in teaching in the length measurement and perimeter topics was examined according to the Knowledge Quartet model. According to the findings of the study, the tasks used during length measurement and perimeter instruction were mostly categorized as low-level tasks. In addition, teacher’s mathematical knowledge in teaching affected the implementation of the tasks.     

Interactionist perspective on negotiation processes of students’ different understandings during small group work on linear algebra

Student group work represents a central learning setting within mathematics programs at the university level. In this study, a theoretical perspective on collaboration is adopted in which the differences between students’ interpretations of a mathematical concept are seen as an opportunity for individual restructuring processes. This so-called interactionist perspective is applied to student group work on linear algebra. The concepts of linear algebra at the university level are characterized by a versatility of different modes of expression and interpretation. For students of linear algebra, the flexible transitions between the different interpretations of linear algebra concepts usually pose a challenge. This study focuses on how students negotiate their different interpretations during group work on linear algebra and how transitions between interpretations might be stimulated or hindered. Video recordings of eight student groups working on a task that required flexible transition between interpretations of homomorphisms were sampled. The recordings were analyzed from an interactionist perspective, focusing on interaction situations in which the participating students expressed and negotiated different interpretations of homomorphisms. The analyses of students’ interactions highlight a phenomenon whereby differences in students’ interpretations remain implicit in group discussions, which constitutes an obstacle to the negotiation process.     

On the use of computational tools to promote students’ mathematical thinking

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. In this snapshot students employ dynamic geometry software to find great mathematical richness around a seemingly simple question about rectangles.

Editor: Uri Wilensky

相似文献   

Are beginning calculus and engineering students adequately prepared for higher education? An assessment of students’ basic mathematical knowledge

Are students transitioning from the secondary level to university studies in mathematics and engineering adequately prepared for education at the tertiary level? In this study, we discuss the prior mathematical knowledge and skills demonstrated by Norwegian engineering (N?=?1537) and calculus (N?=?626) university students by using data from a mathematics assessment administered by the Norwegian Mathematical Council. The assessment examines students’ conceptual understanding, computation skills and problem solving skills on the basis of the mathematics curriculum of lower secondary education. We found that calculus students significantly outperformed engineering students, but both student groups struggled to solve the test, with the calculus and engineering groups scoring an average of 60% and 46%, respectively. Beginning students who fail to master basic skills, such as solving arithmetic and algebra problems, will most likely face difficulties in their further courses. Although few female students enrol in calculus and engineering programmes compared with male ones and are thus underrepresented, male and female students at the same ability level achieved comparable test scores. Furthermore, students reported high levels of intrinsic and extrinsic motivation, and a positive relationship was observed between intrinsic motivation and achievement.     

Classroom interactions that shape the nature of students’ appropriation of a complex mathematical practice

Scholars continue to emphasize the importance of fostering proficiency with mathematical practices as an educational outcome. As teachers attempt to support students in developing these practices, they communicate subtle messages about their nature. However, researchers lack a detailed understanding of the classroom interactions that communicate these messages. To begin to address this gap in the literature, we investigated the relationship between the types of classroom interactions around the mathematical practice of imposing structure and the ways students subsequently engaged in that practice. This led to the identification of three types of classroom interactions that shaped the nature of students’ appropriation of imposing structure: (a) engaging students in the practice, (b) providing different representations of the practice, and (c) reflecting on different instantiations of the practice. Our examination of the nature of these interactions suggests teachers must attend to details as they support students to appropriate mathematical practices in formal learning environments.     

Broadening the sense of ‘dynamic’: a microworld to support students’ mathematical generalisation

In this paper, we seek to broaden the sense in which the word ‘dynamic’ is applied to computational media. Focussing exclusively on the problem of design, the paper describes work in progress, which aims to build a computational system that supports students’ engagement with mathematical generalisation in a collaborative classroom environment by helping them to begin to see its power and to express it for themselves and for others. We present students’ strengths and challenges in appreciating structure and expressing generalities that inform our overall system design. We then describe the main features of the microworld that lies at the core of our system. In conclusion, we point to further steps in the design process to develop a system that is more adaptive to students’ and teachers’ actions and needs.     

Middle school students’ patterning performance on semi-free generalization tasks

This longitudinal study empirically addresses the issue of structure construction and justification among a class of US seventh and eighth-grade Algebra 1 students (mean age of 12.5 years) in the context of novel semi-free pattern generalization (PG) tasks before and after a teaching experiment that emphasized a multiplicative thinking approach to patterns. We compared the students’ PG responses before and after the experiment and found that (1) one source of variability in their abduced structural processing was in part due to an initial conceptual preference toward thinking either in parts or in wholes and (2) a multiplicative understanding of structures significantly aided them in PG conversion (e.g., from the visual to the alphanumeric) and processing (e.g., from nonstandard to standard function-based formulas). Our findings provide both necessary and sufficient conditions for constructing, establishing, and justifying valid structures in the case of (semi-) free figural patterning tasks.     

Learning to notice students’ mathematical thinking through on-line discussions

The goal of this research is to characterize prospective mathematics teachers?? development of professional noticing of students?? mathematical thinking in on-line contexts. Specifically, we are interested in how the participation in on-line discussions, when prospective teachers solve specific tasks, supports the development of professional noticing of students?? mathematical thinking. Findings show that an aspect in which the on-line discussions, as an example of asynchronous collaborative communication interfaces, support this development is related to the role of writing; participating in an on-line discussion plays a significant role since the final written text is functional as regards the activity of interpreting students?? mathematical thinking collaboratively.     

The effect of project-based learning on students’ statistical literacy levels for data representation

The point of this study is to define the effect of project-based learning approach on 8th Grade secondary-school students’ statistical literacy levels for data representation. To achieve this goal, a test which consists of 12 open-ended questions in accordance with the views of experts was developed. Seventy 8th grade secondary-school students, 35 in the experimental group and 35 in the control group, took this test twice, one before the application and one after the application. All the raw scores were turned into linear points by using the Winsteps 3.72 modelling program that makes the Rasch analysis and t-tests, and an ANCOVA analysis was carried out with the linear points. Depending on the findings, it was concluded that the project-based learning approach increases students’ level of statistical literacy for data representation. Students’ levels of statistical literacy before and after the application were shown through the obtained person-item maps.     

Online assessment of students’ reasoning when solving example-eliciting tasks: using conjunction and disjunction to increase the power of examples

ZDM – Mathematics Education - We argue that examples can do more than serve the purpose of illustrating the truth of an existential statement or disconfirming the truth of a universal...     

When time is an implicit variable: An investigation of students’ ways of understanding graphing tasks

In this study, we investigate students’ ways of understanding graphing tasks involving quantitative relationships in which time functions as an implicit variable. Through task-based interviews of students ages 14–16 in a summer mathematics program, we observe a variety of ways of understanding, including thematic or visual association, pointwise thinking, and reasoning parametrically about changes in the two variables to be graphed. We argue that, rather than comprising a hierarchy, these ways of understanding complement one another in helping students discover an invariant relationship between two dynamically varying quantities, and develop a graph of the relationship that captures this invariance. From these ways of understanding, we conjecture several mathematical meanings for graphing that may account for students’ behavior when graphing quantitative relationships.     

Running to keep up with the lecturer or gradual de-ritualization? Biology students’ engagement with construction and data interpretation graphing routines in mathematical modelling tasks

Through a commognitive lens, we examine twelve first-semester biology students’. engagement with graphing routines as they work in groups, during four sessions of Mathematical Modelling (MM). We trace the students’ meta-level learning, particularly as they fluctuate between deploying graphs for mere illustration of data and as sense-making tools. We account for student activity in relation to precedent events in their experiences of graphing and as fluid, if not always productive, interplay between ritualised and exploratory engagement with graph construction and interpretation routines. The students’ construal of the task situations is marked by efforts to keep up with lecturer expectations which allow for changing degrees of student agency but do not factor in the influence of precedent events. Our analysis has pedagogical implications for the way MM problems are formulated and also foregrounds the capacity of the commognitive framework to trace de-ritualization and meta-level learning in students’ MM activity.     

Israeli Jewish and Arab students’ gendering of mathematics

In English-speaking, Western countries, mathematics has traditionally been viewed as a “male domain”, a discipline more suited to males than to females. Recent data from Australian and American students who had been administered two instruments [Leder & Forgasz, in Two new instruments to probe attitudes about gender and mathematics. ERIC, Resources in Education (RIE), ERIC document number: ED463312, 2002] tapping their beliefs about the gendering of mathematics appeared to challenge this traditional, gender-stereotyped view of the discipline. The two instruments were translated into Hebrew and Arabic and administered to large samples of grade 9 students attending Jewish and Arab schools in northern Israel. The aims of this study were to determine if the views of these two culturally different groups of students differed and whether within group gender differences were apparent. The quantitative data alone could not provide explanations for any differences found. However, in conjunction with other sociological data on the differences between the two groups in Israeli society more generally, possible explanations for any differences found were explored. The findings for the Jewish Israeli students were generally consistent with prevailing Western gendered views on mathematics; the Arab Israeli students held different views that appeared to parallel cultural beliefs and the realities of life for this cultural group.