Every child is a GAT child waiting to happen. The primary challenge is getting there.
Motivation is always the biggest issue at any age. I've taken on a few challenges lately across the age spectrum. In one case, I'm pondering how to motivate 3rd grade daughters of math teachers. In another case, it's a 6th grade girl with lots of promise (7th grade in 2 months), who's parents are both adept at advanced math (for a living) but who does not want to do math with her parents. Then there is the 12 year old resident of Math House who's inching his way toward surly rebellious teenager.
The easiest time to motivate kids is right around age 5. The reason I say 'the easiest' is that it's nearly impossible, as I will describe below, and the 6 week disaster that parents will go through, normally, is boot camp for parents on what not to do and how to succeed.
The first step in motivation is to diagnose the problem. Is the parent actually trying to teach math? Stop it. Math is a useless boring topic. Parents need to teach their children soft skills, like doing a bit of math work daily, being comfortable begin totally confused, making mistakes, trying again. The kid can teach herself math. Does the parent expect the child to get anything correct? Stop it. Expectations of any kind are counterproductive.
The internal state of affairs with the child are more challenging. Does the child lack working memory at a suitable level? The working memory I'm shooting for is the 99.7% level. Is the child frustrated for lack of soft skills? At age 5, unless you've been reading to your child 1 or 2 hours a night since birth, working memory and soft skills are probably lacking. How much 'thinking endurance' does the child have during the first practice? I'll size up a child in the first 10 minutes and quietly assign measures like 50% or less along different dimensions. For 5 year olds, I'm OK with zeros across the board. It's a starting point and there's plenty of time to catch up. After 10 minutes, I might have exhausted the child and there's no time to do an actual problem. But there's always tomorrow.
I can't imagine any child being the slightest bit motivated while lacking basic problem tackling skills, so step one is going to be a somewhat painful process. It takes about 6 weeks, and during this time it seems like it's all complaining and no learning. Parents of my inaugural Kindergarten Summer Math Camp laugh that it was more about Cheetos and Skittles than math, but their kids did a lot of math on their own that summer. At home with my own kids it was more painful.
If you start late, it's easier because the child is less likely to cry, but it seems much harder because they are better at articulating 'I hate this.'
I never imagined that a 9 year old girl would not be motivated to do math. Boys, maybe, but certainly not girls. This is the age of Test Prep Math, my primary focus after age 5. Girls love it. The best I can get out of boys is 'these problems aren't lame', but that's good enough for me. This is intentional - the word problems incorporate the way girls learn, which is an approach needed for boys if they want to work at the highest levels. Girls will step out of their comfort zones on the quant problems.
The challenge I have with these two 9 year old girls is that I'm too embarrassed to tell their parents about the books. One teaches college math and the other 5th grade math. I'm reluctant to mention that I invented a revolutionary program that doesn't really teach math but is nonetheless designed to put the kid in the 99% in both math and reading comp. Their kids may be in our program, the parents may be super nice, but by virtue of their backgrounds they are my two top rivals. I made a vague promise that I would meet with them the summer after 4th grade. I figure I'll need about 2 months to crack them into shape. When I'm finished, my 9 year old is going to have some formidable rivals. Game on.
When kids are 5, it's usually silliness and goofiness. I'll make puppets, do math crafts, bring in stuffed animals. Working with kids is fun, but 2nd grade math is pretty boring, so I generally do this for myself. Kids at this age learn on their own as they pick up the skill set. It's an explosion of learning, so all I have to do is find things to keep myself amused during the process. It's a waiting game.
Between 2nd and 4th grade, it's usually me against them. I bring insane, convoluted problems some of which don't have an answer (if you've seen Test Prep Math, you've seen problems like this) and watch the children learn to be surprised and to think. Math can wait. By 4th grade, there will be no math they can't tackle on their own. While the rest of the country is learning decimals and long division, my kids are working out the answers to 3 equations in their brains. It's like a GAT transmorphication machine.
7th grade is a different matter. Somewhere during 6th grade, math becomes math. It's the same enjoyable experience usually, but we're actually dealing with math. The end goal is the same set of skills, but it starts with math, ends with math, and there's a lot of math in between. Even worse, the outcome counts, so it's hard as a parent to have a casual zero expectations attitude toward the process, because parents are thinking 'get ahead in math, do it well on tests, get into a great high school or the right high school curriculum, and then college...'
Yesterday my surly almost-teen-ager announced that he was done with math. This is the bane of parents who teach math, the curse of the oldest child (or the youngest, or the middle child, but in my case the oldest), and the shortcoming of kids who have already chosen a career that has a path through liberal arts and not STEM. On the other 1,000 or so occasions that this has happened in the past, we would do math anyway. But I'm thinking 'How does a parent who doesn't live near the Top Academic Coach in the country get their child to do math?' Time for experiments.
I said nothing, and he started his rebellion by reading non-stop for 3 hours to spite me. For the last few weeks, he's been doing this intentionally and I pretended not to notice. I've been quietly building my arsenal of painting supplies for my counter attack.
After his announcement, I announced that my job as a parent is to prepare my child for their chosen career, a job I take very seriously. I will not fail. Math is not the only way to get there. (It is, actually. I was just posturing.) So I'll give him 2 choices. Either he becomes the top math guy in the country, under our normal rule of 'No Math No Video Games', or we retire all screens until 8th grade and he just does chores instead of math. After giving him a list of cleaning chores, I told him he would paint the interior of the house, starting with the back. Then there are 30 windows on the outside and the garage, or, as the English like to say, the gairej.
He was not amused. For the next 4 hours we had a chore competition, me vacuuming and him painting, both with smug, false satisfied facades. He put way too much paint on the brush and then glanced my way to ensure that I saw there were no drops anywhere. The kitchen trim looks great. It's an old building and there is lots of trim.
At the end of the afternoon, he asked me if he could play video games. 'No', I said, 'No math, no video games. That's the rule in Math House.' But what about all of this painting? he asked. 'Instead of video games and math, from now on you're doing chores all day. That's our deal. I'll give you a few weeks to make your final decision.'
I'm considering what it will be like with a kid who only does chores and no math leading up to the critical year of all A's and 99% on two tests in order to get into high school. It's unprecedented, but it just might work. I think he's all ready prepared. We've been at this non-stop since the summer after 4th grade, which is probably why he needs a break. It's also why I'm interested in next year's 4th graders and next year's 7th graders. I think I have room next year to work on the Chicago project and deal with these other people's problems.
My younger brother came to me and asked 'What is my math?' Your math, I told him, is to get your brother to do his math.
That failed. We walked to a block party so I could track down some of the parents of other 6 graders so I could see what they are up to. 'Your math', I said, 'is to find out where they are serving beer while I figure out what I'm going to do next'.
Sunday, July 23, 2017
Saturday, July 15, 2017
Crushing Math at Grade Level + 2
I have been the proponent of teaching math at a level 2 years ahead your child's grade. There's a way to do this correctly and produce amazing benefits and there's a way to totally screw up the learning process.
Here's a nice article explaining the difference. The article is kind of long and doesn't explain how you actually pull it off, and it's written by a PhD from Ukraine who assumes that our kids are as smart as her kids, which is not even remotely true. Nonetheless, I'm going to have to buy her book and report back here what I've found.
I came up with the idea of Grade + 2 in an act of desperation. We just finished taking the COGAT, and I had another child who needed to study for the COGAT and the OLSAT, and at the time I had no test prep of any type, not even practice books, so I bought Every Day Math Grade 2 and gave it to my child. I speculate that it helped a little with the COGAT, but it helped a lot everywhere else.
The first requirement to go ahead in math is a big working memory. Every single math problem worth doing (as well as all problems of any other type like reading or science) has multiple parts and either new concepts or new techniques. All this newness while trying to solve anything requires a place to store it during the solving, and that's what working memory is. Reading and Read To are good working memory builders. Test Prep Math is a working memory Special Forces Seal Team Six Boot Camp of working memory for older kids. If your child doesn't have working memory yet, each problem goes really slowly and takes multiple re-starts and multiple attempts.
Crying is usually a sign that working memory is still a work in progress. Yelling is a sign that working memory is there but your child is thinking 'why can't I just be average like everyone else'.
The next thing you need to know is that every math problem is really multiple math problems rolled into one. You don't realize this when your child does grade level math because they've learned to do all the steps in one and solve it like a single math problem. With Grade + 2, it's your job to help the child break it down.
The last thing you need to know is that this problem is too hard because your child is not yet adept at 23 + 19 or 12 x 12 or 4.24/15 or whatever but this is part of the problem so you have to figure it out on the spot. The way to do this is to start with the simplest possible problem, like 2 + 2 or 2 x 2, and work your way up from there. In the example of 12 x 12, 2 x 2 takes a few hops to 10 x 10 on the way to 12 x 12, which is really 10 x 10 + 2 x 10 + 10 x 2 + 2 x 2 and requires 4 intermediate steps and a final step.
As I write this, my 8 year old is yelling at me as he tries to solve 15 - [15 - (-5)2]/(-2), which he has no skills to solve, then he looked at my blog and stormed out of the room.
With 12 x 12 and a 6 year old, and it takes us 25 minutes, but the kid barely understood addition to begin with. With problems of this type, the 25 minutes eventually shrinks down to about 5 minutes with a bit of repetition. I chose EDM because there isn't a lot of repetition. Who needs a kid who is good at 12 x 12? That's what calculators are for.
In the end, will your child be good in math at Grade Level + 2? Who cares? The child just went through 6 or 9 months of a) break it down into smaller chunks, b) solve one chunk at a time, c) find a simpler problem and work back from there, d) read it again, e) solve it again, f) check the answer because you've gotten the last 53 out of 53 wrong on the first try. This is not the makings of a calculator. This is the makings of a future 12 year old who's ready for graduate school. This is also 50% or more of the battle on a cognitive skills test.
The last thing to know is that you are not a Ukrainian PhD in math, so your 5 year old isn't going to learn calculus. Even more importantly, your experience on a daily basis is going to look like the following conversation. I asked my kid to do a Grade + 5 problem that had 3 years of unknown material to conquer. The only difference between now and age 5 is that he yells instead of cries and didn't roll up in a ball under the table when he had enough.
Me: 9.9 is about 11% of what number?
Kid: I don't know what decimals or percents are and barely know algebra.
Me: Do it anyway.
Kid: Waaaaa!
Me: Ok, 11% of a number means 11/100 of that number.
Kid: ?
Me: Give me an easier problem than this to do first.
Kid: ?
Me: Ok, 10 is about 10% of what number. 50 is 50% of what number? 20 is 10% of what number? 50% of 100 is ? 25% of 150 is ? ...and on and on and on. We spend about 15 minutes on this.
Me: State this problem as an algebraic equation.
Kid: ?
Me: OK, 'what number' means x. Write it down.
Kid: 9.9x = ?
Me: Why are we doing this?
Kid: You tell me.
Me: No, you tell me.
Kid: No, you tell me.
Me: No, you tell me.
Kid: (eventually on the 9th try with my help) 11/100 x = 9.9. Now what?
I could see after about a week of this that we started to pick up the concept of percent and some insight into decimals. I expect him to flunk the chapter test. Starting with EDM Grade 2, we have flunked our way up to 8th grade math, in the process to flunking our way into lots of other great academic skills and a consistently strong academic performance.
What I get with Grade + 2 is a lot of thinking practice every day. With grade level math, a child is practicing how not to think. Think about that.
Here's a nice article explaining the difference. The article is kind of long and doesn't explain how you actually pull it off, and it's written by a PhD from Ukraine who assumes that our kids are as smart as her kids, which is not even remotely true. Nonetheless, I'm going to have to buy her book and report back here what I've found.
I came up with the idea of Grade + 2 in an act of desperation. We just finished taking the COGAT, and I had another child who needed to study for the COGAT and the OLSAT, and at the time I had no test prep of any type, not even practice books, so I bought Every Day Math Grade 2 and gave it to my child. I speculate that it helped a little with the COGAT, but it helped a lot everywhere else.
The first requirement to go ahead in math is a big working memory. Every single math problem worth doing (as well as all problems of any other type like reading or science) has multiple parts and either new concepts or new techniques. All this newness while trying to solve anything requires a place to store it during the solving, and that's what working memory is. Reading and Read To are good working memory builders. Test Prep Math is a working memory Special Forces Seal Team Six Boot Camp of working memory for older kids. If your child doesn't have working memory yet, each problem goes really slowly and takes multiple re-starts and multiple attempts.
Crying is usually a sign that working memory is still a work in progress. Yelling is a sign that working memory is there but your child is thinking 'why can't I just be average like everyone else'.
The next thing you need to know is that every math problem is really multiple math problems rolled into one. You don't realize this when your child does grade level math because they've learned to do all the steps in one and solve it like a single math problem. With Grade + 2, it's your job to help the child break it down.
The last thing you need to know is that this problem is too hard because your child is not yet adept at 23 + 19 or 12 x 12 or 4.24/15 or whatever but this is part of the problem so you have to figure it out on the spot. The way to do this is to start with the simplest possible problem, like 2 + 2 or 2 x 2, and work your way up from there. In the example of 12 x 12, 2 x 2 takes a few hops to 10 x 10 on the way to 12 x 12, which is really 10 x 10 + 2 x 10 + 10 x 2 + 2 x 2 and requires 4 intermediate steps and a final step.
As I write this, my 8 year old is yelling at me as he tries to solve 15 - [15 - (-5)2]/(-2), which he has no skills to solve, then he looked at my blog and stormed out of the room.
With 12 x 12 and a 6 year old, and it takes us 25 minutes, but the kid barely understood addition to begin with. With problems of this type, the 25 minutes eventually shrinks down to about 5 minutes with a bit of repetition. I chose EDM because there isn't a lot of repetition. Who needs a kid who is good at 12 x 12? That's what calculators are for.
In the end, will your child be good in math at Grade Level + 2? Who cares? The child just went through 6 or 9 months of a) break it down into smaller chunks, b) solve one chunk at a time, c) find a simpler problem and work back from there, d) read it again, e) solve it again, f) check the answer because you've gotten the last 53 out of 53 wrong on the first try. This is not the makings of a calculator. This is the makings of a future 12 year old who's ready for graduate school. This is also 50% or more of the battle on a cognitive skills test.
The last thing to know is that you are not a Ukrainian PhD in math, so your 5 year old isn't going to learn calculus. Even more importantly, your experience on a daily basis is going to look like the following conversation. I asked my kid to do a Grade + 5 problem that had 3 years of unknown material to conquer. The only difference between now and age 5 is that he yells instead of cries and didn't roll up in a ball under the table when he had enough.
Me: 9.9 is about 11% of what number?
Kid: I don't know what decimals or percents are and barely know algebra.
Me: Do it anyway.
Kid: Waaaaa!
Me: Ok, 11% of a number means 11/100 of that number.
Kid: ?
Me: Give me an easier problem than this to do first.
Kid: ?
Me: Ok, 10 is about 10% of what number. 50 is 50% of what number? 20 is 10% of what number? 50% of 100 is ? 25% of 150 is ? ...and on and on and on. We spend about 15 minutes on this.
Me: State this problem as an algebraic equation.
Kid: ?
Me: OK, 'what number' means x. Write it down.
Kid: 9.9x = ?
Me: Why are we doing this?
Kid: You tell me.
Me: No, you tell me.
Kid: No, you tell me.
Me: No, you tell me.
Kid: (eventually on the 9th try with my help) 11/100 x = 9.9. Now what?
I could see after about a week of this that we started to pick up the concept of percent and some insight into decimals. I expect him to flunk the chapter test. Starting with EDM Grade 2, we have flunked our way up to 8th grade math, in the process to flunking our way into lots of other great academic skills and a consistently strong academic performance.
What I get with Grade + 2 is a lot of thinking practice every day. With grade level math, a child is practicing how not to think. Think about that.
Wednesday, July 12, 2017
Back To A Few Problem A Day
Buried in our coat closet is about 3,000 pages of practice test questions that I amassed when I thought this was the path to success. Occasionally I'll meet a kid who is lacking a particular visual skill that the rest of us take for granted, and I'll sift through the stacks until I find a mini-curriculum that address the problem. Mostly, it just sits there gathering dust.
I also have about the same amount of reading comprehension and 10 times as much math material. I'm generally OK with what's available in the reading department, but math curriculum in the US is a total disaster so I've spent more of my personal time focusing on this area, not to mention the fact that I do a lot of math research on my own. I see no path between what I'm doing and the standard math curriculum taught in school beginning at age 5.
My new personal goal is that my kids never get below 99% on a test ever. I don't mean school tests. I'm OK with the occasional 50% on a math test at school because my child is in 4th grade, for example, and we're doing something completely different at home, and my child simply doesn't bother to finish the test at school because it's totally boring. I need that 50% score some where before 6th grade to teach a life lesson, but after that, the bar goes back where it belongs. If you would have been in our house a few years ago, what with me trying to figure out what to do, and the kids demonstrating their lack of skill, you might have wondered how we got to this point.
Each day, my kids do their daily math on their own, and last night it didn't go well. Too many problems, too much stalling and complaining, not enough progress. Today at work I'm sitting here with all electronic devices that I can put in a bag, along with the cords to the bigger devices.
Last night, I instituted a second rule for daily math. The first rule is 'No computer games without math'. Subrule 1.b is 'If I have to nag you, I'm bringing devices and cords to work.' The new rule is that I want them to do any problem that they can find a trick or clever solution strategy and skip the ones that are routine application of known formulas or calculations. They are going to be 'graded' on not only finding a clever solution strategy but also properly classifying the question types. When the children were younger, I would to steer each child toward thinking work and away from useless repetition by choosing each problem or book. But now I just hand them a book and go to work so it's more challenging on all of us.
When I get home, we'll sit down together and discuss the problems. I'm looking for something interesting in the problem. How can this problem be done by cheating instead of calculating? What confusion or trick is the question writer applying to make you get the answer wrong?
All of this is built into Test Prep Math, but now my youngest has stepped beyond this curriculum and we are taking on real math. It's a short step from Test Prep Math to pre-algebra, function theory, and then on to algebra. (I don't consider multiplication, decimals, and long division to be real math.) Right now my soon to be forth grader is plowing through an 8th grade math book and dabbling in algebra.
When Test Prep Math was being written, I was working through the next 6 years of curriculum with the older child and asking the question, 'What skills does a child need so he can tackle the next 6 years of math when he puts down Test Prep Math?' I was hoping for the skills that would allow the child to do it on his own.
It turns out that none of these skills show up in school until much later. It's also the same approach that I devised when beating the COGAT. Teach a child to spend a long time thinking through a single problem, and he can tackle a workbook. Give him a lot of problems in a workbook, and you will likely end up with a calculator and not a thinker. It's not like Test Prep Math is like taking a year off of high grades. We got to 99% pretty quickly, maybe in the first 2 or 3 months. But the 12 year old and I were getting bogged down on the SAT last night and I'm not seeing progress. We need to get back to the thing that got us here in first place, which is fewer problems and more thinking.
I also have about the same amount of reading comprehension and 10 times as much math material. I'm generally OK with what's available in the reading department, but math curriculum in the US is a total disaster so I've spent more of my personal time focusing on this area, not to mention the fact that I do a lot of math research on my own. I see no path between what I'm doing and the standard math curriculum taught in school beginning at age 5.
My new personal goal is that my kids never get below 99% on a test ever. I don't mean school tests. I'm OK with the occasional 50% on a math test at school because my child is in 4th grade, for example, and we're doing something completely different at home, and my child simply doesn't bother to finish the test at school because it's totally boring. I need that 50% score some where before 6th grade to teach a life lesson, but after that, the bar goes back where it belongs. If you would have been in our house a few years ago, what with me trying to figure out what to do, and the kids demonstrating their lack of skill, you might have wondered how we got to this point.
Each day, my kids do their daily math on their own, and last night it didn't go well. Too many problems, too much stalling and complaining, not enough progress. Today at work I'm sitting here with all electronic devices that I can put in a bag, along with the cords to the bigger devices.
Last night, I instituted a second rule for daily math. The first rule is 'No computer games without math'. Subrule 1.b is 'If I have to nag you, I'm bringing devices and cords to work.' The new rule is that I want them to do any problem that they can find a trick or clever solution strategy and skip the ones that are routine application of known formulas or calculations. They are going to be 'graded' on not only finding a clever solution strategy but also properly classifying the question types. When the children were younger, I would to steer each child toward thinking work and away from useless repetition by choosing each problem or book. But now I just hand them a book and go to work so it's more challenging on all of us.
When I get home, we'll sit down together and discuss the problems. I'm looking for something interesting in the problem. How can this problem be done by cheating instead of calculating? What confusion or trick is the question writer applying to make you get the answer wrong?
All of this is built into Test Prep Math, but now my youngest has stepped beyond this curriculum and we are taking on real math. It's a short step from Test Prep Math to pre-algebra, function theory, and then on to algebra. (I don't consider multiplication, decimals, and long division to be real math.) Right now my soon to be forth grader is plowing through an 8th grade math book and dabbling in algebra.
When Test Prep Math was being written, I was working through the next 6 years of curriculum with the older child and asking the question, 'What skills does a child need so he can tackle the next 6 years of math when he puts down Test Prep Math?' I was hoping for the skills that would allow the child to do it on his own.
It turns out that none of these skills show up in school until much later. It's also the same approach that I devised when beating the COGAT. Teach a child to spend a long time thinking through a single problem, and he can tackle a workbook. Give him a lot of problems in a workbook, and you will likely end up with a calculator and not a thinker. It's not like Test Prep Math is like taking a year off of high grades. We got to 99% pretty quickly, maybe in the first 2 or 3 months. But the 12 year old and I were getting bogged down on the SAT last night and I'm not seeing progress. We need to get back to the thing that got us here in first place, which is fewer problems and more thinking.
Wednesday, July 5, 2017
Whole Language Math
Let's start this article with Whole Language Everything and then I'll get back to Whole Language Math.
Whole Language Everything began for me when I noted that 75% of every test score is predicted by the level of vocabulary of the child. Other researchers noted that the child's level of intelligence depends on how many words are used at home and how long the sentences are.
Suppose you have 2 fairly average children at home. If you have a college degree, these kids are probably slightly above average. You're looking at test in the fall for gifted and talented. Take the COGAT for example. It has 3 sections directly related to vocabulary, 3 sections that are quantitative, and 3 sections that are non-verbal shape related.
Here's the strategy that I came up with. We were already doing daily math, so our daily math is going to look a lot like the COGAT instead of something you'd find in a school math text book. This takes care of 6 of the COGAT sections. For these 6 sections, I prescribe covering any vocabulary you can find up to 2 or 3 years beyond grade level. A 4 year old is not going to have much luck with the concept of division or fractions, but this child will know the difference between a rectangular prism and a cube. For age 4, I packaged all of this in Shape Size Color Count, and for age 5 and beyond, the vocabulary is all there in whatever material you use if you just take the time to see it and name it.
The other thing I did was Vocabulary Workshop and the Word Board. We covered Test Prep Phonics, that oddity of phonics and vocabulary, and didn't bother to practice the 3 non-verbal vocabulary sections on the COGAT at all other than to learn the rules with a practice test. My strategy was to crush the 6 sections that vocabulary provides a definite competitive advantage, and the other 3 will take care of themselves.
Our daily math continued way beyond test prep, and it was mainly 2 or 3 hard problems that we spent a lot of time talking through. My kids never practice routine math problem outside of school. Instead, we tackle a new problem that is nearly incomprehensible to start with and talk our way through it. After a few weeks, some of it sticks, some of it doesn't, but it will eventually. I need to start drawing more pictures to look at things visually, but mainly we do a lot of talking.
I recently revised the article "How To Create A Mathematical Genius", which you'll find in the links on the top right of this page. I think this article is almost finished, but I don't cover Whole Language Math. We are currently talking our way through algebra and calculus. Daily math at the pace of 2 or 3 problems a day (which we only do on Saturday's during the school year) just keeps chugging along to a ridiculously advanced level. Every problem involves a discussion of new concepts.
Yesterday for 4th of July Math, I asked my 12 year old to calculate the derivatives of sin(x) and cos(x). I was hoping this would take 3 weeks, but it only took 30 minutes. You'd think a kid who doesn't really understand sin(x) and doesn't understand infinite series and factorials would struggle, but apparently 'struggle' doesn't preclude 'solve'. I never thought we'd end up here.
We spent a lot of time talking through this problem. I have plenty of work to give these kids that they can do on their own that doesn't involve a discussion. But I was struck by the fact our discussion was exactly like the discussions we had at age 4 doing 2 or 3 problems that involved shape transformations.
Many parents define math as a child applying calculation methods to a doable problem and not making mistakes. Under this definition, math requires teaching the child calculation methods so that they can do the next set of problems. With this approach, the child can do a lot of problems correctly and perhaps get A's. I don't want a child who can do a lot of problems correctly and perhaps get A's. I want a child who can figure out next year's math on their own and perhaps go to Stanford or MIT. I want a child who writes really good papers for AP English or AP History some day because they can think at a really high level and has really great verbal skills. If one of my children comes to me in the future and announces 'I want to study Number Theory' then we're going to stop doing Whole Language Math, but until then, math is more talking and less calculating.
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