When I was in school, I was pretty consistently a math underachiever, always scoring in the top percentiles on standardized tests, but rarely succeeding in class. My mother blames this tendency on my second grade teacher, who liked to announce to the whole class when I made a mistake. Some of it may also be due to gender bias -- girls weren't supposed to be good at math. But whatever the reason, it was my own lack of confidence in math that kept me from achieving well. I second guessed myself all the time. Frequently I would look at a problem and know the answer, but without knowing exactly why I knew it. Then, in trying to prove it to myself, I'd make a mistake. It wasn't until I was in sixth grade, while at the American School in London, where my math issues were identified appropriately. I didn't need remedial math. I needed more challenge and confidence. So along with 3 or 4 other kids in my grade and the grade above, we were pulled out of our regular math class and put with Mrs. Heumann.
Mrs. Heumann was one of the very best teachers I ever had. She was tiny and wildly energetic, one of those people who seemed to inhabit her whole body and also several inches of the space beyond. I had thought I'd been thrown into remedial math, but Mrs. Heumann didn't seem to know that. She worked us hard. On the very first day, she took away our pencils and paper. "We're doing math in our heads. Because you need to know that you can." And I could. We all could. And we were good at it. I moved away before the end of that school year and I was very sad to say goodbye to Mrs. Heumann. But amazingly, some of her lessons stuck, especially the one about "you can do it." I still battered my head against the wall sometimes with math, but I kept at it. I was even on my junior high math team for a year. I couldn't believe it.
But after high school calculus, I never took math again. I went to a college without distribution requirements outside the major. The closest I ever came to math again was a microeconomics class that was so bad, I stopped going to class after the first month. The teacher was canned after a single semester and I got the only D on my college transcript. Pretty good, considering I only ever showed up on test day and rarely cracked a book.
But as I see AJ struggling in similar ways, I've been thinking about all this again. AJ has a brain that absorbs higher math concepts readily. He's had a good understanding of complicated issues since preschool. But those things that require memorization or tedious practice often give him enough time to talk himself out of the simple solution and into something more complex and erroneous.
One of the things that can, I think, help students like AJ and like me back then (and maybe me now) are some alternative ways of thinking about the problem. One of the great strengths of the Everyday Mathematics curriculum that AJ's school uses is its support of multiple solving methods. But as the curriculum is actually taught, there are not that many methods endorsed. I've been digging around for other possibilities to help. AJ is very visual and physical, so here are two that have interested us in particular.
1. Finger Math. At one point in my own math struggles, my mother came home from the library with this book, or one very much like it.
I was fascinated. Based on a system used in Korea, Finger Math takes counting on your fingers to a new level by assigning different values to your fingers. The fingers of the right hand are worth 1; the thumb is 5. The fingers of the left hand are worth 10; the thumb is worth 50. This allows you to count to 99 on your fingers. The book also explains how to use fingers for adding, subtracting, multiplying and dividing, although I no longer remember the methods. I need to reacquaint myself with it. Here is a website that explains the same system.
2. Chinese method. Earlier this week, Dedicated Elementary Teacher Overseas posted a video of a Chinese method of addition that fascinated AJ and I. It involves drawing lines to represent the columns of numbers and adding the points of intersection. AJ and I were both fascinated and need to play around with this a little. Here's a video explanation:
Do you have any alternative math methods or tricks you like to use? Fill us in!
Showing posts with label everyday math. Show all posts
Showing posts with label everyday math. Show all posts
Friday, October 16, 2009
Tuesday, July 21, 2009
I Hate Mathematics!
A few days ago, AJ and I made a trip to the library. We have a routine at the library. We conduct any business at the desk (registering for programs, reporting on reading programs, etc.), examine the new books shelf in the children's section, find books to check out on the regular shelves, look for movies if we want one, and then AJ settles down to play on a computer for a few minutes while I head to the adult section to look for books of my own.
When we went a few days ago, AJ and I found a bunch of books for him quickly. He picked up a new book about the moon landing and a book on giant squids. I got him a copy of Marilyn Burns' The I Hate Mathematics! Book. Second grade seems to have caused some problems for AJ's math skills. He was a top notch multiplier when he went in, but the methods confused him and made him focus on the method instead of what he was doing. Consequently, he no longer understands the process of borrowing and makes a lot of mistakes. Consequently, he's getting frustrated and feeling as though he's not good at math anymore.
I had similar problems with math when I was in second grade and The I Hate Mathematics! Book helped me remember that math is fun and challenging and not just like beating your head against a brick wall.
AJ was really excited about the book and sat down at the table to look at it. We were talking about the book when AJ's friend J walked up. J has been in AJ's class for the last two years and they are in Cub Scouts together and they both play a lot of the same sports, although they've never been on the same team.
"Hi, AJ. What's that? 'The I...hate...Math..a..matics...book?' What's 'mathematics'?"
"Hi," AJ said gruffly. "It's just another word for math."
"Oh," said J.
"It a dumb book about math. I don't know what it's doing here." AJ buried it under the giant squid book.
"Cool! Squid!"
I know that AJ has been embarrassed about some of the things he likes, but this is the first time I'd witnessed it so clearly in action. One minute he's telling me about all the cool things he's found in the book. The next he's making fun of it to his friend so his friend doesn't make fun of him. I actually don't think his friend had any intention of making fun of him. I think what sounded to AJ like mockery was really just J struggling to read an unfamiliar word.
This is one of the reasons why I think separate gifted programs are important. It is easier to deal with your sense of difference if you can be different with somebody else. It's easier to appreciate your different interests if there's someone somewhere with whom you can share them.
In the last couple of weeks, AJ has made a friend at camp. He doesn't live very close to us, but AJ and K have been making plans to meet online at Club Penguin, where they can chat and play games remotely. As much as I try to limit AJ's video/computer game times around here, I can also see how things like Club Penguin help him in situations like this. I know the internet helps me find people with like interests to talk to. I'm glad there are places where AJ can go too. Everyone needs to know they aren't alone sometimes.
After we got home from the library that day, AJ pulled out the I Hate Mathematics Book! first and started leafing through it. He found a page of math riddles and started giggling. "Mom, you have to hear this one. Can I ask you some?"
"Hit me. But I bet I'll know the answer."
"You will?"
"This was my favorite book when I was your age."
"It was?" He smiled and started to read.
* * * * *
Marilyn Burns, The I Hate Mathematics! Book (Brown Paper School book)Covello, CA: Yolla Bolly Press, 1975; reprinted by Little, Brown & Co.. The whole Brown Paper series is exceptional, but this is my personal favorite (My Backyard History Book was a close second). Most, if not all, of them are out of print. Marilyn Burns has written other math books for kids, including some more recent picture books. All are worth looking at.
When we went a few days ago, AJ and I found a bunch of books for him quickly. He picked up a new book about the moon landing and a book on giant squids. I got him a copy of Marilyn Burns' The I Hate Mathematics! Book. Second grade seems to have caused some problems for AJ's math skills. He was a top notch multiplier when he went in, but the methods confused him and made him focus on the method instead of what he was doing. Consequently, he no longer understands the process of borrowing and makes a lot of mistakes. Consequently, he's getting frustrated and feeling as though he's not good at math anymore.
I had similar problems with math when I was in second grade and The I Hate Mathematics! Book helped me remember that math is fun and challenging and not just like beating your head against a brick wall.
AJ was really excited about the book and sat down at the table to look at it. We were talking about the book when AJ's friend J walked up. J has been in AJ's class for the last two years and they are in Cub Scouts together and they both play a lot of the same sports, although they've never been on the same team.
"Hi, AJ. What's that? 'The I...hate...Math..a..matics...book?' What's 'mathematics'?"
"Hi," AJ said gruffly. "It's just another word for math."
"Oh," said J.
"It a dumb book about math. I don't know what it's doing here." AJ buried it under the giant squid book.
"Cool! Squid!"
I know that AJ has been embarrassed about some of the things he likes, but this is the first time I'd witnessed it so clearly in action. One minute he's telling me about all the cool things he's found in the book. The next he's making fun of it to his friend so his friend doesn't make fun of him. I actually don't think his friend had any intention of making fun of him. I think what sounded to AJ like mockery was really just J struggling to read an unfamiliar word.
This is one of the reasons why I think separate gifted programs are important. It is easier to deal with your sense of difference if you can be different with somebody else. It's easier to appreciate your different interests if there's someone somewhere with whom you can share them.
In the last couple of weeks, AJ has made a friend at camp. He doesn't live very close to us, but AJ and K have been making plans to meet online at Club Penguin, where they can chat and play games remotely. As much as I try to limit AJ's video/computer game times around here, I can also see how things like Club Penguin help him in situations like this. I know the internet helps me find people with like interests to talk to. I'm glad there are places where AJ can go too. Everyone needs to know they aren't alone sometimes.
After we got home from the library that day, AJ pulled out the I Hate Mathematics Book! first and started leafing through it. He found a page of math riddles and started giggling. "Mom, you have to hear this one. Can I ask you some?"
"Hit me. But I bet I'll know the answer."
"You will?"
"This was my favorite book when I was your age."
"It was?" He smiled and started to read.
* * * * *
Marilyn Burns, The I Hate Mathematics! Book (Brown Paper School book)Covello, CA: Yolla Bolly Press, 1975; reprinted by Little, Brown & Co.. The whole Brown Paper series is exceptional, but this is my personal favorite (My Backyard History Book was a close second). Most, if not all, of them are out of print. Marilyn Burns has written other math books for kids, including some more recent picture books. All are worth looking at.
Friday, February 20, 2009
Midyear Evaluations
This week was midyear parent-teacher conference week at AJ's school. The midyears are by request only, not required. AJ's teacher did not request a conference, but I did. I always do. I can't imagine not taking advantage of a chance to get a better picture of what's going on or to let the teacher know you're paying attention.
My agenda this time was minimal. I wanted to see how some of the organization/behavior issues were going. I wanted to find out what exactly was happening with the school reading groups -- AJ tells us very little and I haven't worked with the reading groups on my volunteer days in a while. But mostly, I wanted to find out what on earth was going on with math.
The news was mostly good -- very good. The behavior issues seem mostly about maturity and the fact that AJ will always take an opportunity to act silly/crazy if someone else is acting that way. This is mostly limited to one other person and they've been separated, so that is going better. We will probably always have organization issues -- AJ's just not a natural. But he's trying. We've been making fewer after school returns to school to find lost homework. He's reportedly keeping his classroom supplies in better order. He has good days and bad days, but there's progress. At home, I see him taking more responsibility for his work. I don't have to remind him quite as much. The things he does daily, he does well. He still struggles with the once-a-week things, since they aren't as securely fixed in his routine.
The reading news was great. I got to see the book he's been working on, one of the "Dear America" series. AJ loves it and it's challenging enough, especially when combined with a series of questions he has to answer at the end which require him to pull out information from the novel and analyze it after he's done reading. This week they'll be starting a Louis Sachar novel that I'm not familiar with called something like "Sideways Tales from the Wayside School." We also got to see a portfolio of his writing, which was wonderful. They are doing a lot more writing than I thought they were -- little comes home, so we didn't realize what they were doing. There were scads of essays, short stories, poems and other types of creative writing.
It turned out, though, that it was good for me to be concerned about math. AJ's teacher showed us a sheet of midyear evaluation scores. Everything was 100% or higher (spelling has extra credit options) except for addition -- that was about 50%. Why was this score so low? Part of it was because AJ didn't read the instructions. When he sees things he thinks are too easy for him, he doesn't bother to make sure he knows what to do. I see this on his homework all the time. But much of it was the opposite problem -- he did read the instructions, but didn't remember that the teacher had told him to do something different. AJ is supposed to multiply most addition worksheets because he doesn't really need the addition practice and he's working on other things. She has reminded him several times, but he doesn't remember. So his score dropped. Now I understand the teacher's point of view here. I know exactly how frustrating AJ can be when you are trying to get him to do something different from the way he's done it before. It take numerous reminders over a fairly long period of time for the behavior to change. This drives me crazy, as I'm sure it drives his teacher crazy. But in this case, I have a little more sympathy with AJ. On the one hand, he's being told to pay more attention to written instructions. On the other, he's hearing that he's not supposed to follow written instructions. I suggested that the multiplication substitution be put in writing, either taped to his desk or, better yet, written on each worksheet.
Meanwhile, we are working with AJ on deciphering word problems. He's got his multiplication and division down, but he doesn't always know how to translate a word problem into a math problem accurately. Yesterday he came home with an assignment to write his own word problems with which he is to try to stump the school principal, who will do one student-created worksheet a week. AJ is very excited about this and I am too. What kid doesn't want to try to be smarter than his teachers and principal? And what a great way to get inside the way word problems work -- instead of yet another work sheet, a creative project. The teacher gets a big gold star for that one.
AJ spent much of the week working on his science fair project with his two friends, which meant I got to spend some time with two of my favorite parents. All three of the kids are definitely performing above average. One of them in ways very similar to AJ. He's in another class. Talking to his mother, I realize how well things really are going this year. Her son is getting minimal reading challenge and only gets challenge spelling when the bring a list in from home, like we did last year. He's getting no extra math at all. The mother of the other boy, who has been in AJ's class for the past two years, agreed that while their teacher this year is not as remarkable as their teacher last year, that she's doing a pretty good job.
The best news I heard at the conference was that when his teacher asks, "Who wants a challenge," he's always the first one with his hand up, shouting, "Me! Me!" This is a big change from last year, where he was often embarrassed about doing something different from everyone else. The difference is that this teacher has integrated challenge assigments seamlessly into her curriculum. In some cases, she offers the challenge assignments to everyone who wants to try them. In part because of class reorganization, AJ gets to work with a partner or in a small group on a number of tasks. The teacher is also using the Everyday Math curriculum in exactly the way I'd hoped she could. Everyday math cycles through a series of topics every year. When they get to a section where AJ is not challenged, she is pulling the assignments from the same topic in the 3rd or 4th grade book. He's still doing the same thing, but at his own level. Consequently, I think AJ is feeling much more a part of the class. This is particularly crucial this year, because he's much more aware of same/different issues than he was last year.
So, there has been definite progress on most fronts since Fall. Good news indeed.
My agenda this time was minimal. I wanted to see how some of the organization/behavior issues were going. I wanted to find out what exactly was happening with the school reading groups -- AJ tells us very little and I haven't worked with the reading groups on my volunteer days in a while. But mostly, I wanted to find out what on earth was going on with math.
The news was mostly good -- very good. The behavior issues seem mostly about maturity and the fact that AJ will always take an opportunity to act silly/crazy if someone else is acting that way. This is mostly limited to one other person and they've been separated, so that is going better. We will probably always have organization issues -- AJ's just not a natural. But he's trying. We've been making fewer after school returns to school to find lost homework. He's reportedly keeping his classroom supplies in better order. He has good days and bad days, but there's progress. At home, I see him taking more responsibility for his work. I don't have to remind him quite as much. The things he does daily, he does well. He still struggles with the once-a-week things, since they aren't as securely fixed in his routine.
The reading news was great. I got to see the book he's been working on, one of the "Dear America" series. AJ loves it and it's challenging enough, especially when combined with a series of questions he has to answer at the end which require him to pull out information from the novel and analyze it after he's done reading. This week they'll be starting a Louis Sachar novel that I'm not familiar with called something like "Sideways Tales from the Wayside School." We also got to see a portfolio of his writing, which was wonderful. They are doing a lot more writing than I thought they were -- little comes home, so we didn't realize what they were doing. There were scads of essays, short stories, poems and other types of creative writing.
It turned out, though, that it was good for me to be concerned about math. AJ's teacher showed us a sheet of midyear evaluation scores. Everything was 100% or higher (spelling has extra credit options) except for addition -- that was about 50%. Why was this score so low? Part of it was because AJ didn't read the instructions. When he sees things he thinks are too easy for him, he doesn't bother to make sure he knows what to do. I see this on his homework all the time. But much of it was the opposite problem -- he did read the instructions, but didn't remember that the teacher had told him to do something different. AJ is supposed to multiply most addition worksheets because he doesn't really need the addition practice and he's working on other things. She has reminded him several times, but he doesn't remember. So his score dropped. Now I understand the teacher's point of view here. I know exactly how frustrating AJ can be when you are trying to get him to do something different from the way he's done it before. It take numerous reminders over a fairly long period of time for the behavior to change. This drives me crazy, as I'm sure it drives his teacher crazy. But in this case, I have a little more sympathy with AJ. On the one hand, he's being told to pay more attention to written instructions. On the other, he's hearing that he's not supposed to follow written instructions. I suggested that the multiplication substitution be put in writing, either taped to his desk or, better yet, written on each worksheet.
Meanwhile, we are working with AJ on deciphering word problems. He's got his multiplication and division down, but he doesn't always know how to translate a word problem into a math problem accurately. Yesterday he came home with an assignment to write his own word problems with which he is to try to stump the school principal, who will do one student-created worksheet a week. AJ is very excited about this and I am too. What kid doesn't want to try to be smarter than his teachers and principal? And what a great way to get inside the way word problems work -- instead of yet another work sheet, a creative project. The teacher gets a big gold star for that one.
AJ spent much of the week working on his science fair project with his two friends, which meant I got to spend some time with two of my favorite parents. All three of the kids are definitely performing above average. One of them in ways very similar to AJ. He's in another class. Talking to his mother, I realize how well things really are going this year. Her son is getting minimal reading challenge and only gets challenge spelling when the bring a list in from home, like we did last year. He's getting no extra math at all. The mother of the other boy, who has been in AJ's class for the past two years, agreed that while their teacher this year is not as remarkable as their teacher last year, that she's doing a pretty good job.
The best news I heard at the conference was that when his teacher asks, "Who wants a challenge," he's always the first one with his hand up, shouting, "Me! Me!" This is a big change from last year, where he was often embarrassed about doing something different from everyone else. The difference is that this teacher has integrated challenge assigments seamlessly into her curriculum. In some cases, she offers the challenge assignments to everyone who wants to try them. In part because of class reorganization, AJ gets to work with a partner or in a small group on a number of tasks. The teacher is also using the Everyday Math curriculum in exactly the way I'd hoped she could. Everyday math cycles through a series of topics every year. When they get to a section where AJ is not challenged, she is pulling the assignments from the same topic in the 3rd or 4th grade book. He's still doing the same thing, but at his own level. Consequently, I think AJ is feeling much more a part of the class. This is particularly crucial this year, because he's much more aware of same/different issues than he was last year.
So, there has been definite progress on most fronts since Fall. Good news indeed.
Tuesday, December 9, 2008
More on Math
AJ and his teacher are continuing to struggle with math. They really just don't understand one another. But now when his teacher doesn't get why he is making mistakes, she has started sending things home to me to go over with him, which allows me to figure out what the problem is and explain it to her, so hopefully communication will continue to improve.
This week, Mrs. F. sent AJ home with a worksheet on a estimating addition, something they've been working on in class. AJ has been struggling with estimation, because he doesn't see the point of the technique, where they round to the nearest 10 and then sort out the ones to get the total. He doesn't like the imprecision of estimation and it takes him longer to do it than it does to add the "normal" way, so he's been assuming he's doing it wrong and keeps coming up with these crazy algorithms that aren't really functional but which explain whichever problem he's working on. Once I explained to AJ that his class was learning a bunch of different ways to add and this was one way, then he was fine. He is a kid who needs to know why he's doing something before he can understand it. I know, because I was a kid like that too.
This morning I was tutoring his reading group. They were looking at an interview between a modern Wampanoag and a pilgrim interpreter from Plymouth talking about the way their respective people did things in the 1620s and creating a Venn diagram based on what they learned (the second grade is very bigg on Venn diagrams). One of the questions asked how many were in their respective settlements and the pilgrim replied "9 score." So we talked about what a score was and I asked if anyone could figure out how much 9 score was. "9x20!" AJ barked out without hesitation. But he was crushed when another kid got the answer before he did. If he can't be first, he doesn't want to participate. While I'm sympathetic to that point of view, I also know how paralyzing such self-expectations can be (talk to me about my decade-old dissertation some time). Still, when such pressure is internal, what's the best way to help ease it?
This week, Mrs. F. sent AJ home with a worksheet on a estimating addition, something they've been working on in class. AJ has been struggling with estimation, because he doesn't see the point of the technique, where they round to the nearest 10 and then sort out the ones to get the total. He doesn't like the imprecision of estimation and it takes him longer to do it than it does to add the "normal" way, so he's been assuming he's doing it wrong and keeps coming up with these crazy algorithms that aren't really functional but which explain whichever problem he's working on. Once I explained to AJ that his class was learning a bunch of different ways to add and this was one way, then he was fine. He is a kid who needs to know why he's doing something before he can understand it. I know, because I was a kid like that too.
This morning I was tutoring his reading group. They were looking at an interview between a modern Wampanoag and a pilgrim interpreter from Plymouth talking about the way their respective people did things in the 1620s and creating a Venn diagram based on what they learned (the second grade is very bigg on Venn diagrams). One of the questions asked how many were in their respective settlements and the pilgrim replied "9 score." So we talked about what a score was and I asked if anyone could figure out how much 9 score was. "9x20!" AJ barked out without hesitation. But he was crushed when another kid got the answer before he did. If he can't be first, he doesn't want to participate. While I'm sympathetic to that point of view, I also know how paralyzing such self-expectations can be (talk to me about my decade-old dissertation some time). Still, when such pressure is internal, what's the best way to help ease it?
Monday, December 1, 2008
Parent-teacher conferences
At our first parent-teacher conference of the year last week, AJ’s teacher Mrs. F handed us a packet of grade 3 and 4 math worksheets that AJ has been assigned to work on when everyone else is doing the regular curriculum. “I’m not sure why he’s having trouble with this,” Mrs. F. said. “It shouldn’t be hard for him. But I find him just sitting there staring at it and not doing it. I even asked him to put a star on the pages he thought were hard and a smiley face on the pages that were easy. But he starred some of the easiest pages.” I told her we’d go over it with him over break and I’d try to get to the bottom of it.
AJ is starting to struggle with the format of school. His teacher this year is much more structured than any he’s had before. I get the impression that he feels like he’s always doing the wrong thing, but I haven’t yet figured out why. His teacher, who is trying to do everything she can to help him, is truly frustrated and perplexed. His test scores are off the charts, but he is having trouble with a number of class activities.
We had trouble working on math over break, a subject that AJ has always dearly loved. But every time we’d sit down to try to look at it, AJ would burst into tears. I have been trying so hard not to let it come to this point. My own love of math was squashed by a clueless (and downright mean) teacher in the second grade. This is what I’ve been afraid of. But eventually, we were able to get past the tears and into the problems. And AJ started to have fun again.
This morning, I sat down and wrote a long email to Mrs. F.:
I think, although I’m not certain, that his teacher is turning him loose with extra work and is not willing or able to spend much time explaining things to him. And I know that AJ is not always willing or able to get up in the middle of class and go ask his teacher what he needs to know. His class is very large and, as generally happens, those who are struggling to work at grade level get more attention than those who are working too far above grade level. But all second graders need help and personal attention, no matter what level they’re at. None of them is independent yet. I didn’t want to come right out and say, “pay more attention to my kid,” because I know she’s doing what she can. But at the same time, she needs and wants to know how to help him. I hope I was diplomatic enough while also being clear.
AJ is starting to struggle with the format of school. His teacher this year is much more structured than any he’s had before. I get the impression that he feels like he’s always doing the wrong thing, but I haven’t yet figured out why. His teacher, who is trying to do everything she can to help him, is truly frustrated and perplexed. His test scores are off the charts, but he is having trouble with a number of class activities.
We had trouble working on math over break, a subject that AJ has always dearly loved. But every time we’d sit down to try to look at it, AJ would burst into tears. I have been trying so hard not to let it come to this point. My own love of math was squashed by a clueless (and downright mean) teacher in the second grade. This is what I’ve been afraid of. But eventually, we were able to get past the tears and into the problems. And AJ started to have fun again.
This morning, I sat down and wrote a long email to Mrs. F.:
AJ and I went over the math packet over break and I tried to get a sense of what had made him star some pages. He also worked on a few pages on his own and we talked them through afterwards. After looking more carefully at the packets, AJ decided that it was all pretty easy for him but mostly not so incredibly easy as to be boring (except for the time pages, at which he rolled his eyes).
I think his stars say more about his difficulty understanding instructions, both those you gave him on starring things, and also the ones on the starred worksheets. He said they are easy now that he knows what they are, but that he didn't know what things like "expanded notation" meant at first. [AJ’s class curriculum is the somewhat controversial Everyday Math program; the packet is drawn from the much more standard Spectrum series; the presentation and some terminology is markedly different.]
He also isn't clear on what the "show your work boxes" are for [each problem has a space on the right margin marked “show your work” -- are they required or are they just there when he needs them? Because he does a lot of the work in his head, if he needs to show his work, someone might need to show him what that means. I did talk to him about how he will at least at some point, need to demonstrate how he figured things out (we talk about this a lot at home, so that shouldn't be a total surprise to him, but he's not used to writing it, and he may balk at it because it slows him down). You'll see how he tried to fill in the "show your work" columns on some of the pages and I think it will give you a good window into how his math brain works. For example, On Lesson 2.3 of the Spectrum Math grade 3 (page 22), the first question gives digits for the various places and he has to figure out what number it spells. The number is 600,903, which he got correctly. In the "show your work" section, he wrote out the number, and then wrote the numbers for each place squished in underneath each digit(100,000, 10,000, etc.). Below that, he wrote "3x3=9" with arrows connecting the 3 and 9 with their twins in 600,903. Then he wrote "3x2=6" and drew arrows between the 3s and 6s. He told me that he thought it was cool that you could make all the digits out of 3, so he decided to show that. Further down the page, where he had to write biggest and smallest numbers made with the digits, he ended up writing the (correct) answers in the "show your work" column and leaving the answer blanks blank. He was so worried about showing his work, that he forgot to write the answers where they were supposed to go.
But AJ also wasn't sure what "hard" meant -- from his perspective, it was too vague. He doesn't always deal well with grey areas. I explained it to him as "hard is something you don't know how to do by yourself and you need someone to show you how to do it." By that definition, the only stars that remained were on the "expanded notation" pages. And once he figured out what that meant, then those stars disappeared as well.
Based on this, I'd like to see him gaining more independence on worksheets like this, being able to carefully read and figure out the instructions for himself. But I also think he may need some spoken words about what to do before each one. It's not so much that he gets it wrong all the time, but that he doesn't trust himself to be getting it right. He seems to expect that he's going to do it incorrectly and wants reassurance.
The other thing it seems like he needs work on, is interpreting word problems. There weren't actually too many examples of that in the packet, but he doesn't trust himself to turn the word problems into equations a lot of the time. He wants constant reassurance. And when the word problems involve subtraction or division, he doesn't always get the order right.
And, perhaps most important, overall, defining things with almost comic precision helps him out a lot. If there is an exception of any kind, he will find it and be confused by it. He hasn't yet learned the psychology of figuring out what the question is asking by what makes sense, not just what is literally stated.
I think, although I’m not certain, that his teacher is turning him loose with extra work and is not willing or able to spend much time explaining things to him. And I know that AJ is not always willing or able to get up in the middle of class and go ask his teacher what he needs to know. His class is very large and, as generally happens, those who are struggling to work at grade level get more attention than those who are working too far above grade level. But all second graders need help and personal attention, no matter what level they’re at. None of them is independent yet. I didn’t want to come right out and say, “pay more attention to my kid,” because I know she’s doing what she can. But at the same time, she needs and wants to know how to help him. I hope I was diplomatic enough while also being clear.
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