That's interesting, what are your reasons for it? How far does it go, when studying quadratics do they learn about b^2 + (-4ac) instead of b^2 - 4ac?

What challenges have you faced, what successes have you had, in getting your kids to "show their work"? I have in mind the way the phrase is used in algebra, but maybe the concept applies in other parts of education too. Even very young kids can learn to solve simple algebra problems like in their heads. "We don't know what A is, we have to find out. 10 minus what number equals 9?" But training a kid to write down must be harder. At the very least they need to know how to use lined paper. It requires a lot of writing, with what to a kid must feel like zero payoff. They have to accept criticism if they make a mistake on any one of those lines. And making a mistake in a deduction like this is more consequential than making a spelling mistake in a paragraph, they may have to redo every line that comes after their mistake. I'm getting ready to start "Algebra 1" with my daughter, in a sort of formal way. What has it been like for you?

We did E, then "Corrective Math Multiplication" and "Corrective Math Division" (two other SRA products), and then F. Before starting E, I saw and to some extent used some excerpts from D that impressed me a lot at this website: http://conceptionofthegood.co.uk/?p=569 My good experience with that (I kept of a diary of it somewhere) and with Reasoning and Writing convinced me to buy the program, piecemeal from ebay. Started with Level E mainly because it was the first that came complete in the mail. We detoured through "Corrective Math" because perhaps in retrospect E was not the best place to start with a 6-year-old: she was doing very sophisticated stuff by the end, but very slowly. Through the 200 days we spent in Level E, she was still skip-counting to figure out 6x8 and 4x4: Levels E and F do not include any "facts practice."

@mathmarm explained to me how to collect all the confusing components of R&W here: Confusing and a little expensive to assemble, but they are really wonderful!

"Half as far from F as they are from the line" suggests the equation (distance to F) = (1/2) (distance to line). It looks like you've instead written (1/2)(distance to F) = (distance to line)!

I might be asking you to stick your neck out, but what are some concrete ways this comes up? Difficult things you wish he'd be willing (able?) to do?

Do you know these lines from Nat King Cole? "The greatest thing you'll ever learn / is just to love and be loved in return." Learning how to learn is a distant second. A crucial aspect of learning how to learn is learning how to take instruction; I think that goes for gifted kids too, though I know that many of them will not pick up this important skill. They lack the opportunity. Kids learn all kinds of things the teacher doesn't intend, if the teaching is not constructed very carefully. For instance, many gifted kids learn in school: there is no point listening to the teacher, I'll ace the exam anyway. The mirror image affects ten times as many kids: there is no point listening to the teacher, I'll fail the exam anyway. Both kids may have learned an ugly truth about the teaching they are getting. But the natural generalization: "I am not teachable," is a mistake. What is going wrong at schools, is that the teaching is not kept "in the zone of proximal development" for almost any child. Not for any sustained period of time. (In the States, some of this can be blamed on the fact that "tracking" is very unpopular with politicians and administrators.) At home, we have tremendous flexibility to keep things in that zone. You and I don't disagree about the value of this at all. What about "efficiency?" The word calls up Frank Taylor and IBM. It doesn't call up a beautiful childhood. But it's not childhood that I want to make efficient, it's instruction. I know it sounds preposterous, given your kids' outcomes, that I would say "Oh, but they could have learned so much more with efficient instruction." Not when my teaching experience amounts to my 2nd grader, since kindergarten, whom I'm also not doing perfectly by. But I guess I really believe this preposterous thing: there is a method of teaching, largely undiscovered, that would enrich everybody. It would enrich children like your sons, children like those that you tutor, and everyone else. And, because it is so efficient, this enrichment could be achieved at the cost of very little time. To uncover how to teach this way, requires looking at the details of "what knowledge is," across all the domains that we want our kids to become knowledgable about. For "knowledge about negative numbers," I think I know what those details are. For "knowledge about Botswana" (or "knowledge about how to write a persuasive essay"), I am certainly in the dark. I think you might not be in the dark, even if you wouldn't in a million years describe your process this way.

You're painting a picture of a kid taking responsibility for his own learning, and going about learning calculus in a thorough and sophisticated way. I've known some whiz kids who are impatient to get to advanced stuff, and sometimes skip things they shouldn't. I'm impressed! But I know I couldn't trust my 8-year-old to do with pie charts what your 16-year-old was doing with calculus. By the way I find this really inspiring and interesting. "How do you do it?" is a different question than "is it desirable to do it?" Maybe your 16-year-old would have been grateful for an extremely efficient course on calculus, even if it required less agency on his part? In this fantasy, don't let "extremely efficient" suggest that any less material is covered but only that it is covered with the maximum clarity, parsimony, and speed. I don't know any program like that, that already exists, for calculus. I do know one for what you might call "K-6 math": it's called "Connecting Math Concepts." I am finishing it up with my daughter in the next few weeks, it worked beautifully. But it took its author several decades to create it. Going forward, I'll have to improvise. Does HG+ stand for "highly gifted"? My view is that the problem of teaching gifted kids is not fundamentally different from the problem of teaching any other kid: both kinds of kids have a lot left to learn. If your 16-year-old is struggling to learn algebra, the teacher's job is to find a way to make it easy for him. If your 16-year-old is struggling to learn quantum field theory, the teacher's job is to find a way to make it easy for him. Of course you might think of the second 16-year-old as more charmed, more likely to succeed even if he doesn't get high-quality teaching. But the nature of high-quality teaching isn't different for the second student than for the first student.

There are two or more meanings of "easy," and I intend only one of them: 1. Something is easy if the kid already knows how to do it 2. Something that the kid doesn't know how to do is easy, if the kid will be able to do it expertly after just a few minutes instruction. I intended meaning 2. As to what it means to learn those easy things "better"? Just because, at the expense of a few minutes of instruction and another few minutes of testing, you can know for sure whether he has learned it or not. (An asterisk there: you can tell whether he's learned it, but not whether he'll remember it, another important problem for the tutor.) But if a kid is given a challenging task, and meets it by some stroke of genius or by determined hard work, I am certainly impressed, and I am sure that the kid has learned a lot. But what exactly he's learned is more unpredictable.

Easy becomes boring when it's the same damn thing over and over. But I don't think that anything that's both novel and easy is boring. In fact I noticed something contrary: when my daughter is acting bored, yawning and slumping and zoning out, it is a very reliable signal that she is confused. Remedy the confusion and she suddenly perks up. (If I ask her to do something that she already knows how to do and has done a thousand times, she doesn't react by getting bored but by getting offended). When this clicked for me I became a much more effective tutor for her, and I also realized the same is true of myself, in my grownup life: when I'm reading something difficult on a subject that is new to me, I might find my attention drifting often or I might even find myself getting sleepy. When it happens, it's very often because the passage I was reading confused me. It used some jargon I wasn't familiar with or something like that and I reacted, not consciously, by getting bored.

I like this mindset. If you know where they are at, then perhaps you can tell if the next thing you are considering teaching them will be hard for them to learn, or easy for them to learn. The hard thing for them to learn is a "challenge," a word that might have an unearned positive ring to it in education. But isn't it intuitive that they will learn the easy thing better and faster? When they learn fast they can learn a lot, and we can hope the hard thing becomes easy before long.

It doesn't sound like you're describing a goal, it sounds like you're describing a process. I also have an aversion to "goals", especially long-term goals, in education or in parenting: I'm afraid of getting my hopes up. I would like the time I spend with my daughter to be beautiful and rewarding now, not to put lots of unpleasant pressure on her in pursuit of an exalted goal. I think of myself as executing a process, not pursuing a goal. But part of that process is "spend n minutes per day in math lessons." We've been doing it for close to a thousand days and those minutes add up. How can she get the most out of them? To me, it seems like an urgent question. Does some portion of those minutes have to be devoted to negative numbers, for 5 years in advance of going through my 13 bullet points? My experience is no. (To be fair, I did spend a week or so almost two years ago introducing negative numbers to my 6-year-old with hand-made manipulatives, inspired by @Not_a_Number's "poker chips." It was fun and she remembers something about it. Perhaps in a subtle way this was fundamental for what she learned since. But I was perplexed that, doing as much math as we do, we hardly ever discussed negative numbers in the intervening two years, not until a few weeks ago when Connecting Math Concepts got around to them.) I don't want to see the skills "put to use" in exercises that only keep the concept from sliding too quickly out of memory. I want to see them put to use (asap) in building something up, in teaching new concepts. My somewhat uncertain sense right now is that there's only one major use of negative numbers in this kind of building up: they allow for a much richer source of examples when introducing coordinate systems and graphs of functions. (That's something my daughter wasn't ready for two years ago, but is now.)

Did you make much math progress as an adult, or does this still trouble you?

The phrase that I used, following Jean, was "linear fashion." I wouldn't use "linear progression" because to my ear it suggests that the kids don't forget and don't ever need remedies. By "linear fashion" I don't mean anything cryptic or metaphorical: only that kids learn in a linear fashion because they experience time in a linear fashion. In time, our lives are a straight line — the obvious fact has nonobvious consequences for education, or at least consequences that many educators have a blind spot about. The main consequence for a tutor is this: if there is a set of things you would like your pupil know, you have to decide what order to present them in. My claim is that the order matters a lot. When you are talking about very large concepts, you have few options for how to order them. Should you teach negative numbers before fractions? I don't have a strong opinion. But when you zoom in on what I am calling "microscopic concepts" you have many many more options for how to order them, and among those options are opportunities for great clarity, efficiency, and acceleration. For example, one of the common core standards for kindergarten is that children are to learn to write numbers from 0 to 20, and in grade one they are to learn to write the rest of the two-digit numbers. (Actually the standard 1.NBT.A.1 is to read and write numerals less than 120). These are both microscopic concepts or microscopic skills: I believe they are presented in the wrong order. I believe it is far more expedient to teach kids (in kindergarten or before) first to read and write two-digit numbers that are not in the teens, and then to treat the teens only when they have mastered the larger numbers. It would save a lot of kids a lot of grief. ("Numbers that start with 1 are tricky. We don't say 'ten six' and we don't say 'tenty six' or "onety six." We say 'sixteen.'") To be concrete about something more advanced, here are the microscopic skills and concepts that I think make up "expertise with negative numbers," at the sixth or fifth grade level. This is cribbed from the Teacher's Guide to "Connecting Math Concepts Level F," which I adore. A weakness of this list (unusual for Connecting Math Concepts) is that it does not indicate how signed numbers can be incorporated into word problems. 1. Being able to combine the values in a problem that adds and subtracts more than one value: (Look at 0+26-15+104-2-8, figure out the total added, figure out the total subtracted, then work the simple subtraction problem) 2. Being able to rewrite and work a problem with signed terms in a different order: (Look at the problem 10 - 2 - 3 = ?. Rewrite it in the new order 2, 3, 10. The correct answer is -2 -3 + 10 = ?.) 3. Being able to locate signed numbers on a number line. (Just being able to put their finger on -5 and +7 when you ask them to.) 4. (skipping some more number line skills that are hard to explain in this format) 5. Being able to indicate which of two signed numbers has greater absolute value ("is farther from zero"). 6. Being able to figure out the magnitude of the answer to a column problem with signed numbers. (If the signs are the same, you add the absolute values, if the signs are different, you subtract the absolute values.) 7. Being able to figure out the sign of the answer to a column problem with two signed numbers. (It's the sign of the one that has larger absolute value) 8. Combining skills 6 and 7 9. (If they know the distributive property already) Apply the distributive property to expressions or column problems like (-15+8-2+3) x 5. 10. Apply the distributive property to expressions or column problems that multiply by a negative number, like (-15+8-2+3) x (-5). (This is a nice way to introduce the sign-flipping rule) 11. Being able to work signed number problems of the form (-4) x (-3). 12. Being able to divide by signed values. 13. Being able to work signed number problems of that add and subtract multiply signed numbers, like (-4) - (-3) + 7 +(-16). If a kid has mastered the prerequisites, she can learn one skill a day in a five-minute session, and practice the previous skills she's learned over another five minutes, and thereby obtain "expertise with negative numbers" over the course of a little more than two hours spread out over a little less than three school weeks. (Of course if she doesn't put that expertise to use right away, she is likely to forget... this is part of my objection to the "spiral curriculum" that I might get on a high horse about later.)

Me neither.

My 8-year-old has the same skills and the same gap. In our case, it's because she and many of her friends have hardly any experience of money. Even at the grocery store she watches me pay with a credit card instead of counting out bills and coins. It was different when I was a kid. I taught her to figure the tax and to compute change not "out in the field" but in our math lessons. But you aren't describing anything that's not linear. You have only proved by example that knowing how many cents in a nickel and how many nickels in a dollar are not prerequisite skills for working with decimals, percentages, and tax. And why should know it be ? When you examine those concepts (decimals, percentages...), even in a rather cursory way, you can see that they have nothing to do with how many cents in a nickel.

I don't know what you mean. Everyone experiences life in a linear fashion; that includes their education. When presenting information to kids (or to anyone), you have to choose what order to present it in. Such choices are very consequential: ordering them in the right way can save an enormous amount of time in our brief and linear lives.

For math, I don't agree. Perhaps because I take a microscopic view of what is a "concept." "Negative numbers" are not a microscopic concept. It could take a long time to become fluent with negative numbers. But, "absolute value of a signed number" is a microscopic concept. It takes only a few minutes for a kid to become fluent with it, and you can test them by presenting just a few problems. (Actually I would divide it into two concepts: "absolute value without the notation" and "absolute value with the notation." The problems "without the notation" have the form: which of +4,+8 is farther from zero? which of +1,-8 is farther from zero? etc. The problems "with the notation" have the form: figure out |-4|, |+9|, etc..) My view: the large "negative numbers" concept is made up out of dozens (at most) of microscopic concepts. A kid can be made fluent with negative numbers very efficiently by identifying these constituent concepts and teaching them explicitly one at a time, to mastery. Language isn't logical, math is logical. Because of this, there are a thousand times as many rules or more that have to be mastered to be fluent in a language as to be fluent in math. Infinitely many of the rules contradict each other, like how to pronounce words that end in "ough." It takes a long time to get used to this network of contradicting rules. The rules in math contradict each other extremely rarely, and usually in very anodyne ways. (An example of this very rare anodyne thing, that I watched out for early on tutoring my 5-at-the-time-year-old: sometimes "x" means times and sometimes it stands for a variable). Instead of contradicting each other, they reinforce each other: mastering one concept is an aid to mastering many others. "Micro" suggests "millionth" but I think there are not millions but only a few thousand of these microconcepts that come up in K-8 education, and, especially since they are mutually reinforcing, that they could be taught over the course of just a couple of years to a large number of children. I might even say to most children.

If they give a beautiful and articulate explanation when you ask, then of course you know your work is done. If they give the kind of answer my 8-year-old is more likely to give, it is quite a bit more ambiguous. My experience is that the capacity for understanding comes far in advance of the capacity for explaining; that goes for kids and adults both. Of course I couldn't prove it to you by your method. I like these examples, agree that a kid who answers them correctly and quickly is likely to understand something well. But I think you still have a lot of digging to do if a kid gives a wrong answer, or a slow one. Side comment on the fractions question: I think 9/7 - ? = 1 or 9/7 + ? = 3 would give more information, specifically because many kids carry around the unfortunate idea that "a fraction is something that is less than one."

The pupil needs to learn both algorithms and concepts, but there is an important asymmetry from the point of view of the tutor. Students accumulate misunderstandings about algorithms and concepts all the time. The teacher has to pre-empt these misunderstandings if possible and remedy them when not possible. To do this effectively, the teacher has to be constantly probing what the kid knows or thinks he knows. Now, how can a teacher tell if what a student knows? You can't reason that, just because you have explained it well, or explained it multiple times, that the student has understood. You have to test them. It is easy to test if they know an algorithm: you just have to watch them execute the algorithm on a range of examples. It is hard to test if they understand a concept. To do it well requires an extremely detailed analysis of the concept, attending to details that are hard for adults who've already mastered the concept to see. My sense (or anyway my complaint) about many professionals who write about "conceptual approaches to teaching math" is that they don't address this difficulty at all. They often are very excited to discuss different creative ideas for teaching a concept, and have nothing to say about how to figure out if a kid has learned the concept or not.

Are you referencing any specific "conceptual curriculums" here? What have you been doing with your 4- and 6-year-old? (Whatever you're doing I certainly didn't mean to criticize it, or you, for using the word "conceptual")

I've grown allergic to the phrase. Some reformers complain that math education is too focused on "procedures" and not enough on "concepts." But my experience so far is that no one who makes this critique can do all three of the following: 1. Assemble a long list of concepts that they would like kids to know. 2. Explain how to tell apart the kids who do and the kids who don't know each concept on the list. 3. Describe teaching procedures that will instill knowledge of the concept in the kid. Common core at least did 1, but their list is shabby. There are "common core exams" but they are designed to give feedback to states and schools, not students. I don't regard them as a good answer to 2. And if they had a theory about 3, I think the evidence is that it hasn't worked. Thank you for posting these!

He's very racist in places. There's no sex or swearing. No women characters. No children ever harmed that I can recall. No God in the stories, lots of monsters. Some characters go mad and commit suicide. The big theme through all the stories is that the universe doesn't care about mankind. I don't personally find the stories very scary: they don't stand up to much scrutiny. All that said, I love many of them.

What is this? I'm not sure I've heard it.

What techniques have you used, or mistakes have you made, when pushing your pupils? Couple of thoughts about pushing my daughter, freshly 8. (Perhaps not very specific thoughts about giftedness, a label that I'm cautious to apply to my kid). One is that when I press and nag, it's mainly to work for longer or to not get distracted. I might henpeck her when she's struggling to stay focused, but if I ever notice she's struggling with the material, I like to back off and find something easier to teach her first. Second thought is a trick that I've gotten a lot out of: a kind of heavy-handed reverse psychology or villain act. If I am very sure that something will be easy for her, I will introduce it by telling her the opposite. "These are hard problems, I bet you won't be able to work these," and make a show of being disappointed when she gets them all right: "What, so quickly? Well, I'll get you with these next problems."