This is the online discussion of teachers as we move through the lesson study process. Lesson Study provides an efficient organization for teachers to collaborate as we learn more about our content and how students learn.
Wednesday, January 28, 2009
Please post here after January 28th.
Whew, thanks for a great discussion this afternoon!
NCTM Article (“Introducing Percents in Linear Measurement to Foster an Understanding of Rational-Number Operations”) Remembering and properly executing calculations with rational numbers is difficult because many kids don’t have a conceptual understanding of rational numbers. The article explained a study that included teaching students a number sense approach so they can move between fractions, decimals, ratios, and percents. Building benchmark umbers first before going to the algorithm was an important idea in the study. Kids were able to figure out many problems dealing with percentages of numbers…like 12.5% of a number by knowing that 12.5% is half of 25%
Using benchmark numbers: 5% = .05 25% = ¼ 50%= ½
Building number sense with percentages (If I can figure out 10%, can I figure out 5%)
After reading/hearing Sue’s blog, we agree that meaning must be attached in order for kids to understand and find meaning.
Ken’s strategy for teaching percent of a whole: (sorry, the boxes won't come out on the blog) 100% is 16 25% 25% 25% 25% 4 4 4 4
Then, we began to break down our developmental story into bigger concepts. We need to get the developmental story organized since there is so much overlap.
Should we find a lesson that focuses on benchmark percents???
Next steps: Let’s group the developmental story according to the bigger concepts. Be on the lookout for % instruction, instruction of benchmark numbers, flexible representation/renaming…what’s the best language? manipulative?
First, I think today's children may be familiar with a "bar" representation of percents from the computer/electronic game environments. I'm sure many of them have seen the "%-completed" window on the screen in different contexts.
Second point may be a bit more significant. Japanese teachers do not consider percents to be numbers. They are puzzled by a problem which asks students to put fractions, decimals and percents on a number line. For Japanese teachers, percents are "relative values." Fractions and decimals can be used to describe relative values, so in such situations, fractions, decimals and percents are interchangeable. However, fractions and decimals can also be used as numbers, while percents are not. 5 is always greater than 3, but 50% isn't always greater than 30% because it all depends on the whole.
Yeh, Tad! I appreciate the Japanese’s teacher perspective. From a 5th grade teacher’s point of view, percents don’t act like other numbers. I can’t multiply or divide with them unless I change them to a decimal. If a percent is a ratio then we are seeing a relationship. Interestingly, my husband noted that the example Tad gave was a problem he frequently encountered in the marketing business. Employees in the hotel world want to combine their group sales occupancy percentage for say 5 years then give the mean as their average group occupany. However, Jan says the whole constantly changes…some years they have 1000 room nights free for group sales and sometimes they have 10,000 group room nights they must sell. People not recognizing the whole and that 30% of 1000 is a lot different from 30% percent of 10,000 has caused him much grief. Not sure I explained this very well, but the bottom line is that adults often don’t think about the meaning of the whole as they look at a percentage.
There were 2 questions presented earlier that we probably need to resolve. PJ asked, 1. Do we teach renaming fractions and decimals to percents at the same time? 2. Percent defines a relationship – comparison: sometimes fractionally and sometimes as a ratio. Students must understand both interpretations to grasp percents beyond 100. Could students be relying more on 1 or the other interpretation and missing the connections?
I am more inclined to teach flexibility between fractions and decimals together first and then percents. What Tad said is an aha moment for me...it makes perfect sense. I'm wondering now if we throw kids off about this very important idea though? I think about the poster we have from Everyday Math that shows the probability meter and has 99%, .99 and 99/100 on the same line. When our whole changes and is not 100, how are kids interpreting it? Percents are just a whole other beast for kids! I don't think I've thought about the fact that there needs to be awareness of what relationship a percent is defining...are we using percent to talk about a fraction or a ratio in this problem? What a strong sense of both concepts and vocabulary necessary for this thinking to take place! I'm sitting here trying to think of a good example problem for each and have given up! On another note, I think kids in upper grades grasp the importance of knowing the whole. If I tell you I'm going to give you 1/2 of my Hershey's Bar, I think they'd want to know how much they're getting and wonder "Well, how big is your Hershey's Bar?" (Wouldn't they?)But then again, I think about what Becky said...if I asked a student "Would you rather have 30% of a Hershey Bar A or 50% of Hershey Bar B?" would they know that the WHOLE Hershey Bar A could be bigger, and therefore they would receive more candy by choosing the 30%? Now I'm just rambling...
Oh, and regarding the Developmental Story, I think it's fine the way you have it Becky...there is SO MUCH this topic (or should I say these topics) covers throughout so many grades that a more clear and concise organization is needed.
Hi all! I believe that I would teach renaming fractions, decimals, and percents at the same time, or teach the percents just following the renaming of fractions and decimals. The reason is that when I look at the 5th grade content limits which will apply to this year and to most extent, next year's FCAT, it says the following for percents: When finding equivalent fractions and decimals, items will be limited to percents that are multiples of 5, up to and including 100.
Floridastandards.org defines Percent as: Per hundred; a special ratio in which the denominator is always 100. The language of percent may change depending on the context. The most common use is in part-whole contexts, for example, where a subset is 40 percent of another set. A second use is change contexts, for example, a set increases or decreases in size by 40 percent to become 140% or 60% of its original size. A third use involves comparing two sets, for example set A is 40% of the size of set B, in other words, set B is 250 percent of set A.
These can be especially confusing for students to make the connection between. This is probably why in the new Standards percents only shows up at 4th grade for these relations: MA.4.A.6.5: Relate halves, fourths, tenths, and hundredths to decimals and percents. And then doesn't appear again until 6th grade where they will be comparing and ordering, renaming and estimating. In 7th grade they will start solving percent problems.
I don't know if this helps or muddies the waters, but I thought it was important to share. I also want to say that Tad makes an excellent point, that most kids are probably familiar with the percentage bar model from their experience with video games. It could be a great way to relate the concept to the students. And my mom said they were a waste of time. I can't wait to tell her about this. :-)
I am starting to think that maybe we should be attacking this from a more primary understanding of percents. It seems to me the problem that I was having with my 5th graders is not even supposed to be taught in 5th grade.
I agree with what Heather said about the candy bar example.. Would children always want 50% of a candy bar because it seems like more..? do the children you teach automatically question what the whole is??? Is the concept of the whole changing (50% of 20 vs. 30% of 100)something that % lessons begin with? Sorry, my 1st grade mind is struggling to undertand all of this.:)
Heather's candy bar example is excellent. And, the bar model is an excellent visual for the children to reference to. I am not sure where in percentages we want to focus our lesson - whith the understanding of what percents are (per 100) or comparing percentages like the candy bar example. Still yet percentages of a number. Much of this requires an understanding of the equivalence or renaming of percentages as a fraction - those benchmark percentages we identified.(Ken's model of 25 percent of 16)
After digesting all of the this (great blogging, by the way) it seems a good thought -- focusing the lesson more on the CONCEPT of percent as relative values as opposed to just teaching how to figure percents. Heather's point about how we display 99% as .99 as 99/100 is well taken. How misleading that must be for the kids! I've always thought of percents as representing a number, but was struck by Tad's input that the Japanese teachers don't view them as numbers, rather relative values. That's much more accurate. It's also a great hook for the lesson (it reminds me of the question, would you rather have a dollar a day for a month or a penny doubled every day for a month - I love the faces when they discover how much money they miss out on by choosing the dollar). The hook for this --would you rather have 30% of candy bar A or 50% of candy bar B (or whatever would be enticing) would be fun, especially when we pull out bar B and they discover it's a little mini candy bar. Would any of them think to ask the size of the bars before making their choice? With what Gary has been sharing, I'm wondering if the state standard decisions have been made based on the discovery that most kids aren't developmentally ready for the complexity of percents until middle school. Perhaps focusing on the relative percent concept will provide a critical foundation for future learning.
Lesson Study Meeting 2
ReplyDelete1/28/09
NCTM Article (“Introducing Percents in Linear Measurement to Foster an Understanding of Rational-Number Operations”)
Remembering and properly executing calculations with rational numbers is difficult because many kids don’t have a conceptual understanding of rational numbers. The article explained a study that included teaching students a number sense approach so they can move between fractions, decimals, ratios, and percents. Building benchmark umbers first before going to the algorithm was an important idea in the study. Kids were able to figure out many problems dealing with percentages of numbers…like 12.5% of a number by knowing that 12.5% is half of 25%
Using benchmark numbers:
5% = .05
25% = ¼
50%= ½
Building number sense with percentages (If I can figure out 10%, can I figure out 5%)
After reading/hearing Sue’s blog, we agree that meaning must be attached in order for kids to understand and find meaning.
Ken’s strategy for teaching percent of a whole:
(sorry, the boxes won't come out on the blog)
100%
is 16
25% 25% 25% 25%
4 4 4 4
Then, we began to break down our developmental story into bigger concepts. We need to get the developmental story organized since there is so much overlap.
* 100- number sense
* ratios/proportional thinking
* fraction- whole
- benchmark
- renaming
* flexible representation – fraction
(renaming) -percent
- decimal
* operations
Should we find a lesson that focuses on benchmark percents???
Next steps:
Let’s group the developmental story according to the bigger concepts.
Be on the lookout for % instruction, instruction of benchmark numbers, flexible representation/renaming…what’s the best language? manipulative?
Just a couple of comments.
ReplyDeleteFirst, I think today's children may be familiar with a "bar" representation of percents from the computer/electronic game environments. I'm sure many of them have seen the "%-completed" window on the screen in different contexts.
Second point may be a bit more significant. Japanese teachers do not consider percents to be numbers. They are puzzled by a problem which asks students to put fractions, decimals and percents on a number line. For Japanese teachers, percents are "relative values." Fractions and decimals can be used to describe relative values, so in such situations, fractions, decimals and percents are interchangeable. However, fractions and decimals can also be used as numbers, while percents are not. 5 is always greater than 3, but 50% isn't always greater than 30% because it all depends on the whole.
Just something to think about.
Yeh, Tad! I appreciate the Japanese’s teacher perspective. From a 5th grade teacher’s point of view, percents don’t act like other numbers. I can’t multiply or divide with them unless I change them to a decimal. If a percent is a ratio then we are seeing a relationship.
ReplyDeleteInterestingly, my husband noted that the example Tad gave was a problem he frequently encountered in the marketing business. Employees in the hotel world want to combine their group sales occupancy percentage for say 5 years then give the mean as their average group occupany. However, Jan says the whole constantly changes…some years they have 1000 room nights free for group sales and sometimes they have 10,000 group room nights they must sell. People not recognizing the whole and that 30% of 1000 is a lot different from 30% percent of 10,000 has caused him much grief. Not sure I explained this very well, but the bottom line is that adults often don’t think about the meaning of the whole as they look at a percentage.
There were 2 questions presented earlier that we probably need to resolve.
PJ asked,
1. Do we teach renaming fractions and decimals to percents at the same time?
2. Percent defines a relationship – comparison: sometimes fractionally and sometimes as a ratio. Students must understand both interpretations to grasp percents beyond 100. Could students be relying more on 1 or the other interpretation and missing the connections?
What do you think?
I am more inclined to teach flexibility between fractions and decimals together first and then percents.
ReplyDeleteWhat Tad said is an aha moment for me...it makes perfect sense. I'm wondering now if we throw kids off about this very important idea though? I think about the poster we have from Everyday Math that shows the probability meter and has 99%, .99 and 99/100 on the same line. When our whole changes and is not 100, how are kids interpreting it?
Percents are just a whole other beast for kids! I don't think I've thought about the fact that there needs to be awareness of what relationship a percent is defining...are we using percent to talk about a fraction or a ratio in this problem? What a strong sense of both concepts and vocabulary necessary for this thinking to take place! I'm sitting here trying to think of a good example problem for each and have given up!
On another note, I think kids in upper grades grasp the importance of knowing the whole. If I tell you I'm going to give you 1/2 of my Hershey's Bar, I think they'd want to know how much they're getting and wonder "Well, how big is your Hershey's Bar?" (Wouldn't they?)But then again, I think about what Becky said...if I asked a student "Would you rather have 30% of a Hershey Bar A or 50% of Hershey Bar B?" would they know that the WHOLE Hershey Bar A could be bigger, and therefore they would receive more candy by choosing the 30%?
Now I'm just rambling...
Oh, and regarding the Developmental Story, I think it's fine the way you have it Becky...there is SO MUCH this topic (or should I say these topics) covers throughout so many grades that a more clear and concise organization is needed.
ReplyDeleteHi all!
ReplyDeleteI believe that I would teach renaming fractions, decimals, and percents at the same time, or teach the percents just following the renaming of fractions and decimals. The reason is that when I look at the 5th grade content limits which will apply to this year and to most extent, next year's FCAT, it says the following for percents: When finding equivalent fractions and decimals, items will be limited to percents that are multiples of 5, up to and including 100.
Floridastandards.org defines Percent as: Per hundred; a special ratio in which the denominator is always 100. The language of percent may change depending on the context. The most common use is in part-whole contexts, for example, where a subset is 40 percent of another set. A second use is change contexts, for example, a set increases or decreases in size by 40 percent to become 140% or 60% of its original size. A third use involves comparing two sets, for example set A is 40% of the size of set B, in other words, set B is 250 percent of set A.
These can be especially confusing for students to make the connection between. This is probably why in the new Standards percents only shows up at 4th grade for these relations: MA.4.A.6.5: Relate halves, fourths, tenths, and hundredths to decimals and percents.
And then doesn't appear again until 6th grade where they will be comparing and ordering, renaming and estimating. In 7th grade they will start solving percent problems.
I don't know if this helps or muddies the waters, but I thought it was important to share. I also want to say that Tad makes an excellent point, that most kids are probably familiar with the percentage bar model from their experience with video games. It could be a great way to relate the concept to the students. And my mom said they were a waste of time. I can't wait to tell her about this. :-)
I am starting to think that maybe we should be attacking this from a more primary understanding of percents. It seems to me the problem that I was having with my 5th graders is not even supposed to be taught in 5th grade.
ReplyDeleteI agree with what Heather said about the candy bar example.. Would children always want 50% of a candy bar because it seems like more..? do the children you teach automatically question what the whole is???
ReplyDeleteIs the concept of the whole changing (50% of 20 vs. 30% of 100)something that % lessons begin with?
Sorry, my 1st grade mind is struggling to undertand all of this.:)
Heather's candy bar example is excellent. And, the bar model is an excellent visual for the children to reference to. I am not sure where in percentages we want to focus our lesson - whith the understanding of what percents are (per 100) or comparing percentages like the candy bar example. Still yet percentages of a number. Much of this requires an understanding of the equivalence or renaming of percentages as a fraction - those benchmark percentages we identified.(Ken's model of 25 percent of 16)
ReplyDeleteAfter digesting all of the this (great blogging, by the way) it seems a good thought -- focusing the lesson more on the CONCEPT of percent as relative values as opposed to just teaching how to figure percents. Heather's point about how we display 99% as .99 as 99/100 is well taken. How misleading that must be for the kids! I've always thought of percents as representing a number, but was struck by Tad's input that the Japanese teachers don't view them as numbers, rather relative values. That's much more accurate. It's also a great hook for the lesson (it reminds me of the question, would you rather have a dollar a day for a month or a penny doubled every day for a month - I love the faces when they discover how much money they miss out on by choosing the dollar). The hook for this --would you rather have 30% of candy bar A or 50% of candy bar B (or whatever would be enticing) would be fun, especially when we pull out bar B and they discover it's a little mini candy bar. Would any of them think to ask the size of the bars before making their choice? With what Gary has been sharing, I'm wondering if the state standard decisions have been made based on the discovery that most kids aren't developmentally ready for the complexity of percents until middle school. Perhaps focusing on the relative percent concept will provide a critical foundation for future learning.
ReplyDelete