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.
Alice and Tad, thank you for helping to clarify elementary understanding of percent. We found,probably because of traditional ways and lack of focus on new standards, that we were teaching far too many “skills in percent”. We are delighted to find we can slow down. We are particularly interested in why percent does not belong on the numberline and fully appreciate Tad’s points. However, a fifth grade teacher noted on the district fraction test was a question requiring students to put numbers in order from smallest to greatest. The numbers included decimals, fractions, and percents. In addition, while we are happy to slow down, we have had feedback from middle school teachers that our students do not perform well on test questions on percents. We need dialogue with the middle school teachers.
The team met this morning from 8:30 – 11:00. Members of the writing team or our teaching partners covered our classes. We will do the same for them on Thursday when the writing team will meet. Ahhh, the blessing of meeting in the morning when we are fresh!
We summarized the most recent blogs and noted 3 possible directions for our lesson: exploring the purpose of percents, tying percents to 100 probably using grids, or working flexibly with benchmark decimals, fractions and percents. We chose to explore the purpose of percents. We reexamined Tad’s blog with the problem of finding the most winning team and Becky’s lesson on the students as coaches and selecting the best shooter among 3 basketball players. One child sunk 3 out of 5 shots, another 14 out of 25 and another 11 out of 20. We honed in on the problem solving strategy of when one needs to compare quantities (ideas) it is helpful to find a part or make a part the same. This led to a discussion of a 3rd grade benchmark for comparing fractions…students are to compare 5/8 and 5/12 by recognizing the numerators are the same so they can now analyze the denominators or when comparing 2/5 and 3/5 students are to recognize the denominators are the same so they must analyze the numerators. Our primary teachers noted when they asked students to compare… the leading question is “what is the same?’
We considered using a problem similar to Becky’s coaching problem but change it to more closely mirror Tad’s winning team strategy. That is 2 of the numbers would have the same number of total shots and 2 would have the same number of shots made thus children would first acknowledge that when one of the numbers is the same, the comparison is easy but if neither quantity is the same, then make one the same. Using 25 or 20 total shots would encourage students to take the numbers to 100. This could be directly linked to “that’s just ow people compare in our world…it is called a percent” This way the lesson moves directly to the value of using percents in our world.
However, the more we spoke the more we realized the value of our students using the problem solving questions “Are some of the numbers the same, then my comparison is easy. If neither quantity is the same how can I make some the same?”. These are questions that 3rd graders could ask themselves as they compare fractions. We are now leaning toward making this the focus of the lesson with the understanding a future lesson would tie the skill to percents or a teacher could tie it to comparing fractions. With this thought in mind we want to give students several experiences in using this skill before we move them to the whole being 100 and percents. The situational story will be about 3 teachers vying for the best player of Guitar Hero. While we know most of our students have played and loooove Guitar Hero, a Wii and Guitar Hero will be brought to school for the class to play as a celebration for the end of FCAT. The lesson will be taught the next day, probably to a 4th grade class. The numbers will represent the number of songs a teacher passes and plays for example, 7 out 15; 7 out of 10; and 8 out of 15. Part of our current mission is to reflect on which numbers to use. Our brains were way too overloaded to delve deeply into numbers this morning.
Especially because our lesson is focusing on this problem solving skill, we believe the questioning is critical hence we reviewed Tad’s presentation on Hatsumon from Conn. We began to identify questions we want students to ask themselves as well as questions found with the lesson’s parts. We noted the following “understanding the task” questions provided stronger and stronger guidance. “What do you know about these numbers?” “How can we compare these numbers?” How can we make something the same?” (Were there other questions we thought appropriate for this situation? I don’t think I got them all.)
Heather posed the question that once we pose the situation, we could ask the students “What questions are you thinking right now?’ This would focus on student need to ask questions as well as present direction for those who may be confused. mmmm
Our goals now are to consider which numbers to use and form questions for the different parts of the lesson.
Diane and Heather would going to try out some situations with their students and report back to us.
Thanks for your patience in reading this epistle to the end. Thanks, especially, for stretching my brain beyond the confines of my head. It now hurts...but that's okay :)
I tried the numbers we discussed yesterday at our meeting.(Thank you Barbara for making that professional development time possible)
The numbers we had chosen were 2 out of 3 1out of 3 2 out of 5
The setting was Wii sports since 100% of the class had seen or played Wii sports. Family members played tennis against the Wii and those were the results in games won. question posed was - which one made the best score?
95% of the students chose 2 out of three, reasoning that that person needed only one more game to win every single game. They were easily able to compare 1out of 3 to 2 out of 3 and conclude that 1 out of 3 definitely was less successful. However, the majority felt that 2 out of 5 was the worst score because that person needed 3 more games to win all games. They could definitely tell that 2/3 was better than 2/5.
Interestingly enough, only one child drew bars and shaded them in to show that 2/5 was better than 1/3 and 2/3 was best score. A few drew circle fractions, but the drawings did not reflect the amounts accurately.
Reflecting on their discussion, I feel that our numbers needed to be different. We had 2 parts the same and two of the total quantities the same.
Thanks for the write-up of what happened in that 2 1/2 hour meeting. Your blog gave my day such a joyful start! How far you have progressed in thinking about percent and this lesson! I’m so glad you were able to work together to arrange the time to meet.
Questions: • Are the numbers being tried with the same grade as the research lesson or with children who are older or younger? Does that make a difference in how you weigh their responses? Or do you expect they would be the same a year or two later? • Is the research lesson now for grade 4? • Since the RL target appears to have evolved, have you changed the goal (unit and/or research lesson) for students? Has it been articulated? What kinds of evidence can show students attain the goal? (Need to know to provide opportunity to show during lesson.) • A sequence for a percent unit has at least partially emerged. Have you written down a sequence of lessons? (I saw related topics earlier but not a unit sequence for the grade.)
Have a productive next meeting and keep thinking!Two aspirin, if necessary. Good luck with the numbers and the Hatsumon. :) Alice
One more thing to think about with language. "Are some of the numbers the same, then my comparison is easy. If neither quantity is the same how can I make some the same?" Each fraction is a single number. What if one denominator has the same digit as the other numerator? Might you think about if the denominators are the same or the numerators are the same...? as a starting point instead?
Okay…so, yesterday was certainly very stimulating. When I got back to my classroom, I “threw” together a problem similar to Becky’s situation, but instead about football quarterback recruits for UF and FSU. (Get a piece of paper…) I gave the situation 11 out of 20 passes completed, 14 out of 25 passes completed, and 13 out of 20 passes completed. I asked the students to work with their groups to decide which recruit they would like on their favorite team.
As I walked around, I heard students doing a few different things: 1) some groups went right to equivalent ratios 11:20=55:100, etc. 2) some groups were talking about how comparing the 11 out of 20 and the 13 out of 20 was simple, but comparing either of those to 14 out of 25 was not so clear-cut. 3) one group of kids was taking the number of passes missed and comparing those, so 9/20 compared to 11/25 and 7/20 … leading them right into the problem the 2nd set of kids was having. 4) some kids even divided the numbers in order to get a decimal…so, 11 divided by 20 is .55, etc.
During our class discussion that followed, one of the kids explained, “See, it’s easy to compare the two that have something the same, like 20 passes tried. But it’s harder to compare 11 out of 25 to the other two because nothing’s the same. So, I don’t know what to do.” One of the kids responded by saying that 100 was a common multiple of both numbers, so we could just change them all to some number of shots out of 100, just like equivalent ratios. He said, “even though there weren’t 100 passes tried, we can pretend that there were and that the recruit completed 55 out of 100 shots.” We talked about that for a minute and that 55 out of 100 expressed a relationship…that 100 shots and 55 shots really didn’t exist. We then put our discovery into words that in order to compare something, we need things to be the same or we need to make something the same. I know this isn’t quite the right language, but that’s the best I could do then! All of this didn’t take very long, maybe 15 minutes.
So, I went on with the lesson to directly teach the kids that people in the world compare things by using per 100 all the time…percents, written %. I gave them some more scenarios...I grabbed the grade book (high interest ;) and they were able to come up with the percents scored on certain assignments by finding equivalent fractions over 100 and turning them into % (I kept the numbers friendly.) Some still insisted on dividing the numerator by the denominator.
At the end of our discussion, one of the kids asked the question, “Can you turn the fraction 15/17 into a percent because I can’t find an equivalent fraction that has 100 as the denominator.” This became their homework problem. I had about 10 kids whose solution was “NO.” Because of the initial reason, but the 14 others remembered that we can turn a fraction into a decimal by dividing the numerator by the denominator. Many of them said “YES” and wrote their answer as .88 or 88/100 or 88%
As I was walking around today while my kids were doing some percent word problems on grids, I started thinking about comparing things. So, I drew on a post-it note 3 flowers (get some markers)… one with 4 red petals and a green stem, one with 4 red petals and a gray stem, and one with 4 purple petals and a green stem. I gave it to one student that was finished and told her to compare these flowers. She said, “well, that’s easy, it’s just like yesterday. There are things that are the same…” and started spouting off the similarities between them and how we would have to change something on one of the flowers to compare it to the other. I wonder if we use something like this to present to kids before we ever jump into numbers if it would help to establish the goal that to compare we need to find things that are the same. Then, it shouldn’t matter what numbers we use, but what the kids notice and question about the numbers. The 5th graders I have have had a lot of experience with fractions and decimals, including comparing fractions; this was their first formal experience with “percent” in math class. I know I put too much into one lesson, but I couldn’t help myself. They were on a roll yesterday
I would really LOVE to get a flow for all of this at some point. My head is spinning.
Thank you for sharing what your students were able to do.
I'm not sure if I wrote this in the blog entries (it was written while back), but, in the Japanese curriculum, there are two situations where situations are compared multiplicatively. One situation is where two like quantities are involved - like the number of passes (complete or attempts) - which leads to the idea of percents. The other situation is where two unlike quantities are involved - like speed, population density, etc. - which are generally referred to as "per-unit quantities."
In both situations, similar types of reasoning can be used. In fact, in the translated textbook series, they look at the per-unit quantities first, so when they get to the unit on percents, they use numbers that are larger. In their newer edition, where percents are discussed first, they use simpler numbers in their introduction.
I think spending a few lessons where students think about the process of reasoning involved - i.e., if one of the quantities are the same, the comparison is easy, thus if none is the same, we can try to make one the same - seems to be a wise investment.
I think it will be great if students understand that whether they create equivalent fractions with 100 as the denominator, use a common multiple as the denominator, compare fractions, completions/attempts, or divide the numerator by the denominator, they are indeed making the total number of pass attempts the same (100, common denominator, or 1 for the last two cases).
Interesting ideas! There are many I have further questions or comments about but I think we first need to get very clear on the goal (Alice mentioned this.) From the discussion, I see two possibilities
Students will understand that comparing quantities when one is the same amount is easy. However, if the quantities aren't the same then you need to make one the same.
or
Students will be able to compare quantities in situations involving ratios. (I'm using the term ratio because I can't think of another. I'm trying to eliminate comparing just one whole number with another whole number. Given our situation, maybe it should be "Students will be able to compare like-quantity situations which involve multiplication.”)
While these 2 goals are intertwined, I think the focus would be different between the two. The former would be a stronger focus on the problem solving process with more conversation about how the process applies across grade levels. It might have elements of Heather’s thoughts about comparing flowers then moving to numbers….not sure…we can work that out. The latter (to me) would utilize strong questioning technique but not focus on how to get students to independently problem solve comparison situations.
So, here are my thoughts. Do you agree? Is there a direction you would prefer to go? Am I missing a direction? I think after we settle the goal we should go to what accomplishment of that goal looks like with students and establish the flow. (I think we have said the flow but just need to get it on paper.) What do you think?
I do agree that we need to settle on a goal. I feel very scattered on the ideas. I am open to any goal that you teachers feel beneficial for students at this level. I don't think I could make an educated guess at this point because I am not familiar with their needs and weeknesses.
I think I would like to see a lesson addressing the firstgoal:
Students will understand that comparing quantities when one is the same amount is easy. However, if the quantities aren't the same then you need to make one the same.
This is something that we all need to address in our classrooms. Becky, Ken, Sue, Heather: if you feel strongly about focusing on the other goal, then lets go there.
Wow! So many good ideas.. my head is spinning a bit. As for the first goal( students will understand comparing quantities when onme amount is the same=easy).. heather's students seem to understand that it was more difficult to compare #'s of completed passes because one qb threw more passes than the others... Is this something that you think all students understand or realize? Or would a problem like this stump most children right away? It would seem beneficial to make sure that children understand this completely...I see value in both goals. It almost seems liek we would want to start with the first goal and then that would naturally lead into the more complex lessons.(comparing situations.. That being said, I'm agreeable to whatever the intermediate teachers feel there is the most need. You guys definitely know better where your children tend to break down...
First, I think you do not really mean or want to use the language of comparison of two aquantities that are the same. Too much room for confusion and ambiguity. I think you may want to refer to rational numbers. But if you want to use quantities, they are not the same unless each is just another way of naming the other: 1/2, 2/4. I believe what you want to refer to is that the numbers must have a common referent or base to be compared (common denominator, same whole, comparing number of children to number of children (although you could compare number of children to number of pencils and find you have enough for only 85% of the children): everyone has an equal number of chances or a level playing field etc. I think the child in Heather's (HW?) class who described pretneding that everyone had 100 chances came as close as you can to what children would understand and the extewnsion to grading papers--kids know there weren't 100 aquestions on each one--was a nice example also. But try not to leave ambiguity about "sameness"--it is not the quantities that are the same but there is a common referent for the comparison to be valid.
From Gaea Leindhardt paper: >>Based on our analysis of why percent is hard and what students seem to know intuitively, we believe that the following components constitute a solid understanding of percent: (a) understanding the concepts of portions, multiples, and express these multiplicative relationships in the language of percent; (b) recognizing that because percent is an intensive relationship between quantities, implicit referents exist and must be searched out; (c) developing a system of flexible representations for different percent meanings; and (d) having available any number of workable procedures, including the use of benchmarks, proportional thinking strategies, and formal proportions. We also believe that instruction in percent should begin with work on understanding the relational language of percent rather than its procedural connections to decimal or fractional notations. When initial instruction emphasizes notational conversion, students may never realize the existence of the implicit relationships and comparisons described by percent.<<
We struggled with the wording and certainly appreciate ideas on being specific and mathematically correct. We even tried using terms like "useful common quantity".
How about: Students will understand it is easy to compare rational numbers with a common referent. However, if you do not have a common referent then you must create one.
We will have to translate this to kid friendly language. We need to be sure we all understand the meaning of "rational number" and "referent". I think it will be a challenge to translate "referent" to kid friendly language. Primary teachers, we really need you help on this as well as the wording used in comparing.
I was thinking of what mastery of this goal would look like. Would it be similar to our "understanding the equal sign goal" in which different scenarios are presented and students must apply their understanding?
Steph, even if most students can tell which qback was the most successful, I don't think many can verbalize their reasoning clearly. I think it's important to verbalize reasoning to be able to use an idea later. I agree with you that we need to make sure students have masterful understanding of this idea before progressing. Becky
Wording here is indeed rather tricky. I probably started the language, "if one quantity is the same..." That works in the context of per-one quantity like crowdedness and speed since those situations DO involve 2 different quantities.
Percents situations - or those situations that can lead to percents - involve ratios of the same quantity. What we are comparing is "relative quantities," though that's not a common phrase, either. In each scenario, what we have is the number being compared (e.g., 15 completed passes) and the base of comparison (e.g., 25 total pass attempts). So, if either the number being compared or the base of comparison is the same, then comparing two scenarios will be done easily by looking at the other number. To determine percents, we ask if the base were 100, what would be the compared number?
I prefer not to use part-whole here since percents can be calculated on situations like population increase etc.
I'm here and reading each day, trying to digest all of this information. I'm looking forward to our next meeting when we can sort this out all together, as I'm having difficulty making any useful contributions at this point. Just want you to know I'm still here, and still thinking, for whatever good that is doing any of us. Sue Tabor :)
Alice and Tad, thank you for helping to clarify elementary understanding of percent. We found,probably because of traditional ways and lack of focus on new standards, that we were teaching far too many “skills in percent”. We are delighted to find we can slow down. We are particularly interested in why percent does not belong on the numberline and fully appreciate Tad’s points. However, a fifth grade teacher noted on the district fraction test was a question requiring students to put numbers in order from smallest to greatest. The numbers included decimals, fractions, and percents. In addition, while we are happy to slow down, we have had feedback from middle school teachers that our students do not perform well on test questions on percents. We need dialogue with the middle school teachers.
ReplyDeleteThe team met this morning from 8:30 – 11:00. Members of the writing team or our teaching partners covered our classes. We will do the same for them on Thursday when the writing team will meet. Ahhh, the blessing of meeting in the morning when we are fresh!
We summarized the most recent blogs and noted 3 possible directions for our lesson: exploring the purpose of percents, tying percents to 100 probably using grids, or working flexibly with benchmark decimals, fractions and percents. We chose to explore the purpose of percents. We reexamined Tad’s blog with the problem of finding the most winning team and Becky’s lesson on the students as coaches and selecting the best shooter among 3 basketball players. One child sunk 3 out of 5 shots, another 14 out of 25 and another 11 out of 20.
We honed in on the problem solving strategy of when one needs to compare quantities (ideas) it is helpful to find a part or make a part the same. This led to a discussion of a 3rd grade benchmark for comparing fractions…students are to compare 5/8 and 5/12 by recognizing the numerators are the same so they can now analyze the denominators or when comparing 2/5 and 3/5 students are to recognize the denominators are the same so they must analyze the numerators. Our primary teachers noted when they asked students to compare… the leading question is “what is the same?’
We considered using a problem similar to Becky’s coaching problem but change it to more closely mirror Tad’s winning team strategy. That is 2 of the numbers would have the same number of total shots and 2 would have the same number of shots made thus children would first acknowledge that when one of the numbers is the same, the comparison is easy but if neither quantity is the same, then make one the same. Using 25 or 20 total shots would encourage students to take the numbers to 100. This could be directly linked to “that’s just ow people compare in our world…it is called a percent” This way the lesson moves directly to the value of using percents in our world.
However, the more we spoke the more we realized the value of our students using the problem solving questions “Are some of the numbers the same, then my comparison is easy. If neither quantity is the same how can I make some the same?”. These are questions that 3rd graders could ask themselves as they compare fractions. We are now leaning toward making this the focus of the lesson with the understanding a future lesson would tie the skill to percents or a teacher could tie it to comparing fractions. With this thought in mind we want to give students several experiences in using this skill before we move them to the whole being 100 and percents.
The situational story will be about 3 teachers vying for the best player of Guitar Hero. While we know most of our students have played and loooove Guitar Hero, a Wii and Guitar Hero will be brought to school for the class to play as a celebration for the end of FCAT. The lesson will be taught the next day, probably to a 4th grade class. The numbers will represent the number of songs a teacher passes and plays for example, 7 out 15; 7 out of 10; and 8 out of 15.
Part of our current mission is to reflect on which numbers to use. Our brains were way too overloaded to delve deeply into numbers this morning.
Especially because our lesson is focusing on this problem solving skill, we believe the questioning is critical hence we reviewed Tad’s presentation on Hatsumon from Conn.
We began to identify questions we want students to ask themselves as well as questions found with the lesson’s parts. We noted the following “understanding the task” questions provided stronger and stronger guidance. “What do you know about these numbers?” “How can we compare these numbers?” How can we make something the same?” (Were there other questions we thought appropriate for this situation? I don’t think I got them all.)
Heather posed the question that once we pose the situation, we could ask the students “What questions are you thinking right now?’ This would focus on student need to ask questions as well as present direction for those who may be confused. mmmm
Our goals now are to consider which numbers to use and form questions for the different parts of the lesson.
Diane and Heather would going to try out some situations with their students and report back to us.
Thanks for your patience in reading this epistle to the end. Thanks, especially, for stretching my brain beyond the confines of my head. It now hurts...but that's okay :)
Becky
I tried the numbers we discussed yesterday at our meeting.(Thank you Barbara for making that professional development time possible)
ReplyDeleteThe numbers we had chosen were
2 out of 3
1out of 3
2 out of 5
The setting was Wii sports since 100% of the class had seen or played Wii sports. Family members played tennis against the Wii and those were the results in games won. question posed was - which one made the best score?
95% of the students chose 2 out of three, reasoning that that person needed only one more game to win every single game. They were easily able to compare 1out of 3 to 2 out of 3 and conclude that 1 out of 3 definitely was less successful. However, the majority felt that 2 out of 5 was the worst score because that person needed 3 more games to win all games. They could definitely tell that 2/3 was better than 2/5.
Interestingly enough, only one child drew bars and shaded them in to show that 2/5 was better than 1/3 and 2/3 was best score.
A few drew circle fractions, but the drawings did not reflect the amounts accurately.
Reflecting on their discussion, I feel that our numbers needed to be different. We had 2 parts the same and two of the total quantities the same.
Becky and team,
ReplyDeleteThanks for the write-up of what happened in that 2 1/2 hour meeting.
Your blog gave my day such a joyful start!
How far you have progressed in thinking about percent and this lesson!
I’m so glad you were able to work together to arrange the time to meet.
Questions:
• Are the numbers being tried with the same grade as the research lesson or with children who are older or younger? Does that make a difference in how you weigh their responses? Or do you expect they would be the same a year or two later?
• Is the research lesson now for grade 4?
• Since the RL target appears to have evolved, have you changed the goal (unit and/or research lesson) for students? Has it been articulated? What kinds of evidence can show students attain the goal? (Need to know to provide opportunity to show during lesson.)
• A sequence for a percent unit has at least partially emerged. Have you written down a sequence of lessons? (I saw related topics earlier but not a unit sequence for the grade.)
Have a productive next meeting and keep thinking!Two aspirin, if necessary.
Good luck with the numbers and the Hatsumon.
:)
Alice
One more thing to think about with language.
ReplyDelete"Are some of the numbers the same, then my comparison is easy. If neither quantity is the same how can I make some the same?"
Each fraction is a single number.
What if one denominator has the same digit as the other numerator?
Might you think about if the denominators are the same or the numerators are the same...? as a starting point instead?
Okay…so, yesterday was certainly very stimulating. When I got back to my classroom, I “threw” together a problem similar to Becky’s situation, but instead about football quarterback recruits for UF and FSU. (Get a piece of paper…) I gave the situation 11 out of 20 passes completed, 14 out of 25 passes completed, and 13 out of 20 passes completed. I asked the students to work with their groups to decide which recruit they would like on their favorite team.
ReplyDeleteAs I walked around, I heard students doing a few different things: 1) some groups went right to equivalent ratios 11:20=55:100, etc. 2) some groups were talking about how comparing the 11 out of 20 and the 13 out of 20 was simple, but comparing either of those to 14 out of 25 was not so clear-cut. 3) one group of kids was taking the number of passes missed and comparing those, so 9/20 compared to 11/25 and 7/20 … leading them right into the problem the 2nd set of kids was having. 4) some kids even divided the numbers in order to get a decimal…so, 11 divided by 20 is .55, etc.
During our class discussion that followed, one of the kids explained, “See, it’s easy to compare the two that have something the same, like 20 passes tried. But it’s harder to compare 11 out of 25 to the other two because nothing’s the same. So, I don’t know what to do.” One of the kids responded by saying that 100 was a common multiple of both numbers, so we could just change them all to some number of shots out of 100, just like equivalent ratios. He said, “even though there weren’t 100 passes tried, we can pretend that there were and that the recruit completed 55 out of 100 shots.” We talked about that for a minute and that 55 out of 100 expressed a relationship…that 100 shots and 55 shots really didn’t exist. We then put our discovery into words that in order to compare something, we need things to be the same or we need to make something the same. I know this isn’t quite the right language, but that’s the best I could do then! All of this didn’t take very long, maybe 15 minutes.
So, I went on with the lesson to directly teach the kids that people in the world compare things by using per 100 all the time…percents, written %. I gave them some more scenarios...I grabbed the grade book (high interest ;) and they were able to come up with the percents scored on certain assignments by finding equivalent fractions over 100 and turning them into % (I kept the numbers friendly.) Some still insisted on dividing the numerator by the denominator.
At the end of our discussion, one of the kids asked the question, “Can you turn the fraction 15/17 into a percent because I can’t find an equivalent fraction that has 100 as the denominator.” This became their homework problem. I had about 10 kids whose solution was “NO.” Because of the initial reason, but the 14 others remembered that we can turn a fraction into a decimal by dividing the numerator by the denominator. Many of them said “YES” and wrote their answer as .88 or 88/100 or 88%
As I was walking around today while my kids were doing some percent word problems on grids, I started thinking about comparing things. So, I drew on a post-it note 3 flowers (get some markers)… one with 4 red petals and a green stem, one with 4 red petals and a gray stem, and one with 4 purple petals and a green stem. I gave it to one student that was finished and told her to compare these flowers. She said, “well, that’s easy, it’s just like yesterday. There are things that are the same…” and started spouting off the similarities between them and how we would have to change something on one of the flowers to compare it to the other.
I wonder if we use something like this to present to kids before we ever jump into numbers if it would help to establish the goal that to compare we need to find things that are the same. Then, it shouldn’t matter what numbers we use, but what the kids notice and question about the numbers. The 5th graders I have have had a lot of experience with fractions and decimals, including comparing fractions; this was their first formal experience with “percent” in math class. I know I put too much into one lesson, but I couldn’t help myself. They were on a roll yesterday
I would really LOVE to get a flow for all of this at some point. My head is spinning.
Heather,
ReplyDeleteThank you for sharing what your students were able to do.
I'm not sure if I wrote this in the blog entries (it was written while back), but, in the Japanese curriculum, there are two situations where situations are compared multiplicatively. One situation is where two like quantities are involved - like the number of passes (complete or attempts) - which leads to the idea of percents. The other situation is where two unlike quantities are involved - like speed, population density, etc. - which are generally referred to as "per-unit quantities."
In both situations, similar types of reasoning can be used. In fact, in the translated textbook series, they look at the per-unit quantities first, so when they get to the unit on percents, they use numbers that are larger. In their newer edition, where percents are discussed first, they use simpler numbers in their introduction.
I think spending a few lessons where students think about the process of reasoning involved - i.e., if one of the quantities are the same, the comparison is easy, thus if none is the same, we can try to make one the same - seems to be a wise investment.
I think it will be great if students understand that whether they create equivalent fractions with 100 as the denominator, use a common multiple as the denominator, compare fractions, completions/attempts, or divide the numerator by the denominator, they are indeed making the total number of pass attempts the same (100, common denominator, or 1 for the last two cases).
Interesting ideas! There are many I have further questions or comments about but I think we first need to get very clear on the goal (Alice mentioned this.) From the discussion, I see two possibilities
ReplyDeleteStudents will understand that comparing quantities when one is the same amount is easy. However, if the quantities aren't the same then you need to make one the same.
or
Students will be able to compare quantities in situations involving ratios. (I'm using the term ratio because I can't think of another. I'm trying to eliminate comparing just one whole number with another whole number. Given our situation, maybe it should be "Students will be able to compare like-quantity situations which involve multiplication.”)
While these 2 goals are intertwined, I think the focus would be different between the two. The former would be a stronger focus on the problem solving process with more conversation about how the process applies across grade levels. It might have elements of Heather’s thoughts about comparing flowers then moving to numbers….not sure…we can work that out.
The latter (to me) would utilize strong questioning technique but not focus on how to get students to independently problem solve comparison situations.
So, here are my thoughts. Do you agree? Is there a direction you would prefer to go? Am I missing a direction?
I think after we settle the goal we should go to what accomplishment of that goal looks like with students and establish the flow. (I think we have said the flow but just need to get it on paper.)
What do you think?
Err... I forgot to sign the last post. It's from Becky
ReplyDeleteI do agree that we need to settle on a goal. I feel very scattered on the ideas. I am open to any goal that you teachers feel beneficial for students at this level. I don't think I could make an educated guess at this point because I am not familiar with their needs and weeknesses.
ReplyDeleteI think I would like to see a lesson addressing the firstgoal:
ReplyDeleteStudents will understand that comparing quantities when one is the same amount is easy. However, if the quantities aren't the same then you need to make one the same.
This is something that we all need to address in our classrooms. Becky, Ken, Sue, Heather: if you feel strongly about focusing on the other goal, then lets go there.
Wow! So many good ideas.. my head is spinning a bit. As for the first goal( students will understand comparing quantities when onme amount is the same=easy).. heather's students seem to understand that it was more difficult to compare #'s of completed passes because one qb threw more passes than the others... Is this something that you think all students understand or realize? Or would a problem like this stump most children right away? It would seem beneficial to make sure that children understand this completely...I see value in both goals. It almost seems liek we would want to start with the first goal and then that would naturally lead into the more complex lessons.(comparing situations..
ReplyDeleteThat being said, I'm agreeable to whatever the intermediate teachers feel there is the most need. You guys definitely know better where your children tend to break down...
Here goes try #4. I keep getting disconnected.
ReplyDeleteFirst, I think you do not really mean or want to use the language of comparison of two aquantities that are the same. Too much room for confusion and ambiguity. I think you may want to refer to rational numbers. But if you want to use quantities, they are not the same unless each is just another way of naming the other: 1/2, 2/4. I believe what you want to refer to is that the numbers must have a common referent or base to be compared (common denominator, same whole, comparing number of children to number of children (although you could compare number of children to number of pencils and find you have enough for only 85% of the children): everyone has an equal number of chances or a level playing field etc.
I think the child in Heather's (HW?) class who described pretneding that everyone had 100 chances came as close as you can to what children would understand and the extewnsion to grading papers--kids know there weren't 100 aquestions on each one--was a nice example also. But try not to leave ambiguity about "sameness"--it is not the quantities that are the same but there is a common referent for the comparison to be valid.
From Gaea Leindhardt paper:
>>Based on our analysis of why percent is hard and what students seem to know intuitively, we believe that the
following components constitute a solid understanding of percent: (a) understanding the concepts of portions, multiples, and express these multiplicative relationships in the language of percent; (b) recognizing that because percent is an intensive relationship between quantities, implicit
referents exist and must be searched out; (c) developing
a system of flexible representations for different percent meanings; and (d) having available any number of
workable procedures, including the use of benchmarks,
proportional thinking strategies, and formal proportions.
We also believe that instruction in percent should begin with work on understanding the relational language of
percent rather than its procedural connections to decimal or fractional notations. When initial instruction
emphasizes notational conversion, students may never realize the existence of the implicit relationships and comparisons described by percent.<<
Alice
We struggled with the wording and certainly appreciate ideas on being specific and mathematically correct. We even tried using terms like "useful common quantity".
ReplyDeleteHow about:
Students will understand it is easy to compare rational numbers with a common referent. However, if you do not have a common referent then you must create one.
We will have to translate this to kid friendly language. We need to be sure we all understand the meaning of "rational number" and "referent". I think it will be a challenge to translate "referent" to kid friendly language. Primary teachers, we really need you help on this as well as the wording used in comparing.
I was thinking of what mastery of this goal would look like. Would it be similar to our "understanding the equal sign goal" in which different scenarios are presented and students must apply their understanding?
Steph, even if most students can tell which qback was the most successful, I don't think many can verbalize their reasoning clearly. I think it's important to verbalize reasoning to be able to use an idea later. I agree with you that we need to make sure students have masterful understanding of this idea before progressing.
Becky
Is it a common unit?
ReplyDeleteWould kids understand that we have to think of the quantities using a common unit?
Alice
Wording here is indeed rather tricky. I probably started the language, "if one quantity is the same..." That works in the context of per-one quantity like crowdedness and speed since those situations DO involve 2 different quantities.
ReplyDeletePercents situations - or those situations that can lead to percents - involve ratios of the same quantity. What we are comparing is "relative quantities," though that's not a common phrase, either. In each scenario, what we have is the number being compared (e.g., 15 completed passes) and the base of comparison (e.g., 25 total pass attempts). So, if either the number being compared or the base of comparison is the same, then comparing two scenarios will be done easily by looking at the other number. To determine percents, we ask if the base were 100, what would be the compared number?
I prefer not to use part-whole here since percents can be calculated on situations like population increase etc.
Does this make sense?
I'm here and reading each day, trying to digest all of this information. I'm looking forward to our next meeting when we can sort this out all together, as I'm having difficulty making any useful contributions at this point. Just want you to know I'm still here, and still thinking, for whatever good that is doing any of us. Sue Tabor :)
ReplyDeleteSorry - that last post is from Sue, not my 10 year old daughter, Haley. Somehow my password is coming up as my name -- I'm not blog proficient yet.
ReplyDeletetest
ReplyDeletetesting
ReplyDelete