本研究的主要目的在於探討國小兒童分數減法學習層次與皮亞傑式認知能力之關 係。研究對象為花蓮地區五、六年級學生215名，受試依其在皮亞傑式認知能力測驗之得分 分為具體操作期、過渡時期及形式操作期三個群組，再以每個群組兒童在分數減法測驗（利 用Intraconcept方法分析產生）之作答樣式，利用Walbesser指數及Ordering Theory方法分析並產生有效的學習層次，此等層次乃是兒童取向之學習層次，故能配合兒童之自然學習序列，而達到成功的學習。研究的結果發現： 　　1.五年級兒童尚有85.6％停留在具體操作期，真正進入形式操作期者僅佔1.8％；六年 級兒童尚有58.7％停留在具體操作期，真正進入形式操作期者僅佔12.5％；五、六年級兒童 合併則尚有72.6％停留在具體操作期，真正進入形式操作期者僅有7.0％。 　　2.過渡時期和形式操作期兒童對於各類型之分數減法運算均已相當純熟，其學習層次也 就不存在了。 　　3.具體操作期兒童取向之學習層次確實存在，且各類型分數減法之層次關係與現行國小 數學科教材出現之順序略有差異。 　　4.分析兒童取向之學習層次時，利用Phillips等學者提出之intra-concept方法，能引入一些利用傳統的邏輯性之工作分析方法無法引入之變項（例如：質數、合數、倍數關係等） 。 　　5.單獨由題目難度順序排列，無法產生有效的學習層次。 根據本研究之發現，研究者並提出若干建議，以供編寫教材及教學之參考。

　　　　　Given that the probability of an event happened is 5/13, we put it into practice for 7 times, then the frequency of appearing once exactly. is one of the numbers 0,1,2,3,4,5,6,7. We can ascertain that the frequency of 3 times is the most possible, because of 3/7 is closest to 5/13among {0/7,1/7,2/7,...7/7,} . Therefore 3/7 is a possible probability with 5/13. Furtheremore, we extend the practice to another field {6,7,8,9}. These possible probabilities can be:　2/6, 3/7, 3/8, 4/9. In these possible probabilities the given value might approximate as follows: |2/6-5/13|=0.051282 |3/7-5/13|=0.043956 |3/8-5/13|=0.009615 |4/9-5/13|=0.059823 Obviously, in the experiment of intervals [6,9], 3/8 is the optimum approximation of probability with 5/13. The above research will extends to solve optimum rationalizing for approximation. For example, it limits the range of denominators among 50-100 , finding the closest fraction with π. Under general conditions, if given any real number ψand intervals [a,b], we hope to find a rational number f/e, so that |f/e-ψ|=min |p/q-ψ| , and that for all, e,q ε[a,b] and denote that e,f,p,q are non-zero integers. To solve the above problem efficiently, even though the example given of solving the problem can't be found by analizing it in the computer's program, I take the following steps: 1.Give up the way to solve it by only using a formula and seek for a method of approximation to get to the points. 2.Geometrize numbers and interpret by lattice points. 3Extend the concept of convergents. 4.Establish and combine the model of Optimum Rationalizing for Approximation to get the destination. 5.Design the computer's program by the model of mathematics given, to test for accuracy and speed.