绝对价值不平等 ——高中生的解决方案和误解外文翻译资料

问题

1

2

3

4

5

6

7

8

正确

35

22.9

42.1

56.8

43.2

43.9

61.4

43.6

4级

60.2

47.8

61.7

82.6

76.6

67.7

93.1

74.1

5级

45.5

33.3

50.3

67.6

57.2

53.8

70.5

56.4

共计

错误

45

47.5

32.5

18.2

32.2

35

16.8

31.4

4级

37.3

44.2

35.8

14.9

21.9

31.3

14.4

20.9

5级

41.8

46.1

33.9

16.8

27.8

33.5

15.8

27

共计

空白

20

29.6

25.4

25

24.6

21.1

21.8

25

4级

2.5

8

2.5

2.5

1.5

1

2.5

5

5级

12.7

20.6

15.8

15.6

15

12.7

13.7

16.6

共计

chi;2（2）

46.27

48.81

47.999

49.69

69.14

49.89

39.74

49.

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Absolute value inequalities: high school studentsrsquo; solutions and misconceptions

Nava Almog amp; Bat-Sheva Ilany

Abstract Inequalities are one of the foundational subjects in high school math curricula, but there is a lack of academic research into how students learn certain types of inequalities. This article fills part of the research gap by presenting the findings of a study that examined high school studentsrsquo; methods of approaching absolute value inequalities, their common mis- takes, misconceptions, and the possible sources of these mistakes and misconceptions. The research study used two tools—a questionnaire and personal interviews. The questionnaire was given to 481 students in the 10th and 11th grades in Israel who studied mathematics at intermediate and advanced levels. It was administered after the students had studied inequal- ities. Thirty-two students were interviewed in order to find their ways of thinking and the sources of their errors. The main types of mistakes that students consistently made when solving absolute value inequalities were found. Based on the studyrsquo;s findings, teachers can understand studentsrsquo; thought processes and use this understanding to conduct remediation and enhance mathematics instruction.

Keywords Absolute value inequalities . Inequalities . Errors . Mistakes . Misconceptions

1 Introduction

There is wide agreement that it is important for teachers to be familiar with their studentsrsquo; ways of thinking about mathematical concepts, both correct and incorrect, and to be aware of possible causes of their studentsrsquo; common mistakes. Such knowledge significantly contrib- utes to the instruction process (Karsenty, Arcavi amp; Hadas, 2007; Even amp; Tirosh, 2002). The creation of a body of knowledge about ways of thinking, common mistakes, and previous conceptions that students have about the subjects they learn at school can contribute to teachersrsquo; pedagogical content knowledge. A teacher who takes account of the previous knowledge and ways of thinking of his or her students can develop appropriate activities for them (Even amp; Tirosh, 1995) and can implement a teaching model that emphasizes concep- tual understanding (Frank, Kazemi amp; Battey, 2007).

Understanding the principles and usage of inequalities is of fundamental importance in mathematics. In a symposium about inequalities, George Polya claimed “Inequalities play a role in most branches of mathematics and have widely different applications” (Shisha, 1967). Inequalities are woven into a variety of mathematical topics, including algebra, trigonometry, linear programming, and investigation of functions.

Boero and Bazzini (2004) claim: “In most countries, inequalities are taught in secondary school as a subordinate subject (in relationship with equations), dealt with in a purely algorith- mic manner that avoids, in particular, the difficulties inherent in the concept of function. This approach implies a “trivialisation” of the subject, resulting in a sequence of routine procedures, which are not easy for students to understand, interpret and control. As a consequence of this approach, students are unable to manage inequalities which do not fit the learned schemas”. The situation in Israel is quite similar. Linear and quadratic inequalities are taught for about 10 h in junior high school. In high school, algebraic inequalities are taught for another 10 h, primarily to advanced mathematics levels. In general, teachers present algebraic solutions for conven- tional inequalities, and teach only the logical and formal aspects of inequalities. The graphical approach is generally presented only when students study quadratic inequalities.

Only a few studies address studentsrsquo; ways of thinking about solving inequalities (e.g., Tsamir amp; Almog, 1999, 2001; Boero, Bazzini amp; Garuti, 2001; Tsamir amp; Bazzini, 2001, 2002, 2003). These studies deal especially with linear, quadratic, and rational inequalities.

Researchers indicate that students make mistakes related to the logical connectors and and or. For example, when solving the task , students wrote, without any logical connection between them, and without reaching a final answer. Other students wrote or (Tsamir, Almog amp; Tirosh, 1998; Tsamir amp; Almog, 2001). Another example is studentsrsquo; responses to. Students wrote ;, failing to provide a logical connection between the two expressions or to address a case in which the factors are negative (Kroll, 1986). These researchers found that the source of these mistakes is a misunderstanding of logical connectors, which results in students switching the two logical connectors and and or.

Another type of error diagnosed in research was caused by studentsrsquo; generalization of their knowledge of solving equations to solve inequalities. A tendency to overgeneralize is a central cause of studentsrsquo; mistakes in many mathematical topics (Matz, 1982; Fischbein, Deri, Nello amp; Marino, 1985; Fischbein, 1975). Research indicated some ways in which the equation model incorrectly influences studentsrsquo; approach to solving inequalities:

amp;Failure to change the direction of the inequality sign when multiplying or dividing by a negative number. For example, solving the inequality: in the same way as solving the equation: (Kroll, 1986).

amp;Multiplying both sides of a rational inequality by the denominator. For example, solving the inequality: in the same way as solving the equation: (Tsamir et al., 1998).

amp;Solving the quadratic inequality: ;in the same way as solving the equation:(Kroll, 1986).

amp;Overgeneralization of the rule to (Kroll, 1986).

In addition, some research has found that there are epistemological obstacles in the teaching of the concept “absolute value” (Thomaidis, 1995; Thomaidis amp; Tzanakis, 2007).

Students have difficulty understanding the meaning of the concept “absolute value”. There

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