Диспозиция учителей средней школы к интеллектуальному вызову и её влияние на практику преподавания и успеваемость учащихся Текст научной статьи по специальности «Науки об образовании»

SECONDARY MATHEMATICS TEACHERS’ DISPOSITION TOWARD CHALLENGE AND ITS EFFECT ON TEACHING PRACTICE AND STUDENT

PERFORMANCE

Y. Valverde, M. Tchoshanov

ДИСПОЗИЦИЯ УЧИТЕЛЕЙ СРЕДНЕЙ ШКОЛЫ К ИНТЕЛЛЕКТУАЛЬНОМУ ВЫЗОВУ И ЕЁ ВЛИЯНИЕ НА ПРАКТИКУ ПРЕПОДАВАНИЯ И УСПЕВАЕМОСТЬ

УЧАЩИХСЯ

Й.Вальверде, М.Чошанов

Это исследование направлено на изучение диспозиции учителей математики к интеллектуальному вызову, представленному задачами с повышающимся уровнем сложности и ее связи с практикой преподавания и успеваемостью учащихся. Два учителя математики старших классов на юго-западе США были отобраны для данного кейс-исследования из группы учителей-магистрантов. Учителя были проинтервьюированы до и после планирования и проведения серии уроков по решению задач с повышающимся уровнем сложности. Дополнительно были проанализированы работы учащихся с целью изучения взаимосвязи между диспозицией учителя и успеваемостью учащихся. В работе применялись смешанные (качественные и количественные) методы исследования для поиска ответа на главный вопрос: в какой степени диспозиция учителя к интеллектуальному вызову влияет на практику преподавания и уровень обученности учащихся и какова природа этой связи?

Ключевые слова: принятие и избежание интеллектуального вызова, диспозиция, математика в старшей школе, уровень обученности.

Background and Significance

Avoidance of challenge is a response mechanism built up by individuals who lack motivation to face an obstacle because they do not excel (Fincham, Hokoda, & Sanders, 1989). The attribution theory has been studied extensively and has led to a valuable understanding of avoidance of challenge when faced with a mathematical problem (Bosh & Bowers, 1992). Attributions are defined as “the internal explanation individuals devise to explain their success or failure at a given task” (Grimes, 1981). This theory has been proven successful through several studies by establishing a direct influence between an individual’s perception of a certain situation and the individual’s consequent

reaction. Understanding what attributions are made by the mathematics teacher with regards to his or her success or failure in a subject is an important step in identifying the disposition toward challenge.

Teachers who avoid challenge are characterized by, “their tendency to attribute failure to external factors rather than effort, tend to show decrements in performance following failure” (Fincham, Hokoda, & Sanders, 1989). Teachers who suffer from anxiety, self degenerate themselves instead of attributing failure to a lack of effort. The greater one’s coping ability, the greater one’s ability to adjust to situations where they are evaluated such as those encountered in schooling and teaching (Gersten, Chard, Jayanthi, Baker, Morphy, & Flojo, 2009). Research also shows that teaching anxiety, including anxiety caused by challenge, is related to low student performance (Montgomery & Rupp, 2005).

Methodology

Sample and Instrument

The research sample included two high school mathematics teachers and their students. Arturo and David (names are changed for the purpose of anonymity) are experienced in-service high school teachers (more than 5 years of teaching experience) studying toward their Master’s Degree as Instructional Specialist in Mathematics Education in one of the southwestern U.S. universities.

In order to investigate teachers’ disposition toward mathematical challenge and their Algebra-1 students’ reaction, we developed a scale, which determined mathematics teachers’ acceptance or avoidance of mathematical challenges. We administered two problem solving interviews (before the lesson and after the lesson) to both teacher participants. The following is a sample of the pre-interview, which consisted of the task and two supportive questions:

1. Rabbit and Turtle run a 80 meter “over and back” race from a starting point to a tree (40m), then back to the starting point again. Rabbit’s speed over is 4 m/s and back is 8 m/s. Turtle’s speed both ways is 6 m/s. Who will win the race and why?

2. How challenging was the Task-1 for you? Rate it on a scale from 1 to 5 (1 - lowest challenge, 5 - highest challenge). Explain why.

3. How likely you will use the Task-1 in your own classroom? Rate it on a scale from 1 to 5 (1 - less likely, 5 - most likely). Explain why.

The following is the sample of the post-interview with the same task, question 2, and modified question 3:

1. Rabbit and Turtle run a 80 meter “over and back” race from a starting point to a tree (40m), then back to the starting point again. Rabbit’s speed over is 4 m/s and back is 8 m/s. Turtle’s speed both ways is 6 m/s. Who will win the race and why?

2. How challenging was the Task-1 for you? Rate it on a scale from 1 to 5 (1 - lowest challenge, 5 - highest challenge). Explain why.

3. Have you used the Task-1 (or modification of the Task-1) in your teaching? If -yes, how challenging was the Task-1 for your students? Rate it on a scale from 1 to 5 (1 -lowest challenge, 5 - highest challenge). Explain why.

Each interview consisted of three tasks with increased level of challenge based on the concept of the weighted average as presented below:

Task 1. Rabbit and Turtle run a 80 meter “over and back” race from a starting point to a tree (40m), then back to the starting point again. Rabbit’s speed over is 4 m/s and back is 8 m/s. Turtle’s speed both ways is 6 m/s. Who will win the race and why?

Task 2. Rabbit and Turtle run a 80 meter “over and back” race from a starting point to a tree (40m), then back to the starting point again. Rabbit’s speed over is r1 m/s, back is r2 m/s, and his average speed is 6m/s. Turtle’s speed both ways is 6 m/s. Would Rabbit win the race? Why or why not?

Task 3. Rabbit and Turtle run d meter “over and back” race from a starting point to a tree (d/2), then back to the starting point again. Rabbit’s speed over is r1 m/s and back is r2 m/s. Turtle’s speed over is r3 m/s and back r4 m/s. Rabbit and Turtle have equal average speeds. Would Rabbit win the race? Specify conditions under which Rabbit could win.

Interviews were analyzed using qualitative methods, such as analysis by meaning coding and finding common themes through careful examination of the data via theoretical lens of positioning theory and disposition descriptors. To analyze the information collected through the interviews, a coding sheet was created, where data was organized in related coding categories.

After teacher participants completed the first interview, they were asked to produce and deliver a lesson in their classroom where students would be taught the same content they were tested on. When analyzing the lesson plan the following was looked for: activities that provided students a strong understanding of the mathematical concept, clear student-oriented objectives, and assessment. Student work was collected to examine the effect of teacher disposition toward challenge on student performance.

Results and Discussion

Below we analyze the results of the study broken by three major categories: 1) Teacher interview; 2) Lesson plan; and 3) Student work.

Arturo’s Interview

The following reports were obtained before the lesson was delivered. Arturo was asked to solve the task and answer the follow up questions rating each on a scale from 1 to 5 (1 - lowest challenge/likelihood, 5 - highest challenge/likelihood).

Table 1: Arturo’s pre-interview responses on questions 2 and 3

Question Task 1 Task 2 Task 3

How challenging was it for you? 3 5 4

How likely you will use it in your classroom? 5 5 4

The table below represents data obtained after the lesson was delivered. Arturo was asked to solve the task and answer the follow up questions rating each on a scale from 1 to 5 (1 - lowest challenge/likelihood, 5 - highest challenge/likelihood).

Table 2: Arturo’s post-interview responses on questions 2 and 3

Question Task 1 Task 2 Task 3

How challenging was it for you? 3 4 5

How challenging was it for your students? 4 4 Did not use it

Before the lesson, Arturo reported the task 2 as possessing highest level of challenge for him. While developing his lesson plan, Arturo decided to slightly modify the context for tasks 1 and 2: instead of Rabbit and Turtle race he decided to use teams of students’ race as a context of the tasks. Analysis of student work revealed that as a result of the mathematical challenge perceived by the teacher, students did not attempt to solve the same task. On items where Arturo found the lesson of high mathematical difficulty, students also experienced this unpleasant emotional state of finding the task challenging and in turn, performed poorly. On another task - task 3, Arturo reported that the task was challenging for him and decided not to expose the students to the material.

This avoidance of mathematical challenge evidently affected students’ engagement towards the task. It was observed that the students work and efforts on the same task resembled that of their instructor. Arturo’s students were characterized by their lack of drive to continue facing an obstacle. We argue that such students’ discouragement may have been modeled by the teacher’s disposition toward challenge. As evidenced by Arturo’s students’ work, they suffered anxiety, became easily discouraged, the task became distorted; students experienced difficulty concentrating and sustaining motivation. In the same manner, students self-degenerated by comments like “el huevudo” as their name implying that they were not smart enough or good students instead of attributing failure to a lack of effort. The table below depicts the students’ products compared to those of their instructors.

Figure 1 below depicts Arturo’s solution on the task 1 and the work of one of his average performing students on the modified task 1.

Arturo’s work on task-1

Task-1. Rabbit and Turtle run ^8CTme)er "over and back" race frornjytarting point toatree (40m), then back to the starting point again. Rabbit's speed overisM m/s. and backi/8m/{ Turtle's speed both ways is G m/s. Who will win the race and why?

L\ 4-S

-10s

vHT

wj- - -5s ^(■5 — iTs

fciok.4

— s

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= IMS

Average performing student’s work from Arturo’s class

Figure 1: Arturo’s and his average performing student’s work on the task 1

Arturo demonstrates that the task 1 represents a middle level challenge for him and he is capable to correctly solve the task. However, no explanations were provided during the first interview as to why he believes that Turtle will win the race. Consequently, when Arturo takes the same task and presents it to the classroom, the result is that the average performing student (whose work is shown above) avoids the challenge by simply answering that the first team will be the winner of the race and shows no calculations or reasoning behind his logic. The lack of guidance and elaboration of his teacher lead the student to feel confused and avoid attempts to completely solve the task. Interestingly, as the same student progresses through the lesson his explanations become even weaker and his motivation and interest in the lesson has dropping further.

David’s Interview

The following reports were obtained before the lesson was delivered by David on a scale from 1 to 5 (1 - lowest challenge/likelihood, 5 - highest challenge/likelihood). Table 3: David’s pre-interview responses on questions 2 and 3

Question Task 1 Task 2 Task 3

Question Task 1 Task 2 Task 3

How challenging was it for you? 1 3 3

How likely you will use it in your classroom? 5 1 3

The table below represents David’s scores on questions 2 and 3 after the lesson was delivered (l is lowest and 5 is the highest challenge/likelihood).

Table 4: David’s post-interview responses on questions 2 and 3

Question Task 1 Task 2 Task 3

How challenging was it for you? 3 2 4

How challenging was it for your students? 5 4 5

Before the lesson, David reported one of the tasks - task 4 - as most challenging for

him. In spite of the mathematical challenge perceived by the teacher, the lesson was delivered to the students. David decided to keep the same context for the tasks. In turn, students solved every task and demonstrated high competency level throughout the lesson. It was observed that the students work and efforts on the same task also resembled that of their teacher.

David’s work on task-1 Task-1. Rabbit and Turtle run a 80 meter "over and back" race from a starting point to a tree (40m), then back to the starting point again. Rabbit's speed over is 4 m/s and back is 8 m/s. ■ Turtle's speed both ways is 6 m/s. Who will win the race and why?

Average performing student’s work from David’s class Question 1. Before doing any calculations, make a prediction: who do you think will win the race? Explain.

Figure 2: David’s and his average performing student’s work on the task 1

After analyzing student work, it was concluded that students in David’s class had significantly higher performance than those in Arturo’s class mainly due to the

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acceptance of challenge demonstrated by David. During the first interview, David first calculates the velocity of both competitors in the race and demonstrates his degree of precision in his response. As compared to Arturo’s response, David is not hesitant about his calculations and explanations. David reports that the task was challenging for him as well. However, in spite of the challenge, he decides to take the challenge back into his classroom and present it to his students. Such confidence modeled by the teacher encouraged students (the work of one presented above in figure 2) outperform counterparts from Arturo’s class. Overall, the student response from David’s class is correct and more sophisticated than the one presented by student in Arturo’s class.

Students’ Performance in Arturo’s Class

Further analysis of student’s work in Arturo’s class showed that students had stopped answering questions and not attempting to resolve the problems presented as they progressed through the lesson. The student’s work became gradually unorganized and overall messy. It was concluded that student had lost their interest in the task. Also, it was observed that students were confused and when faced with lack of appropriate guidance, became discouraged. The figure below depicts the students’ disengagement in the mathematics activities as the lesson progressed.

Sketch: Sketch a distance-time graph for the relay race between both teams and label the graph

clearly.

Figure 3: Samples of average performing student work from Arturo’s class

Students’ Performance in David’s Class

Analysis of average student work in David’s class showed that students had become engaged and elaborated their mathematical explanations and their reasoning as they progressed through the lesson. The figure below depicts the students’ engagement as the lesson progressed.

What geometric concept that you have studied in the past represents the concept of the product of two quantities? Draw a picture that would represent this concept. ,

<&■ r2

C\^

Question 3. Create a distance-time graph for the situation. Sketch Rabbit and Turtle's race on the same coordinate system below. Label your graph clearly. .

TTWve. \

Figure 4: Samples of average student work from David’s class

Throughout of the analysis, it became evident that students in David’s class had been exposed to a lesson that provided them with confidence, good explanations and high quality of instruction. To obtain confidence and to strengthen the claims made, the lesson plans and delivery were also examined.

Arturo’s Lesson Plan Design and Delivery

Arturo had planned that the content of the lesson would focus on connection between the concept of weighted average and algebra through the use of graphic and visual representations. However, no graphical or visual representations were provided in the lesson plan. Arturo also planned to connect the arithmetic, harmonic and geometric means within the lesson to foster students’ in-depth understanding of algebra. Unfortunately, analysis of student work did not reveal understanding of the concept of

mean. Instead, students withdrew the task and ceased their efforts to completely solve the tasks provided. Arturo had planned to conduct four activities. Only three activities took place in the classroom including pre and post assessment. Overall, the lesson plan was not fulfilled in the classroom. Graphs and visuals that were planned to be utilized during class to support students’ understanding were not employed after all. The coordinate plane provided to students was not helpful for students and created confusion. The challenge that the lesson provided was not embraced by the instructor and as a result the students became disengaged.

David’s Lesson Plan Design and Delivery

David had planned that the content would also be focused on connection between the concept of weighted average and algebra through graphic and visual representations. Graphics and visual representations were provided. David’s lesson plan also aimed to connect the arithmetic, harmonic and geometric mean to improve students’ understanding of the concept. Analysis of student work showed a persistent attempt to explore the concept of mean from multiple perspectives. David had planned to conduct four activities including pre and post assessment. All the activities were delivered to the class. The lesson plan was fulfilled in the classroom. Graphs and visuals that were planned to be utilized during class in order to support students’ understanding were fully employed. The coordinate plane with the grid provided to students was useful for students and facilitated the graphing process. The challenge that the lesson provided was embraced by the instructor and as a result the students became more engaged with mathematics.

Conclusions

A common obstacle that teachers face when teaching mathematics is the students’ fear and anxiety towards the abstraction that characterizes the subject. This is commonly rooted in past failures in the subject matter. The purpose of this study is to determine whether their teachers’ disposition towards challenge is another contributing factor. The

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main intent of study is to examine whether students’ lack of willingness to face a mathematical challenge is modeled and transferred from their instructors and to determine whether teachers’ disposition to confront challenge is related to students’ performance.

Continuous failure leading to students’ discouragement can be reduced through a concentrated effort from the teacher. Only when the teacher is able to reduce and cope with the anxiety towards mathematical abstractions can the problem of student discouragement towards challenging mathematical tasks be overcome. This study demonstrated that students of teachers, who tend to avoid mathematical challenges and become easily discouraged, produce students who will experience the same problems. Such students also experience difficulty sustaining motivation or engagement with the task.

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3) Elton, L. R. B. (1971). Aims and Objectives in the Teaching of Mathematics to non-Mathematicians. International Journal of Mathematical Education in Science and Technology, 2(1), 75-81. doi:10.1080/0020739710020107

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doi:10.1080/14794800008520134

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