小学数学凑十法与破十法

数学凑十法与破十法 小学一数学的计算内容主要包括:10以内的加减法、20以内的加减法、100以内的加减法三部分内容。其中，10以内的加减法和20以内的进位加法在第一册出现，20以内的退位减法和100以内的加减法(只包括口算部分)在第二册出现。 执教过一年级数学的老师对于这部分内容很熟悉，也一定了解“20以内的减法”的基本算理——“破十法”。在旧版教材中，“破十法”被摆在十分明显的位置，并通过例题的解法演示，一步一步地引领学生掌握。在课标实验教材中，“20以内的减法”从第1课“十几减几”开始，“十几减几”中出现的算式和例题都是个位够减的，并不涉及退位问题。第2课“11减几”则涉及退位减，并在这一课出现“破十法”。教材中出现情景“聪聪买皮球”，从11个皮球中取走3个，引导学生在生活情景中发现数学问题，并想办法解决。具体操作是运用学具“摆一摆、算一算”，并展示书中主人公“红红”和“聪聪”的办法，红红直接从11中数出3个，还剩8个;聪聪则使用“破十法”，把11分成10和1，先从10中取走3个，剩下7个，再加上1个等于8个。新教材中的这样的是符合“新课标”理念的，体现了“数学学习从生活实际出发”，“让学生学习有价值的数学”，“设计学生主动进行观察、实验、猜测、验证、推理与交流等数学活动”的思想。 可是，笔者在教学过程中发现了这样一个问题:当我们让孩子动手实践，鼓励他们“用自己的方法算时，教材中重要的“破十法”备受冷落，几乎没有学生选择这种思路解题。大多数孩子都选择了红红的temperature piping and equipment, avoid oil pipeline, avoiding vibration equipment, principles of construction of beautiful and neat easy to install. 5.1.5.2 from field instrumentation and valves field junction boxes to the single line slot cable using pipe, 220VAC power cord shall be full pipe laying, indoor coaxial cables concealed and outdoor wear buried. 5.1.5.3 bending cold bending method should be used to protect pipe, bending angle should not be less than 900, when wearing non-armoured cable and laid, bend radius should not be less than 6 times the pipe diameter, when while wearing armored cable and concealed, bend radius should not be less than 10 times the pipe diameter. Bend protection tube there should be no dents, cracks, single right-angled bend protection tube should not be more than two. 5.1.5.4 between protective tube and between protective tube and connector, threaded connections should be used, it tube end screw thread length should not be less than 1/2 of the fitting, and ensures the electrical connection. 5.1.5.5 protection tube should line up, ... 5.1.7 5.1.7.1 check tubing for pressure piping laying pipe, components, materials and specifications meet the design requirements, and have quality certificates, before laying should be cleared by welding and bending the welding slag, rust, chip or a foreign body, and preservative treatments. 5.1.7.2 in pressure piping welding welder must have a valid welder certificate. 5.1.7.3 tubing under the conditions to meet measurement requirements should be as short as possible, General length should not be more than 15m, pipe bending cold bending, cutting machine, minimizing bending and cross, and there shall be no sharp and complex bending, bending radius of not less than 3 times the pipe diameter. 5.1.7.4 pipe and process equipment, piping, or building surface distance is greater than 50mm, not be laid parallel to the oil, combustible, explosive 办法——直接从11中数出3个，得到剩下的8个。情争之下，老师只好亲自出马，在投影下演示了聪聪的“巧妙算法”，教材中的重点内容——破十法。然而，问题并未由此解决，当老师再次尊重学生感受，让孩子评价两种算法的优劣时，新的问题又出现了:所有的孩子都认为“红红的办法”更好，没人愿意主动接受教材中的“破十法”。 笔者认为，课堂中出现的这一现象颇有代表性，反映了课改实验中一个十分重要的问题——教材编写者和教学实施者对学生已有经验的忽视。仍以“破十法”的教学内容为例，我们在教材编写和教学设计中犯了以下两个错误: 其一，把握学生心理特点不准确。一年级学生以形象思维为主，逻辑思维能力低下。对于他们而言，11减3的最好办法就是从11中一个一个地数出3个。而且事实证明，他们的方法只需要一步就能完成，确实比先算10减3，再算3加1简单。在这样两种算法之间，孩子自然而然地会选择前者。 其二，教材设计不符合学生认知规律。11是10的“近邻”，以11为退位减法的开始可以降低教材难度，便于学生理解。3则是10以内一个较小的数，先算 11减3也是为了降低教材难度。可是，我们这样一味地考虑降低难度，恰恰剥夺了学生尝试其他方法的兴趣和刺激。试想，如果教材中先出现11减5，5在10 以内不算一个小数，直接用11去减，学生需要一个思考的过程。在思考的过程中，他们就可能尝试并发现新的方法。实验教材中，也出现了11减5，却是在11 nnections should be used, it tube end screw thread length should not be less than 1/2 of the fitting, and ensures the electrical coube should not be more than two. 5.1.5.4 between protective tube and between protective tube and connector, threaded connectiangled bend protection t-less than 10 times the pipe diameter. Bend protection tube there should be no dents, cracks, single rightshould not be less than 6 times the pipe diameter, when while wearing armored cable and concealed, bend radius should not be armoured cable and laid, bend radius-od should be used to protect pipe, bending angle should not be less than 900, when wearing noner cord shall be full pipe laying, indoor coaxial cables concealed and outdoor wear buried. 5.1.5.3 bending cold bending methAC powto install. 5.1.5.2 from field instrumentation and valves field junction boxes to the single line slot cable using pipe, 220Viping and equipment, avoid oil pipeline, avoiding vibration equipment, principles of construction of beautiful and neat easy temperature plosiveocess equipment, piping, or building surface distance is greater than 50mm, not be laid parallel to the oil, combustible, expnd prs, and there shall be no sharp and complex bending, bending radius of not less than 3 times the pipe diameter. 5.1.7.4 pipe apossible, General length should not be more than 15m, pipe bending cold bending, cutting machine, minimizing bending and cros rt asr must have a valid welder certificate. 5.1.7.3 tubing under the conditions to meet measurement requirements should be as shobending the welding slag, rust, chip or a foreign body, and preservative treatments. 5.1.7.2 in pressure piping welding welde ng andials and specifications meet the design requirements, and have quality certificates, before laying should be cleared by weldion. 5.1.5.5 protection tube should line up, ... 5.1.7 5.1.7.1 check tubing for pressure piping laying pipe, components, mater2 减3之后，这样的设计无法进一步打开学生的思路，学生会受到原有经验和先入为主效应的影响，教材中的重点——破十法，也就无从突破了。 笔者认为，在“课改实验”中，无论是教材的编写者，还是教学的执行者，仅有一些前卫的思想和理论是不够的。好的教学，必须是有的放矢的，必须是面向学生，面向全体学生的。尊重学生，就应该尊重他们的理解和经验，尊重他们的思考的发现，更要尊重他们的心理特点和认知规律。只有我们真正了解学生，才能够真正实现“让每个孩子在生活情景中，学到有价值的数学”。 进位加的计算方法是“凑十法”，退位减的计算方法是“破十法”，即掌握“凑十法”与“破十法”是熟练计算20以内、100以内的加减法的基础和关键。 一、凑十法 “凑十法”是计算20以内进位加法的基础和核心，它的过程很复杂，简言之是:先分，再凑十，最后把零头相加。具体来说，分之前的观察十分重要，首先决定凑哪个数，分哪个数，要凑成十的那个数还缺几，就从另一个数分出几，这个思维过程包含量不变的数学思想和最优化思维。这里的最优化思维是思考问题从最好、最简洁的角度，因此这种“凑十法”就是让学生计算起来既容易想，又不容易出错的过程。“凑十法”虽然分三步，但这个过程之前的观察思维是复杂的，是“凑十法”过程的酝酿和成形，所谓的“胸有成竹”。例如:9+7，它的“凑十法”有两种不同的形式， layin ensures the electrical connection. 5.1.5.5 protection tube should line up, ... 5.1.7 5.1.7.1 check tubing for pressure pipingnector, threaded connections should be used, it tube end screw thread length should not be less than 1/2 of the fitting, and angled bend protection tube should not be more than two. 5.1.5.4 between protective tube and between protective tube and con-right bend radius should not be less than 10 times the pipe diameter. Bend protection tube there should be no dents, cracks, singleconcealed, armoured cable and laid, bend radius should not be less than 6 times the pipe diameter, when while wearing armored cable and -bending cold bending method should be used to protect pipe, bending angle should not be less than 900, when wearing non.5.3 t cable using pipe, 220VAC power cord shall be full pipe laying, indoor coaxial cables concealed and outdoor wear buried. 5.1beautiful and neat easy to install. 5.1.5.2 from field instrumentation and valves field junction boxes to the single line slo temperature piping and equipment, avoid oil pipeline, avoiding vibration equipment, principles of construction ofthe oil, combustible, explosiveel to e diameter. 5.1.7.4 pipe and process equipment, piping, or building surface distance is greater than 50mm, not be laid parallinimizing bending and cross, and there shall be no sharp and complex bending, bending radius of not less than 3 times the pipine, muirements should be as short as possible, General length should not be more than 15m, pipe bending cold bending, cutting machssure piping welding welder must have a valid welder certificate. 5.1.7.3 tubing under the conditions to meet measurement reqin prehould be cleared by welding and bending the welding slag, rust, chip or a foreign body, and preservative treatments. 5.1.7.2 g pipe, components, materials and specifications meet the design requirements, and have quality certificates, before laying s3 1、 9 + 7=16 2、 9 + 7=16 第一种思考的过程是:凑9分7，凑9成十缺1，需要从7中分出1，还剩下6。具体过程是:?把7分成1和6，?1与9凑成10，?10再加7剩下的6是16。第二种思考的过程是:凑7分9，凑7成十缺3，需要从9中分出3，还剩下6。具体过程是:?把9分成6和3，?3与7凑成10，?10再加9剩下的6是16。 以上两种形式的“凑十法”，如果只看算式想凑十过程，即抽象又复杂，如旱地里拔葱，很难，结合实物、学具让学生动手摆摆，在学生的头脑中留下的是生动有趣的实物图象，如:上面的9+7，让学生左边摆9个黄圆形，右边摆7个红圆形，然后把1个红圆形移到左边成为10个，学生口述过程是:7分成1和6，9加1得10，10加6得16;或者左边的3个黄圆形移到右边，右边成为10个，学生口述过程是:9分成3和6，3加7得10，10加6得16。 理解了“凑十法”的过程，就是明白了“凑十法”的算理，接着采用各种形式巩固“凑十法”。一是看图填方框，如: 9 + 5=? 9 + 5=?, 二是根据教材提供的习题，圈圈、算算，三是根据教材提供的习题，看图列式计算等。 r must have a valid welder certificate. 5.1.7.3 tubing under the conditions to meet measurement requirements should be as shobending the welding slag, rust, chip or a foreign body, and preservative treatments. 5.1.7.2 in pressure piping welding welde ng andials and specifications meet the design requirements, and have quality certificates, before laying should be cleared by weldion. 5.1.5.5 protection tube should line up, ... 5.1.7 5.1.7.1 check tubing for pressure piping laying pipe, components, maternnections should be used, it tube end screw thread length should not be less than 1/2 of the fitting, and ensures the electrical coube should not be more than two. 5.1.5.4 between protective tube and between protective tube and connector, threaded connectiangled bend protection t-less than 10 times the pipe diameter. Bend protection tube there should be no dents, cracks, single rightshould not be less than 6 times the pipe diameter, when while wearing armored cable and concealed, bend radius should not be armoured cable and laid, bend radius-od should be used to protect pipe, bending angle should not be less than 900, when wearing noner cord shall be full pipe laying, indoor coaxial cables concealed and outdoor wear buried. 5.1.5.3 bending cold bending methAC powto install. 5.1.5.2 from field instrumentation and valves field junction boxes to the single line slot cable using pipe, 220Viping and equipment, avoid oil pipeline, avoiding vibration equipment, principles of construction of beautiful and neat easy temperature plosiveocess equipment, piping, or building surface distance is greater than 50mm, not be laid parallel to the oil, combustible, expnd prs, and there shall be no sharp and complex bending, bending radius of not less than 3 times the pipe diameter. 5.1.7.4 pipe apossible, General length should not be more than 15m, pipe bending cold bending, cutting machine, minimizing bending and cros rt as4 学生掌握了“凑十法”的算理后，利用各种形式，采用各种手段，达到脱口而出的程度，从而形成技能，这一步决定着学生的熟练程度，只有多加练习，才能形成熟练的口算能力。一般每堂课都拿出一些时间练习计算，天天练，日久也就克服了这个难点，做得又对又快。 二、破十法 20以内的退位减法的计算方法，比进位加法多一些，因为有进位加法做基础，所以教材上只介绍了“做减法，想加法”，教材上出现的20以内的退位减法的例题是以插图的形式出现的，如:12—9，12个风筝用虚线圈起9个，学生一看就知道得3个，做一做中，用小棒摆摆或者圈圈，总之，都是以学生喜闻乐见的形式出现的。但是，如果20以内的进位加法不过关，做20以内的退位减法就会有影响，因此，除了做减法想加法外，我向学生介绍了典型的方法——用“破十法”来计算20以内的退位减法。 由于学生已经学会了复杂的“凑十法”，所以在学习“破十法”时，相对简单些。“破十法”的思维过程是:先分，后减，再加。即先从被减数分出一个10，再从10中减去减数，所得的差与被减数剩下的合起来。如:12 — 9=3，先把12分成10和2，10减9得1，1再加2得3。 2 1 ine, muirements should be as short as possible, General length should not be more than 15m, pipe bending cold bending, cutting machssure piping welding welder must have a valid welder certificate. 5.1.7.3 tubing under the conditions to meet measurement reqin prehould be cleared by welding and bending the welding slag, rust, chip or a foreign body, and preservative treatments. 5.1.7.2 g pipe, components, materials and specifications meet the design requirements, and have quality certificates, before laying slayin ensures the electrical connection. 5.1.5.5 protection tube should line up, ... 5.1.7 5.1.7.1 check tubing for pressure pipingnector, threaded connections should be used, it tube end screw thread length should not be less than 1/2 of the fitting, and angled bend protection tube should not be more than two. 5.1.5.4 between protective tube and between protective tube and con-right bend radius should not be less than 10 times the pipe diameter. Bend protection tube there should be no dents, cracks, singleconcealed, armoured cable and laid, bend radius should not be less than 6 times the pipe diameter, when while wearing armored cable and -bending cold bending method should be used to protect pipe, bending angle should not be less than 900, when wearing non.5.3 t cable using pipe, 220VAC power cord shall be full pipe laying, indoor coaxial cables concealed and outdoor wear buried. 5.1beautiful and neat easy to install. 5.1.5.2 from field instrumentation and valves field junction boxes to the single line slo temperature piping and equipment, avoid oil pipeline, avoiding vibration equipment, principles of construction ofthe oil, combustible, explosiveel to e diameter. 5.1.7.4 pipe and process equipment, piping, or building surface distance is greater than 50mm, not be laid parallinimizing bending and cross, and there shall be no sharp and complex bending, bending radius of not less than 3 times the pip5 三、100以内的加减法 100以内的加减法包括整十数加、减整十数，两位数加、减一位数和整十数，以上这些内容教材上都是以图画的形式出现，在一个个故事情节中学习。其中，重点是两位数加、减一位数和整十数，通过操作小棒，成捆的与成捆的相加减，单根的与单根的相加减，从而让学生理解十位上的数相加减，个位上的数相加减的算理。两位数加、减一位数和整十数中，两位数加、减一位数的进位加、退位减是教学的难点，它计算的基础和核心还是“凑十法”与“破十法”，采用的具体措施是让学生操作小棒，掌握方法。 1、两位数加一位数的进位加 如:教学24+9时，左边摆2捆零4根小棒，右边摆9根，怎样把它们合起来,9根加4根是13根，即1捆零3根，这一步就是“凑十法”的过程，接着，13根加2捆是3捆零3根。然后，不摆小棒，看着算式24+9直接口述计算过程，即:4加9得13，13加20得33。 2、两位数减一位数的退位减 如:教学36—8，先摆出3捆零6根小棒，然后拿走8根，怎样拿,?拿走8根的一种方法是，先拿走散的6根，再从一个整捆中拿走2根，剩下的8根与原来的其它两捆合起来是28根。?拿走8根的另一种方法是:散的6根减8根不够减，需与1捆合起来是16根，从16根里减去8根得8根，这一步就是“破十法”的应用，然后8根与原来的2捆合起来是28根。第一种方法容易理解，但过程相对麻烦些，第二种方法用破十法计算，也容易理解，过程也相对简单些， ials and specifications meet the design requirements, and have quality certificates, before laying should be cleared by weldion. 5.1.5.5 protection tube should line up, ... 5.1.7 5.1.7.1 check tubing for pressure piping laying pipe, components, maternnections should be used, it tube end screw thread length should not be less than 1/2 of the fitting, and ensures the electrical coube should not be more than two. 5.1.5.4 between protective tube and between protective tube and connector, threaded connectiangled bend protection t-less than 10 times the pipe diameter. Bend protection tube there should be no dents, cracks, single rightshould not be less than 6 times the pipe diameter, when while wearing armored cable and concealed, bend radius should not be armoured cable and laid, bend radius-od should be used to protect pipe, bending angle should not be less than 900, when wearing noner cord shall be full pipe laying, indoor coaxial cables concealed and outdoor wear buried. 5.1.5.3 bending cold bending methAC powto install. 5.1.5.2 from field instrumentation and valves field junction boxes to the single line slot cable using pipe, 220Viping and equipment, avoid oil pipeline, avoiding vibration equipment, principles of construction of beautiful and neat easy temperature plosiveocess equipment, piping, or building surface distance is greater than 50mm, not be laid parallel to the oil, combustible, expnd prs, and there shall be no sharp and complex bending, bending radius of not less than 3 times the pipe diameter. 5.1.7.4 pipe apossible, General length should not be more than 15m, pipe bending cold bending, cutting machine, minimizing bending and cros rt asr must have a valid welder certificate. 5.1.7.3 tubing under the conditions to meet measurement requirements should be as shobending the welding slag, rust, chip or a foreign body, and preservative treatments. 5.1.7.2 in pressure piping welding welde ng and6 但需要有扎实熟练的20以内的退位减法作基础，否则，计算正确率低，或计算速度特别慢，现在，在教学中，主要应用第二种口算方法，即用的基本方法还是“破十法”。 从印象中，现行教材与传统教材有一些差别，用“凑十法”计算20以内的加法时，介绍了两种本质相同形式不同的方法，有利于学生发散思维能力的拓展，传统的教材只介绍了一种，思路相对狭窄些。而传统的教材用“破十法”计算20以内的退位减法时，例题很多，可能因为这种方法难理解的缘故，而现行教材主要是看减法想加法，所以只要用一个例题引出计算方法就可以，因此，现行教材在教学20以内的退位减法时，干脆利落，就纯粹计算部分只出现了一个例题。 用“凑十法”、“破十法”来计算20以内、100以内的进位加和退位减，是教学中强调的重点，大部分学生掌握得较好，算得又对又快，但是就是有个别学生不习惯于用这种方法，他们喜欢数手指或列竖式(现在教材没有出现竖式，可能是家长教的)来计算，虽然速度慢了些，也能算对，其实，随着他们年龄的增长，随着练习的增多，即使学不会“凑十法”、“破十法”，他们的计算能力也能逐步提高。从另一方面讲，他们虽然算的慢，但在活生生的现实中会算会用，并也表现出一些灵性，能积极、灵活、敏捷的应对各种问题，这样的学生是不是也应该被认可，正如数学课程所指出的:“不同的人在数学上能得到不同的发展。” nector, threaded connections should be used, it tube end screw thread length should not be less than 1/2 of the fitting, and angled bend protection tube should not be more than two. 5.1.5.4 between protective tube and between protective tube and con-right bend radius should not be less than 10 times the pipe diameter. Bend protection tube there should be no dents, cracks, singleconcealed, armoured cable and laid, bend radius should not be less than 6 times the pipe diameter, when while wearing armored cable and -bending cold bending method should be used to protect pipe, bending angle should not be less than 900, when wearing non.5.3 t cable using pipe, 220VAC power cord shall be full pipe laying, indoor coaxial cables concealed and outdoor wear buried. 5.1beautiful and neat easy to install. 5.1.5.2 from field instrumentation and valves field junction boxes to the single line slo temperature piping and equipment, avoid oil pipeline, avoiding vibration equipment, principles of construction ofthe oil, combustible, explosiveel to e diameter. 5.1.7.4 pipe and process equipment, piping, or building surface distance is greater than 50mm, not be laid parallinimizing bending and cross, and there shall be no sharp and complex bending, bending radius of not less than 3 times the pipine, muirements should be as short as possible, General length should not be more than 15m, pipe bending cold bending, cutting machssure piping welding welder must have a valid welder certificate. 5.1.7.3 tubing under the conditions to meet measurement reqin prehould be cleared by welding and bending the welding slag, rust, chip or a foreign body, and preservative treatments. 5.1.7.2 g pipe, components, materials and specifications meet the design requirements, and have quality certificates, before laying slayin ensures the electrical connection. 5.1.5.5 protection tube should line up, ... 5.1.7 5.1.7.1 check tubing for pressure piping7