فصلنامه علمی

نوع مقاله: مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشکده علوم پایه، دانشگاه تربیت دبیر شهید رجائی، تهران ، ایران

10.22061/tej.2020.5808.2284

چکیده

پیشینه و اهداف: درک مفاهیم ریاضی بدون تأکید بر استدلال غیر‌ممکن است و جنبه ابزاری و رویه‌ای پیدا می‌کند و چنان‌چه ریاضیات به جای مجموعه‌ای از رویه‌ها، به عنوان یک علم مستدل یاد گرفته شود، راحتتر می‌تواند بازآفرینی شود. از طرف دیگر، هدف هر سیستم‌ آموزشی آماده کردن دانش‌آموزان برای زندگی اجتماعی است؛ به طوری که آنها قادر باشند به وظایف روزمره خود به عنوان یک شهروند به خوبی عمل کنند. در این راستا آنها باید بتوانند با استدلال‌هایی که ارائه می‌کنند، خود و دیگران را قانع نمایند. اما، دانش‌آموزان در ارتباط با درک و فهم از استدلال و اثبات در ریاضی و همچنین در ارزیابی درستی آنها با مشکلات گسترده‌ای روبرو هستند. بنابر این، بررسی ارزیابی دانش‌آموزان از درستی و اعتبار استدلال‌های ریاضی و استفاده از این استدلال‌ها در متقاعد کردن خود و دیگران از اهمیت شایانی برخوردار است و شایسته توجه و پژوهش‌های بیشتر می‌باشد. هدف پژوهش حاضر، بررسی توانایی دانش‌آموزان پایه یازدهمدر ارزیابی استدلال‌های ریاضی است تا نقاط قوت و ضعف دانش‌آموزان مشخص گردد.
روش‌ها‌: این پژوهش به روش زمینه‌یابی انجام گرفته است. جامعه آماری، دانش‌آموزان پایه یازدهم شهر زنجان و نمونه، 393 نفر از دانش‌آموزان پسر و دختر به روش تصادفی خوشه‌ای از مدارس تیزهوشان، نمونه، شاهد و دولتی می‌باشند و انتخاب نمونه به گونه‌ای بود که تمامی سطح‌های دانش‌آموزی را شامل گردد. ابزار اندازه‌گیری، یک آزمون محقق‌ساخته است که شامل 3 مسأله در موقعیت‌های آشنا، کاملاآشنا و ناآشنا بود که برای هریک از این سه موقعیت، پاسخ‌هایی به دانش‌آموزان ارائه شد تا در بین آنها مشخص کنند که کدام پاسخ‌ را برای متقاعد کردن خود و کدام را برای متقاعد کردن دوستان و بالاخره کدام را برای دریافت بهترین نمره از معلم انتخاب می‌کنند. برای تجزیه و تحلیل داده‌ها، روش‌های‌ آمار توصیفی و آمار استنباطی (آزمون خی دو) مورد استفاده قرار گرفت.
یافته‌ها: نتایج بیانگر آن بود که دانش‌آموزان توانایی ارزیابی استدلال‌های ریاضی را ندارند و در بیش از 60 درصد موارد علاقه خاصی به استفاده از روش‌های صوری دارند. از پاسخ‌های انتخابی دانش‌آموزان برای قانع ساختن خود و دوستان می‌توان نتیجه گرفت هر چقدر موقعیت ناآشناتر باشد؛ دانش‌آموزان به ملاک‌های مورد قبول برای پذیرش یک استدلال منطقی کمتر توجه می‌کنند. عملکرد دانش‌آموزان برای دریافت نمره کامل از معلم، بیانگر توجه آنها به پاسخ‌های نمادین درست و نادرست افزایش یافته و به نظر می‌رسد شکل ارائه برای آنها مهم‌تر است و آنها توانایی تشخیص محتوای اثبات صوری درست از نادرست را ندارند؛ ولی در تشخیص استدلال‌های نامعتبر در موقعیت آشنا درک بهتری دارند. نتایج نشان می‌دهد جنسیت در برخی موارد روی عملکرد دانش‌آموزان تأثیر می‌گذارد.
نتیجه‌گیری: می‌توان گفت که شیوه آموزش فعلی در ریاضیات نتایج قابل توجهی در حیطه استدلال و اثبات به همراه نداشته است و لازم است شیوه‌های آموزش و محتوای کتب درسی بازنگری گردد. نتایج این تحقیق می‌تواند مورد استفاده سیاست‌گذاران تعلیم و تربیت و مولفان کتب درسی قرار گیرد تا با آگاهی از دیدگاه دانش‌آموزان به استدلال‌های ریاضی به جایگاه آن در کتاب‌های ریاضی توجه  ویژه‌ای نمایند و شاید با تغییر نحوه تالیف کتب درسی گامی اساسی برای رفع مشکلات بردارند. همچنین معلمان ریاضی با آگاهی از عملکرد دانش‌آموزان در زمینه استدلال و اثبات، می‎توانند به نقاط ضعف و قوت دانش‌آموزان خود در فرایند اثبات‌های ریاضی پی‌ببرند و بد‌فهمی آن‌ها در این زمینه شناسایی کنند.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

The ability of 11th grade students to evaluate mathematical reasoning

نویسندگان [English]

  • N. Yaftian
  • M. R. Ansari

Department of mathematics, Faculty of Sciences, Shahid Rajaee Teacher Training University , Tehran, Iran

چکیده [English]

Background and Objectives: Understanding mathematical concepts is impossible without emphasizing reasoning and takes on instrumental and procedural aspects, and can be more easily recreated if mathematics is learned as a reasoned science instead of a set of procedures. On the other hand, the goal of any educational system is to prepare students for social life; So that they can perform their daily duties well as a citizen. In this regard, they must be able to convince themselves and others with the reasoning they present. However, students face widespread difficulties in understanding reasoning and proof in mathematics as well as in assessing their correctness. Therefore, it is important for students to evaluate the correctness and validity of mathematical reasoning and to use these reasoning to convince themselves and others and deserves further attention and research. The purpose of this research was to study the ability of 11th grade students to evaluate mathematical reasoning to identify the strengths and weaknesses of students..
Methods:The present study was conducted by survey method.The statistical population consisted of the 11th grade students in Zanjan and the sample includes 393 boy and girl students selected by random cluster sampling from the gifted, exemplary public, Shahed and public schools and the sample was selected to include all levels of students.. The research instrument is a researcher-made test consisting of 3 problems in familiar, completely familiar and unfamiliar situations.  Students were provided with some responses for  each of these three situations to determine  which responses can be selected to convince themselves,  which ones can be chosen to convince friends, and finally which ones can be selected to get the best score. Descriptive and inferential statistics (Chi-square test) were used for data analysis.
Findings: The findings indicated that students were not capable of evaluating mathematical reasoning and in more than 60% of cases they were particularly interested in using formal methods. Selecting the  responses to persuade themselves and friends in more unfamiliar situations indicated that students paid less attentionto to accepted criteria  for accepting a logical reasoning. Students' performance to get the best score from the teacher indicated that their attention to  correct and incorrectsymbolic responses has increased,  the form of presentation seems to be more important to them,. Although they are not able to distinguish formal proof content from the false one, they have ea better understanding for distinguishing invalid reasoning in the familiar situations. The results showed that in some cases gender  influenced  students' performance.
Conclusion: It can be said that the current teaching method in mathematics has not had significant results in the area of reasoning and proof. Therefore, it is necessary to review the teaching methods and the content of the  textbooks. The results of this research can be used by education policy makers and textbook authors to pay special attention to the situation of reasoning in mathematics textbooks by being aware of students' views on mathematical reasoning, and perhaps by changing the way textbooks are written, a fundamental step to solve difficulties. Also, by being aware of students' performance in the field of reasoning and proof, math teachers can identify the strengths and weaknesses of their students in the process of math proofs and identify their misconceptions in this field.

کلیدواژه‌ها [English]

  • mathematics Education
  • Reasoning
  • Proof
  • 11th grade students

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