پديد آورندگان :
اميني‌فر، الهه نويسنده استاديار گروه آموزش رياضي , , مجيدي‌فر، مريم نويسنده دانشجوي كارشناسي ارشد آموزش رياضي , , صالح صدق‌پور، بهرام نويسنده استاديار گروه علوم تربيتي ,
كليدواژه :
يادگيري , تانژانت , دايرهي مثلثاتي , زواياي مثلثاتي
چكيده فارسي :
هدف از پژوهش حاضر، بررسي رابطه‌ي بين فهميدن دايرهي مثلثاتي و مولفه‌هاي تشكيل دهنده‌ي آن بود. در اين راستا، ابتدا آزموني شامل 67 سوال مطابق با جدول هدف ـ محتواي طبقهبندي بلوم طرح شد. با استفاده از ضرايب تميز، دشواري و عدم هماهنگي دروني سوالات، 41 سوال نامناسب حذف گرديد. آزمون نهايي با 26 سوال بر روي 147 نفر از دانش‌آموزان سال دوم متوسطه رياضي‌فيزيك و علوم‌تجربي مدارس شهر زنجان اجرا، و مقدار آلفاي كرونباخ براي هر مولفه محاسبه گرديد. با استفاده از نتايج به‌دست آمده، مدل تجربي دانش يادگيرنده تدوين گرديد. تجزيه و تحليل روابط بين مولفه‌ها با استفاده از روش تحليل مسير نشان داد كه رابطه‌ي مستقيم معناداري بين دانش زواياي مثلثاتي با فهميدن دايرهي مثلثاتي وجود دارد. بين فهميدن زواياي مثلثاتي با فهميدن تانژانت؛ و فهميدن زواياي مثلثاتي با فهميدن دايرهي مثلثاتي نيز رابطه‌ي مستقيم معناداري وجود دارد. هم‌چنين رابطه‌ي مستقيم معناداري بين فهميدن تانژانت با فهميدن دايرهي مثلثاتي وجود دارد. نتايج اين پژوهش نشان دادند كه مولفهي دانش زواياي مثلثاتي با واسطه‌گري مولفهي فهميدن زواياي مثلثاتي و مولفهي فهميدن تانژانت باعث افزايش مولفهي فهميدن دايرهي مثلثاتي در يادگيرنده مي‌شود.
چكيده لاتين :
Introduction
The role of mathematics in explaining the phenomena and universe is not hidden from anybody. Conceptual understanding with knowledge of the facts and procedural skills is an important part of mathematics proficiency. Learning and teaching mathematics is a psychological aspect in which considerable improvement cannot be achieved, unless the learning methods are known. However, learning will occur when the learner is able to understand the concepts of each topic and apply them to solve real world problems. Augmentation of student’s learning and performance in mathematics is considered as one of the main issues discussed in the field of mathematics education. Understanding how students learn this subject can help mathematics teachers in choosing teaching methods. Realistic understanding of how students learn and carry out mathematics problems enables teachers to adopt scientific right decision in opting curriculum topics, rearranging content and teaching methods and attempting to eliminate students’ learning obstacles. Therefore, it is essential to design and develop a curriculum to suit the learner’s cognition and mentality. Paying attention to the educational goals and how students learn mathematical subjects assists executive programmers to design and modify the educational content based on the internal attributes of the learner.
Trigonometric circle is one of the important subjects in mathematics which students find problematic to learn. Studies show that teaching activities in mathematics classes have failed to create understanding of the trigonometric circle. Sometimes educational contexts cause gaps in their understanding of trigonometric circle. It can be created by inaccurate definitions in the textbooks or inappropriate applications that are used to teach concepts. Consequently students’ understanding can be promoted by recognizing the process of learning and investigating factors that affect their learning. Designing a model of causal relations in accordance with teaching objects can provide an effective solution for curriculum designers and as a result it can increase the performance and quality of learning. The aim of the current study was to present a structural model for teaching and learning of trigonometric circle. To do this, learning objectives of trigonometric circle were determined and based on them and also prerequisites, feedback and analysis of data, the model was designed.
Research question
Is there any relationship between the ‘understanding of trigonometric circle’ and components which compose it including ‘knowledge of trigonometric angles’, ‘understanding of trigonometric angles’, ‘knowledge of trigonometric circle’ and ‘understanding of tangent’?
Method
This research is a correlational study. The statistical population included all the students who were studying in grade 10 in the fields of experimental science and mathematics–physics in the city of Zanjan. The statistical sample employed in the study was accessible sampling.
At the first stage, a test consisting of 67 items was designed considering the goal-content table and Bloom’s taxonomy. The items in three homogeneity tests (a, b, and c) were prepared for use in a pilot study. Then the test was administered to three groups of 38 students. The reliability was determined through Cronbach’s alpha (0.85). By using discrimination and difficulty indices, and internal consistency of the test items, 41 inappropriate questions were deleted. The test with 26 items was administered to 147 students. Then Cronbach’s alpha was calculated for each factor: knowledge of trigonometric angles (?=0.67); understanding of trigonometric angles (?=0.63); understanding of tangent (?=0.58); and understanding of trigonometric circle (?= 0.72). The empirical model of learner’s knowledge was designed based on the results.
Results
Examination of the relationships with path analysis showed a good fit of the data for the experimental model. There was not a meaningful relationship between the components of ‘knowledge of trigonometric angles’ and ‘understanding of trigonometric circle’ (b=-0.18, t=2.50, p < 0.05). But there was a meaningful relationship between the components of ‘understanding of trigonometric angles’ and ‘understanding of tangent’ (b=0.30, t=3.81, p < 0.01); and also between ‘understanding of trigonometric angles’ and ‘understanding of trigonometric circle’ (b=0.30, t=3.88, p < 0.01). Furthermore, there was a meaningful relationship between the components of ‘understanding of tangent’ and ‘understanding of trigonometric circle’ (b=0.20, t=2.56, p < 0.05).
Conclusion
The results about the effectiveness of ‘knowledge of trigonometric angles’ on ‘understanding of trigonometric angles’ rejected the research hypothesis. This indicates that there is not a relationship between these two variables, but in the whole model, this relationship is meaningful. Analysis of the data show that ‘understanding of trigonometric angles’ impacts ‘understanding of tangent’. Therefore, the research hypothesis is confirmed. It reveals that ‘understanding of trigonometric angles’ is a base for ‘understanding of tangent’. Analysis of the data also show that ‘understanding of trigonometric angles’ impacts ‘understanding of trigonometric circle’ and as a result ‘understanding of trigonometric angles’ enhances ‘understanding of trigonometric circle’ in students. The component of ‘understanding of tangent’ impacts ‘understanding of trigonometric circle’. Thus ‘understanding of tangent’ augments ‘understanding of trigonometric circle’.
In summary, the results of this study show that the component of ‘knowledge of trigonometric angles’ with the mediation of ‘understanding of trigonometric angles’ and ‘understanding of tangent’ increases ‘understanding of trigonometric circle’ in students.
Keywords: learning, trigonometric circle, trigonometric angles, tangent.
