Strengthening estimation teaching has become an important part of the new curriculum reform. However, due to the particularity of estimation teaching itself, its diversity in problem-solving strategies and the uncertainty of calculation results, estimation teaching has become one of the teaching contents that teachers are confused about.
Confusion 1: Teachers don’t pay enough attention to estimation and lack of guidance.
Our teachers have always paid more attention to accurate calculations, and the teaching materials are also organized in this way. As a result, many teachers do not pay attention to allowing students to understand the meaning of estimation and experience the value of estimation. They only allow students to briefly discuss the methods and results of estimation according to the procedures of the textbook, thus making estimation "tasteless" and teaching it tasteless. , cannot be abandoned. What’s the point of such an estimate?
Confusion 2: Students have little awareness of estimation and weak ability
For estimation teaching, the teaching materials are all about calculation activities centered on solving specific problems. In actual teaching, we require students to first figure out whether to estimate or calculate accurately. Most students discovered a pattern in the process of solving problems: any question with words such as "estimate, approximately" is an estimate. However, although the teaching content of some textbooks contains words such as "approximately", there is no need to estimate it. It makes students confused and teachers also feel very embarrassed in teaching.
Confusion 3: The teaching materials do not reflect enough on estimation and the evaluation is insufficient
Various versions of the teaching materials have significantly increased the estimation content and are more systematic than before. However, I personally feel that the overall arrangement is not very effective and the arrangement is not reasonable and permeable enough. , so arranging an estimation class because of calculations does not pay enough attention to estimation teaching, and the arrangement of teaching materials is somewhat stylized and lacks rationality. In addition, there is a lack of evaluation of students' estimation teaching and estimation ability. In daily exercises and tests, estimation content is rarely included, which makes it even less likely to attract the attention of teachers and students.
has the following thoughts on several misunderstandings in estimation teaching:
Misunderstanding 1: "The evaluation standard for estimation is single" thinking
Simple calculation questions basically follow the "rounding" method. Unifying estimated answers and understanding estimates as random calculations are two extremes. Many times, when we talk about estimation, we naturally think of the "rounding" method. Indeed, we still need this method for more explanations of estimation. For example, when calculating the question 58×34, there are many calculation methods:
(1) 58 is regarded as 60, 34 is regarded as 30, and the product is 1800.
(2) 58 is regarded as 60, 34 is regarded as 35, and the total is 2100.
(3) 58 is regarded as 60, 34 remains unchanged, and the total is 2040.
(4) 58 remains unchanged, 34 is regarded as 30, and the total is 1740.
It can be seen that the estimation methods are diverse. When conflicts arise between diversified evaluations and standard answers, teachers do not need to pursue a unified answer, as long as it is reasonable, rather than having a unified standard answer.
Misunderstanding 2: Thinking about actuarial calculation first and then estimation
When many students do estimation questions, they will first calculate the accurate value and then use the accurate value to estimate. This fact shows that some students do not understand how and why to estimate. Such an estimate is obviously contrary to the original intention of the estimate. The so-called estimation should be a rough estimate of the calculation results within a certain range. Its original intention is to quickly find the accurate value within the allowable range when the exact value is not required or it is difficult to find the exact value. It is related to There aren't many necessary connections to the correct value. The reasons are: First, some students do not understand the meaning of estimation and think that using correct values to estimate is a clever method. Overall, I still don’t understand why we need to estimate. Second, some of the questions are very simple to calculate in writing. Students feel that they can directly get the correct value without estimating, so why do they need to estimate.Third, we usually learn to calculate accurate values first. Students have the preconceived impression that estimation is also such a calculation. In fact, estimation is more demanding than ordinary calculations. It challenges thinking more and requires students to have a good sense of numbers and flexibly use all calculations. Learning is just the ability.
Misunderstanding 3: Thinking about estimating for the sake of estimating
When teaching estimation application questions, how to guide students to choose whether it is necessary (necessary) to estimate based on the specific situation? There are no words such as "approximately" in the question, so many students will not choose to estimate. For example: "Fifth grade students are going on an autumn outing. Each set of tickets and tickets is 49 yuan. A total of 104 sets of tickets are needed. How much money should be prepared to buy tickets?" If this question does not appear in the estimation class, then there must be many students who choose to calculate directly. , how many students would think of using estimates? It is safe to say that if a student chooses to calculate accurately, the teacher will also correct him correctly. So, here, the key is to see how the teacher creates this situation. The teacher can remind me that this is a spring outing. Look, if the teacher brings exactly 5,096 yuan, is it reasonable? This guides students to explore together this type of questions in life that require estimates but the words "approximately" are hidden in the questions. The estimated results are reasonable as long as they can meet the needs of the actual problem.
Sometimes, there are these words in the question, but there is no need to estimate. For example: "A silkworm spins approximately 1,500 meters of silk. Xiaohong raised 6 silkworms. How many meters of silk do they spin?" It can be seen from the number in the question that it is obviously inappropriate to use 1,500 as 2,000 and then use the estimate. In fact, this At this time, students can understand that 1500 meters is already an approximate number, and there is no need to estimate it into other numbers for calculation. The result 1500×6=9000 meters is already an approximate number.
Questions similar to this. For example: Xiaojun walks about 65 meters per minute, and it takes him about 8 minutes to walk from home to school. How many meters is it from his home to the school? In this question, 65 meters and 8 minutes are already approximate numbers, and the calculation of 65×8 is also an approximate value. There is no need to make an estimate, but the result is indeed the approximate number of .
Under the analysis of such exercises, students gradually understand that some questions with words such as "approximately" are actually the results obtained through precise calculations, which are still approximate numbers rather than precise numbers, thus truly understanding the role of estimation teaching in daily life. the meaning of existence.
Strategies for optimizing estimation teaching:
Strategy 1: Feel the value of estimation
Hua Luogeng once said: "One of the reasons why people have the impression that mathematics is boring, mysterious and difficult to understand is that it is divorced from reality." So it should start from Students are familiar with life situations and things they are interested in, so that they can realize that estimates are around them, feel the practicality of estimates, and develop a sense of familiarity with estimates.
1. Seize the opportunity of estimation
During teaching, teachers should further explore the teaching materials of estimation and seize the opportunity of estimation teaching. For example, when teaching "Estimation by Addition and Subtraction", a sample question is given: Mom takes 100 yuan to the store to buy the following daily necessities: 28 yuan for a thermos, 43 yuan for a kettle, and 24 yuan for a water cup. Does mom bring enough money? During the exchange, one student said: "28+43=71 (yuan). It costs about 70 yuan to buy a thermos and kettle. The remaining money is enough to buy a water cup." Another student said: "28+43+ 24=95 (yuan), 95 yuan does not exceed 100 yuan, so the money brought by my mother is enough." This lesson is arranged before the written arithmetic teaching, and it can even be advanced to the second volume of the first grade to include "adding and subtracting whole tens." Learn after counting. In this way, since most students are not very proficient in the calculation of general two-digit addition and subtraction of two-digit numbers, they will be willing to choose the estimation method and understand the necessity of estimation.
2. Creating an estimation situation
When teaching for the first time, we can also create one or several uncertain quantities, which will help change the students' concept that accurate calculations must be made everywhere, so as to understand the necessity of estimation. Take the lesson just now, for example, you can deliberately blur the last numbers (i.e. 28, 43) of the prices of thermos bottles and kettles when the example questions are presented.At this time, students are unable to perform precise calculations, and their thinking naturally shifts from actuarial calculation to estimation. For example: the price of a hot water bottle can be regarded as 30 yuan, the price of a kettle can be regarded as 40 yuan, 30+40=70 (yuan), and a water cup of 24 yuan is enough.
Another example: let students know something like 124×3=? When will such an estimate appear? You can formulate a question like this: There is a big price reduction on down jackets in the mall. A Yaya down jacket is 124 yuan. Mom wants to buy three pieces. Without counting, can you estimate how much money mom needs to bring? When solving, you can set up such a link:
①What do you ask for when you ask your mother how much money you need to spend? (The student’s answer is to find out what three 124s are);
② Can you make equations? (Student answer 124×3=);
③ Explore the estimation method. Students should explore independently. When summarizing, it is clear that 124 is regarded as about one hundred, and one hundred multiplied by 3 is more than three hundred. At this time The answer is more than a few hundred; if the answer is more than a few hundred or tens, then it should be handled like this: think of 124 as more than one hundred and twenty, and multiplying more than one hundred and twenty by 3 is more than three hundred and sixty. In this way, students can experience estimation teaching and understand it.
strategy two: understand the estimation method
There are many estimation methods, but not all estimation methods are correct. Wrong estimation methods can lead to wrong decisions, which illustrates the difference between successful and unsuccessful estimation strategies. Although this is more complicated than ordinary word problems, students can understand it through effective guidance.
1. Understand the general strategies of estimation
Mathematical estimation should be based on principles. It must be in the form of oral calculation. Within the allowed range, the simpler the better. But its formation does not happen overnight. It requires teachers to pay attention to "estimation" teaching, and it also requires teachers to cultivate it effectively and scientifically over a long period of time. During teaching, teachers should guide students to understand the general strategies of estimation based on the teaching content. The first is the simplicity strategy. That is, simplifying numbers into simpler forms to make calculations easier. For example, 8725 can be shortened to 8700. Second is the conversion strategy, which simplifies the problem even further. For example, when estimating the answer to 53+49+46, a good estimator might convert the problem into a multiplication problem. Such a student would think: 50 times 3 is 150, so the answer is almost 150.
2. Master the basic methods of estimation
The most important thing in estimation teaching lies in the guidance of estimation methods. When teaching, we can guide students to master some basic estimation methods:
| ① rounding estimation method. When performs the four arithmetic operations on decimals, it retains the addend, minuend, subtrahend, factor, dividend, and divisor to integers based on the "rounding method", and then calculates the approximate number. For example, 3.14×7.21, students can estimate that their product is about 21 based on 3×7=21, and then calculate the accurate result.
②Digital estimation method. When calculates the multi-digit multiplication of integers and division , based on the number of digits of the factor, dividend, and divisor, estimate the number of digits the product or quotient is. The number of digits of the product is equal to the sum of the digits of the two factors or one less than the sum, and the number of digits of the quotient is equal to the number of digits of the dividend minus the number of digits of the divisor or the difference is one more than this difference. For example, 456×64, students can deduce that its product is four or five digits based on this experience.
③ Rule-based estimation method. estimates based on relevant laws in teaching. For example, when calculating decimal multiplication and division, one factor (except zero) can be less than 1 and the product is less than another factor; one factor is greater than 1 and the product is greater than the other factor. If the divisor is greater than 1, the quotient is less than the dividend; if the divisor is less than 1, the quotient is greater than the dividend.
④ Contact the actual estimation method. For example, the number of animals, the number of trees, and the number of chartered boats must be integers; airplanes fly much faster than people walk; the germination rate and attendance rate cannot exceed 100%, etc.
⑤Use the small method to estimate the big method or use the big method to estimate the small method. When estimating in , if the number is too large or too small and difficult to estimate, first estimate the number of units, and then estimate the number that is too large (or too small) based on the number of units, that is, the overall number.For example, to estimate the weight of a peanut, we can first estimate the weight of 100 peanuts, and then divide it by 100 to estimate the weight of a peanut.
strategy three: Cultivate the habit of estimation
Everyone uses estimation frequently in life, but students do not have the habit of estimation during study. This is because learning and application are separated. In order to cultivate students' estimation awareness, learning and application must be combined. In teaching, I skillfully use the "three changes" to effectively improve students' estimation skills.
1. Change "dispensable" to "ubiquitous". , estimation has been an optional content in primary school mathematics textbooks for a long time, and because this part of the content is presented in a relatively simple format, it fails to reflect the complete intention of cultivating students' estimation ability and serves as an optional Dispensable role. Therefore, teachers should use teaching materials creatively and create opportunities for students to make estimates even in areas where estimation is not required in the original teaching materials, activate the teaching materials, and make students feel that estimation is everywhere. In the fields of mathematics and algebra, estimation penetrates into every aspect of calculation. If the teacher guides students to estimate before the system calculation, they can analyze the approximate range of the value of the solution. For example, when calculating 2613÷13, students tend to miss the 0 in the middle of the quotient. If we estimate first, 2600÷13=200, so the quotient of 2613 divided by 13 must be more than 200. Estimation is carried out during calculation. For the four mixed operation questions, during the calculation process, it is necessary to observe whether the order of operations is correct, and to estimate the results of each separate operation to see whether it conforms to the relevant rules of calculation. Estimating after calculation is to compare and analyze whether the calculated value is within the estimated value range or consistent with objective reality, so as to determine whether there are any errors in the calculation process.
2. Change "Don't want to estimate" to "Like to estimate". teachers should create realistic, interesting and challenging scenarios in teaching so that students can gradually experience the feasibility of estimation. With the rapid development of science and technology, complex calculations can be completed by computers and counters. There are many life events that are impossible and unnecessary to accurately calculate. Teachers generally require students to check calculations, which is completely necessary. However, some teachers require students to use written calculations based on inverse operations or to perform strict calculations again no matter what the problem is. This not only increases the burden on students, but also makes them rigid. In fact, some errors are easy to find using estimates. Don't use written calculations to check errors for every question. In teaching, these functions of estimation can be used to attract students' attention, so that students can change from passive to active in estimation.
3． Change "single estimate" to "multiple estimate". In teaching, on the one hand, students are encouraged to estimate the results before solving problems. On the other hand, after each completion of the work, rough calculations are carried out using the estimation method. Thirdly, students are encouraged to estimate in real life, such as estimating the shopping cost after shopping, and they are encouraged to record it in the form of a small mathematical diary. This allows students to use their own understanding of the book and its relationships to grasp the calculation results, and try to use different methods to self-judge the rationality of the estimation results according to the needs of the problem. It can be said that it kills three birds with one stone. The first is to improve the accuracy of homework; the second is to consolidate estimation skills and cultivate estimation awareness; and the third is to allow students to feel the charm of mathematics and the joy of learning in life.
When estimating becomes a habit, I discovered a phenomenon among students: many students are always proud of themselves for their closest estimate. For example: Xiaohong’s mother buys a washing machine for 3,025 yuan and a rice cooker for 204 yuan. How much does it cost? The students exchanged their own algorithms: Algorithm 1: Think of 3025 as 3000 and 204 as 200, which is about 3200 yuan; Algorithm 2: Think of 3025 as more than 3030 and 204 as less than 200, which is about 3230 yuan; Algorithm 3: Treat 3025 as 3050, 204 as 200, and the result is 3250 yuan. Through comparison, we believe that 3230 is closest to the accurate value.
The indicator of learning mathematics well is not the ability to solve problems, but the ability to use mathematics. When should estimation be used and when should not be used? This is an issue that teachers should pay attention to. The refinement of estimation methods is something that teachers and students need to explore together. Problem, but after mastering the method, it is very important to discern when we solve practical problems. When to estimate and when not to estimate, being able to discern is the sign of students' estimation awareness.
Estimation is inseparable from students’ number sense. This feeling is supported by “awareness”. The current problem of “estimation” teaching is not to tell children the estimation method and let students follow the rules. Rules", but can students think of such methods when solving problems? This is commendable and is also the key to whether students truly understand the significance of estimation.
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