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1.
In this article, we present the results of a research study that explores secondary students’ capacity to perform translations of algebraic statements between the verbal and symbolic representation systems through the lens of errors. We classify and compare the errors made by 2 groups of students: 1 at the beginning of their studies in school algebra and another 1 completing their studies on algebra in compulsory education. This comparison allows us to detect errors which require specific attention in instruction due to its persistence and to identify errors that disappear as students advance in their study of algebra. The results and conclusions have a pedagogic value to inform instruction and also lead to backed conjectures and research questions to push forward research on student’s translation capacity and students’ knowledge of algebraic symbolism.  相似文献   

2.
In this study, we explored the relationship between prospective teachers’ algebraic thinking and the questions they posed during one-on-one diagnostic interviews that focused on investigating the algebraic thinking of middle school students. To do so, we evaluated prospective teachers’ algebraic thinking proficiency across 125 algebra-based tasks and we analyzed the characteristics of questions they posed during the interviews. We found that prospective teachers with lower algebraic thinking proficiency did not ask any probing questions. Instead, they either posed questions that simply accepted and affirmed student responses or posed questions that guided the students toward an answer without probing student thinking. In contrast, prospective teachers with higher algebraic thinking proficiency were able to pose probing questions to investigate student thinking or help students clarify their thinking. However, less than half of their questions were of this probing type. These results suggest that prospective teachers’ algebraic thinking proficiency is related to the types of questions they ask to explore the algebraic thinking of students. Implications for mathematics teacher education are discussed.  相似文献   

3.
Educational Studies in Mathematics - This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in...  相似文献   

4.
In this study, we investigated the impact of computer spreadsheets on the problem solving practices of students for math story problems, and more specifically on the transition from arithmetic to algebraic reasoning, through the construction of algebraic expressions. We investigated the relationships among the students’ prior knowledge and skills, the verification processes, and the effectiveness of the problem solving tasks. For identifying the factors involved in the problem solving process and their role, in our analysis we employed the Structural Equation Modeling (SEM) approach. We mainly focused on math story problems and on students of tertiary education with little prior experience on the use of computers and spreadsheets. Analysis of the data indicates that spreadsheets can support the transition from arithmetic to algebraic reasoning and this transition is influenced by prior skills of the students relevant to the interaction with the interface (enter formula skills), and the students’ frequency of verification of the solution.  相似文献   

5.
Miroslav Lovric 《PRIMUS》2018,28(7):683-698
We discuss teaching and learning situations that surfaced when computer programming and mathematics were brought together in a course where students write computer code to explore mathematics problems. Combining programming and mathematics creates a rich ecosystem which, on top of traditional mathematics activities (writing solutions, proofs, etc.), offers simulation and experimentation, invites discussions about structure, requires logic and testing strategies, and handles mathematics objects with an added feeling of reality. Focusing on novice and inexperienced programmers, we look for answers to the practice-oriented question, “How do students reason through their difficulties when using programming to explore a mathematics problem?” Following literature review and methodology, we build the programming model, which we use to study students' experiences as they approach a mathematical problem by writing computer code. Our research is based on analyzing students' in-class work and class notes, author's observations of students working on their computers, and his interactions with students in class and elsewhere. In the four case studies that we present we touch upon students' difficulties in working with complex conditional statements and recurrence relations. As well, we discuss cases where resolving a programming issue demands posing and answering mathematical questions.  相似文献   

6.
In this paper, we present a cognitive analysis of the relationship between the argumentation process leading to the construction of a conjecture and its algebraic proof in solving Calendar Algebra problems. To solve this kind of problem, students encounter two sources of potential difficulties: the shift from using arithmetic in the argumentation to using algebra in the proof and the shift from an inductive argument towards a deductive proof. Thus, the aims of this article are to describe these cognitive difficulties and to show how students overcome them. Methodologically, we compare students’ problem solving process corresponding to three problems presented in the first four lessons of a teaching experiment. The analysis and comparison between these three resolution processes is performed using Toulmin’s model.  相似文献   

7.
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8.
Abstract

The objective of this study was to identify cognitive apprehensions used by fifth- and sixth-grade students (10–12-year-olds) when answering far generalization questions in two problems of visual pattern generalization. A total of 81 students solved two linear generalizing problems, presented in two different configurations, in a succession of figures (square tables or trapezoid tables). The results showed that students used different types of cognitive apprehensions to solve problems and that these apprehensions sometimes changed according to the configuration of the sequence of figures. This finding indicates that configurations could determine apprehensions used by students, which in some cases led to the emergence of algebraic thinking. In addition, difficulties in modifying apprehension and a lack of coordination between spatial and numerical structures could explain some students’ difficulties in far generalization.  相似文献   

9.
Structure sense: The relationship between algebraic and numerical contexts   总被引：1，自引：0，他引：1  
Several researchers suggest that students' difficulties with the algebraic structure are in part due to their lack of understanding of structural notions in arithmetic. They assume that the algebraic system used by students inherits structural properties associated with the number system with which students are familiar. This study explored this assumption. In an attempt to discover whether wrong interpretations of the algebraic structure found in an algebraic context occur in a purely numerical one, we interviewed 53 sixth-graders individually. The assessment confirms the assumption: students' difficulties with the algebraic structure were found in purely numerical contexts. However, the study also confirms two, seemingly, contradictory observations. On the one hand, the students' interpretations of the structures of the expressions were very consistent; that is, the same tendencies were found in many students' answers. In this sense the students' behaviour was consistent. On the other hand, it was clearly observed that the same student may give an incorrect answer in one context and a correct answer in another. In this sense, it often seemed that the students were inconsistent.This revised version was published online in September 2005 with corrections to the Cover Date.  相似文献   

10.
Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and other symbols (e.g. “+”, “=”) made it possible to achieve what is considered one of the greatest cultural accomplishments in human history, namely, the constitution of a symbolic algebraic language and the concomitant rise of symbolic thinking. Because of their profound historical ties with natural language, the emerging syntax and meanings of symbolic algebraic language were marked in a definite way by the syntax and meanings of the former. However, at a certain point, these ties were loosened and algebraic symbolism became a language in its own right. Without alluding to the theory of recapitulation, in this paper, we travel back and forth from history to the present to explore key passages in the constitution of the syntax and meanings of symbolic algebraic language. A contextual semiotic analysis of the use of algebraic terms in 9th century Arabic as well as in contemporary students' mathematical activity, sheds some light on the conceptual challenges posed by the learning of algebra.  相似文献   

11.
汉英语言中表示颜色的词语十分丰富。我们不仅要观察它们本身的基本意义。更要洞察它们的象征意义,因为它们的象征意义在不同民族语言中有不同的特点．这种不同文化之间颜色象征意义的差异是由于各自民族的文化历史背景、审美心理的不同而产生的。是在社会发展、历史沉淀的产物,是一种永久性的文化现象．本文试图从中西文化对比的角度,去分析、探究几个典型颜色词的象征意义。  相似文献   

12.
In the seventeenth and eighteenth centuries, mathematicians and physical philosophers managed to study, via mathematics, various physical systems of the sublunar world through idealized and simplified models of these systems, constructed with the help of geometry. By analyzing these models, they were able to formulate new concepts, laws and theories of physics and then through models again, to apply these concepts and theories to new physical phenomena and check the results by means of experiment. Students’ difficulties with the mathematics of high school physics are well known. Science education research attributes them to inadequately deep understanding of mathematics and mainly to inadequate understanding of the meaning of symbolic mathematical expressions. There seem to be, however, more causes of these difficulties. One of them, not independent from the previous ones, is the complex meaning of the algebraic concepts used in school physics (e.g. variables, parameters, functions), as well as the complexities added by physics itself (e.g. that equations’ symbols represent magnitudes with empirical meaning and units instead of pure numbers). Another source of difficulties is that the theories and laws of physics are often applied, via mathematics, to simplified, and idealized physical models of the world and not to the world itself. This concerns not only the applications of basic theories but also all authentic end-of-the-chapter problems. Hence, students have to understand and participate in a complex interplay between physics concepts and theories, physical and mathematical models, and the real world, often without being aware that they are working with models and not directly with the real world.  相似文献   

13.
This paper explores some of the ambiguities inherent in the notions of generality and genericity, drawing parallels between natural language and mathematics, and thereby obliquely attacking the entrenched view that mathematics is unambiguous. Alternative ways of construing 2N, for example, suggest approaches to some of the difficulties which students find with an algebraic representation of generality. Examples are given to show that confusion of levels is widespread throughout mathematics, but that the very confusion is a source of richness of meaning.  相似文献   

14.
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15.
The aim of this study is to scrutinize the characteristics of conceptual meaning making when students engage with virtual worlds in combination with a spreadsheet with the aim to develop graphs. We study how these tools and the representations they contain or enable students to construct serve to influence their understanding of energy resource consumption. The data were gathered in 1st grade upper-secondary science classes and they constitute the basis for the interaction analysis of students?? meaning making with representations. Our analyses demonstrate the difficulties involved in developing students?? orientation toward more conceptual orientations to representations of the knowledge domain. Virtual worlds do not in themselves represent a solution to this problem.  相似文献   

16.
For many students the ideas and methods of algebra appear obscure and mysterious, their sense and purpose unclear, and their applicability to anything genuinely real or interesting very remote. Students often fail to acquire an understanding of the key concepts, despite their inherent simplicity. Even when they gain the notion of variables, expressions and equations, students often lack the strategic knowledge required to motivate and direct the global planning and detailed execution of an attack on a problem. These conceptual and strategic difficulties are compounded by the needs for precise performance of the arithmetic and symbolic operations required in manipulating expressions. Extended operations like subtracting an expression from both sides of an equation or expanding a product of three terms, are very difficult for beginning students. Their buggy performance in carrying out the detailed manipulative work greatly confounds and frustrates their acquisition and assimilation of the most important and central ideas.In an effort to confront these difficulties and show how they can be overcome, we are developing a Logo-based introductory algebra course for sixth graders. Our approach has three major components: work on Logo programming projects in algebraically rich contexts whose content is meaning ful and compelling to students, the use of algebra microworlds with concrete iconic representations of formal objects and operations, and the introduction of the algebra workbench, an expert instructional system to aid students in performing extended algebraic operations.The algebra workbench will employ a set of powerful symbolic manipulation tools for performing the standard manipulations of high school algebra. It will have two main modes of use: demonstration mode, which uses an expert tutor program to solve algebra problems incrementally, explaining its strategy and its step by step operations in straightforward terms along the way; and practice mode, in which the student tries to solve a problem with the assistance of the tutor, which performs the operations requested by the student at each step and which can be called at any point to advise the student of the correctness of a step, to perform or explain any step, to evaluate the student's solution, or to perform a problem that she poses.These powerful aids make it possible to effectively separate out the difficulties in performing the formal and manipulative aspects of algebra work from those encountered in learning the central conceptual and strategic content. Distinctly different kinds of instructional tools and activities-Logo programming, expert tutors, or algebra microworlds-can thus be brought to bear where each is most appropriate and effective.  相似文献   

17.
Textbooks in applied mathematics often use graphs to explain the meaning of formulae, even though their benefit is still not fully explored. To test processes underlying this assumed multimedia effect we collected performance scores, eye movements, and think-aloud protocols from students solving problems in vector calculus with and without graphs. Results showed no overall multimedia effect, but instead an effect to confirm statements that were accompanied by graphs, irrespective of whether these statements were true or false. Eye movement and verbal data shed light on this surprising finding. Students looked proportionally less at the text and the problem statement when a graph was present. Moreover, they experienced more mental effort with the graph, as indicated by more silent pauses in thinking aloud. Hence, students actively processed the graphs. This, however, was not sufficient. Further analysis revealed that the more students looked at the statement, the better they performed. Thus, in the multimedia condition the graph drew students’ attention and cognitive capacities away from focusing on the statement. A good alternative strategy in the multimedia condition was to frequently look between graph and problem statement, and thus to integrate their information. In conclusion, graphs influence where students look and what they process, and may even mislead them into believing accompanying information. Thus, teachers and textbook designers should be very critical on when to use graphs and carefully consider how the graphs are integrated with other parts of the problem.  相似文献   

18.
In this study, we investigate the meaning students attribute to the structure of mathematical induction (MI) and the process of proof construction using mathematical induction in the context of a geometric recursion problem. Two hundred and thirteen 17-year-old students of an upper secondary school in Greece participated in the study. Students’ responses in 3 written tasks and the interviews with 18 of them are analyzed. Though MI is treated operationally in school, the students, when challenged, started to recognize the structural characteristics of MI. In the case of proof construction, we identified 2 types of transition from argumentation to proof, interwoven in the structure of the geometrical pattern. In the first type, MI was applied to the algebraic statement that derived from the direct translation of the geometrical situation. In the second type, MI was embedded functionally in the geometrical structure of the pattern.  相似文献   

19.
在阅读和讲授文言文的过程中,人们常常会遇到一些注释问题与词汇难点。不疏通字词,便不能准确地理解原文。审文例是义训的重要手段,对于解释词义有着极为重要的作用。在中学文言文教学中恰当地运用审文例的方法,有助于我们探求词义,理解语义。  相似文献   

20.
This study looks at a mixed ability group of 21 Year 5 primary students (aged 9–10 years old) who had previously never had formal instruction using letters to stand for unknowns or variables in a mathematics context; nor had they been introduced to formal algebraic notation. Three lessons were taught using the computer software Grid Algebra where they began working with formal notation and were solving linear equations with some degree of success by the end of the lessons. The teaching was such that nothing was explained or justified by the teacher explicitly. The students appeared either not to meet, or to overcome quickly, some of the difficulties identified within previous research studies. They demonstrated remarkable confidence working with complicated linear algebraic expressions written in formal notation. A key feature of the software activities was that formal notation continually needed to be used and interpreted, and the software provided neutral feedback which enabled the students to educate their interpretation of the notation.  相似文献   

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