- 學位論文
論Marshall對四則運算應用問題作分類的發現-一個隱性建議使用規則的實驗
Marshall Revisited – A study of the effects of logical analytical abilities on classification abilities
摘要
針對數學問題加以類型化或是其餘分類方式，不論是以情境、運算等模式來分類，都是透過已歸納好的題目類型，輔助學習者在嘗試解題的過程中，利用同類概念的解題方式，培養學習者將數學概念轉化成解題的能力，但學習數學是無法靠記憶來死背，當學習者的認知被固定類型的題目所箝制，不能達到真正的理解、有意義的學習。 換句話說，學習不僅是概念的改變更應是認知結構的改變。學習者會在認知發展的階段來建構個體有組織的抽象概念架構，靠同化（assimilation）及調適（accommodation）兩個互補的歷程，獲得概念架構上的改變；所以，學習者必定根據某些原理、概念或者是事實等來組織學習者的知識，所以如果能事先對這些原理、概念或者是事實給予某種歸類再給學習者建構，學習者是否因環境所給予的開放性架構而獲得高層次的思考呢？ 本論文主要是針對國小文字應用題目提出靈活的題目解題方式，設計基本的題型方面訓練，通過這些基礎訓練，能進一步鞏固基礎訓練的效果，將所學的數學基礎知識逐步轉化成綜合運用的能力，並針對題目提供一些數學上的概念及規則去引導學生瞭解、熟悉每個題型的特點，強化分類練習，使其對概念點能夠觸類旁通，鞏固、彌補了基礎訓練中的知識，而在進階的分類練習中，在不斷地"認識---實踐---再認識---再實踐"的解題過程中，使學習者對題型由感性認識上升到理性認識，發現概念間關聯性，組織學習者的概念架構，理解問題，以彈性應用知識成功地解題，並建立對數學能力有用的知識類型。
並列摘要
Past research on 「classifications of elementary word problems」 has an important finding. Subjects generally classify problems according to the underlying arithmetic operations (addition, subtraction, multiplication, division). One exception to this is when the subjects are elementary students that have poor math scores, in which case the subjects would classify word problems by their respective contexts. It may be possible (as Marshall demonstrated) to train subjects how to do the “right” classification. However, from the perspectives of constructivism-style learning, we do not consider trainings of this kind meaningful, because what is of importance is how the subjects make classifications themselves and not how we can teach them “the right classification method”. Learning is a complex mental activity. It not only involves changes in concepts, but also involves admissible changes in cognitive structures. Supposedly, this is all accomplished through assimilation and accommodation. But whether it is assimilation or accommodation, the ability of making classifications obviously play an important role in it. Just how a learner classifies things? He/she must be doing it according to some principles that he/she considers “useful” or important. If we can somehow influence the subjects in his/her choices of such principles, wouldn’t it affect his/her classification behavior? With that in mind, we did an experiment on classifications of elementary word problems. The subjects were forty primary school students, ranging from the fourth grade through the sixth grade. First, we obtained their original classifications of twenty word problems in the pretest. Then, we suggested to these students (via training) how different wordings may be “translated into” various logical sentences (there are nine logical sentences altogether). After the training, we asked the students to classify the same set of word problems again in the posttest. The results are somewhat interesting. In general, the “good” students still classify the word problems according to the underlying arithmetic operations (i.e., there is no change in their classification behavior). However, there are subtle changes in their classification behaviors when the word problem in question is actually more complex than what they originally had thought. We attribute this “recognition of the inherent problem complexity” to the effect of the training (in logical way of thinking). Another interesting result is that the “bad” students changed their classification behavior after the training. Originally, these students classified the word problems according to their respective contexts (a finding that is consistent with previous research). But after the training (in logical way of thinking), these students started to classify the problems according to the underlying arithmetic operations.