If using a standard algorithm, students must understand why it works and be able to explain it. Assessment, particularly diagnostic assessment, allows you to assess student understanding. Because "standard algorithms require an understanding of regrouping, exchanging 10 in one place-value position for 1 in the position to the left - or the reverse, exchanging 1 for 10 in the position to the right" - students should demonstrate strong number sense and an understanding of base-ten thinking. In a diagnostic assessment, using models or manipulatives, you can ask students to describe each step taken to solve a problem. Probing questions can be asked if students don't provide enough explanation or their reasoning appears fuzzy. If the assessment uses a written record of student work, a graphic organizer that encourages students to document each step taken is a beneficial tool. "If students are not seeing the ten as a unit, you may need more work on place-value activities."
Standard algorithms can be confusing for students, so it is important that the assessment the teacher chooses really checks the students' understanding. Standard algorithms are tricky because students tend to focus on digits, as opposed to whole number value, and this thought process can lead to mistakes and misunderstanding. It is like trying to complete a puzzle without knowing what the picture of the whole puzzle is supposed to look like and there are a bunch of extra pieces from another puzzle. Without proper understanding of the big picture, students will more easily make unhelpful mistakes. I like your suggestion for the kinds of assessments that could work to check this understanding, Dean. I think using models will help a lot with learning this type of mathematical concept, but how would you do this for large numbers, I wonder? ~Meghan Ehle
Standard algorithms are the formal problem students expect when they receive a math problem. As a teacher, it is important to prepare assignments that encourage students to use inventive strategies to show their work. Inventive strategies are the creative, flexible use of numbers to and strategies that students “invent” that helps them to understand the problem they are asked to solve. Invented strategies can be anything, including orienting numbers differently, to beginning with the largest parts of the numbers instead of the smallest. The idea behind this way of learning standard algorithms is that the students should learn in a way that they can explain to you, not the other way around. I used this concept in a class of 5th grade math students today. They were working on multiplying mixed fractions. When I think of how I may have learned how to do this problem, it would have been with a standard algorithm. Something I “did”, but didn’t always understand. Today, the students and I discussed how they understood the problem. They were asked to teach me, instead of me teaching them. One student explained her math in I way that I found completely unexpected. While it’s not how i would have approached the problem, the value was in the fact that SHE understood the problem and could see it in her mind. Students who understand the mental computation of the problems are less likely to be need reteaching, their inventive strategies serve them well on standardized test and overall they make less mistakes.
I like that you imply that the standard algorithm should be introduced last or at least in the mix of other approaches. Instead of spending weeks on introducing, reviewing, and practicing that standard algorithm, students can play around with different approaches and become more comfortable with the numbers and their number values. Much like you have said, students will become more flexible thinkers when they are aware of these different approaches.
This Chapter begins by expressing the need for students to be presented with multiple strategies in order to solve addition and subtraction problems rather than just a single method. This requires students to think flexibly about numbers and and the best ways to approach a problem depending on the context. I like how in the reading it emphasized the importance of invented strategies over standard algorithms because it is most important that the student, who is the one using them, actually understands what they are doing. That being said, the standard algorithms may be more useful at times so they should also be taught. It is crucial when teaching students standard algorithm to assess their understanding because you want to ensure that they are understanding the process and that the procedure makes sense, otherwise they are simply memorizing a series of steps. As with so many aspects of math, students need to be able to explain their thinking and express why a strategy works even for the standard algorithms. Assessing students in this way allows the students to share their ideas and reinforce their own understanding.
If using a standard algorithm, students must understand why it works and be able to explain it. Assessment, particularly diagnostic assessment, allows you to assess student understanding. Because "standard algorithms require an understanding of regrouping, exchanging 10 in one place-value position for 1 in the position to the left - or the reverse, exchanging 1 for 10 in the position to the right" - students should demonstrate strong number sense and an understanding of base-ten thinking. In a diagnostic assessment, using models or manipulatives, you can ask students to describe each step taken to solve a problem. Probing questions can be asked if students don't provide enough explanation or their reasoning appears fuzzy. If the assessment uses a written record of student work, a graphic organizer that encourages students to document each step taken is a beneficial tool. "If students are not seeing the ten as a unit, you may need more work on place-value activities."
ReplyDeleteStandard algorithms can be confusing for students, so it is important that the assessment the teacher chooses really checks the students' understanding. Standard algorithms are tricky because students tend to focus on digits, as opposed to whole number value, and this thought process can lead to mistakes and misunderstanding. It is like trying to complete a puzzle without knowing what the picture of the whole puzzle is supposed to look like and there are a bunch of extra pieces from another puzzle. Without proper understanding of the big picture, students will more easily make unhelpful mistakes. I like your suggestion for the kinds of assessments that could work to check this understanding, Dean. I think using models will help a lot with learning this type of mathematical concept, but how would you do this for large numbers, I wonder?
ReplyDelete~Meghan Ehle
Standard algorithms are the formal problem students expect when they receive a math problem. As a teacher, it is important to prepare assignments that encourage students to use inventive strategies to show their work. Inventive strategies are the creative, flexible use of numbers to and strategies that students “invent” that helps them to understand the problem they are asked to solve. Invented strategies can be anything, including orienting numbers differently, to beginning with the largest parts of the numbers instead of the smallest. The idea behind this way of learning standard algorithms is that the students should learn in a way that they can explain to you, not the other way around. I used this concept in a class of 5th grade math students today. They were working on multiplying mixed fractions. When I think of how I may have learned how to do this problem, it would have been with a standard algorithm. Something I “did”, but didn’t always understand. Today, the students and I discussed how they understood the problem. They were asked to teach me, instead of me teaching them. One student explained her math in I way that I found completely unexpected. While it’s not how i would have approached the problem, the value was in the fact that SHE understood the problem and could see it in her mind. Students who understand the mental computation of the problems are less likely to be need reteaching, their inventive strategies serve them well on standardized test and overall they make less mistakes.
ReplyDeleteI like that you imply that the standard algorithm should be introduced last or at least in the mix of other approaches. Instead of spending weeks on introducing, reviewing, and practicing that standard algorithm, students can play around with different approaches and become more comfortable with the numbers and their number values. Much like you have said, students will become more flexible thinkers when they are aware of these different approaches.
ReplyDeleteThis Chapter begins by expressing the need for students to be presented with multiple strategies in order to solve addition and subtraction problems rather than just a single method. This requires students to think flexibly about numbers and and the best ways to approach a problem depending on the context. I like how in the reading it emphasized the importance of invented strategies over standard algorithms because it is most important that the student, who is the one using them, actually understands what they are doing. That being said, the standard algorithms may be more useful at times so they should also be taught. It is crucial when teaching students standard algorithm to assess their understanding because you want to ensure that they are understanding the process and that the procedure makes sense, otherwise they are simply memorizing a series of steps. As with so many aspects of math, students need to be able to explain their thinking and express why a strategy works even for the standard algorithms. Assessing students in this way allows the students to share their ideas and reinforce their own understanding.
ReplyDelete