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1a:T13f3,こんにちは！RIZと申します。
問題集の問題は解けるけれど初見の問題では解けなくなるということですね。
まずとても当たり前の話をしますが、数学は問題文から解答を考えなければなりません。現在の、問題集の問題は解けるけれども初見の問題では手が止まってしまうというのは、単に問題集の答えを覚えているだけに他なりません。そこで、今回は初見の問題でも解けるようにするためにはどのようにすれば良いかについてお話しします。
前提として、数学の公式や定義はしっかり学習しているとします。もし質問文に書かれている数学用語というのがこうした公式や定義であるなら、定義はまずしっかり覚えてください。そして公式についてはできれば丸暗記するより、導出できるようにしたほうが良いです。ただもう時間があまりないので最悪丸暗記でもいいですが、導出できるようにすることで、なぜその公式が成り立つのか理解できるので覚えやすくもなりますし、もし忘れてしまっても対応できるようになるのでおすすめです。例えば三角関数の2倍角とか3倍角なんかは加法定理とか、数3ですがド・モアブルの定理などから簡単に導出できますよね。加法定理を毎回導出するのは流石に面倒ですが、2倍角や3倍角を加法定理から導出するのは少しの時間でできますよね。このようにあまり覚えていなくても簡単に導出できる公式はなるべく導出できるようにした方が良いです。
さて、話を戻しますが、以上のように公式や定義が頭に入っていることを前提として、初見の問題でどのように対処するべきかについてお話しします。まず冒頭でもお話ししたように、数学は問題文だけから解答を考えなければなりません。そこでまず、問題文の条件に着目します。条件というのはいろいろあります。例えばnを自然数とするとか、x、yが円の方程式を満たしているとか、垂直に交わるとか、さまざまです。他にも、直接的には書かれていないけれども重要な条件もあります。例えば与えられた式が対称式であるとかです。こうした条件から、解答を考えていきます。例えば上の例で言えば、nを自然数として、かつnに関する命題が与えられて証明しなさいといった問題であれば、自然数かつ証明問題であることから数学的帰納法が浮かびますし、x、yが円の方程式を満たしていて、かつx、yの2変数からなる関数の最大最小を考えたい時、xとyが円の方程式を満たすという条件から、θを媒介変数としてx、yをcosθとsinθで置くとかが考えられます。他にも、垂直に交わるという条件があれば、例えばその垂直に交わる直線の傾き同士の積は−1とか、内積0とか、あるいは図形的に三平方の定理を利用することも可能かもしれません。以上のように、条件を見たときにいろいろなことが考えられるようになることで、初見の問題で同じような条件が出てきたときに対応できます。もちろん入試問題というのは問題集には載っていない初見の問題である場合がほとんどです。なので普段解いている問題と全く同じでないのは当たり前ですが、条件に関して言えば部分的に共通していますよね。なのでこうしたことが想起できるようになれば、初見の問題でも対応できるようになるわけです。しかしこのように、条件を見てそこから解法を想起するというのは初見では無理ですよね。それを問題集から学ぶわけです。つまり、ただ問題を解いて、解けなかったら答えを見て覚えて終わりではなく、解法を見たとき、それが「なぜ」そうなるのかを考えます。そして、もし自分が初見でその問題を解くとしたら、まず問題文のどの条件に着目するのかを考えます。このようにすることで、解法のストックを増やしていくわけです。とにかく、解答を見たものでも初見だったらどうするのか、そして「なぜ」そうするのかまで説明できるようになることで、初見の問題でも、それまでストックした解法の引き出しから解法を想起でき、対応できるようになるわけです。なのでまずは今までやった問題集で、問題文のどの条件に着目して、「なぜ」その解答になるのか考えながら学習するようにしてみてください。以上になります。ご質問などありましたらコメント欄の方でお願いします！2:["$","main",null,{"className":"px-4 pt-4 pb-4","children":["$","div",null,{"className":"max-w-3xl mx-auto w-full","children":[["$","div",null,{"className":"mb-8","children":["$","$L7",null,{"href":"https://unilink-app.onelink.me/isbO/h6xeh63x?advice=_UQpU3EBTqPwDZPuw48a","target":"_blank","children":["$","$L8",null,{"src":"/images/web_to_app_banner.jpg","width":3660,"height":1500,"sizes":"100vw","style":{"width":"100%","height":"auto"},"alt":"UniLink WebToAppバナー画像","className":"mb-4 rounded"}]}]}],["$","h1",null,{"className":"text-xl font-semibold mb-2","children":"和積の勉強はしますか？"}],["$","div",null,{"className":"flex justify-between mb-4","children":[["$","div",null,{"className":"text-left text-xs text-caption","children":["クリップ(",2,") コメント(",0,")"]}],["$","div",null,{"className":"text-right text-xs text-caption","children":"4/7 14:42"}]]}],["$","div",null,{"className":"coach-mark 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mb-2","children":"回答"}],["$","div",null,{"className":"mb-4","children":["$","$L15",null,{"adviserImageUrl":"https://firebasestorage.googleapis.com/v0/b/unilink-48e75.appspot.com/o/images%2Fs_3983601E69944A10B5BABE77BE0575D4.jpg?alt=media&token=55ab099c-25ec-46a6-a1fb-9ca3e0620ce2","adviserName":"ねこまる","adviserDepartment":"名古屋大学理学部","adviceId":"_UQpU3EBTqPwDZPuw48a"}]}],["$","div",null,{"className":"coach-mark mb-4","children":"すべての回答者は、学生証などを使用してUniLinkによって審査された東大・京大・慶應・早稲田・一橋・東工大・旧帝大のいずれかに所属する現役難関大生です。加えて、実際の回答をUniLinkが確認して一定の水準をクリアした合格者だけが登録できる仕組みとなっています。"}],["$","div",null,{"className":"mb-8","children":[["$","div",null,{"className":"leading-loose whitespace-pre-wrap mb-4","children":[["$","div","advice-part-0",{"children":[null,"三角関数の和積のことですかね？\n\n以下三角関数の和積についての話となりますので間違っていたら申し訳ないです。\n\n私はあまり覚える必要は無いと思います。\nというのも、和積の公式は自分で導けますし何より覚える量が結構多いので私は覚えませんでした。\n三角関数の倍角の公式ならまだしも、和積を使う場面というものは頻繁に出てくるものではありませんので覚えるのも大変かなと思います。\n\n一度和積の公式の導出方法を調べ、何回か練習すればそれで良いと思います。正直和積を覚えるくらいであれば他の英単語なり漢字なり覚えた方が賢明です。\n確かに和積の公式を覚えていた方がアドバンテージにはなりますが、二次試験で出たとしても小問1つに出てくるかどうか。恐らく共通テストで出題される時は誘導がつくので暗記の心配をする必要は無いと思いますよ。"]}]]}],["$","div",null,{"className":"mb-4","children":["$","$L16",null,{"adviserImageUrl":"https://firebasestorage.googleapis.com/v0/b/unilink-48e75.appspot.com/o/images%2Fs_3983601E69944A10B5BABE77BE0575D4.jpg?alt=media&token=55ab099c-25ec-46a6-a1fb-9ca3e0620ce2","adviserName":"ねこまる","adviserDepartment":"名古屋大学理学部","adviceId":"_UQpU3EBTqPwDZPuw48a","numberOfFan":16,"clipsAvg":13,"adviceRateAvg":4.666666666666667,"profile":""}]}],["$","div",null,{"children":["$","$L7",null,{"href":"https://ck.jp.ap.valuecommerce.com/servlet/referral?sid=3364577&pid=884970531&vc_url=http%3A%2F%2Fshingakunet.com%2F%3Fvos%3Dnrmnvccp0000100","rel":"nofollow","target":"_blank","children":["$","$L8",null,{"src":"/images/document_request_banner.jpg","width":3660,"height":1500,"sizes":"100vw","style":{"width":"100%","height":"auto"},"alt":"UniLink 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text-caption line-clamp-2 mb-1","children":"　数弱で浪人した者です。私は、質問者様が現在の勉強法を継続される事を断固支持します。確かに時間が余計に掛かる道ですが、それで問題を解くのが少しでも楽になる事を体感されているのは素晴らしいことですし、それが正しい勉強法です。\n\n　さて、この勉強法で間に合うかどうかですが、定理公式が出てきた度に取り組めば受験に間にあわないなんてことにはならないでしょう。むしろ焦って暗記に走る方が何倍も危険です。受験直前になってもなおうわべだけで分かったつもりになっているというレベルの知識は、入試本番では使い物になりません。そんな知識だけで受験に挑むのは落ちに行ってるようなものです。数学はそんなに甘くありません。数学は身につけるもの、そして、身につけるには自分の手を動かして理解していく事を繰り返すしかありません。証明を忘れてしまったら何度でも復習して下さい。私もこれを幾度となく繰り返しました。また、有名な定理公式の導出方法＝証明を知っているとあっさり解ける、なんて問題も整数分野などではよくあります。\n\n　一つお勧めは、問題の解答を見てとっぴな解法だなあと感じることがあれば、それは問題の基礎的な部分が分かってない証拠だと疑ってみることです。例えば数学が全くできない人に問題を解説してあげるとき、自分では当たり前に感じている箇所で、\"なんでそうやんの？\"と聞かれたことはないですか？普段の問題の解説集も同じで、解法が自分にとってとっぴに見えてしまったら、その問題の要求するレベルに達してないと判断して間違いないです。質問者様の質問には直接関係ないですが、私が受験経験から学んだことですのでお伝えしておきます。\n\n　学問では回り道に見えることが結局は王道です。私は予備校でそれを痛感させられました。めんどくさそうで遠ざけていた定理公式の証明を自分の手で行なって初めて習得できた実感をえました。質問者様の強みは今すでにこの遠回りの威力を知っておられるということです。どうか自分を信じこの努力を続けて下さい。健闘を祈ります。"}],["$","div",null,{"className":"flex 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"公式の証明について"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"こんにちは！回答させていただきます。\n\n公式の証明を覚えているとどう役に立つかということですが、正直、受験に合格するという観点では公式の証明問題が解ける以上のメリットはあまりないです！\n公式の証明では、受験数学のセオリーからみれば特殊な考え方を使うものが多く、考え方が他の問題に役立つ事も少ないのです。\n数学という学問を修める意味では、公式の証明を理解していることは重要だと思いますが。\n\nしかし、本番で公式の証明問題が解けるという一点だけで、覚える理由としては十分ではないでしょうか？\n実際の入試でそういった問題が出ているわけですし。4完を狙うなら公式の証明問題は落とせませんしね！\n\n余談ですが、三角関数の和積の公式とか、ベクトルの内積を使った三角形の面積の公式とかを、もし暗記せずにテスト中に導こうと思ってるなら、それはダメですよ！時間がもったいないですから。これはマジです！\n\n長文失礼しました。頑張ってくださいね！\n\n\n\n\n"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 24","className":"text-subPrimary mr-1","children":["$undefined",[["$","path","0",{"fill":"none","d":"M0 0h24v24H0V0z","children":[]}],["$","path","1",{"d":"M12 6c1.1 0 2 .9 2 2s-.9 2-2 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"証明や導出がすごい気になってしまう"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"三味線さん、はじめまして。\n\nお気持ちはすごく分かります。\nたしかに解答の細かいところに疑問を持ったり、その都度公式を導出していると参考書の進むペースは遅くなってしまいますが、その分、質は高くなると思うので全然良いことだと思いますし、むしろそうするべきだと思います。\n\nよく言われる「数学は理解」という言葉は、なぜその公式を使ったのか、なぜその解法で解くのか、なぜその変換を行うのか、もっと細かいことで言うと、なぜその順に解答を記述するのかといったことを理解することです。\n\n「数学は暗記」という言葉もたまに聞きますが、これは単純に英単語みたいに暗記すると言うことではなくて、どうしてこの解法を使うのかを理解した上でどうゆう問題が出たらどの解法を使うのかを暗記すると言うことです。\n仮に理解の過程を飛ばして暗記だけすると、少し問題の形が変わっただけで解法が思い浮かばないということになってしまいます。\n\nそして理解を深めるためには、三味線さんのように細かいところにも疑問を持って問題を解くのが一番の近道です。公式は導出ができる方が理解度ははるかに上がりますし、たまにある公式の導出に基づいた問題なんかも出題されることもあります。\nまた質問文中のことで触れると、なぜ置換積分はこうゆう形でするのか、一次独立とは何か、解答に使われている言葉の意図、こういったことに疑問をもって考えるのはとても良いことだと思います。確認しても忘れてしまうのは人間なので仕方ないことで、確認してその時に理解したことをノートなんかに纏めておきましょう。次に同じような疑問が出た時にノートを見返すことで少しずつ定着して力になっていくはずです。\n私の場合だと2.3回では定着せず、5回とか10回その都度見返すことで定着し始めた感じだったので、忘れているから力になっていないと焦らずに、自分のペースで頑張ってください！\n\n応援しています☺️\n"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"数学　共通テスト　8割"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"共通テストでそれくらいの成績なのであればおそらく基礎事項を理解しきっていないのだと思います。教科書の復習からはじめましょう。あと一年しかないのに教科書なんて…と焦るかもしれませんが基礎事項を理解しきってからは点数がグンと上がるので点数が上がるまで耐えましょう。逆に教科書の基礎事項を理解しきらずに点数を取ることはむずかしいです。\nここで基礎事項を習得する上でのアドバイスをしますが、①なるべく定理は証明から理解して暗記する量を減らす、②教科書の問題を解きながら習得していく、③暗記が難しいものは語呂合わせを活用する、という3つのことを意識してほしいです。\n①に関して、三角関数の和積をそのまま暗記するのではなく、加法定理から導けるようにしておく、という例が挙げられます。このようにすると問題を解くときに少し時間はかかるかもしれませんが使っていくうちに早くできるようになります。\n②の理由は、数式は暗記するだけでは意味がなく、使えなければならないこと、使ううちに数式を覚えられるようになるということがあります。\n③について、語呂合わせの力は強力です。私も受験生時代はその力にたすけられました。インターネットで覚えたい事項の語呂合わせを検索すれば割と出てきます。あなたが覚えにくいことはみんなも覚えにくいからです。\nあなたの合格を願っています。がんばってください！"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 24","className":"text-subPrimary 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"数ⅲで積分 焦ってます。"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"まずは基本的な計算が出来るようになりましょう。\nその為にも教科書の問題で良いので、部分積分や置換積分の章の問題をこなして「部分積分と置換積分のどちらをすれば良いか分かれば計算ができる」という状態にします。\nそしてここからが多くの人が悩む、どこをどう置換するのか？いつ部分積分や置換積分をするのか？という問題です。\n基本的に置換積分や部分積分の目的は複雑な関数の積分を、既知の積分に置き換えるor変形するという事です。上の計算問題をこなしていく内に、どんな形の積分なら計算できるか感覚的に分かると思います。\nそのように解ける形の積分を自分なりに頭の中で整理してどの形に変形できるかな？と考えます。\n基本的には難しい所や邪魔なところを置換するorxのべき乗やlogを消したいから部分積分する…といった理解でも正直問題はないです。中には双曲線関数など数学的に重要な例や深い視点に立てば自然な置換も存在しますが、これは誘導が着くはずなのであまり気にしなくて良いです。\n纏めると、まずはきちんと計算が出来るようになる事と計算が出来る積分形を自分なりにしっかり理解する事が肝要だと思います。\n拙い文章ですが、役に立てれば幸いです。"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 24","className":"text-subPrimary 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"数学の解法暗記"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"こんばんは。\r\n\r\n高校の数学は、おっしゃる通り、中学までの数学と比べると、様々か角度からのアプローチができるようになります。ですが、（少し厳しいことを書くかもしれませんがお許し下さい）名古屋大学を受験するにあたって、解法を一つしかわかっていないようでは、合格への道はかなり遠いと思います。\r\n\r\nといいますのも、名古屋大学の数学の入試は文系理系問わず、試験当日全員に、問題冊子、解答用紙に加えて、数学公式集が配布されます。（もちろん公式集には全ての公式が掲載されているわけではありませんが）数学の入試で、公式集が配布されるということは、つまり、「ただ単に、公式に代入して、答えが求められる」ことのできる人を大学が求めているわけではないでしょうし、そのような人が有利な採点はなされないという大学側からのメッセージではないかと思います。\r\n\r\nこのように考えますと、解法を何通り覚えたかではなく、なぜその公式・定理を使うのかということの方が大切だと思います。ただし、いきなりなぜその公式・定理を使うのかということを意識するとハードルが高すぎる可能性もありますので、まずは、複数解法のある問題に関しては、どの解法が最も計算が楽かや、どの解法が最もミスをしにくいかというような意識で、最終的には「解き方を暗記する」のではなく「なぜその公式・定理を使うのか」というような意識で数学を学習していくといいのではないかと思います。\r\n\r\nまだ３年生の5月です。現段階で、駿台模試でC判定をお持ちであれば、このままの調子で勉強していけば、合格できると思いますよ。頑張ってください。\r\n"}],["$","div",null,{"className":"flex 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"どうしても理解できない問題に出会った時"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"京大文学部所属のvenusと申します。よろしくお願いします！\n\n端的に答えると、丸暗記は避けるべきです（という回答が来るのは質問者さんも想定されていたと思いますが笑）。\n\nここでいう丸暗記とは、「どうしてその考え方・解法を使うのか」を全く理解しないままに、解答の文言を暗記する、という意味です。\n\n一方で、「解法暗記」という言葉も存在しますが、これは、その解法で上手くいく理屈を理解した上での暗記です（と思っています）。\n\n根本的に違うのは、類題に対応できるか否かという点です。理屈を理解していれば、「この問題は前の問題と本質的に同じだから、同じ理屈を当てはめられそうだ」という思考になるのですが、ひとつの問題の解答の文言だけを暗記している場合は、「なんだこの問題、見たことないぞ」で終わってしまうのです。\n\nでは解説を読んでも理解できない場合はどうすれば良いか、ですが、誰かに教えてもらう、これがベストです。友人でも先生でも良いと思います。本の解説は、著者により取捨選択されているので、洗練されすぎて逆に分からないということは往々にしてあります。\nその問題を解ける人に、一から解説してもらう、あるいは、解説にある表現を補足してもらうことで、突然腑に落ちることがあり得ます。\n\nもう一つ、その問題が、いわゆる「悪問」、受験生が誰も解けていないような問題、である可能性があります。ただ、それは先生に解説を頼んだ上での判断になると思います。先生が、「この問題は解ける必要はないよ」と言った場合は、その言葉をそのまま受け取ってください。\nそれでもモヤっとする場合、その対処法を私は知りません笑、ごめんなさい。\n\nとりあえず、自分一人で理解できないことはよくあるので、信頼できる人に教えを乞いましょう！それもテクニックです。\n質問者さんの健闘を祈っています！"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 24","className":"text-subPrimary 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"わからない問題にかける時間について"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"それは分野によって異なります。\n例えば 微分積分の問題は15分程度考えてわからなかったら答えを見ても良いと思います。\nなぜなら 微積はわりとワンパターンなので覚えたら終いだからです。\nそれに比べて 整数問題はワンパターンでは解けません。なのでじっくり考えるべきです。 どうしてもわからない時はその問題を一旦解くのをやめて、時間をおいて考えてみてください。 意外とわかったりします。\n\n数学の偏差値を上げるためには 勉強の際 一問を一問で完結させないことがポイントです。 そのためには 問題を解いたら その類題も解いてみたり、難しい問題が出て来たら どこの発想がなくて解けなかったのかしっかり分析することがひつようです。\nそしてもし過去問演習や模試の復習でわからない問題が出て来たら、 解答をすぐに見るのではなく、 思考のフローチャートを書いてみてください。\n\n具体的にいうならば\n三角関数の問題を解く際\n㊀グラフ㊁加法定理㊂変換公式\n\n→㊂でいこう\nCosだけの式になったから\n㊀tで置換する㊁因数分解する㊂tanに変換してみる\nなどなどと 樹形図のように思考回路を記すんです。\nするとどの状況でどの発想が足りなかったのかが明確になり、次にも繋がる勉強になります。やってみてください。"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 24","className":"text-subPrimary 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