فصلنامه علمی

نوع مقاله: مقاله پژوهشی

نویسندگان

1 گروه ریاضی، دانشکده علوم پایه، دانشگاه آزاد اسلامی، واحد تهران مرکزی، تهران، ایران

2 گروه ریاضی، دانشکده علوم ریاضی، دانشگاه خوارزمی، تهران، ایران

چکیده

پیشینه و اهداف: الگوهای شکلی ظرفیتی بی­نظیر برای ارتقای تفکر تابعی دارند. تعمیم الگوها در ریاضیات مدرسه ای مسیری برای ارتقای تفکر تابعی در نظر گرفته می شود. مفهوم متغیر یکی از مفاهیم مطرح در تعمیم الگوهاست. توجه به الگوهای شکلی فرصتی را فراهم می­کند تا دانش­آموزان معنای متغیرها و چگونگی استفاده از آنها را درک کنند. استدلال نیز به عنوان مفهوم مرکزی در تعمیم الگوها مطرح است. تعداد متغیر، یکی از مشخصه­هایی است که در تکالیف تعمیم الگو مطرح شده است و لیکن تمام تحقیقات مطرح شده در رابطه با الگوهای خطی و درجه­ی دومِ یک­متغیره بوده است. این تحقیق با هدف شناسایی طرح­واره­های پیش­نیاز در تعمیم الگوهای شکلی دو متغیره به انجام رسید. چنانچه از مفهوم دو متغیره پیداست درک فضای سه­بعدی  پیش­نیازی برای درک الگوهای دو متغیره و تعمیم آنهاست. در این الگوها به جای یک متغیر مستقل، دو متغیر مستقل وجود دارد که به طور همزمان تغییر می­کنند و بر متغیر وابسته اثر می­گذارند. درک این سه­تایی­ها نیاز به توسعه­ی طرح­واره­ی فضای R2  به فضای R3 دارد که از نظر شناختی مرحله­ی پیچیده­ای نیست و نیاز به بازسازی طرح­واره­ی موجود ندارد. 
روش‌ها: تحقیق حاضر بخشی از یک تحقیق گسترده است که به روش تحقیق کمی-کیفی (آمیخته) انجام شده است. چهارچوب تحقیق، چهارچوب APOS با به کار گیری چرخه­های تدریس فعالیت گروهی- بحث کلاسی- تمرین درخانه (ACE) می­باشد. این تحقیق در سه مرحله انجام گرفت. در مرحله­ی اول تجزیه تکوینی مقدماتی برای تعمیم الگوهای شکلی دو متغیره، با استفاده از پیشینه­ی تحقیق، تحلیل خود مفهوم و تجربیات محقق طراحی شد. در مرحله­­­ی دوم، جامعه­ی آماری، دانش آموزان پایه­ی هفتم مدارس دولتی شهرستان ملکان به تعداد 493 دانش آموز بودند. مطابق با فرمول تعیین حجم نمونه­ی کوکران، تعداد 220 نفر دانش آموز دختر و پسر پایه هفتم شهرستان ملکان (آذربایجان­شرقی)، درترم دوم سال تحصیلی 2018 میلادی، در آزمون اولیه شرکت کردند. آزمون شامل 7 تکلیف با موضوع الگوهای شکلی (یک­متغیره، دومتغیره) بر اساس چهارچوب APOS  طراحی شد و روایی آزمون توسط سه آموزشگر ریاضی و چهار معلم مجرب بررسی و  مورد تایید قرار گرفت. پایایی آزمون و هماهنگی درونی سؤالات با یافتن ضریب آلفای کرونباخ و آلفای68/0 تأیید گردید. مدت زمان پاسخگویی حدودا 90 دقیقه بود. مرحله­ی سوم تحقیق با اخذ رضایت از اداره آموزش و پرورش شهرستان و مدیران مدارس و اولیای دانش­آموزان جمعا با 19 دانش­آموز داوطلب آغاز شد. این مرحله به صورت کیفی در سه چرخه­ی تحقیق انجام گرفت.
یافته ­ها:  با استفاده از تجزیه و تحلیل پاسخ­های دانش­آموزان بر اساس چارچوب APOS  و  اجرای سه چرخه­ی تحقیق با روش تدریس ACE، تجزیه تکوینی نهایی شد و نقائص دانش­آموزان در طرح­واره­های پیش­نیاز  استدلال، فضای سه­بعدی R3 و متغیر در تعمیم الگوهای شکلی دو متغیره، شناسایی و کدگذاری شدند. اغلب دانش­آموزان درک خوبی از کار با دو متغیر در بخش عبارات جبری داشتند. ولیکن با ورود به بحث الگوی  شکلی دو متغیره در نامگذاری متغیرها  و به کارگیری درست آنها در جایگاه مستقل و وابسته  و جایگذاری مقادیر مشکلاتی بروز می­دادند.
نتیجه­ گیری: با شناسایی سازه­های ذهنی دانش­آموزان در تعمیم الگوها، مسیر یاددهی و یادگیری آن، هموارتر خواهد شد.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Enhancing functional thinking: Identifying the prior schemas of seventh grade students in generalization of two-variable figural patterns

نویسندگان [English]

  • R. Afkhami 1
  • N. Asghary 1
  • A. Medghalchi 2

1 Department of Mathematics, Faculty of sciences, Islamic Azad University, Central Tehran Branch, Tehran, Iran

2 Department of Mathematics, Faculty of Mathematical sciences, Kharazmi University, Tehran, Iran

چکیده [English]

Background and Objectives: The figural patterns have a unique capacity to enhance functional thinking. The patterns generalization in school mathematics is considered as a way to promote functional thinking. Variable is one of the concepts in patterns generalization. Paying attention to figural patterns provides an opportunity for students to understand the meaning of variable and how to use it. Reference is also a central concept in patterns generalization. The number of variables is one of the characteristics that has been proposed in the pattern generalization tasks, but all the research has been related to one variable, linear and quadratic patterns. The aim of this study was to identifying the prior schemas in generalization of two-variable figural patterns. As regard to the concept of two variables, understanding three-dimensional space is a prerequisite for understanding and generalizing two-variable patterns. In these patterns, instead of one independent variable, there are two independent variables that change simultaneously and affect the dependent variable. Understanding these patterns requires the development of the R2 space scheme to R3 space, which is not a cognitively complex step and does not require the reconstruction of the existing scheme.
Methods: The present research is part of a broad research which is done using quantitative-qualitative (mixed) research method. The research framework is APOS theory and based on the use of ACE (Activities, Class discussions and Exercises) teaching cycles. This research was conducted in three steps. In the first step, initial genetic decomposition for generalization of two-variable figural patterns was designed using the background, self-concept analysis and researchers’ experiences. It includes the prior schemas for generalization. In the second step, from the total 493 students of Malekan city (in East Azerbaijan) as the statistical population of research, a sample of 220, 7th grade students were selected based on the Cochran formula for determination of sample size. Then, a test that includes 7 tasks was designed based on APOS framework. The validity of the test was confirmed by three experts in mathematics education and four experienced teachers. Internal consistency of questions was estimated with Cronbach’s alpha and reported to be 0.69. Students responded the test at 90 minutes. The third step of research began with 19 students, with permission from the education and training office of Malekan, and school principals and parents of students. This step is done in three cycles.
Findings: Using the analysis of students' responses to this test based on the APOS framework and doing three cycles of the research were conducted with the teaching method of Activity-Class Discussion-Exercise(ACE) with 19 students; genetic decomposition was finalized in this way, and defects of students in reference schema, R3 schema and variables schema as prior schemas in generalization of two-variable figural patterns were identified and encoded. Most of students had a good understanding of working with two variables. However in the context of generalization of two-variable figural pattern revealed many difficulties at the naming of variables,  and using independent and dependent variables in  proper positionConclusion: Identifying the mental constructs of students in generalizing patterns eases better teaching and learning.
Conclusion: By identifying the mental structures of students in generalizing patterns, the path of teaching and learning will be smoother

کلیدواژه‌ها [English]

  • Functional Thinking
  • Generalization
  • APOS Theory
  • Two-variable figural pattern

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