Math curriculum for toddlers?

[sonya_post]

Hi again Sonya,

I've copied the email I sent earlier down below to provide a bit more continuity.  You have asked a number of really good questions, I will try to get to them all in some sort of fashion.

You asked whether my books (KTMs) can be used alone to prepare a student.  You will need some source of problems outside of my books - that can be you writing things down or saying them aloud, that can be a website (of which there are a lot these days) that will generate some problems of the type you need, that can be a workbook, or that can be a curriculum.  The KTM's are meant to explain everything and lay things out for someone who is going to be teaching math.  They give all of the little steps and they give suggestions for games or math activities - however, they are not workbooks.  AoPS and I have talked about creating some workbooks to go along with the KTM's, but so far nothing has come of it.  AoPS has some ideas that they are working on developing over the next two to three years to provide materials to younger children, but at this point that is still some distance off.  No doubt some good things will be available at AoPS in a few years when your newborn is a little older, but it would be reckless at this point for me to speculate on what exactly they will have.  To continue with your question...  AoPS currently starts at Algebra I.  There are a few areas that the current KTM's do not cover that someone should see before Algebra 1: negative numbers, decimal numbers, working with square roots and radicals, and a bit more work with variable expressions (combining like terms and such).  I have a half written KTM4 that covers many of those topics, but I am afraid that is not much use to you.  So, there is that gap.  I would not be surprised to see a pre-algebra course from AoPS in the not-so-distant future, but that is pure speculation on my part.

You wrote "I want something my child will enjoy doing."  No matter which program you adopt, you have a lot you can do to make it fun and varied.  If the book has too many boring problems, only assign a small fraction of them.  Sprinkle in some games, puzzles, or activities from my books or other sources.  Introduce them to ancient number systems and have them solve problems using those number systems.  Emphasize understanding over repetition, but sprinkle in 5 minutes of repetition every day.  Ask them to estimate answers to see if they really understand what they are doing and what they are working with.  Ask them to check answers (occasionally) to reinforce the relationships between inverse operations ( add vs subtract and multiply vs divide).  I am puzzled by what you wrote when you said "I am afraid I ruined a perfectly good math student with Saxon." - what happened?  I am guessing math became just an endless sequence of workbook pages that had no sense of fun, challenge, or relevance.  Teaching is so varied, I am reluctant to try to give you anything that might sound like THE ANSWER - for some children, the social aspect of things is crucial, and for them just doing problems together is all it takes to make it fun.  Others like puzzles, others like real world problems, and some just love workbook pages.

I took a quick look over at the MEP stuff.  No, I had not heard of it.  It is a little bit tricky to get a sense of the whole thing from the snapshots I looked at.  I like the wide variety of types of problems that they present and that they make many of the early arithmetic problems into puzzles (perhaps later ones too?).  I also liked the teaching philosophies they mentioned in one of their videos, but I don't know how that plays out in terms of the instructional material they give you.

Well, perhaps that is enough of my going on and on for now.  I hope that some of this has been helpful to you.

best wishes, Chris

[sonya_post]

Dr. Wright (Chris) has agreed to join us here. I thought I'd offer a welcome before he arrives and say thanks. I don't think he wants to feel like a curriculum hawker, but I don't think that should be an issue as there are several curriculum hawkers on here and we actually like it when they show up and talk to us.

Anyway Chris, just hoping you can offer your thoughts on teaching toddlers math  and perhaps the immediate obstacles you think that would be encountered.

[nee1]

Thank you so much for inviting him over, Sonya. I was about sending you a pm to request that you invite him over. I so much wish that he shares his insights with us. I've ordered the Dr Wright book, thanks again to Keri for mentioning them (the preview looked quite good). And Keri, apart from those two books you mentioned, please do you have any recommendations of other excellent books for ideas on teaching ? I'm still awaiting your response.

Sonya, you are a huge fan of AoPs. Can you give some insights why you like them so much? I re-read your insights on aangeles thread. Can you go into much more detail here? Do you think a child on AoPs will have a solid grounding and understanding of the 'why' behind the math? Please elaborate and post as much as you want on this topic. I think you've made me a fan of AoPs too, hence I'm very keen to hear your insights. Thank you so, so much.

Welcome Dr Wright. We're happy to hear your insights.

[Chris Wright]

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

[seastar]

Welcome Chris. It's great to have someone with your expertise and ideas on here. I have just ordered your book and I cannot wait to get started with it. I really like your idea of frequent low-key exposure and I'll definitely be using your idea of the two piles of food - thanks!

[Kerileanne99]

Nee1-
Yikes! I just wrote out a quite long post, opened another window on the IPad to copy a link, and returned to find everything deleted!  Aarggghh! On the other hand, I think I have finally figured out why! Has anyone else noticed that in a similar situation their open post is deleted if another post is added during this time?! I get the red warning, you another post has been added to this thread, you may wish to review before posting message--but the box is BLANK >

Okay, back to business at hand...
I now have quite a collection of this type of. Book as I find they are the perfect SUPPLEMENTS to working with Alex...I will try to give you my two cents on some of my favorites.

I am quite excited to have Dr. Wright's thoughts on teaching toddlers math as I found Kitchen Table Math 1
 extremely useful in getting started.  That being said, if you are looking for an open-and-go book for math
game ideas for toddlers? I wouldn't suggest this one...the biggest benefit for me was that KTM taught ME
how to understand how to approach teaching Alex. Things that I assumed would be intuitive just are not,
when you can better see the logic of their thoughts.  Indeed, I think that incorporating the information
gleaned from the book truly helped facilitate Alexs love of numbers and math. Instead of just thinking she
didn't 'get it', I was able to see it from her perspective better, and to even anticipate some of the seemingly
random answers she would come up with
He does have some truly fantastic ideas for very practical and low-cost activities and specific games as well, and almost all can be modified for toddlers.  I really have to credit his book with making me feel confident to adapt and incorporate my own ideas and games as well.

The Peggy Kaye Games for Math book? Definitely not a book that can be used on its own, and I did have
a laugh at the fact that she thought some math concepts simply beyond a toddlers range until certain ages,
but I REALLY like some of the ideas she has. Ugh, but it is written in a narrative style thatI found actually
slightly annoying after reading KTM.  If I had read it first, it would have bben more beneficial, as the
narrative style seems to be designed to explain how kids think about math..of which KTM does a better job
 of.  If budget is limited, I would give this one a miss.  I think somewhere I typed out a list of my favorite ideas and games taken from it if that would be helpful.

There is another book called Family Math that has a lot of fun game ideas as well, still worth having.
http://www.amazon.com/Family-Math-Jean-Kerr-Stenmark/dp/0912511060/ref=pd_sim_b_2
Really a lot of fun with good ideas as well.

And of course, I still think that Marshmallow Math is the book to start with for toddlers, mostly due to the
style and simplicity.  You could use it as open and go, but as I have previously suggested, we found it much more effective to read ALL of these books in their entirety and then incorporate bits throughout out day.  
We never found a more linear-style to our liking, and would have seriously slowed her down. She Begs for little blocks of math time, we just rarely if ever sit down for a lesson. For us, it is about a two minute discussion here, a quick game of GoFish and determining who has the most/least pairs, that sort of thing!

Of course, we still use RS as our spine with lots of extra resources as fun supplements...

[Chris Wright]

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

[Kerileanne99]

Thanks so much for joining us here Chris.  I can understand your reluctance but members on this particular forum will be incredibly grateful for your input, and not see it as a plug at all! Especially since you have the experience of teaching your OWN children.  There just really is no substitute for that

Book 1 was, as I mentioned, incredibly helpful in getting me started. I am awaiting books 2 and 3, as we are working on lots of things like time, money, and logic...because of the style of the books, I am not really worried that they be geared towards her age group (she is 2 years 9 mo), but rather to help me plan.  Not having seen 2 and 3, I am wondering what your take on this would be.
Thanks again...
Keri

[Chris Wright]

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]

Oh, see, you already fit in here SO well!
The parents on this forum see age requirements and 'suggestions' as a personal challenge!  As in, how can I find a fun, exciting, no-pressure way of teaching this material at a fraction of recommended age!

I am thrilled to hear that they can all be used together! And I think you might well be pleasantly suprised at what many of the toddlers in this forum routinely do
On this forum there are kids doing levels of math at 4 that kids in the US don't touch on until at least middle school, and LOVE every minute of. Alex seems to enjoy math and numbers, but there are several doing so much more!
And yes, we started teaching Alex multiplication through Right Brain methods just after she could add single digits competently.  We read a MathStart book called Doubling the Ducks, and it just seemed natural.  I think it took about 3 days to teach her 0's, 1's, 2's, and 4's facts.
She uses an AL Abacus for much work, tally sticks,counters, and primarily number stamps to do problems to avoid turning it into a writing lesson.
She can count forward and backward as high as she wants, and we have taught place value to the millions and billions although she cannot DO anyhing with them but read them She just likes big numbers and would add another wooden number to the row and read it for her whilst she giggled hysterically. Eventually we bought her a calculator so she could enter them herself.
Skip counting by 2's, 5's, 10's, time to the quarter-hour, some measurement in various units.
She learned volume and liquid measurements in the bathtub with measuring cups and spoons, plastic beakers, graduated cylinders, that sort of thing.
Fractions? Not so much. She can figure out how many pieces are required for a group, and identify what to call the fraction, and can do simple addition and subtraction with it, as well as understanding the relationship to the whole...but no luck for example, understanding that 6/8 is the same as 3/4...
Our biggest success has really come with geometry though. Like many kids in this forum, concepts like polygons, parallel, perpendicular, symmetry, 3d solids, angles, etc., is much easier to introduce when they are tiny...

Anyway, I really mention this as a baseline...there are kids all over the board here, although not a single one that would be considered of average math knowledge by age Some are doing a lot more and some are doing less focus on math.
I would probably say that the work we have dome with Alex is somewhere in the middle...

Not to mention that a majority of tots here can read on their own LONG before 'formal' education age!

I think you can see why a book written by a math prof who doesn't believe in chronological progression and has taught this with his own kids would be highly appreciated here!

[Chris Wright]

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

[nee1]

Hi Chris,
You are most welcome to the forum. Please do not feel constrained, we are a really friendly and open-minded group.  I ordered your books today, thanks to Keri for mentioning them. And please post as much as you want, I want to learn from your expertise.


Keri,
I normally type out my posts in Notepad or Word before pasting them on the forum, to ensure it does not get wiped out. I've had the bad experience of losing a very long post. You may use that option too, as I do not want to lose your insights anymore. Your recommendations mean a lot to me, and I do not want to miss a single sentence of what you wanted to say.

I decided to skip the Games for Math book, in favour of the Dr Wright KTM books 1 - 3. Thanks for your insights. I wanted to skip Family Math too since I've ordered the 3 KTM books. What do you think? I checked amazon for preview of pages of Family Math, but there was none. Thank you for your insights.

[Kerileanne99]

Chris-
That actually does sound like a lot of fun! She really likes to 'play' with numbers, and we play a version of 'eye spy' math in the world. I bet teaching her 'animal math' would be fun! In an effort to excite her with regard to languages, we taught her to count in ten+ languages (only up to ten-twenty) I didn't know if it would interest her in other languages or just end up being a strange 'party trick'.  But it actually worked really well to teach her that the quantity is all that matters, regardless of the name...and because of this we had no trouble teaching her to count 'the math way' for RightStart. (as in ten-1, ten-2, so on...)

For reading, many of her favorite story/picture books are math-related, and we have amassed quite a collection of those as she reads them again and again.
(here is her current favorite: http://www.amazon.com/The-Adventures-Penrose-Mathematical-Cat/dp/1884550142/ref=sr_1_6?ie=UTF8&qid=1347144894&sr=8-6&keywords=The+adventures+of+Primrose)

I am sure that you have seen the Donald in MathMagic Land video, but she would watch it all day if allowed.  Where else can toddlers learn about the Golden Ratio, math in architecture, and octaves, as well as mathematical formulas for playing Billiards?!

Nee1-
How very kind of you! I actually do most of my posting via the IPad, but I am going to have to get in the habit of doing longer posts on the computer
I think you are going to have your hands full for now, with plenty of resources! I would skip the Family Math book for now and spend the money on playing cards, fun board games, and if you don't have it, the first MathTacular video! It is fun and will also give you lots of inspiration.  Not to mention the kiddos love to watch it again and again!

[Mandabplus3]

Hi Chris, welcome to the forum! So great to have you here.
Firstly you lot talk a whole lot while I am sleeping!    

Chris feel free to plug your book here. We invited you to   Probably it would be a good idea for you to pop in a link for your preferred purchase method or we will all use Keri's amazon link.
I would love to read up on the mental strategies children are developing as they learn concepts like one to one correspondence. Is most of that type of discussion in your first book? Or do we get those thoughts and disruptions in all three books? I often stumble across kids who just look so blank in the face when I know they are ready for the next step in the math sequence. If I could pinpoint their thinking at this time I could much more easily " talk" then through it to success.

Age guidelines mean very little to us here. Actually in general I think they are quite off. When I did my
teaching degree I was taught one to one correspondence can't be expected from a child until 6-8 years old! At the time I had a three year old who had it mastered with no help from me, so I questioned a lot of the expectations from then on. This is a great forum for questioning the norm!  
Keri, I have been inspired to teach my kindy kids "time" thanks for that, wouldn't have thought about it had you not said Alex was doing it. But really if they know their number 1-12 what's stopping them? ( oh yes that right ME!   ) I can assure you time is NOT in their curriculum...hmmmm i will creatively link it in
Chris what are your thoughts on memorizing math facts before possibly understanding the concepts behind them? A few parents have asked this on the forum. Te children here are quite capable of memorizing their times tables well before they are able to manipulate objects/numbers to demonstrate that they understand them. Most of the parents decided to teach both and reinforce any weak understandings later. What do you think?
Keri, just post the half finished post and edit it after with the link. It's called cheating! 

[Mandabplus3]

Jaykob LOVES mathtacular! Will watch it from start to finish if I let him! LOVES IT!

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