How can we help students think about very small place values such as thousandths and millionths in the same way we get students to think about very large place values such as millions and billions?
I think that the concept of millions, and millionths, is hard for anyone to grasp, not just elementary-aged students! A children's book that I like, "How Much is a Million" by David Swartz and Steven Kellogg, explains big numbers in a tangible way. Though children today may be familiar with the idea of millions and billions, I never hear them talking about millionths or billionths. In chapter 17, Van De Walle et al. write about instructing students in not rounding solutions to two decimal points. Rounding decimals so that all students ever see are tenths and hundredths is not helpful in their development of true number sense. Enormous numbers are a part of the math we do and the vocabulary we use (in explaining how far away the moon is or how much the national debt is), and infinitesimal numbers should be as well, especially because these numbers are a key to how decimals work. I think it is important to get students thinking about what is in between decimal places, and help them realize that there are always numbers in between any two consecutive numbers or decimals. Chapter 17 includes two neat activities that explore this idea on page 416: “Close Decimals” and “Zoom.” Such small numbers could be put into the context of scientists measuring parts per million of a chemical, or cell thickness, or NASA needing to precisely measure a microinch (one-millionth of an inch) so as not to blow up another one of their telescopes. This will be helpful because if students are more aware of the “density (p. 416)” of decimals, they will be better able to decide whether .12 is closer to .131456 or .14, which apparently is a common error students make when they don’t understand decimals. I get this in theory, but my question is: how can you possibly create visual models for such small numbers? Students really can’t "see" this, can they? If so much of math learning is enhanced by visual models, and even dependent upon them, how accurately can you teach something so conceptual as a millionth? Another bothersome thing with teaching this is that there is no way to show “infinity” that I can think of, and I wouldn't even want to try because while it is something I like to think about, I do not have a clear grasp on it myself. How can we expect elementary-aged students to get this?
Building a solid foundation of 'more reasonable' numbers would be a good starting point: thousands, hundreds, tens, ones . tenths, hundredths, thousandths. Numbers right of the decimal point should also be represented and understood as fractions: 1/10, 1/100, and 1/1000.
With an understanding of these foundational values, one could then explore the symmetry of the number line and place values (revealed in the fractions). If we can imagine large numbers in the millions and billions, we should be able to explain, through symmetry, that there are equally small numbers like millionths and billionths.
Using the place value chart as a tool to understand the 10-to-1 relationship could also help. This idea of 10-to-1 could be developed using squares of 100, strips of 10 and squares of 1, and using those same piece as a square of 1, strips of tenths, and squares of hundredths.
This chapter makes me feel uneasy about my own understanding of decimals and their value so I imagine for younger children understanding small place values can be a huge challenge!
I like the idea of using the 10-to-1 relationship in both directions. Students who have a strong understanding that base-ten models can continue to grow bigger and bigger (10 cubes make 1 strip, 10 strips make 1 square etc) can begin to explore how these base-ten models can grow smaller.
If students have developed a strong understanding of fractions they can begin to connect these small numbers to decimals. Where would that fraction fall on a number line between two numbers? How are 1/1000 and 0.001 alike? Do you think this is a large number or small number? Would you want 1/1000 of a pizza?
Tiffany brings up a great point. Connecting to students existing understanding of fractions allows them to see how decimals relate directly to fractions. Using base ten models, with whole squares,tens and hundredths pieces to as a visual representation of the decimals. If I was teaching this, I would begin with reviewing common fractions of 1/2 or 1/4, how they relate to decimals. I might also use a number line to show 0-1 and break out fractions and then show as decimals. Another take away from this chapter that I could see using is the grid which shows hundreds, tens, ones, decimal, tenths, hundredths. This grid shows how to the left of the decimal is a fraction and to the right is a whole number.
Reading Meg's response about millionths and then reading Dean's comments about starting with smaller numbers makes me think about how much time we have had to spend building the idea that our number system is built on base 10. The sixth graders should have a solid knowledge of the base ten system but have proven to us in Essex that many students do not have a true meaning for the building block of our base ten number system. Moving the decimal point to the left to understand ten percent of a number or understanding what a 100 percent meaning as far as one is concerned. Making the numbers real with dinner or grocery bills helped but leading the students to think, estimate, and talk about what they knew helped a great deal. The students needed prompting and were not practiced in thinking or talking about the numbers. Although the students have a basic knowledge about metric measurements they looked at me strangely when I asked them how much snow we got in centimeters or meters. Using the decimal point to explain how many centimeters of snow we got with a unit of meters (.76 meters of snow) threw a wrench into the conversation. Just starting ratios it was nice to see the lights go on when we had to convert a ratio to a unit ratio using one (divide by 3/3) and hear the students yell out that was similar to when we tried to determine equivalent fractions! I have not understood why some students have not been able to understand that they can add a zero onto the right of a decimal number without changing the number (.2300). Any suggestions other than any models or fraction comparison?
In order for students to understand huge and often difficult numbers to represent like millions and thousands they have to have a solid understanding of place value and regrouping. Students being flexible with regrouping is introduced through a variety of methods, most notably through the use of base 10 models and making 10 with 1 units.This is the same process used when thinking about thousandths and millionths. I believe that like many topics in school, in order for students to understand the concept of millionths and thousandths, those numbers need to be made tangible. Using tools like base ten models provide an opportunity for students to physically see that it would take millions of tinies to make a super square. Also providing students with real life examples of these concepts would help them bring context to the math. For instance how many grains of rice to fill a 1 ounce cup? How many small cups to make a 20 lb. bag? etc.
I think that the concept of millions, and millionths, is hard for anyone to grasp, not just elementary-aged students! A children's book that I like, "How Much is a Million" by David Swartz and Steven Kellogg, explains big numbers in a tangible way. Though children today may be familiar with the idea of millions and billions, I never hear them talking about millionths or billionths. In chapter 17, Van De Walle et al. write about instructing students in not rounding solutions to two decimal points. Rounding decimals so that all students ever see are tenths and hundredths is not helpful in their development of true number sense. Enormous numbers are a part of the math we do and the vocabulary we use (in explaining how far away the moon is or how much the national debt is), and infinitesimal numbers should be as well, especially because these numbers are a key to how decimals work. I think it is important to get students thinking about what is in between decimal places, and help them realize that there are always numbers in between any two consecutive numbers or decimals. Chapter 17 includes two neat activities that explore this idea on page 416: “Close Decimals” and “Zoom.” Such small numbers could be put into the context of scientists measuring parts per million of a chemical, or cell thickness, or NASA needing to precisely measure a microinch (one-millionth of an inch) so as not to blow up another one of their telescopes. This will be helpful because if students are more aware of the “density (p. 416)” of decimals, they will be better able to decide whether .12 is closer to .131456 or .14, which apparently is a common error students make when they don’t understand decimals.
ReplyDeleteI get this in theory, but my question is: how can you possibly create visual models for such small numbers? Students really can’t "see" this, can they? If so much of math learning is enhanced by visual models, and even dependent upon them, how accurately can you teach something so conceptual as a millionth? Another bothersome thing with teaching this is that there is no way to show “infinity” that I can think of, and I wouldn't even want to try because while it is something I like to think about, I do not have a clear grasp on it myself. How can we expect elementary-aged students to get this?
Building a solid foundation of 'more reasonable' numbers would be a good starting point: thousands, hundreds, tens, ones . tenths, hundredths, thousandths. Numbers right of the decimal point should also be represented and understood as fractions: 1/10, 1/100, and 1/1000.
ReplyDeleteWith an understanding of these foundational values, one could then explore the symmetry of the number line and place values (revealed in the fractions). If we can imagine large numbers in the millions and billions, we should be able to explain, through symmetry, that there are equally small numbers like millionths and billionths.
Using the place value chart as a tool to understand the 10-to-1 relationship could also help. This idea of 10-to-1 could be developed using squares of 100, strips of 10 and squares of 1, and using those same piece as a square of 1, strips of tenths, and squares of hundredths.
This chapter makes me feel uneasy about my own understanding of decimals and their value so I imagine for younger children understanding small place values can be a huge challenge!
ReplyDeleteI like the idea of using the 10-to-1 relationship in both directions. Students who have a strong understanding that base-ten models can continue to grow bigger and bigger (10 cubes make 1 strip, 10 strips make 1 square etc) can begin to explore how these base-ten models can grow smaller.
If students have developed a strong understanding of fractions they can begin to connect these small numbers to decimals. Where would that fraction fall on a number line between two numbers? How are 1/1000 and 0.001 alike? Do you think this is a large number or small number? Would you want 1/1000 of a pizza?
Tiffany brings up a great point. Connecting to students existing understanding of fractions allows them to see how decimals relate directly to fractions. Using base ten models, with whole squares,tens and hundredths pieces to as a visual representation of the decimals. If I was teaching this, I would begin with reviewing common fractions of 1/2 or 1/4, how they relate to decimals. I might also use a number line to show 0-1 and break out fractions and then show as decimals. Another take away from this chapter that I could see using is the grid which shows hundreds, tens, ones, decimal, tenths, hundredths. This grid shows how to the left of the decimal is a fraction and to the right is a whole number.
DeleteReading Meg's response about millionths and then reading Dean's comments about starting with smaller numbers makes me think about how much time we have had to spend building the idea that our number system is built on base 10. The sixth graders should have a solid knowledge of the base ten system but have proven to us in Essex that many students do not have a true meaning for the building block of our base ten number system. Moving the decimal point to the left to understand ten percent of a number or understanding what a 100 percent meaning as far as one is concerned. Making the numbers real with dinner or grocery bills helped but leading the students to think, estimate, and talk about what they knew helped a great deal. The students needed prompting and were not practiced in thinking or talking about the numbers. Although the students have a basic knowledge about metric measurements they looked at me strangely when I asked them how much snow we got in centimeters or meters. Using the decimal point to explain how many centimeters of snow we got with a unit of meters (.76 meters of snow) threw a wrench into the conversation.
ReplyDeleteJust starting ratios it was nice to see the lights go on when we had to convert a ratio to a unit ratio using one (divide by 3/3) and hear the students yell out that was similar to when we tried to determine equivalent fractions! I have not understood why some students have not been able to understand that they can add a zero onto the right of a decimal number without changing the number (.2300). Any suggestions other than any models or fraction comparison?
In order for students to understand huge and often difficult numbers to represent like millions and thousands they have to have a solid understanding of place value and regrouping. Students being flexible with regrouping is introduced through a variety of methods, most notably through the use of base 10 models and making 10 with 1 units.This is the same process used when thinking about thousandths and millionths. I believe that like many topics in school, in order for students to understand the concept of millionths and thousandths, those numbers need to be made tangible. Using tools like base ten models provide an opportunity for students to physically see that it would take millions of tinies to make a super square. Also providing students with real life examples of these concepts would help them bring context to the math. For instance how many grains of rice to fill a 1 ounce cup? How many small cups to make a 20 lb. bag? etc.
ReplyDelete