绝对值不等式: 高中生的解决方案和误解外文翻译资料

任务

1

2

3

4

5

6

7

8

|x|gt;3

|xminus;2| lt;1

|x| 1 gt;0

|x| ge;0

|x| le;0

|x| gt;0

|x| lt;0

|x| 1 le;0

正确

35

22.9

42.1

56.8

43.2

43.9

61.4

43.6

Level 4

60.2

47.8

61.7

82.6

76.6

67.7

83.1

74.1

Level 5

45.5

33.3

50.3

67.6

57.2

53.8

70.5

56.4

Total

不正确

45

47.5

32.5

18.2

32.2

35

16.8

31.4

Level 4

37.3

44.2

35.8

14.9

21.9

31.3

14.4

20.9

Level 5

41.8

46.1

33.9

16.8

27.8

33.5

15.8

27

Total

空白

20

29.6

25.4

25

24.6

21.1

21.8

25

Level 4

2.5

8

2.5

2.5

1.5

1

2.5

5

Level 5

12.7

20.6

15.8

15.6

15

12.7

13.7

16.6

Total

chi;²

46.27

48.81

47.999

49.69

69.14

49.89

39.74

49.773

P lt;0.00001

任务

1

2

3

4

5

6

7

8

|x|gt;3

|xminus;2|lt;1

|x| 1gt;0

|x| ge;0

|x|le;0

|x|gt;0

|x|lt;0

|x| 1le;0

立即解决

没有代数 35.1 0 49.3 59.7 53 45.3 66.9 55.6

运算

使用公式

8.2

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本科毕业设计（论文）

外文翻译

附：外文原文

Absolute value inequalities: high school studentsrsquo; solutions and misconceptions

作者：Nava Almog amp; Bat-Sheva Ilany

国籍：Israel

出处：Educational Studies in Mathematics, 2012, Vol.81 (3), pp.347-364.

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 ； Errors .；Mistakes

1 Methodology

1.1 Sample

In Israeli high schools, algebraic inequalities are mainly taught to advanced mathematics levels. Even so, discussions are usually limited, emphasizing the practical algorithmic perspective of algebraic manipulations. There are three levels of learning mathematics: levels 3, 4, and 5. The students in levels 4 and 5—the intermediate and advanced levels, respectively—comprise the population of the research. For this research study, we chose a representative sample of 481 students (280 students from level 4 and 201 students from level 5) from 13 high schools. All of the students had already finished studying algebraic inequalities, including absolute value inequalities. It can be assumed that these students have received a traditional teaching of the subject.

1.2 Research tools

This research study used two tools—a questionnaire and personal interviews—in order to assess studentsrsquo; knowledge of, and approaches to solving, absolute value inequalities.

Questionnaire: The questionnaire consisted of eight absolute value inequality tasks. Most items were intentionally chosen to be solvable without algebraic manipulations, to make the problem more apparent, although in class the students learned to solve tasks using algebraic manipulation. We chose tasks that students had not been given in class, with the intention to test studentsrsquo; understanding. The solution for most tasks on the questionnaire should be immediately obvious to anyone who understands the concept of absolute value. The tasks on the questionnaire included a variety of types of results:

Tasks where the result is an infinite solution set and the complementary set is also infinite. These included the following:

Task 1: |x| gt;3 (the solution is: {x/xgt;3 or xlt;minus;3})

Task 2: |xminus;2| lt;1 (the solution is: {x/1lt;xlt;3})

Tasks where the result is R. These included the following:

Task 3: |x| 1gt;0

Task 4: |x|ge;0

Tasks where the result is related to only one value. These included the following:

Task 5: |x| le;0 (the solution is: {x/x00})

Task 6: |x| gt;0 (the solution is: {x/xne;0})

Tasks where the result is ϕ. These included the following:

Task 7: |x| lt;0

Task 8: |x| 1 le;0

The questions were given to the students in a random order. The purpose of the questionnaire was to assess:

The percentage of students who correctly solve absolute value inequalities after learning the subject.

In what ways students solve absolute value inequalities, focusing on the following two issues:

（a）Do students use graphs in order to solve absolute value inequalities? If so, do they use them correctly and effectively? Literature reports that the usage of graphs for solving inequalities is highly effective (Dreyfus amp; Eisenberg, 1985; Tsamir amp; Reshef, 2006; Abramovich amp; Ehrlich, 2007).

（b）Do students use the number line in order to solve absolute value inequalities? If so, do they use it correctly and effectively?

（c） What mistakes do students typically make when solving absolute value inequalities?

Interviews: We conducted 32 semi structured personal interviews with students who answered the questionnaire (18 students from level 4 and 14 students from level. After the questionnaires were analyzed, we chose which students to interview based on their written answers. We chose students who presented nonstandard answers, either correct or incorrect, or students who gave only their final result and did not elaborate upon their process of solving the task. The purpose of the interviews was to discuss the studentsrsquo; approaches to the tasks in order to fully comprehend their ways of thinking and identify the causes of both their correct and incorrect solutions.

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