Between Algebra and Geometry: The Dual Nature of the Number Line (original) (raw)
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We analyze the statistical distribution of the answers given by 2 nd to 10 th graders to a set of number line problems. To structure our analysis of students' misconceptions, we identified three clusters of problems related to the number line. Our analysis shows that neglecting one of the main features of the number line can be a potential cause for misconceptions. By further exploring the students' mistakes, we found that children ignore either the geometric or the algebraic nature of the number line, making inappropriate decisions within the problem context. The errors in the problems treated within this paper originate from lack of understanding of the dual nature of the number line and are persistent over time.
Students' misconceptions in middle school mathematics : an honors thesis (HONRS 499)
2004
Middle school is a significant period in students' mathematical development, a period in which they crystallize their understanding of mathematical concepts and procedures and make the unconscious decision as to whether they will be successful in math. The project is intended to be an instructional resource providing teachers of middle school mathematics with information about common mathematical misconceptions held by students. The data for this research is taken from actual student work on the applied skills sections of the 2002 6 th and 8 th grade ISTEP+ Mathematics Assessments. The misconceptions are categorized according to specific standards as defined by NCTM, and corresponding Indiana Academic Standards are identified. lowe a great thanks also to Donna Biggs and Marilyn De Weese for your willingness to allow me to use the test data and your copy machine and also for making room for me to come in to the school frequently to work on my research.
Number line in the history and the education of mathematics
2020
In modern mathematics curricula in primary and secondary education, number line is an important supervisory tool for understanding many concepts, such as different types of numbers, equations, and more. The use of the number line is supported by a large number of researchers, but there are also studies showing that students find it difficult to use. Although the concept of number line is important for teaching and there is a great deal of debate about its use, as far as we know, there are very few systematic studies that examine the epistemological development of some components regarding the concept of number line throughout history and correlate this development by learning this concept from the students. However, there are no studies that examine the concept of historical development of the number line as a whole or relate it to student behavior. In this paper, therefore, the first attempt has been made to examine the overall development of the concept of number line in the history of mathematics. We have therefore studied the historical evolution of the concept of number line and divided it into periods, according to the characteristics of this evolution. It seems that based on the slow mathematical integration of the concept of number line at the end of the 19th century, but also on some other critical points in the four historical periods that we have analyzed, some of the difficulties that students encounter when using it are likely to be epistemological obstacles.
Charting the role of the number line in mathematical development
Frontiers in Psychology, 2013
Individuals who do well in mathematics and science also often have good spatial skills. However, the predictive direction of links between spatial abilities and mathematical learning has not been firmly established, especially for young children. In the present research, we addressed this issue using a sample from a longitudinal data set that spanned 4 years and which includes measures of mathematical performance and various cognitive skills, including spatial ability. Children were tested once in each of 4 years (Time 1, 2, 3, and 4). At Time 3 and 4, 101 children (in Grades 2, 3, or 4 at Time 3) completed mathematical measures including (a) a number line task (0-1000), (b) arithmetic, and (c) number system knowledge. Measures of spatial ability were collected at Time 1, 2, or 3. As expected, spatial ability was correlated with all of the mathematical measures at Time 3 and 4, and predicted growth in number line performance from Time 3 to Time 4. However, spatial ability did not predict growth in either arithmetic or in number system knowledge. Path analyses were used to test whether number line performance at Time 3 was predictive of arithmetic and number system knowledge at Time 4 or whether the reverse patterns were dominant. Contrary to the prediction that the number line is an important causal construct that facilitates learning arithmetic, no evidence was found that number line performance predicted growth in calculation more than calculation predicted number line growth. However, number system knowledge at Time 3 was predictive of number line performance at Time 4, independently of spatial ability. These results provide useful information about which aspects of growth in mathematical performance are (and are not) related to spatial ability and clarify the relations between number line performance and measures of arithmetic and number system knowledge.
THE NUMBER LINE AS A REPRESENTATION OF DECIMAL NUMBERS: A RESEARCH WITH SIXTH GRADE STUDENTS
Based on model on representations, in this paper we examine 12 year old students' understanding of the concept of decimal numbers. For this reason the study was conducted with the use of three kinds of tests related to decimal numbers. These three tests involved recognition, representation and translation tasks. In particular, the idea of the number line as a geometrical model is being discussed in respect to representations and translation between different representations. The application of the implicative statistical method demonstrated a compartmentalization of the different tasks and this signifies that there is a lack of coordination between recognition, representation and translation in decimal numbers.
The Number Line: An Auxiliary Means or an Obstacle?
International Journal for mathematics teaching and learning, 2010
The aim of this paper is to investigate the ways in which the number line can function in solving mathematical tasks by first graders (6 year olds). The main research question was whether the number line functioned as an auxiliary means or as an obstacle for these students. Through analysis of the 32 students‟ answers it appears that the number line functions both as an auxiliary means and as an obstacle with the latter occurring in the majority of cases.