You notice a student is estimating by doing the computation and rounding the answer. Why might the student be using this strategy? What experiences might you plan to improve the student's ability to estimate?
Chapter 13 and 12 for that matter are classic chapters and at the heart of the debate about fact fluency. Example after example in the two chapters prove that students are able to successfully manipulate numbers and calculate if they have good number sense and flexibility of thinking. These two ideas are important because many of my students that I currently work with do not have good number sense and are still very concrete thinkers which make these strategies hard for some of the students to apply. Others with good number sense and a developed ability to manipulate what they are given for numbers has equally demonstrated that they can use strategies mentioned in the book. I often seen evidence of students using pictures or models of tens or of large groupings, rounding numbers or decomposing numbers to multiply. The student mentioned in the question above may be developmentally not there yet both in his being to concrete and an ability to decompose numbers. I would work with the student to better use a knowledge of our base ten system to round and then more easily estimate. (19 x 42....round to 20 x 40 so the student will multiply 2 x 4 and add two 00 to estimate 800) These kind of problems make excellent warm up or entry problems that came be shown under a ELMO as model work for other students. It encourages student participate, math talk and reasoning with numbers.
I like the concept of teaching better rounding skills through the base ten system. You could also use a number line like suggested in the writing to help grasp the visual concept of rounding. These exercises might be beneficial to help students estimate better. As a teacher and after reading this chapter, I understand estimating to be a skill in which students understand how numbers work. I like the suggestion of supporting rounding knowledge with the task of asking whether a number (shown briefly on projector) is higher or lower than a computation. I have not worked with rounding or had a student show me their computation and rounding the answer, but look forward to the opportunity.
For so long the idea of one correct answer has been inappropriately emphasized in math classrooms. Therefore, students may have a difficult time with estimating, thinking that there must better estimates than others. The student in this scenario may be uncomfortable with estimates and keen on finding that best answer/estimate. Furthermore, "good estimators tend to employ a variety of computational strategies they have developed over time." Students with less-developed computational strategies may find it challenging to estimate.
To support this student in developing estimating skills, you might work with them individually starting with rounding individual numbers to the nearest 10s or other benchmark numbers (number sense). Progressing to operations, the student should continue to be encouraged to use these benchmarks to represent the various parts of the problem, and strategically think how these might simplify the calculation. In the classroom, student strategies should be shared with everyone, so everyone realizes there are multiple ways to estimate an answer. I should also accept a range of estimates, not judging one estimate better than another for some reason. The book suggests using acceptable ranges within which an answer might fall.
Once standard algorithm are introduced might you have students solve problems AND ask them to judge the appropriateness of their answer by showing an estimate?
I like that idea, Dean, of estimating and showing a final answer. This method would help them become better estimators, and be able to justify and have confidence in their abilities to do so. Having invented strategies for estimating requires students to have a good sense of the operations they are using to solve problems as well as a good number sense. I agree that perhaps this student needs more lessons on the operations themselves, as well as more practice with base ten numbers and methods of estimating. Maybe just letting them know that estimates are simply that - estimates, and they are designed to help you make an educated guess as to what the answer should be, may help them become more comfortable with the process. You could make it a game, and see how close the initial estimates are to the actual answer they get after they solve it. Maybe this way they could invent their own methods to use that are more helpful to them in thinking about the estimation of numbers.
Meg, I like your idea about incorporating an opportunity for students to share their different or invented strategies. I think as in all areas if math it is important to differentiate our approaches to a problem. I also think that a game is a fun chance to put this skill into action. Giving the students a certain range for the correct answer is another way to reinforce the idea of an estimate over an exact answer.
A student who is using this strategy has not fully developed their understanding of flexible and fluent thinking with whole numbers. It was often talked about throughout both chapters that students want to find the best answer. With estimation, different thinking will produce different answers, not a right or wrong answer.
The base ten system can be a great tool in helping students develop a stronger understanding of estimation. Students also need exposure to different estimation experiences so that they can find out which method of estimation works best.
I think a good starting point is asking students for information about the number but not asking for the answer. The book uses this examples and suggests using prompting questions such as 'Is it more than or less than x'. Once this student feels more confident in their ability to use estimation, move on to other activities such as 'What was Your Method'. In this activity students are given the estimation answer to a problem and asked questions about why that estimate was used. Students can focus on the various estimation techniques and decide which ones are better approaches. Students can collaborate and discuss what they come up with and compare.
Responding to Tiffany's response concerning students base tens knowledge. Experience has repeatedly shown that the students who are struggling often lack the knowledge of how to best use a good base ten knowledge or even possibly a lack of practical reasoning that our number system is built on a base ten system. I have found students this year in 6th grade that did not really understand that our system is a base ten system. More students did not think about using rounding up or down or taking part of a number in order to use the base ten knowledge to help estimate or calculate would be a good strategy. Number discussions daily seem to be a good idea and best practice in understanding students level of knowledge.
Chapter 13 and 12 for that matter are classic chapters and at the heart of the debate about fact fluency. Example after example in the two chapters prove that students are able to successfully manipulate numbers and calculate if they have good number sense and flexibility of thinking. These two ideas are important because many of my students that I currently work with do not have good number sense and are still very concrete thinkers which make these strategies hard for some of the students to apply. Others with good number sense and a developed ability to manipulate what they are given for numbers has equally demonstrated that they can use strategies mentioned in the book. I often seen evidence of students using pictures or models of tens or of large groupings, rounding numbers or decomposing numbers to multiply. The student mentioned in the question above may be developmentally not there yet both in his being to concrete and an ability to decompose numbers. I would work with the student to better use a knowledge of our base ten system to round and then more easily estimate. (19 x 42....round to 20 x 40 so the student will multiply 2 x 4 and add two 00 to estimate 800) These kind of problems make excellent warm up or entry problems that came be shown under a ELMO as model work for other students. It encourages student participate, math talk and reasoning with numbers.
ReplyDeleteI like the concept of teaching better rounding skills through the base ten system. You could also use a number line like suggested in the writing to help grasp the visual concept of rounding. These exercises might be beneficial to help students estimate better. As a teacher and after reading this chapter, I understand estimating to be a skill in which students understand how numbers work. I like the suggestion of supporting rounding knowledge with the task of asking whether a number (shown briefly on projector) is higher or lower than a computation. I have not worked with rounding or had a student show me their computation and rounding the answer, but look forward to the opportunity.
DeleteFor so long the idea of one correct answer has been inappropriately emphasized in math classrooms. Therefore, students may have a difficult time with estimating, thinking that there must better estimates than others. The student in this scenario may be uncomfortable with estimates and keen on finding that best answer/estimate. Furthermore, "good estimators tend to employ a variety of computational strategies they have developed over time." Students with less-developed computational strategies may find it challenging to estimate.
ReplyDeleteTo support this student in developing estimating skills, you might work with them individually starting with rounding individual numbers to the nearest 10s or other benchmark numbers (number sense). Progressing to operations, the student should continue to be encouraged to use these benchmarks to represent the various parts of the problem, and strategically think how these might simplify the calculation. In the classroom, student strategies should be shared with everyone, so everyone realizes there are multiple ways to estimate an answer. I should also accept a range of estimates, not judging one estimate better than another for some reason. The book suggests using acceptable ranges within which an answer might fall.
Once standard algorithm are introduced might you have students solve problems AND ask them to judge the appropriateness of their answer by showing an estimate?
I like that idea, Dean, of estimating and showing a final answer. This method would help them become better estimators, and be able to justify and have confidence in their abilities to do so. Having invented strategies for estimating requires students to have a good sense of the operations they are using to solve problems as well as a good number sense. I agree that perhaps this student needs more lessons on the operations themselves, as well as more practice with base ten numbers and methods of estimating. Maybe just letting them know that estimates are simply that - estimates, and they are designed to help you make an educated guess as to what the answer should be, may help them become more comfortable with the process. You could make it a game, and see how close the initial estimates are to the actual answer they get after they solve it. Maybe this way they could invent their own methods to use that are more helpful to them in thinking about the estimation of numbers.
ReplyDeleteMeg, I like your idea about incorporating an opportunity for students to share their different or invented strategies. I think as in all areas if math it is important to differentiate our approaches to a problem. I also think that a game is a fun chance to put this skill into action. Giving the students a certain range for the correct answer is another way to reinforce the idea of an estimate over an exact answer.
DeleteA student who is using this strategy has not fully developed their understanding of flexible and fluent thinking with whole numbers. It was often talked about throughout both chapters that students want to find the best answer. With estimation, different thinking will produce different answers, not a right or wrong answer.
ReplyDeleteThe base ten system can be a great tool in helping students develop a stronger understanding of estimation. Students also need exposure to different estimation experiences so that they can find out which method of estimation works best.
I think a good starting point is asking students for information about the number but not asking for the answer. The book uses this examples and suggests using prompting questions such as 'Is it more than or less than x'. Once this student feels more confident in their ability to use estimation, move on to other activities such as 'What was Your Method'. In this activity students are given the estimation answer to a problem and asked questions about why that estimate was used. Students can focus on the various estimation techniques and decide which ones are better approaches. Students can collaborate and discuss what they come up with and compare.
Responding to Tiffany's response concerning students base tens knowledge. Experience has repeatedly shown that the students who are struggling often lack the knowledge of how to best use a good base ten knowledge or even possibly a lack of practical reasoning that our number system is built on a base ten system. I have found students this year in 6th grade that did not really understand that our system is a base ten system. More students did not think about using rounding up or down or taking part of a number in order to use the base ten knowledge to help estimate or calculate would be a good strategy. Number discussions daily seem to be a good idea and best practice in understanding students level of knowledge.
Delete