Quod Erat Demonstrandum

2010/06/30

空間感

Filed under: Junior Form Mathematics — johnmayhk @ 6:35 下午

同工問:「你點樣教 3D figures 呢個 topic?」我告之曰:「我也覺得難教,只用『題海戰』,用實物軟件示範等等或叫學生剪剪摺摺。」

就以 net 這個題目為例,學生多數看到「終點」:一係 3D 圖(在 2D 上表達出來的 3D 圖),一係 2D net,但由 3D 轉化成 2D(或 2D 轉化成 3D)的變化過程,卻沒有顯示(起碼在教科書中沒有,算不算鷹架不足?),但過程往往是重要的。

隨便舉一道考題:今年(2010 年)北京中考數學試題

圖片摘自:中考網
http://zhongkao.eduu.com/e/20100608/4c0dde340cb8b.shtml

我們可以如何教學生面對諸如此類的題目?我相信就算平日又用實物又用軟件,部份學生始終沒法想像其「變化過程」。有沒有比較有系統和有效的方法「教授」和「提升」空間感?

軟件是很有用,比如我亂擬一題:參考以下摺紙圖樣

當摺成多面體後 A 點和哪些點重疊?

一用軟件,顯示變化過程,答案一目了然,見:

問題是,如何使學生在他腦海中也可以同樣地產生該變化過程,以應付此等紙筆考核?

當然,你或會說我舉此例是誇張了,但我想指出,就算上述的北京中考題,對某些學生可能也是誇張的。

類似情況:如何教人想像四維幾何物體?對於一些人,也是很誇張的吧,但也是有方法的。

當然,已有不少教授三維立體的教本,也看過坊間一本頗厚的書,當中非常多的考題嘗試幫助同學理解;但題海法真的是好的學習方法嗎?

另外,有些學生根本不用教,面對立體問題沒有難度,這種天生(?)的空間感根本不是授課員教出來。

猶記得教我教數的老師說:「你係數學老師,唔係話畀我聽你唔識教下話?」對不起,我真的仍在探索。

還有些問題可以問,例如:學習這些東西是否真的和銜接進一步的中學數學知識密切相關?

[OT] 數碼危機:如果學校和 IT school 「終止合約」,那麼放在當中的圖片、文件就會消失,到時本網誌和舊討論區的有關連結通通死 link(情況如教城把 2005 年前的東西鏟掉,現在在討論區的 GeoGebra files 又似乎看不到),除了逐條連結改動,還有沒有好方法?儘管我寫和上載的東西是無聊廢話。

10 則迴響 »

  1. 空間感我真係唔勁,成日 D 3D 圖都睇唔出佢點畫….
    btw~呀sir,今年 AL app D人點呀?想知多D

    迴響 由 Carmen — 2010/06/30 @ 9:10 下午 | 回覆

    • 唔明現在很多同學可以輕易接觸 3D games 和 3D movies,但 3D sense 似乎掌握得不好。

      今年 AL app 有 100% 及格,但…只能用 T_T 代表我的心情。

      迴響 由 johnmayhk — 2010/06/30 @ 9:48 下午 | 回覆

  2. What about pure math?

    迴響 由 Justin — 2010/06/30 @ 11:54 下午 | 回覆

    • Pure 真係 handsfree(免提)了。

      迴響 由 johnmayhk — 2010/07/01 @ 8:25 上午 | 回覆

  3. 不知道老師影片中的示範,是利用哪個 3D 軟體做成的呢?
    尚請賜教,謝謝。

    迴響 由 羅驥韡 — 2010/07/01 @ 8:35 上午 | 回覆

    • 我用的是多年前的舊版

      poly32.exe

      不知道最 up-to-date 的 version 如何?

      迴響 由 johnmayhk — 2010/07/01 @ 8:39 上午 | 回覆

  4. John Sir,
    In fact I can’t handle 3D topic very well, when students ask me, i sometimes can’t solve their problems within a short period of time…Can u suggest some ways for me to improve the teaching skills of 3D???

    迴響 由 Sam — 2010/07/01 @ 11:07 上午 | 回覆

    • May be some 3D problems require not only spatial sense, but also logical thinking etc. And it is somehow difficult to tackle. I am still thinking of good ways to teach the topic efficiently (though it is not necessary for me to teach this topic directly because I’m not a math teacher of junior forms). Up to now, I still believe that showing the process of ‘changing’ (from 2D to 3D etc.)explicitly during the demonstration (using real objects or software) is quite reasonable (but don’t know whether it is effective or not) in understanding. And presenting some ‘counter-examples’ may be useful: students may be familiarized with platonic solids and their nets, but how about ‘incorrect nets’ (so to speak)? Give examples. There are lots of exploration activities concern 3D shapes and nets (I read some materials when I was studying in cu), just do some (for fun) if time is available. Well, I am not experienced in teaching this topic and that’s why I wrote this stuff, hoping for hearing more ideas from readers! Thank you for leaving comments.

      迴響 由 johnmayhk — 2010/07/01 @ 6:06 下午 | 回覆

  5. in fact, i am now using all sorts of resources to teach 3D, and i am still finding rooms for improvements, i think sometimes co-geom is also a difficult topic to teach students, as teachers are not sure if students understand about the formula(especially the normal form)!!!

    迴響 由 Sam — 2010/07/03 @ 7:07 下午 | 回覆

  6. 用teleport 成個站/page copy
    落黎得唔得

    迴響 由 路人甲 — 2010/07/04 @ 2:00 下午 | 回覆


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