How can students learn to write two- and three-digit numbers in a way that is connected to the base-ten meanings of ones and tens or ones, tens, and hundreds?
Yes, Yes, Yes it will help. I could not agree more with what you have said. Friday I had a student that is being considered for advancing in his mathematics because he is often found reading but does well most times. This student was talking about the number of people in a population and did not understand moving the decimal place on spot would represent 10 percent. He thought that one percent of 310 is 31, and published his report. A student that is always in trouble for hyper-activity corrected him and I should not admit but I enjoyed watching as the "trouble" student corrected the other student with a great description of why in the base ten system we only have to move the decimal point to know what ten percent or one percent would be. It is interesting when we switch from models/rods to money said they never thought about money's representation before. The concepts of what is one, what is ten, and what is 100 seem to not often be truly understood or thought of by the students I work with at school.
I have to admit, I may not have completely understood it myself quite the right way before. I noticed this the other day when my son, the 1st grader, came home with his math homework (they get one worksheet a week to do with folks at home). It was exactly this lesson about ones, tens, and hundreds place values. When he didn't get my explanation of how to do it the first time, I realized I only knew this information because some teacher told me it was so. It's not like I don't get it now, because I do (and I can "do" percents, for the most part). It is more that I was not allowed to question the rules of mathematics or how my teachers taught it before. That was the habit with which I was taught math. I do not even recall ever using counters or other physical objects to aid in my understanding. Visualizing and working with mathematical concepts symbolically is somewhat new to me; I always got it before and did well, but now I wonder how much was memorization and using the prescribed pattern of my teachers to find the solution. I am excited to try this stuff out in my classroom!
Once students have a strong understanding of base-ten meanings, they can begin to expand their horizons and experiment with writing two and three digit numbers. The base-ten concepts include making groups of tens and using leftovers to arrive at a number versus counting by ones; students must understand the relationship between groups of tens and ten ones. These relationships can be reinforced by using consistent base-ten language. For example, saying 25 as two tens and five ones. Students also need to have an understanding of place-value notation. Students can then move onto understanding two-digit and three-digit number names. Using base-ten materials can and should be used when reinforcing these oral names. To understand the written symbols, students can use base-ten blocks and place-value mats. This will allow the student to break down a large number with base-ten concepts and show their understanding of place-value within a number.
As Tiffany makes clear in her blog post, it is imperative to develop place ten understanding by explaining the relationships early and often. To the youngest learner, always connecting numbers to the tens and ones will reinforce this learning. I find the ten frames are a fun way to count and group. As described in the reading, using these ten-frame manipulatives as visuals for students to see the grouping of ten ones for every decade of counting, solidifies the understanding and integration of the place value concepts of grouping.
Using physical models to show base-ten concepts helps students understand the concept of "tens" as a set of ten units. The most effective types of models to show this are those that the students can actually group themselves with single pieces (ex. bundles of sticks). As students begin making groups of ten the base ten language can be introduced to help students connect the concept of place value. Once students understand this they can move on to grouping tens to make 100. As students verbalize the numbers they are counting using base ten language they will then be able to correctly write the number. Without this understanding they may write the number 21 as 201.
Manipulatives are a powerful tool here, and continue to be used through 5/6th grade. Understanding that a cube of 1000 can be divided into flats of 100, and flats of 100 can be divided into sticks of 10, etc. represents mental flexibility when dealing with place values (obviously this is in the higher grades). Equally important to the manipulatives and verbal acuity is writing numbers and understanding the relationship between the manipulatives, the words, and the actual written number. I've worked with a number of 5/6th graders who don't necessarily have trouble with math, but struggle with the movement of the decimal point and what that means to the magnitude of the number they are working with.
Yes, Yes, Yes it will help. I could not agree more with what you have said. Friday I had a student that is being considered for advancing in his mathematics because he is often found reading but does well most times. This student was talking about the number of people in a population and did not understand moving the decimal place on spot would represent 10 percent. He thought that one percent of 310 is 31, and published his report. A student that is always in trouble for hyper-activity corrected him and I should not admit but I enjoyed watching as the "trouble" student corrected the other student with a great description of why in the base ten system we only have to move the decimal point to know what ten percent or one percent would be. It is interesting when we switch from models/rods to money said they never thought about money's representation before. The concepts of what is one, what is ten, and what is 100 seem to not often be truly understood or thought of by the students I work with at school.
ReplyDeleteI have to admit, I may not have completely understood it myself quite the right way before. I noticed this the other day when my son, the 1st grader, came home with his math homework (they get one worksheet a week to do with folks at home). It was exactly this lesson about ones, tens, and hundreds place values. When he didn't get my explanation of how to do it the first time, I realized I only knew this information because some teacher told me it was so. It's not like I don't get it now, because I do (and I can "do" percents, for the most part). It is more that I was not allowed to question the rules of mathematics or how my teachers taught it before. That was the habit with which I was taught math. I do not even recall ever using counters or other physical objects to aid in my understanding. Visualizing and working with mathematical concepts symbolically is somewhat new to me; I always got it before and did well, but now I wonder how much was memorization and using the prescribed pattern of my teachers to find the solution. I am excited to try this stuff out in my classroom!
DeleteOnce students have a strong understanding of base-ten meanings, they can begin to expand their horizons and experiment with writing two and three digit numbers. The base-ten concepts include making groups of tens and using leftovers to arrive at a number versus counting by ones; students must understand the relationship between groups of tens and ten ones. These relationships can be reinforced by using consistent base-ten language. For example, saying 25 as two tens and five ones. Students also need to have an understanding of place-value notation. Students can then move onto understanding two-digit and three-digit number names. Using base-ten materials can and should be used when reinforcing these oral names. To understand the written symbols, students can use base-ten blocks and place-value mats. This will allow the student to break down a large number with base-ten concepts and show their understanding of place-value within a number.
ReplyDeleteAs Tiffany makes clear in her blog post, it is imperative to develop place ten understanding by explaining the relationships early and often. To the youngest learner, always connecting numbers to the tens and ones will reinforce this learning. I find the ten frames are a fun way to count and group. As described in the reading, using these ten-frame manipulatives as visuals for students to see the grouping of ten ones for every decade of counting, solidifies the understanding and integration of the place value concepts of grouping.
ReplyDeleteUsing physical models to show base-ten concepts helps students understand the concept of "tens" as a set of ten units. The most effective types of models to show this are those that the students can actually group themselves with single pieces (ex. bundles of sticks). As students begin making groups of ten the base ten language can be introduced to help students connect the concept of place value. Once students understand this they can move on to grouping tens to make 100. As students verbalize the numbers they are counting using base ten language they will then be able to correctly write the number. Without this understanding they may write the number 21 as 201.
ReplyDeleteManipulatives are a powerful tool here, and continue to be used through 5/6th grade. Understanding that a cube of 1000 can be divided into flats of 100, and flats of 100 can be divided into sticks of 10, etc. represents mental flexibility when dealing with place values (obviously this is in the higher grades). Equally important to the manipulatives and verbal acuity is writing numbers and understanding the relationship between the manipulatives, the words, and the actual written number. I've worked with a number of 5/6th graders who don't necessarily have trouble with math, but struggle with the movement of the decimal point and what that means to the magnitude of the number they are working with.
Delete