Keri - I have always thought that finding math books that can be read as stories would be a particularly good way for the parents who may not be comfortable with math to spend some time with their children enjoying math together. I looked briefly at the snippets that Amazon has of the Adventures of Penrose the Mathematical Cat. The chapter on binary numbers was amusingly introduced, but it seemed awfully brief for someone (child or parent) seeing it for the first time. Do you find you need to augment the topics a lot, or is there more background in the book in places that I didn't happen to see? I would love to see more titles from your favorite math story reading list.

Mandabplus3 - Thanks for the welcome. Thanks for considering buying my books - I make about the same amount of money pretty much any way you buy them - they are available from Amazon, AoPS, or from Rainbow Resource, or from me. I only handle bulk orders (I'm not really set up for handling orders for the most part). Rainbow Resource has a considerably better price on my books than anyone else (not sure how their business model works). My daughter, also my webmistress, makes a little bit of money if you buy the books through my web site (www.drwrightskitchentablemath.com), but it's no big deal.

I'm afraid I don't have much more to say about one-to-one counting than what I wrote in section 1.1 of my first book. In that section, which I think is excerpted on Amazon, I mention that learning to count involves at least five ideas - 1-to-1 (that numbers can be associated with things in a 1-1 fashion), the numbers are ordered, the last number when counting is how many things there are, counting is universal (it doesn't matter which objects you are counting and it doesn't matter if the collection is a mixture of things), and it doesn't matter what order you count things in. I found these ideas being discussed in a math education book and tried to distill them in my book. To answer your question - I would say almost all of the discussions in my books are higher level than that particular discussion - I was talking about these ideas at the start of book 1 because I wanted to make sure that parents understood how much a child had to learn, and I also wanted to emphasize to them that they had to wait for their child to get to the right point in their development - that these particular topics mostly could not be hurried (though obviously by doing things like counting in front of your child you are providing them with experiences which will fuel their development). In other places in the books I do talk about what makes certain topics tricky or what some of the stumbling blocks are, but those discussions aren't as oriented toward development steps as that discussion in 1.1. By the way, I am a bit astonished that anyone would suggest that the idea of 1-to-1 correspondence wouldn't usually be mastered before age 6. However, people have also said that Algebra shouldn't be done before sometime in Middle School because children are not able to think in terms of abstract quantities before then - clearly that is a rule that doesn't hold for every child.

As for memorizing facts... What you wrote "Most of the parents decided to teach both and reinforce any weak understandings later." sounds okay to me, though I suppose it depends on just how much the children are simply memorizing something that is meaningless to them. The danger of memorizing alone is obviously that the child will have no idea what they are talking about (as in, no feeling for the results) and also that it could become a grind (and therefore a turn-off concerning math). If both of those pitfalls are avoided, then I don't think there is a problem. In KTM1 I try to take a very structured approach to learning the math facts, and I think that method has two advantages in general - it helps the child see math relationships and thereby gain an appreciation that cool things are going on with the numbers, and it makes learning the math facts more of a puzzle and less of a straight drill. To take an easy example, in teaching multiplying by 5's one can push the importance of knowing how to multiply by 10 (and its relationship to place value) and knowing how to take halves of things - not only might that make learning the 5's easier, but by making a task easier it inspires the child to consider the relationship of 5, 10, and half as something useful (rather than as just another random piece of information).

- Chris

What fun. That is great that so many of these children are getting to do what is appropriate for them rather than what is generally expected of their age. And yes, I am pleasantly surprised by all of the cool mathematical things you describe that your child can do. It does sound like the ideas in KTM2 & 3 are about right to start looking at.

One fun thing that you might look at, that emphasizes fresh perspectives on place value, is to look at number bases. I call that Animal Math in KTM2. By representing numbers in bases other than 10, and doing arithmetic in those bases, it allows a child to see that the regrouping isn't some automatic thing that always involves 10, that it is dependent on the base you are using. And of course, that flexibility with regrouping also pays off when doing addition and subtraction with mixed numbers and with different kinds of non-metric measurements. One thing that makes it fun is to think of it as the arithmetic that is done by animals with different numbers of "digits" - people use base 10 because we have 10 fingers, but a starfish would want to use base 5 - that sort of thing.

I also introduced primes, powers, and prime factorizations very early because they are so fundamental to understanding how multiplication and division work. There are, of course, many things you can be introducing, but those are some of my favorites (though it was a bit awkward when my children were in kindergarten and wanted to practice their arithmetic in other bases with their teacher who had no idea how to do it).  - Chris

I think you have the right attitude about not worrying about your child's age. It makes so much more sense to see what they seem to be ready for and try it out. I was a little sad to have to organize my books so much by topic - it makes it seem as though one should master all of the material in one chapter before moving on to the next. However, my earlier organization, which was organized more by the order I did things with my own children, was way too confusing for my early parent volunteers to read. Similarly, not all of book 1 needs to be mastered before doing books 2 and 3 (which were initially intended to be a single book before the folks at AoPS convinced me that it would have been an absurdly large book). Ages were assigned to my books just for marketing purposes - I don't believe in those age ranges, but something had to be put there so that shoppers would have a simple way of guessing whether the content was appropriate (I have had a few amusing conversations with parents on both sides of the: KTM1 should not be for children of ages 2-8, I have an 8 year-old and KTM1 is much too easy/hard for my child).

All that said, it would be quite surprising if a three year-old was ready for the material in KTM 2 or 3. Place value is the main doorway concept to KTM2 - when your child is ready to start thinking about adding two two-digit numbers, that will be the time for KTM2. I suppose you could look at the "counting higher than 100" before then, and you could do the more complex fraction work also - both of which are in KTM2. Even though the topics in KTM3 are generally less arithmetic, they still involve more sophisticated ideas than the typical very young child is ready for. All of which reminds me of a incidental conversation I had one day with a local elementary school teacher in a coffee shop - I was asking why they believed in holding off doing multiplication until after doing multi-digit addition and subtraction. I only mention this in that the order that things "should" be done in is seen very differently by different people.

- Chris

Kerileanne99 - Thanks for the good thoughts about my books. Their usefulness to you was what I was aiming at, so I am glad that it has worked for you.

Oh, by the way, please call me Chris. When AoPS and I were casting about for a title for the books, my wife came up with KTM, but AoPS suggested modifying it to Dr. Wright's KTM for two reasons - they thought it would make the title of the book more personal (which seemed right to all of us) and they are friends of mine and knew how much I dislike using a title with my name (a little friendly ribbing on their part).

Speaking of AoPS, I am also a huge fans of theirs. Historically, their materials have been aimed at the highly motivated and high achieving math students, and they have great resources for that crowd. Everyone over at AoPS is super good at math, loves working with children, and loves thinking about math education - they are very nice people. In addition to being the original publisher for my first book and generously putting a lot of excellent editing into all three of my books, they have recently come out with the start of their Beast Academy materials for younger children. I have looked those over, but I have not used them with a child so I don't know how well they actually work in practice. Their Beast Academy is their effort to be part of math education for children younger than middle school.

And yes, as Sonya said, I don't want to come off as someone promoting my books here or promoting AoPS either. In addition to being a bit of an introvert, I often don't comment on threads because I am concerned about seeming like I am trying to sell books rather than simply commenting about math education ideas.

- Chris

Hi Sonya and friends,

Thanks for asking for my thoughts. In reading the last few pages of this discussion it sounds like folks are going in some excellent directions. I was not familiar with Marshmallow Math, so it was very interesting to me to go take a quick glance at what can be seen on Amazon. It looks like it came out around the time of my first book and has some similar ideas in its approach. I'll probably order a copy one of these days to get a better idea exactly what is in the book.

To state the obvious that you seem to all already know, the key thing with little ones is frequent, low-key exposure. Just counting out loud every time you are in a situation that can be counted is a great way to get things going. As your child starts getting familiar with the counting, vary it by counting backwards sometimes or counting by 2's sometimes. Ask your child to compare the sizes of two small numbers of things so they are thinking about quantities - for example, have two small piles of something they like to eat and ask them which pile they would rather have. Simple things like that. All of which you are probably already doing.

Anyway, I am not entirely sure what I should be commenting on, so perhaps I'll stop there and see where this leads. Best wishes - Chris