出版資訊 (11/2003)

台師大數學系 洪萬生教授輯

○《中等教育》雙月刊第五十四卷第五期『數學專號』

  本期(200310月號)有五篇與數學教育有關的文章,目錄如下:

1.〈評《高中數學》第一冊第一章的『邏輯概念』內容〉(洪萬生撰)

2.〈線對稱概念發展之問卷設計〉(左台益、陳天宏撰)

3.〈學習單教案設計的一個例子-圓錐曲線〉(蘇惠玉撰)

4.〈數學史融入『對數』教學活動的實作心得〉(陳啟文撰)

5.〈數學焦慮之研究〉(蔡文標撰)

其中第134篇都涉及HPM,其『中、英文摘要』轉載如下,歡迎大家批評指教。

 

摘要:

•〈評《高中數學》第一冊第一章的『邏輯概念』內容〉(洪萬生撰)

        本文針對高中數學教師極感困擾的高一課程『邏輯單元』提出評論,內容涵蓋六種版本,分別是『三民版』、『翰林版』、『康熙版』、『龍騰版』、『南一版』以及『正中版』。這些教科書編者在編寫此一單元時,對於邏輯學知識本身及其如何納入數學教材,欠缺比較正規的認識或理解,因此,他們在書寫時運用了相當多的『非形式』用語,從而也無助於教師與學生釐清數學 vs. 邏輯的關係。本文採取歷史進路,凸顯『命題』與『論證』的意義,然後,再據以評論上述各教科書的缺失或不足。鑒於本文所指陳的一些問題,筆者建議『邏輯』單元移到高一上課程的『附錄』,或者乾脆移到高三下學期。至於修訂的主要內容,則請編者或修訂者務必認真地區分『有效邏輯推論』與『數學命題真假』。同時,邏輯用詞也應該精確,以免喪失它們被引進高中數學教材的美意。

關鍵詞:高中數學、邏輯單元、命題、論證

        This article is devoted to a critical review on the “logic” topic of the current senior high school mathematics textbooks in Taiwan.  Among the six editions of the textbooks published by six different publishers, their editors fail to understand, in a “formal” way, what is logic about and how logic should be incorporated into mathematics curriculum.  This is due apparently to their use of “informal” terminology in the presentation of the topic.  Therefore, it does no help to student’s understanding of logic and its relation to mathematics.  In this article, I adopt a historical approach to explore meaning of proposition and argumentation, which is, in turn, used to comment on the topic of the textbooks.  As a conclusion, I urge the editors in their revisions of the textbooks to move the topic either to become an appendix to the textbooks used for the first semester of the senior high school students or that used for the last semester of the senior high school students.  As for the content recommended for revision, I also suggest that the editors should seek to clarify the distinction between “valid logical inference” and “truth of mathematical proposition”.  In addition, logic terms should also be presented precisely in order to make sense of the introduction of logical concepts to the mathematical textbook.  

Key words: senior high school mathematics, logic topic, proposition, argumentation.

 

•〈學習單教案設計的一個例子-圓錐曲線〉(蘇惠玉撰)

在現行高中數學課程中,由於直角座標系的引入,以代數符號操弄為主的解析幾何成為主流,導致學生對幾何概念的瞭解,變得零散瑣碎。本文以圓錐曲線的教學為例,藉著學習單的設計,將圓錐曲線的相關史料引入,譬如阿波羅尼斯《錐線論》中的「正焦弦」觀念,結合「圓錐截痕」與「圓錐曲線方程式」這兩個表徵。本學習單的目標,在於提供學生對於幾何物件的學習另一角度的思考,並期望學生因而對於所學的幾何物件,有較全面性的瞭解。最後,我們也附上實施後學生的回饋與筆者的使用建議,以供大家參考。

關鍵詞:數形合一,圓錐截痕,正焦弦。

In senior high school mathematical curriculum, with the introduction of coordinate system, manipulating algebraic symbolism comes to dominate the core in the learning of analytic geometry.  Students’ appreciation of geometrical concepts turns out to be messy and trifling.  This article takes for example the teaching of conic sections.  By designing worksheets, teachers can introduce the historical material about conic sections to students.  Using the object “latus rectum” in Conics of Apollonius, we can connect “conic sections” -- geometrical representation of the curve, with “the equation of conic sections” -- algebraic representation of the curve.  The goal of these worksheets is to help students think from another angle about geometrical entities, with the hope that students can come to understand a whole picture of the subject.  This article concludes with the feedbacks from my students and my suggestions on how to use these worksheets.

Key words: Unity of geometry and algebra, conic sections, latus rectum.

 

•〈數學史融入『對數』教學活動的實作心得〉(陳啟文撰)

    數學教學的進行方式,是從簡單到複雜、從具體到抽象。為此,數學教師關心的問題總是:如何將外在的世界帶入課堂上﹖什麼是最好的方法,能讓學生認為數學是一種可以推理和探索的語言﹖一般來說,教師能找到好教材的話,對他們而言,設計教學活動將是容易的。不過,在高中數學的某些概念的教學上,有時可能會遭遇一點困難。

    舉例來說,利用『對數』來做計算,是一種過去式的學習動機了。今天在高科技計算機的發展下,教師並沒有足夠的理由,說服學生學習對數,以解決他們在生活中可能面對的一些問題。儘管教師強調對數如何在數學及科學應用上扮演重要的角色,相信學生也只能在大學相關的高等科目才會了解到。除此之外,在現在的教科書中,常直接地將對數定義為指數函數的反函數以現在的術語表示,即「若a x = y,則稱x是以a為底數時,y的對數」。如此形式化的定義,可能使得學生感到對數其實與人類的生活並無相關!在這種情形下,似乎會使上述的兩個問題變得更難回答。就我的觀點,將數學史整合融入數學教學中,應該是很好的另類選擇。

    在本篇文章中,我提出一個筆者曾在課堂上使用過的教法,目的在使學生知道對數概念的演進,以及數學家如何用採取不同的步驟來做代數運算。如果太早給予現在的代數符號,有時會讓學生望而卻步。基於此一考量,我不在引入對數的一開始就給予學生現在的代數符號表徵,而是將重心放在等差及等比的對應關係,僅在最後才用對數的代數符號來做銜接。

關鍵詞:數學教學、對數、數學史、等差與等比級數

    The teaching of mathematics has a structure that proceeds from the simple to the complex and from the concrete to the abstract.  In so doing, math teachers’ chief concern is always how to bring the outside world into the classroom and what is the best way to get students to think of math as a language that can be used to reason and explore.  Usually, if teachers can find some good teaching materials, it will be easier for them to plan teaching activities.  However, some trouble may happen to the teaching of some concepts in senior high school mathematics course.

    For instance, the calculation of logarithm is now a thing of past.  When teaching the concept of logarithm, due to the development of hightech calculator, teachers don’t have sufficient reasons to motivate students to use logarithm to solve problems which they will encounter in their life.  Even though teachers stress that logarithmic function has played an important role not only in mathematics but also in its application to science, students will not realize this until they study advanced courses in college.  Moreover, the formal definition of the concept “logarithm” in textbooks directly defined as an inverse function of exponential functionin our modern terminology, if a x = y, then the logarithm of y to the base a is x, may cause students to feel that logarithm has nothing to do with human life.  Thus, this situation seems to lead the abovementioned questions to become much less easy to answer.  From my viewpoint, integrating the history of mathematics into the teaching should be a good alternative.

    In this article, I present a teaching sequence which has been undertaken in my classroom whose purpose is to make students know the evolution of the concept of logarithm and the different steps which mathematicians have taken to operate algebraic symbols.  My teaching sequence is centered on the correspondence between arithmetic and geometric progressions.  Instead of launching the students into the modern algebraic symbolism from the very startsomething that often discourage many of themalgebraic symbols are only introduced at the end.

Key words: mathematics teaching, logarithm, history of mathematics, arithmetic and geometric progression