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.
Tuesday, March 10, 2009
Please post here after March 10
Thanks for a great meeting Friday. I find myself pondering those qualities of strong mathematicians and how to explicitly teach them. That's what lesson study is about!
When students look at situations of games won versus games played to determine the best player, we know some children may subtract to find the better player. Diane and I both experienced this in our classes. We know they are not seeing these as ratios with a multiplicative relationship, but then why should they when they haven’t experienced instruction on ratios? ( This is my jab for teaching the idea of ratios early.) For example: Player A won 8 out of 12 thus needs to win 4 more to win all games. Player B won 4 out of 12 thus needs to win 8 more to win all games. Player C won 8 out of 20 thus needs to win 12 more win all games. We know most students will say A is better than B since they played the same number of games and A won more. However, some will say B is better than C because he only needs to win 8 more games. We are puzzling over the direction to take students who have this misconception. Here are thoughts for 3 directions. I would appreciate your opinions. 1) Is it fair to subtract games won from games played when Player C played more games than Player B? (This needs an elaboration sentence, but I’m not sure what.) 2) Suppose they played one more game, can you predict whether they would win or lose? No, so we cannot say if Player B played 8 games he would win them all. (I don’t see how this one plays out but I thought I would mention it in case some of you do.) 3) Let’s try your strategy with another set of numbers: Player D won 1 out of 2 games. Player E won 99 out of 100 games. They both only need to win 1 more game so they are equal in playing ability. No? If you can win 99 out of 100 games you must be a very strong player. Winning 1 out of 2 is good but not great.
We know they are changing the relationship between the 2 numbers but I'm not sure of how to get that across or if we can. My sense is that it must be simple since it isn't a focus of the lesson. Thanks, Becky
What you want kids to recognize is that everyone needs a level playing field before you can compare their performance. Supposition 2: Even if he played and won the next 8 games he still would not have won all his games. He now would have won 12 out of 20. How would they feel if some kids had 20 questions on a test and others only had 8? Would that be fair? Could you move from there to the idea of finding a way to imagine everyone has had the same number of opportunities? Long ago people decided to imagine what if all the opportunities or everything is imagined as 100; that would be an easy way to make such a comparison. 50 out of 100 is one-half, 4 out of 8 is one-half, 12 out of 24 is one-half. It's similar to looking at equivalent fractions, but in this case we want 100 to be the basis of comparison. When thinking about part/whole a denominator of 100 is called hundredths. When we compare how much of something is how much right or won or accomplished we call it percent. Per cent means for every 100 and provides a common basis for comparison. Does that make any sense?
Once someone has lost ANY games they can never make it to having won them all.
I've rambled but maybe something will prompt another thought.
I like the direction Becky's 3rd thought takes...seems like it fits well with the "adaptive reasoning" questioning we were talking about at our last meeting. (By the way, I'll post the summary.) Seems like this would be a logical way to think about their subtraction strategy...i can even picture a quick drawing to represent such a scenario. This way of thinking allows kids with any experience in fractions to understand that 1 out of 2 is NOT the same as 99 out of 100. I hesitate to move to the "out of 100" situation yet since the focus is really to get them to find any common unit of comparison. However, this would be helpful for a next lesson, perhaps the day after? Just a thought...depending on the class, perhaps they've had experience with ratios? Are more teachers teaching this now b/c of Thinking Math?
Summary of Friday’s (3/6) meeting. Please add/edit. I feel like it's VERY choppy...
To start our meeting, Becky gave us some good news! According to the latest TIMSS report, students in the US are doing much better in 4th and 8th grades. This is an effect of teaching for understanding! Yeah us!
We started our LS discussion by agreeing that there is NO WORD for the word we’re looking for! We will use “common unit of comparison.” The word unit will refer to labeling and the number.
We’d like to keep in the back of our minds throughout this lesson where it would fall in a series of (perhaps) 3 or 4 lessons within a unit. What comes next? Perhaps we can get some lesson springboards started for later teaching.
We looked at the words in a couple of Japanese lesson goals. These were “Students will think about…” and “Students will understand the merit of having…” As a team, we really like these goals, but agree that they are hard to measure. Stephanie suggested by adding “by justifying” to the goals that then they become measureable. We also discussed the Japanese “human being” goals that are added into their lessons. They see the “whole child.” We’d like to keep these types of goals in mind.
After looking over Tad’s slide “Purposes of Questioning,” we want our lesson to focus on “strategic competence” where students are able to develop foresight in formulating or choosing strategies to solve math problems. We also like the “adaptive reasoning” questioning…the capacity for logical thought, reflection, explanation, and justification. An overarching goal came out of this discussion that “Students can think about an efficient strategy while considering the whole problem.”
For our Lesson Study lesson, we are focused not only on student learning, but teacher growth as well. We would like to become stronger questioners throughout our lessons, not just one or two “big” questions at the end of the lesson.
We next began the task of putting the flow of the lesson together. We know our hook will be Guitar Hero, and a problem something like “ Mrs. Pittard passed 9 out of 13 songs. Mrs. LeJeune passed 10 out of 13 songs. Mr. O’Brien passed 9 out of 11 songs.” Stopping here to get the kids to think…telling them to “Think about the information you know so far. What are you thinking?” (Here we inserted some probing questions if the students don’t realize that things are the same somehow. “What is the same about the problem? How could we make this easier for you to compare?”)
Some problems have come up with the best numbers to use: * What if kids begin to use an incorrect subtraction strategy to find out which player is the best…a misconception about the ratio (For example, 7 out of 10 is -3 and 16 out of 20 is -4, so for that reason, 7 out of 10 is better than 16 out of 20 because 7 is only 3 away from 10.) Should we have something, a drawing or model of some kind prepared to prove them wrong if this happens? * What numbers are best to use (we will be blogging this)?
I like the direction Becky's 3rd thought takes...seems like it fits well with the "adaptive reasoning" questioning we were talking about at our last meeting. (By the way, I'll post the summary.) Seems like this would be a logical way to think about their subtraction strategy...i can even picture a quick drawing to represent such a scenario. This way of thinking allows kids with any experience in fractions to understand that 1 out of 2 is NOT the same as 99 out of 100. I hesitate to move to the "out of 100" situation yet since the focus is really to get them to find any common unit of comparison. However, this would be helpful for a next lesson, perhaps the day after? Just a thought...depending on the class, perhaps they've had experience with ratios? Are more teachers teaching this now b/c of Thinking Math?
I agree with Heather in that the third scenario is a good direction to go in. That would help dispell the subtraction strategy. Great thinking Becky. I also agree with not going to the out of 100 concept yet. That would be a next step lesson. We do want them to understand the common unit of comparison.
At this point, my understanding is we've decided this will be a 4th grade lesson. The main goal is for the children to find something the same before trying to compare. While we ultimately want them to see percentages as a way to do this, I think introducing anything about "100" in this lesson, or even the following day might be premature. I'm concerned that by introducing the 100 concept too soon, we will move them away from the goal of this lesson too quickly, before they've had a chance to really digest and practice the concept. We already have kids, even in fifth grade, that automatically take everything to 100 to compare. Don't we won't them to see that often this method is making more work for themselves than necessary? I also like the third scenario that Becky provided. It makes an obvious distinction to help dispell the subtraction theory that the kids should be able to quickly relate to and see. Yet it keeps us at our goal of "finding something the same" as a means for comparing and ordering.
I think the idea of comparing the difference is exactly the kind of "error" that we want in this lesson. I think you can deal with the error in many different ways, but the best one (or ones) is what might come from other students. I doubt all other students will be convinced by this particular reasoning, so they will come up with something.
One thing you may want to do is to make sure everyone understands what the difference means. The difference in this context is the number of losses (assuming there is no "tie" in this situation). So, what this child is saying is that B is a better team because C has more losses. Once this is understood, someone might also say, "But C has more wins than B does."
The important point here is that neither wins or losses by themselves cannot determine the better team. We have to consider wins (or losses, I suppose) AND the number of games played in order to determine a better team. So, this error may be so valuable to help students understand this idea that you may want to consider suggesting it even if no one actually comes up with it.
What Tad says makes perfect sense. It sounds so logical to point out that students must consider both wins and losses to understand... Do you think we should be point that out when working with the 1st set of #'s(8 of12, 4 of 12 and 8 of 20) that the subtraction/difference method isn't going to help students solve this problem, or should we wait until after the 2nd set of #'s?
My sense is that it's such an important idea that it should be with the first set. As Tad noted, if the kids don't bring it up, I think we should. What do you all think? Is the lesson getting too long?
Thank you, Tad and Alice, for stretching out thoughts. We have a rough draft of the lesson which we are working on. We'll post it next week. Becky
I agree that it should be part of the first set. If none of the students go there on their own, perhaps we can design our questioning to guide them to ponder it without having to point it out. Is there a way we can we test run the draft lesson to see if it's too long? It might be, yet all three sets of numbers are critical. I'm not sure how to shorten it without losing a key element.
I would also agree that it should come out in the first set. I'm not sure if the students will bring out the number of losses in this situation. If the lesson was about a sport or typical game, then I think the concept of difference (dealing with of the number of losses) would come up. But with Guitar Hero, which I haven't played so I'm not sure, I believe you said it talks about the number of songs passed. So the students may not associate it as a "loss". If that's the case, then perhaps it should be brought up with creative questioning.
All three sets of numbers are crucial. The first set has two common units of comparison, the second has one common unit of comparison, and the third has none. Thus, the children will have to find a way to make a common unit to compare. To cut any of the three out would make the lesson less powerful.
When students look at situations of games won versus games played to determine the best player, we know some children may subtract to find the better player. Diane and I both experienced this in our classes. We know they are not seeing these as ratios with a multiplicative relationship, but then why should they when they haven’t experienced instruction on ratios? ( This is my jab for teaching the idea of ratios early.)
ReplyDeleteFor example:
Player A won 8 out of 12 thus needs to win 4 more to win all games.
Player B won 4 out of 12 thus needs to win 8 more to win all games.
Player C won 8 out of 20 thus needs to win 12 more win all games.
We know most students will say A is better than B since they played the same number of games and A won more. However, some will say B is better than C because he only needs to win 8 more games.
We are puzzling over the direction to take students who have this misconception.
Here are thoughts for 3 directions. I would appreciate your opinions.
1) Is it fair to subtract games won from games played when Player C played more games than Player B? (This needs an elaboration sentence, but I’m not sure what.)
2) Suppose they played one more game, can you predict whether they would win or lose? No, so we cannot say if Player B played 8 games he would win them all. (I don’t see how this one plays out but I thought I would mention it in case some of you do.)
3) Let’s try your strategy with another set of numbers: Player D won 1 out of 2 games. Player E won 99 out of 100 games. They both only need to win 1 more game so they are equal in playing ability. No? If you can win 99 out of 100 games you must be a very strong player. Winning 1 out of 2 is good but not great.
We know they are changing the relationship between the 2 numbers but I'm not sure of how to get that across or if we can. My sense is that it must be simple since it isn't a focus of the lesson.
Thanks,
Becky
What you want kids to recognize is that everyone needs a level playing field before you can compare their performance. Supposition 2: Even if he played and won the next 8 games he still would not have won all his games. He now would have won 12 out of 20. How would they feel if some kids had 20 questions on a test and others only had 8? Would that be fair?
ReplyDeleteCould you move from there to the idea of finding a way to imagine everyone has had the same number of opportunities? Long ago people decided to imagine what if all the opportunities or everything is imagined as 100; that would be an easy way to make such a comparison. 50 out of 100 is one-half, 4 out of 8 is one-half, 12 out of 24 is one-half. It's similar to looking at equivalent fractions, but in this case we want 100 to be the basis of comparison. When thinking about part/whole a denominator of 100 is called hundredths. When we compare how much of something is how much right or won or accomplished we call it percent. Per cent means for every 100 and provides a common basis for comparison. Does that make any sense?
Once someone has lost ANY games they can never make it to having won them all.
I've rambled but maybe something will prompt another thought.
Alice
I like the direction Becky's 3rd thought takes...seems like it fits well with the "adaptive reasoning" questioning we were talking about at our last meeting. (By the way, I'll post the summary.) Seems like this would be a logical way to think about their subtraction strategy...i can even picture a quick drawing to represent such a scenario. This way of thinking allows kids with any experience in fractions to understand that 1 out of 2 is NOT the same as 99 out of 100.
ReplyDeleteI hesitate to move to the "out of 100" situation yet since the focus is really to get them to find any common unit of comparison. However, this would be helpful for a next lesson, perhaps the day after?
Just a thought...depending on the class, perhaps they've had experience with ratios? Are more teachers teaching this now b/c of Thinking Math?
Summary of Friday’s (3/6) meeting. Please add/edit. I feel like it's VERY choppy...
ReplyDeleteTo start our meeting, Becky gave us some good news! According to the latest TIMSS report, students in the US are doing much better in 4th and 8th grades. This is an effect of teaching for understanding! Yeah us!
We started our LS discussion by agreeing that there is NO WORD for the word we’re looking for! We will use “common unit of comparison.” The word unit will refer to labeling and the number.
We’d like to keep in the back of our minds throughout this lesson where it would fall in a series of (perhaps) 3 or 4 lessons within a unit. What comes next? Perhaps we can get some lesson springboards started for later teaching.
We looked at the words in a couple of Japanese lesson goals. These were
“Students will think about…” and “Students will understand the merit of having…”
As a team, we really like these goals, but agree that they are hard to measure. Stephanie suggested by adding “by justifying” to the goals that then they become measureable. We also discussed the Japanese “human being” goals that are added into their lessons. They see the “whole child.” We’d like to keep these types of goals in mind.
After looking over Tad’s slide “Purposes of Questioning,” we want our lesson to focus on “strategic competence” where students are able to develop foresight in formulating or choosing strategies to solve math problems. We also like the “adaptive reasoning” questioning…the capacity for logical thought, reflection, explanation, and justification.
An overarching goal came out of this discussion that “Students can think about an efficient strategy while considering the whole problem.”
For our Lesson Study lesson, we are focused not only on student learning, but teacher growth as well. We would like to become stronger questioners throughout our lessons, not just one or two “big” questions at the end of the lesson.
We next began the task of putting the flow of the lesson together. We know our hook will be Guitar Hero, and a problem something like “ Mrs. Pittard passed 9 out of 13 songs. Mrs. LeJeune passed 10 out of 13 songs. Mr. O’Brien passed 9 out of 11 songs.” Stopping here to get the kids to think…telling them to “Think about the information you know so far. What are you thinking?” (Here we inserted some probing questions if the students don’t realize that things are the same somehow. “What is the same about the problem? How could we make this easier for you to compare?”)
Some problems have come up with the best numbers to use:
* What if kids begin to use an incorrect subtraction strategy to find out which player is the best…a misconception about the ratio (For example, 7 out of 10 is -3 and 16 out of 20 is -4, so for that reason, 7 out of 10 is better than 16 out of 20 because 7 is only 3 away from 10.) Should we have something, a drawing or model of some kind prepared to prove them wrong if this happens?
* What numbers are best to use (we will be blogging this)?
I like the direction Becky's 3rd thought takes...seems like it fits well with the "adaptive reasoning" questioning we were talking about at our last meeting. (By the way, I'll post the summary.) Seems like this would be a logical way to think about their subtraction strategy...i can even picture a quick drawing to represent such a scenario. This way of thinking allows kids with any experience in fractions to understand that 1 out of 2 is NOT the same as 99 out of 100.
I hesitate to move to the "out of 100" situation yet since the focus is really to get them to find any common unit of comparison. However, this would be helpful for a next lesson, perhaps the day after?
Just a thought...depending on the class, perhaps they've had experience with ratios? Are more teachers teaching this now b/c of Thinking Math?
I agree with Heather in that the third scenario is a good direction to go in. That would help dispell the subtraction strategy. Great thinking Becky.
ReplyDeleteI also agree with not going to the out of 100 concept yet. That would be a next step lesson. We do want them to understand the common unit of comparison.
At this point, my understanding is we've decided this will be a 4th grade lesson. The main goal is for the children to find something the same before trying to compare. While we ultimately want them to see percentages as a way to do this, I think introducing anything about "100" in this lesson, or even the following day might be premature. I'm concerned that by introducing the 100 concept too soon, we will move them away from the goal of this lesson too quickly, before they've had a chance to really digest and practice the concept. We already have kids, even in fifth grade, that automatically take everything to 100 to compare. Don't we won't them to see that often this method is making more work for themselves than necessary? I also like the third scenario that Becky provided. It makes an obvious distinction to help dispell the subtraction theory that the kids should be able to quickly relate to and see. Yet it keeps us at our goal of "finding something the same" as a means for comparing and ordering.
ReplyDeleteI think the idea of comparing the difference is exactly the kind of "error" that we want in this lesson. I think you can deal with the error in many different ways, but the best one (or ones) is what might come from other students. I doubt all other students will be convinced by this particular reasoning, so they will come up with something.
ReplyDeleteOne thing you may want to do is to make sure everyone understands what the difference means. The difference in this context is the number of losses (assuming there is no "tie" in this situation). So, what this child is saying is that B is a better team because C has more losses. Once this is understood, someone might also say, "But C has more wins than B does."
The important point here is that neither wins or losses by themselves cannot determine the better team. We have to consider wins (or losses, I suppose) AND the number of games played in order to determine a better team. So, this error may be so valuable to help students understand this idea that you may want to consider suggesting it even if no one actually comes up with it.
What Tad says makes perfect sense. It sounds so logical to point out that students must consider both wins and losses to understand...
ReplyDeleteDo you think we should be point that out when working with the 1st set of #'s(8 of12, 4 of 12 and 8 of 20) that the subtraction/difference method isn't going to help students solve this problem, or should we wait until after the 2nd set of #'s?
My sense is that it's such an important idea that it should be with the first set. As Tad noted, if the kids don't bring it up, I think we should. What do you all think? Is the lesson getting too long?
ReplyDeleteThank you, Tad and Alice, for stretching out thoughts. We have a rough draft of the lesson which we are working on. We'll post it next week.
Becky
I agree that it should be part of the first set. If none of the students go there on their own, perhaps we can design our questioning to guide them to ponder it without having to point it out. Is there a way we can we test run the draft lesson to see if it's too long? It might be, yet all three sets of numbers are critical. I'm not sure how to shorten it without losing a key element.
ReplyDeleteI would also agree that it should come out in the first set. I'm not sure if the students will bring out the number of losses in this situation. If the lesson was about a sport or typical game, then I think the concept of difference (dealing with of the number of losses) would come up. But with Guitar Hero, which I haven't played so I'm not sure, I believe you said it talks about the number of songs passed. So the students may not associate it as a "loss". If that's the case, then perhaps it should be brought up with creative questioning.
ReplyDeleteAll three sets of numbers are crucial. The first set has two common units of comparison, the second has one common unit of comparison, and the third has none. Thus, the children will have to find a way to make a common unit to compare.
ReplyDeleteTo cut any of the three out would make the lesson less powerful.