Our recent work with some students working to find out how they feel they are "smart" at math has me thinking of one of my kids at my school, and how he views himself as a mathematician.
I have been spending some time with him during math and am trying to explore some of his ideas about what it means to him to be "smart" at doing mathematics. Watching him work and talking to him about what strategies he uses to get his solutions, I am often struck as how well he seems to grasp the concepts we are going over. I have this reaction because his scores on assessments often do not prove this out.
The more I watch and probe, the more I am coming to the realization that he feels people who are "smart" at math are the fastest to complete their work. He can explain his mathematical thinking, but is often interrupted in his explanation by the discovery of a calculating error made in haste. This happens frequently. It also carries over to his assessment work.
Knowing what I do about him, I am trying to instill in him two simple ideas in hopes he can alter his views of what mathematicians do: First, I am working to get him to believe that math is about his process, his strategy, and in the end his accurate results - speed has nothing to do with it. I want him to be transparent with what he is doing on the paper so he doesn't trip himself up by going to fast and taking shortcuts.
Second, his confidence is often shaken by these errors he makes because of rushing. He doesn't understand why he is getting things wrong when he feels he knows what to do. I have had to do some re-building of his esteem. I ask him: "what strategies do you know that will help you solve this?" He takes me through his steps on a problem and we find out where things are going wrong together (almost always in simple miscalculations, not concepts). Part of this rebuilding is to show him that a major part of doing math is knowing how to find and fix your mistakes.
Now that I know where his struggles lie, I am often stopping by to remind him that "you know the stuff, just take your time and be accurate. Don't worry about how long it takes you to get there, just get there carefully. You can do this!"
It seems almost too simple, but now that I have an idea of what is tripping it up and we are addressing it, I have hopes he will do better. It's a reminder for me that kids do tend to inherently think that faster means better in math, and it can lead to them not seeing themselves as "good at math" when they truly are.
Tuesday, February 28, 2012
Monday, February 13, 2012
All together now
Adults and little kids seemingly love to count out loud with one another. We all did as a class, anyway. And my son gets a kick out of choral counting with mom and me (all the way to 20!). But so far my meager evidence from our Tuesday site doesn't prove that out with the elementary school kids.
Maybe the numbers didn't challenge them enough, or maybe they were weirded out by the odd, tall man who wanted them to loudly skip-count with him, but the kids seemed disinterested to say the least. I'd love to get another chance to see if I could make it more exciting for those students, but it's probably not in the cards.
That being said, I think I would like to try it with my kids who are used to me. They got a kick out of the Quick Images, so I think they may go for this too. I know they would actually really get into the patterns that become revealed, and it's a given that they love to hear themselves talk, so the "out loud" part would be a plus.
Since they just began to add and subtract fractions, I think it would be a good "math facts" booster to have them count up by simple fractions to start, like 2/3 and 3/4, then maybe something like 7/8 or 5/16just to be mean as a fun, challenging way to stretch their thinking.
I am interested to see how well they do and what sort of patterns we can find within the numbers if we try it out. Will keep you posted...
Maybe the numbers didn't challenge them enough, or maybe they were weirded out by the odd, tall man who wanted them to loudly skip-count with him, but the kids seemed disinterested to say the least. I'd love to get another chance to see if I could make it more exciting for those students, but it's probably not in the cards.
That being said, I think I would like to try it with my kids who are used to me. They got a kick out of the Quick Images, so I think they may go for this too. I know they would actually really get into the patterns that become revealed, and it's a given that they love to hear themselves talk, so the "out loud" part would be a plus.
Since they just began to add and subtract fractions, I think it would be a good "math facts" booster to have them count up by simple fractions to start, like 2/3 and 3/4, then maybe something like 7/8 or 5/16
I am interested to see how well they do and what sort of patterns we can find within the numbers if we try it out. Will keep you posted...
Saturday, February 11, 2012
Bask in the beautiful chaos
I sometimes feel like the primary grades student teachers are plotting against me. I see them down in their first floor hall, talking and laughing, sharing "something" that must be very funny, and just collaborating in general.
Then there's me, all alone up on the second floor, Hangin' with Mr. Cooper (+2 nerd skill points if you remember that show). The advantage of being the lone intern upstairs is that I am asked/ allowed/ called upon to hop into other classrooms from time to time, which is cool. I get to see similar teaching techniques and styles used in different ways. Here are a couple of those educational nuggets:
In our classroom the "turn & talk" is used in multiple ways, but mostly as a means of creating a full table group discussion (and also creating noise). This is one of my favorite moments, when the silence of direct instruction explodes into 26 different voices and ideas. The room saturates with the thoughts of the students.
Yet across the way, I have seen it used mainly as a tool for quiet partner sharing. Still the same number of ways of understanding being expressed, but more calm. It all has to do with the teacher comfort level with bringing the kids back, and how well they know their kids can handle the freedom of open discussion.
Then there's the gathering on the floor and lining up for the halls. One of the classrooms nearby is very structured, and those kid have assigned seating in the front of the room for gatherings and designated spots in line. Yep, all lined up in rows, perfectly designed for efficiency. This does work well, but it has removed all choice from the students - and in turn all sense of ownership and responsibility for making good decisions.
We roll a little different in 205. Kids can choose their arrangements, but know to make a good choice of where they are - "Are you going to be able to keep quiet and stay focused by this neighbor?", "Is this a good place for you?". It sometimes requires 10 seconds of rearranging on our part, but I think that the students becoming self sufficient and learning about putting themselves in positive situations is a valuable life lesson. There won't always be adults around to do it for them.
Seeing the different strategies used by the teachers I am with makes me realize the my CT does certainly induce situations that are more "messy", but life is very chaotic and by exposing the kids to these situations and giving them the tools to learn how to handle it themselves, I feel we are better serving them in the long run.
Then there's me, all alone up on the second floor, Hangin' with Mr. Cooper (+2 nerd skill points if you remember that show). The advantage of being the lone intern upstairs is that I am asked/ allowed/ called upon to hop into other classrooms from time to time, which is cool. I get to see similar teaching techniques and styles used in different ways. Here are a couple of those educational nuggets:
In our classroom the "turn & talk" is used in multiple ways, but mostly as a means of creating a full table group discussion (and also creating noise). This is one of my favorite moments, when the silence of direct instruction explodes into 26 different voices and ideas. The room saturates with the thoughts of the students.
Yet across the way, I have seen it used mainly as a tool for quiet partner sharing. Still the same number of ways of understanding being expressed, but more calm. It all has to do with the teacher comfort level with bringing the kids back, and how well they know their kids can handle the freedom of open discussion.
Then there's the gathering on the floor and lining up for the halls. One of the classrooms nearby is very structured, and those kid have assigned seating in the front of the room for gatherings and designated spots in line. Yep, all lined up in rows, perfectly designed for efficiency. This does work well, but it has removed all choice from the students - and in turn all sense of ownership and responsibility for making good decisions.
We roll a little different in 205. Kids can choose their arrangements, but know to make a good choice of where they are - "Are you going to be able to keep quiet and stay focused by this neighbor?", "Is this a good place for you?". It sometimes requires 10 seconds of rearranging on our part, but I think that the students becoming self sufficient and learning about putting themselves in positive situations is a valuable life lesson. There won't always be adults around to do it for them.
Seeing the different strategies used by the teachers I am with makes me realize the my CT does certainly induce situations that are more "messy", but life is very chaotic and by exposing the kids to these situations and giving them the tools to learn how to handle it themselves, I feel we are better serving them in the long run.
Sunday, February 5, 2012
Adventures in subitizing
The cool thing about doing Quick Images with 5th graders is that you can explain the purpose of it to them. They understand when you explain the concept of children's mathematical development moving from counting single dots at a time, toward more sophisticated ideas like clumping dots together, or recognizing 5 dots immediately based on their knowledge of that pattern from using dice. They also enjoy saying the word "subitize" as much as I do.
The down side of QI's and 10 year-olds is that the patterns I had to use were not at all difficult for them. They enjoyed explaining their thinking, hearing other ideas of how to "see" the solution, and getting to play a fun game, but I found that on day one they weren't really stretching their thinking.
So on day two I modified one of the cards at the end of a string of 3. We were working on patterns of 4 (four dot clusters). At this point the kids have all put on their cool jackets, popped the collars, and feel like they can breeze through the quick images - having fun, but not working too hard. I chuckled inside when I heard the collective "hey!", "what?", "was that..?", and other shrieks of doubt when I snuck this doozy in on them:
Really nothing earth-shattering, and most of them got the answer just as quick as with the others once they refocused, but I could tell they had to do some more mental gymnastics to find this one out. I only had time to get the solution to this one from a couple of students, but it was fascinating nonetheless. Rather than the typical counting up, adding on, or simple multiplication, this problem required multiple processes. Here is how they did it:
"I recognized the shapes as groups of 4, and saw there were blocks of 8, so I started to just multiply those blocks of 8 times the 4 times across. But then I noticed some were missing, so I looked at it as 3 groups of 4 on each side (as circled above) and got 3 x 4 = 12, and there were 2 of those, so I had 24. Then I added the 2 little groups of 3 each, so 6 more. So I got 30."
"I noticed there were two dots missing right away, so I knew that. I saw it as groups of 4, and it was 4 groups across and 2 groups high, so there were 8 groups of 4. That means I had 32, and I know it was 2 less than that from the 2 that were missing, so it was really 30."
When this latter solution was re-voiced, we settled on an equation that looked like this: 2 x 4= 8 (groups), x 4 (dots/group) = 32, - 2 (the dots know to be missing) = 30.
These kids have some pretty sophisticated strategies for solving these puzzles. They have asked me when we are going to do some more. I do have some plans to try some even more complicated dot puzzles like these. I have and idea to do some with half-dots since we are working on adding fractions, and when we start our intro to geometry we are going to do some where they are given 10 little interlocking cubes on their desk and shown a quick image of a 3-D shape that they must replicate with their pieces.
These kids already have the ability to remember and manipulate the images they see in their minds - that is why some of these patterns are easier for them. I am extremely eager to see how they might handle the more complex spatial manipulation that comes with the geometric shapes, and how they will group different parts of the shapes together - subitizing with geometry, and actually using the manipulatives in front of them to bring the shapes to life.
The down side of QI's and 10 year-olds is that the patterns I had to use were not at all difficult for them. They enjoyed explaining their thinking, hearing other ideas of how to "see" the solution, and getting to play a fun game, but I found that on day one they weren't really stretching their thinking.
So on day two I modified one of the cards at the end of a string of 3. We were working on patterns of 4 (four dot clusters). At this point the kids have all put on their cool jackets, popped the collars, and feel like they can breeze through the quick images - having fun, but not working too hard. I chuckled inside when I heard the collective "hey!", "what?", "was that..?", and other shrieks of doubt when I snuck this doozy in on them:
Really nothing earth-shattering, and most of them got the answer just as quick as with the others once they refocused, but I could tell they had to do some more mental gymnastics to find this one out. I only had time to get the solution to this one from a couple of students, but it was fascinating nonetheless. Rather than the typical counting up, adding on, or simple multiplication, this problem required multiple processes. Here is how they did it:
"I recognized the shapes as groups of 4, and saw there were blocks of 8, so I started to just multiply those blocks of 8 times the 4 times across. But then I noticed some were missing, so I looked at it as 3 groups of 4 on each side (as circled above) and got 3 x 4 = 12, and there were 2 of those, so I had 24. Then I added the 2 little groups of 3 each, so 6 more. So I got 30."
"I noticed there were two dots missing right away, so I knew that. I saw it as groups of 4, and it was 4 groups across and 2 groups high, so there were 8 groups of 4. That means I had 32, and I know it was 2 less than that from the 2 that were missing, so it was really 30."
When this latter solution was re-voiced, we settled on an equation that looked like this: 2 x 4= 8 (groups), x 4 (dots/group) = 32, - 2 (the dots know to be missing) = 30.
These kids have some pretty sophisticated strategies for solving these puzzles. They have asked me when we are going to do some more. I do have some plans to try some even more complicated dot puzzles like these. I have and idea to do some with half-dots since we are working on adding fractions, and when we start our intro to geometry we are going to do some where they are given 10 little interlocking cubes on their desk and shown a quick image of a 3-D shape that they must replicate with their pieces.
These kids already have the ability to remember and manipulate the images they see in their minds - that is why some of these patterns are easier for them. I am extremely eager to see how they might handle the more complex spatial manipulation that comes with the geometric shapes, and how they will group different parts of the shapes together - subitizing with geometry, and actually using the manipulatives in front of them to bring the shapes to life.
Friday, February 3, 2012
Feedback loop
As I teach more and more within my classroom, I am starting to get more and more feedback from my CT. It has all been really positive and helped push me to want to take on more challenges and try new things with our students.
Here is a typical post lesson recap for us (whenever we can find some of that all-precious time): It usually starts with me talking about what I felt went well, and what I noticed that could use some polishing. Then I ask a question or two about what she would normally do when she sees the lesson going a certain way, or how she deals with those little things that pop up. I am usually concerned about one or two things that I noticed, and try to go through my thinking during the lesson, and talk about why I made some of the choices that I did. I know, me over-thinking things, can you believe it?
Case in point: yesterday I was doing some math problems with my kids on their little white boards. They are starting to learn how to add and subtract fractions. Some students really get it already, and some are still a little fuzzy. Two kids in particular are struggling, so I had spent the previous day working with them quite a bit. As I was giving them the problems I would wait a bit for the kids to "show me they were done", and they were ready to present by raising their boards. I so desperately wanted those two kids to raise their boards. The tricky part is if I linger too long to wait for those kids, I lose the other 20 or so. It's a balancing act. As the tasks progressed, I still kept waiting for those two boards to raise. I could see their brains working, but I could also see the eyes of the other kids starting to glaze over. As we came to the last of our questions, I was determined to hold out and let those kids get their thoughts out of their heads, give them that extra time to put the answer onto their board and raise it high.
The seconds seemed to stretch on forever (it's amazing how 30 seconds feels like 25 minutes when standing in front of that board). Finally, a miracle! The boards came up in the air, and a smile came to their faces. I even chose one of them to explain their thinking, and they did so flawlessly. My decision to hold out at the risk of losing some of the kids who were farther ahead seemed justified - I now had a student who previously was in doubt, feeling and thinking of themselves as a mathematician, full of confidence.
But still, I am wracked with guilt that the kids who understood the math had to wait an extra ten seconds. I feel like I have let them down. This is when my CT chuckles and says "Huh, I thought that went really well - you gave ______ a chance to show her classmates what she knows. She seemed excited, that was great. But you know what you could do? You can write some difficult 'challenge problems' over on the side of the board for those kids who want the extra work - that'll keep them busy while we let the kids who need the extra time work through the math". Genius, pure genius. She always seems to have the most simple, perfect solution for my concerns ready and waiting for me.
I guess this is my long way of saying that the best feedback I get from my CT is the acknowledgement that I am noticing the right things and making sound choices. I am seeing the tricky areas that I'm not so sure about and am not afraid to admit I don't think something was working. I appreciate that she is eager to share her ideas on how I could get those sticky parts unstuck, and how she lets me know that every teacher runs into those same "difficult spots".
The feedback I get allows me to enter each experience with a fearlessness and willingness to try things out; to not be so worried about doing something wrong, but excited to do those things that I think are right.
Here is a typical post lesson recap for us (whenever we can find some of that all-precious time): It usually starts with me talking about what I felt went well, and what I noticed that could use some polishing. Then I ask a question or two about what she would normally do when she sees the lesson going a certain way, or how she deals with those little things that pop up. I am usually concerned about one or two things that I noticed, and try to go through my thinking during the lesson, and talk about why I made some of the choices that I did. I know, me over-thinking things, can you believe it?
Case in point: yesterday I was doing some math problems with my kids on their little white boards. They are starting to learn how to add and subtract fractions. Some students really get it already, and some are still a little fuzzy. Two kids in particular are struggling, so I had spent the previous day working with them quite a bit. As I was giving them the problems I would wait a bit for the kids to "show me they were done", and they were ready to present by raising their boards. I so desperately wanted those two kids to raise their boards. The tricky part is if I linger too long to wait for those kids, I lose the other 20 or so. It's a balancing act. As the tasks progressed, I still kept waiting for those two boards to raise. I could see their brains working, but I could also see the eyes of the other kids starting to glaze over. As we came to the last of our questions, I was determined to hold out and let those kids get their thoughts out of their heads, give them that extra time to put the answer onto their board and raise it high.
The seconds seemed to stretch on forever (it's amazing how 30 seconds feels like 25 minutes when standing in front of that board). Finally, a miracle! The boards came up in the air, and a smile came to their faces. I even chose one of them to explain their thinking, and they did so flawlessly. My decision to hold out at the risk of losing some of the kids who were farther ahead seemed justified - I now had a student who previously was in doubt, feeling and thinking of themselves as a mathematician, full of confidence.
But still, I am wracked with guilt that the kids who understood the math had to wait an extra ten seconds. I feel like I have let them down. This is when my CT chuckles and says "Huh, I thought that went really well - you gave ______ a chance to show her classmates what she knows. She seemed excited, that was great. But you know what you could do? You can write some difficult 'challenge problems' over on the side of the board for those kids who want the extra work - that'll keep them busy while we let the kids who need the extra time work through the math". Genius, pure genius. She always seems to have the most simple, perfect solution for my concerns ready and waiting for me.
I guess this is my long way of saying that the best feedback I get from my CT is the acknowledgement that I am noticing the right things and making sound choices. I am seeing the tricky areas that I'm not so sure about and am not afraid to admit I don't think something was working. I appreciate that she is eager to share her ideas on how I could get those sticky parts unstuck, and how she lets me know that every teacher runs into those same "difficult spots".
The feedback I get allows me to enter each experience with a fearlessness and willingness to try things out; to not be so worried about doing something wrong, but excited to do those things that I think are right.
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