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べつに、高所から「数学書の読み方を教えてしんぜよう」ということではなく(そんなん、ぼくのほうこそ教えてほしい)、あくまで「ぼくはこういうふうに数学書を読んでいる」というメモとして。

以下、箇条書きで、ぼくが数学書を読むさいに気をつけていること/やっていることを、ぼくが個人的に「重要だな」と思われる順番で書いていきます。

1. 定義をきちんとおさえる
「定義をきちんとおさえる」なんてのは死ぬほど当たり前のことなんですが、ただ、数学書というのはおうおうにして最初のほうは「そんなん、自明じゃん」と思われるようなことが書いてあるがゆえに、その「そんなん、自明じゃん」というノリをいつまでも引きずっていい加減に(というのはつまり、定義をきちんとおさえず、流し読み感覚で)読み進めると、いつしか「ぜんぜん分からん」となっていることがぼくはとても多いんですね。そこで、そういう仕儀に陥らないため、どんなにそれが自明に思われようと、定義は一つひとつインデックス・カードに書き写し(そのさい、その定義が出てくる典型例をカードの裏側に書くことが多い)、それらを随時見返して、定義が「身体になじんだ」と言えるほどになるように努めています。つまり、英単語を覚えるのに単語カードを使うのとおなじ要領です。(ちなみに、「これは重要」と思われる定理にかんしてもインデックス・カードに書き写し、カードの裏側には証明を書いて、これらにかんしてもときおり見返し、定理のステートメントを身体になじませることはもとより、証明の概略だけでも思いうかぶように努力だけはしています)

2. 演習問題はぜんぶやる
有名な言葉に「解ける演習問題はやるな。解けない演習問題をやれ」というものがありますが、ある演習問題が解けるかどうかはそれを解くまでは分からないわけで、これはけっきょく「演習問題はぜんぶやる」ということに帰着する、というどうでもいいツッコミはともかく(というか、このツッコミどころこそが、この言葉のおもしろさのひとつ、なわけですが)、「うーん、分からん」と詰まってしまった演習問題を「うーん、分からん」のまま放置しないというのは、「定義を『身体になじんだ』と言えるほどに身につける」のとならんで、重要なわりにそれをつつがなく遂行するのはなかなかむずかしいので、いくども拳拳服膺すべき言葉だと思います。

さて、ぼくの場合、解けない演習問題には、つぎのように取り組んでいます。まず、ノートに問題文そのものを書き写す。そして、そこに出てくる定義や定理を写したインデックス・カードをわきにならべて(このように、複数の定義や定理を一望できるというのが、定義や定理をインデックス・カードに書き写すことの利点のひとつでもあります)、それらをにらみつつあれやこれや試行錯誤する。そうした試行錯誤のすえ、解けなかった演習問題が解ければ万々歳なのですが、初見で「うーん、分からん」と詰まってしまった演習問題はいつまでも「うーん、分からん」という状態にとどまりつづけることも多く、そういう場合、答えがその本に書いてあればそれをノートに書き写し、そこにいろいろ書き足したり書き換えたりしながら、その答えを「完膚なきまでに理解した」と言える状態にまで持っていくよう努力します。また、ある時点で「完膚なきまでに理解した」と思っても、最初「うーん、分からん」となった問題はいつしかふたたびまた「うーん、分からん」となっていることが多いので、そういう問題にかんしては、その問題を書き写した箇所にでっかい赤丸でもしておいて、時間をおいて何度か解いてみるということができればサイコーです(ただ、これはぼくもそれほどつつがなくやりおおせているというわけではない)。

【追記】上で、「演習問題に解答は付いておらず、そして自力では解けない」という場合についてふれていませんでした。この場合、いちばん手っとりばやいのは人に聞くことです。しかし、独習している人などはとくに、まわりに聞けるような人がいないということも多からんと思われますので(もっとも最近では、Twitter などを通じて、数学の質問を気軽にできるような人との交流もそれほど敷居の高いものではなくなってきていますので、そうした場を活用するのも手です)、その場合、「4. おなじ主題について書かれた本を何冊か手もとにおいておく」で用意することをおすすめすることになる「おなじ主題について書かれた何冊かの本」をひっくり返して、解法のヒントを探ってみるとよいでしょう。「ヒント」どころか、ほとんどおなじ問題が解答付きで載っていることも少なくありません。ともあれ、自力では解けなかった問題にかんしては、その解答が本に載っていたのであれ、人に聞いたのであれ、あるいは他の本に載っていたのであれ、その解答をノートに書き写し、それを「完膚なきまでに理解した」と言えるまで試行錯誤し、しかるのち、しばらくしてからまたその問題を解いてみる、というのが重要であることには変わりありません。それでは、ある演習問題の解答が本に載っておらず、さりとて人にも聞けず、他の本を開いてみても解答あるいはヒントは見いだせなかったらどうするか? その場合、「この問題が解けなかった!」ということを一目瞭然になるようにしておき(ぼくの場合、問題を書きこんだわきと、そしてそのページの上部にでっかい赤丸を付けておきます)、少し時間をおいてから再チャレンジしてみます。そうすると、ふしぎにすんなり解けたりするものです。

3. 行きつ戻りつしながら読む
定義や定理というのは、それが提示された時点ではおうおうにしてあまりピンとこないものです。それでも、「何がピンとこないか」をしっかりと保持しつつ先を読み進めると、それら定義や定理が、具体例や、あるいは他の定義や定理で用いられているのを見ることで「なるほど!」となることがけっこうある。ただ、いくら読み進めてもそうした定義や定理についてピンとこなければ、それらピンとこない定義や定理が出てきたページ以前に書かれていることの理解に問題があると考えられますので、その場合はそれらピンとこない定義や定理が出てきたページ以前に戻ります。そのさい、ピンとこない定義や定理が出てきたページを起点に、どれぐらい読み進めれば(あるいは戻れば)いいのかはいちがいには言えませんが、読み進める場合、10ページを超えても定義や定理がピンとこなければ、それはほぼ確実にピンとこない定義や定理が出てきた箇所以前の理解に問題があると考えられます。そして、その場合、ピンとこない定義や定理が出てきた箇所以前に戻るわけですが、ピンとこない定義や定理をピンとこさせるためには、ときに数十ページ戻らなければならないということもありますので、戻ることをおそれない(あるいは、めんどくさがらない)ようにしましょう。

4. おなじ主題について書かれた本を何冊か手もとにおいておく
おなじ主題について書かれた本を何冊か手もとにおいておくというのは、かならずしも必須ではないですが、それでも、ひじょうに有用な場合が多いので(たとえば、主軸で読んでいる本ではピンとこなかったことがらが、別の本の記述ですっきりと納得がいったり、あるいは、あることがらにかんして、その理解自体はつつがなくできているのだけど、そのことがらについての別の本の記述を読むことで、それについての視野が広がったりする)、そのようにすることをつよくすすめます。その場合、主軸に読んでる本と同程度のもの、やや程度がさがるもの、そして、その主題にかんして「基本書」と目されるもの(つまりは、程度の高いもの)をそれぞれ1冊づつ用意しておくといいでしょう。ただ、ある主題について書かれた本が何冊も手もとにあるからといって、それらをとっかえひっかえするのではなく、じっくりと主軸として読むのはあくまで1冊にするべきです(ある主題について書かれた本をあれやこれやと渡り歩いても、けっきょくは何も身につかないことが多いように思われますので)。

【追記】こちらの記事に上でふれたインデックス・カードやノートの写真が載っていますので、よろしければどうぞ。
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