從建模觀點初探學生關於資料之數學推理

Inquire Students' Mathematical Reasoning about Data on a Modeling Perspective

陳蕙茹(Hui-Ju Chen) ； 柳賢(Shian Leou)

《花蓮教育大學學報》 26期 (2008/05) Pp. 21-47

https://doi.org/10.7073/JNHUE.200805.0021

摘要

將生活中的資料轉換成數學形式之數學推理是進行資料分析所需之能力。然而，此種能力之養成至今似乎未被重視。建模活動，不僅涉及到數學化過程，也可以培養學生對資料進行選擇、分析、應用之能力(Doerr & English, 2003)以填補目前數學教學上的不足。雖然，學生建模的能力有限，但經由建模活動中的探索，有助於學生進行情境的推理、數學化、解釋與溝通(Lin & Yang, 2005)。故，本研究嘗試從建模觀點來探討國一學生之資料推理。研究者參考Doerr與English(2003)的研究設計三個引模活動，透過縱貫分析學生在各題建模循環中所出現的解題類型，再橫向分析各題的解題類型，以及分析學生在三個引模活動中所呈現之數學化過程。
根據研究結果所得的結論如下。學生於建模活動中的解題表現類型有：將「名次」視為「標籤」，與將「名次」視為「量」兩種。其中將資料量化的運算類型為：「次數加總」、「百分比」與「佔幾等份」；將資料量化的表徵為：「文字敘述」、「數學算式」、「圖形與表格」。同時研究者也發現學生先使用文字表徵後，再使用數字或算式表徵，最後才出現圖、表之表徵。在圖、表的呈現順序中發現，表格出現於長條圖之後。此外，在學生於建模活動中，也出現了水平、垂直數學化過程。數學化過程的轉變包括了兩種特徵：「數學運算方式逐漸簡化」、「數學表徵方式逐漸多元化」。最後，研究者對引模活動的設計與研究者的角色進行反思，並根據研究結果提出對數學教學與對未來研究之建議。

關鍵字

數學建模 ； 資料推理 ； 數學化

並列摘要

Students' mathematical reasoning abilities transferring the information from the daily life to the mathematical world are necessary when they analyzed data. However to develop the ability was not emphasized in mathematical teaching today. Modeling activity which involved not only the process of mathematizing but also the selection, analysis and application of data (Doerr & English, 2003) is useful to improve students' abilities about data reasoning. Although students can not do modeling well by themselves, they could learn how to reason, interpret, and communicate data through exploring them in the modeling activity (Lin & Yang, 2005). The purpose of this study was to inquire students' reasoning ability about data modeling. Three model-eliciting activities were designed according to Doerr and English (2003). We presented a detailed analysis of students' problem-solving in every model-eliciting activity and across the sequence of tasks. We also analyzed the process of mathematizing at the same time.
To sum up, we found out two kinds of problem-solving behaviors in students' modeling activity. In the first kind, students viewed ranks as labels, but they viewed ranks as quantities in second one. Among the second kind of problem-solving behaviors, we found two operational styles including ”summing up”, ”transferring into percentages”, and ”being part of the whole”; we found three kinds of mathematical presentations including words, formulas well as figures and tables at the same time. Two characteristics we found in the process of mathematizing included simplifying the operations and multiplying mathematical presentations. The conclusion would provide some implications for today's mathematical pedagogy and further studies.

並列關鍵字

mathematical modeling ； data reasoning ； mathematizing