作者：教學科-組別1

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解鎖密碼「五五學習方陣」

文/新北市私立及人小學 吳月貴老師

一、前言

　　「五五學習方陣」是從一個簡單的數學小遊戲，歷經多年的教學應用及改良，成為今日有多版本的學習遊戲。「基礎版」是簡易的五五數學方陣，承襲當年看到的原始設計，只是將它廣泛應用於高年級數學領域各單元，作為單元複習或多單元統整複習，皆有不錯的成效；「進階版」的進階程度，來自誘答的複雜性，對學生而言，深具挑戰性；「統整版」是筆者發展的教學設計，融入多領域的知識內涵，讓學生藉由遊戲，熟練各領域的概念，藉此增進各項能力。每屆實施此教學活動，學生總是興致高昂，具有高度的完成動力，有助於對習題的精熟練習。

 

二、文獻探討

　　筆者利用五五方陣設計一連串延伸的教學方陣，起源於大約三十年前在報紙副刊上看到的數學小遊戲。後來在各種數學遊戲書籍中所見到的許多「方陣」的數學遊戲，皆是指在每個格子中填上一個自然數，使直、橫、斜加起來都變成相同的數目，有三方陣、四方陣、五方陣等不同的數學益智遊戲(1)。有些書籍內容則將它命名為「魔方陣」，它根源於中國古時的「幻方」(2)。

　　有一篇來自智利的4-5年級數學教學活動，名稱為「恆星益智遊戲。乘法遊戲」(3)。遊戲方式是利用12塊三角形重新拼排成六角星形，每塊三角形的邊上包含兩個乘法和一個答案，將兩塊三角形放在一起，使乘法和答案緊挨著，最後會出現一個漂亮的六角星形。這種利用兩個邊緊挨著「題目-答案」組合的遊戲方式，即為本文學習方陣的解鎖方式。

 

三、動手做數學方陣

　　製作五五數學方陣需要一些訣竅和時間，先以三三數學方陣為例，說明它的製作方式及需要注意的小妙招。後文的五五方陣或其他方陣則依此法衍生。

步驟一：畫「空白表格」，如下圖：

　　先畫好一個 3格×3格的三三數學方陣，英文字母是決定數字卡的方向。此圖是在PPT上繪製出的表格。

步驟二：製作「解答」，如下圖：

　　依序將設計好的算式及答案，填在相鄰的邊上，此為「解答」。例如：「26+34=60」，「26+34」置放在數字卡E的下方，「60」置放在數字卡B的上方；「8×17=136」，「8×17」置放在數字卡E的右方，「136」置放在數字卡A的左方。選擇PPT繪製，便是因為圖文皆可旋轉角度，方陣裡的算式或答案可以依照構思作排列。

步驟三：製作「題目」，如下圖：

　　將解答沿著方陣的經緯線剪開，成為一張張如下圖的數字卡，再重新隨機安置，變成「題目」。製作題目時需要注意英文字母的方向。

 

四、數學方陣在教學上的應用

(一)「基礎版」五五數學方陣

　　學生拿到數學方陣的題目後，先沿著方陣的經緯線剪開，成為一張一張的數字卡，再按照相鄰為正確的題目及答案的組合來重新拼排。

　　當學生嘗試幾次遊戲後，會發覺有些數字卡的上下、左右方有一些空白處，進而察覺該數字卡應該置放於方陣的旁邊或四個角落，而加速他們完成數學方陣，不過並不影響學生對數學基礎概念複習的初衷。此款能加速完成的數學方陣設計為「基礎版」。

　　在多年設計五五數學方陣的經驗中，任何單元的數學概念皆可入題，只要題目不過長，以計算題為佳，畢竟當初設計的起點就是為了加強學生的計算能力。此款「基礎版」共需要40組題目和答案。

▲ 圖A

　　在同一個五五數學方陣中，可以選擇同一個單元入題，當作單元複習，以上圖圖A為例，那是五年級上學期「多位數的乘除法」單元，遊戲設計目的即為供學生增進該單元的計算能力；也可以多個單元入題，當作總複習，以下圖圖B為例，那是融合五年級下學期「分數」、「體積」、「容積」、「時間的計算」等單元的設計。

　　下圖圖B是「基礎版」的題目，下圖圖C是它的解答。在學生完成方陣組合的過程中，曾經詢問老師:為什麼有時候同一份五五數學方陣上會出現重複的英文字母？老師笑而不答，只告訴學生:那是提醒作答者要將數字卡擺正！事實上，那是別有用意的設計。

　　在圖C解答中，第一橫排的英文字母是「ORDER」，第二橫排的英文字母是「CANDY」，第三橫排的英文字母是「PIZZA」，第四橫排的英文字母是「SNACK」，第五橫排的英文字母是「SUGAR」，它的設計是方便老師批閱或對答案，還可以順便檢驗學生對英文單字的認識。

▲ 圖B

▲ 圖C

(二)「進階版」五五數學方陣

　　下圖圖D是五年級下學期「小數的乘除法」單元，任何一張數字卡的上下、左右方都有題目或答案，也就是學生不能再心存僥倖想快速完成，他們必須一題一題計算，先找出兩兩相對的「題目-答案」組合，再依照線索拼排成正確的方陣。此款必須按部就班計算的數學方陣設計為「進階版」。

▲ 圖D

　　此款「進階版」共需要40組位於經緯線上正確的題目和答案，和另外10組分別散置在周圍邊上混淆作答的題目和答案。這10組題目和答案的誘答性愈高，愈能增加完成此方陣的難度，教師可以依照學生程度進行設計。

(三) 跨領域的「統整版」五五學習方陣

　　「統整版」為融合數學、國語、自然、社會、英文等多項領域而設計的進階版五五學習方陣。要選擇哪些領域、那些單元入題，以教學目的導向，可靈活運用，增添教學趣味，提高學習成效。

　　下圖圖E是融合數學、國語的「基礎統整版」，因為在某些數字卡的周圍仍可見未安排題目或答案，所以作答者能先行大致安排方陣周圍的數字卡，接著再逐步破解、尋找每一組答案。

　　下圖圖F是融合數學、國語、社會、自然、英文的「進階統整版」，因為每一張數字卡的周圍都有題目或答案，作答者必須逐步破解、尋找每一組答案，因為其中有些誘答的題目或答案干擾遊戲進行，所以還要兼顧排列位置，最後才能完成拼圖。這款設計頗受學生喜愛，因為可以檢驗自己對課程內容的複習程度，還能紓解評量考前的緊張壓力。

▲ 圖E

▲ 圖F

 

五、後語

　　經過多年的教學應用，發覺完成方陣需要多項能力，從最基礎的計算能力，到高階的組合邏輯，尤其是誘答性高的進階版數學方陣。進行方式可以採取以單元複習為目的，讓學生熟練該單元的概念或計算技能；有時則是期中、期末評量前的「減壓」複習，學生透過它更加精熟學習重點，但卻以為只是遊戲而樂於完成；我們還嘗試運用在大型的親子數學闖關遊戲中，讓學生組隊，藉由分組合作學習，完成解鎖五五數學方陣的任務，家長可從旁協助小組成員，進而了解自己孩子的能力高低，是一舉多得的活動及教材設計。

　　在設計五五數學方陣的過程中，製作解答、重新安排題目的步驟相對簡易，真正繁複的部分是在電腦PPT上的作業。未來若能寫出程式設計軟體，教師只需要選擇想製作的學習方陣種類，例如：二二方陣、三三方陣、四四方陣、五五方陣……，再鍵入所需要的題組，即可自動、隨機組成基礎版、進階版、統整版的學習方陣，甚至變成學習遊戲軟體，讓學生可以在家操作、複習功課，相信必能促使學生樂在學習中。

 

(點我下載PPT)

六、參考書目

1. 文明豐編著（1987）。《增進計算能力的數學遊戲》。臺北：益群書店。

2. Claudia Zaslavsky（2002）。《生活科學28民俗數學遊戲》。陳昭蓉譯。臺北：遠哲科學教育基金會。

3. Pérez ，M.C. （2016）。JUEGO PUZZLE ESTELAR. JUEGO DE MULTIPLICAR（恆星益智遊戲。乘法遊戲）。（2022-8-21）
    https://www.aulapt.org/2016/02/13/juego-puzzle-estelar/

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