Math Woes (and some reading/language discussion beginning on p.4)

[PeterPan]

48 minutes ago, Cake and Pi said:

Maybe! I have place value chips. Each place value has its own color, and on one side the units all have "1" the tens all have "10" etc. He understands trading. I've played games with him where he can trade me ten yellow 1 chips for a green 10 chip, and he gets that and thinks its hilarious for some reason. But I haven't gotten him to connect that back to physical representations of numbers, like base-ten blocks, the abacus, or c-rods, or to written (or place-value card) two-digit numbers. 

I have a Lakeshore Learning addition kit we did where you start with trays for each of the place value coins (1s, 10s, 100s) and work through a series of cards with 2 digit then 3 digit addition. Maybe it went up to 4? I don't remember. We worked through it *very slowly* like maybe 2 cards a day, and with time were finally doing "superman" man, hehe, where we would combine them.

That's really great that he enjoys doing the trades!! That means you're hitting him where he is, at that point where it's challenging and engaging. Good stuff!!

https://www.lakeshorelearning.com/products/math/place-value/hands-on-addition-regrouping-kit/p/FF297

[Not_a_Number]

1 minute ago, PeterPan said:

I have a Lakeshore Learning addition kit we did where you start with trays for each of the place value coins (1s, 10s, 100s) and work through a series of cards with 2 digit then 3 digit addition. Maybe it went up to 4? I don't remember. We worked through it *very slowly* like maybe 2 cards a day, and with time were finally doing "superman" man, hehe, where we would combine them.

That's really great that he enjoys doing the trades!! That means you're hitting him where he is, at that point where it's challenging and engaging. Good stuff!!

https://www.lakeshorelearning.com/products/math/place-value/hands-on-addition-regrouping-kit/p/FF297

Did this help? 🙂 I'm always curious what works for a wide range of kids, and unfortunately, I don't have access to a wide range of kids. 

[kirstenhill]

2 hours ago, BaseballandHockey said:

Yes to all this!

I agree 100%.

I don't think that you can separate math from language.  I do think that many kids with language disabilities need experiences where mathematical ideas are presented with low language demands, and then need language layered on top of that.  Whereas many kids without language disabilities can have both an idea, and the words for the idea introduced together.  

 

I think I might need to start a spin off thread in this direction! My 2E 9 year old is getting so frustrated with how not remembering/understanding various math terms is slowing him down in math progress.   I have probably defined the word "exponent" (just for example) dozens of times and he understands how to do math with exponents but forgets that "exponent" is the word that matches with the concept "multiplying a number by itself that many times".  But other than being his personal math dictionary, I haven't made a lot of progress in really remediating the weakness.

[PeterPan]

4 minutes ago, kirstenhill said:

I think I might need to start a spin off thread in this direction! My 2E 9 year old is getting so frustrated with how not remembering/understanding various math terms is slowing him down in math progress.   I have probably defined the word "exponent" (just for example) dozens of times and he understands how to do math with exponents but forgets that "exponent" is the word that matches with the concept "multiplying a number by itself that many times".  But other than being his personal math dictionary, I haven't made a lot of progress in really remediating the weakness.

My dd had word retrieval issues, so we made a list of the words for her to have in front of her for math while we worked together. In your case, you'd want the word plus some meaningful picture or symbol. You could take pictures of your manipulatives demonstrating the concept or use google image.

[Cake and Pi]

2 hours ago, Not_a_Number said:

Do you think he'd be able to associate a "twenty one" (said out loud) with two green chips and a single chip?

Nope. The chip trading is just a memorized rule. He's memorized that he can trade the ten 1-chips for a 10-chip (but not vice versa) and that's exactly where the meaning ends.

2 hours ago, Not_a_Number said:

it sounds like his own natural model is not length-based. So then I'd probably try to spin off from the "quantity" model

What would that look like? How would you present such a model? Would my home-made c-rods (that essentially show a quantity of glued-together cm cubes) fit in that model, or do you think they are more likely to cause confusion in a quantity-based model?

[Not_a_Number]

1 minute ago, Cake and Pi said:

Nope. The chip trading is just a memorized rule. He's memorized that he can trade the ten 1-chips for a 10-chip (but not vice versa) and that's exactly where the meaning ends.

OK, let me make sure I have this clear, because I'm not sure I'm understanding. As in, you think he wouldn't be able to remember the name? Because ultimately, I don't think there IS any meaning in place value other than the fact that you can trade one unit for another one. Place value is a convention that's not any more meaningful than the fact that you can trade chips. 

Maybe let me see if I can dig a bit deeper here 🙂 . What would you like his understanding of place value to look like? Would you like him to be able to see something like 

* * * * * * * * * * 

* * * * * * * * * * 

* * * 

on the abacus or in the C-rods or something and to be able to quickly say "twenty three"? Or would you like him to be able to take the written number 23 and measure this out on the abacus? What kind of proficiency are you looking for? 

If he looks at a 31, do you think he'd be able to get out 3 greens and 1 yellow out? Or is the issue that you don't think he'd connect the 3 greens and 1 yellow to any numbers in singles? 

 

1 minute ago, Cake and Pi said:

What would that look like? How would you present such a model? Would my home-made c-rods (that essentially show a quantity of glued-together cm cubes) fit in that model, or do you think they are more likely to cause confusion in a quantity-based model?

I'm really not sure, because I've never used C-rods. Honestly, I think that model would look a lot like @BaseballandHockey's games, which are all very quantity-based. I've used a quantity model with both my kids so far, which means that I don't ask them any lengths (unless they can count the units), don't use number lines to add or subtract, and let them count if they absolutely have to, although I heavily encourage not counting if they can do something else, like arrange the things we're counting in a recognizable pattern or split it up. But my kids are obviously mathy, and you know what to do with mathy kids yourself... I really wish I had more experience with this, because I'm good at experimenting, and I'd have probably had lots of experiments to report 😕 .

[PeterPan]

11 minutes ago, Cake and Pi said:

Nope. The chip trading is just a memorized rule. He's memorized that he can trade the ten 1-chips for a 10-chip (but not vice versa) and that's exactly where the meaning ends.

Is he interested in playing store? We made or had play money and ds loved to play store. And it was sort of ridiculous, like we were all business start ups and you could either buy each others' goods or outright buy their business, hehe. So you started with hundreds of dollars, had a bank. You could sneak in trades.

[Not_a_Number]

Just now, PeterPan said:

Is he interested in playing store? We made or had play money and ds loved to play store. And it was sort of ridiculous, like we were all business start ups and you could either buy each others' goods or outright buy their business, hehe. So you started with hundreds of dollars, had a bank. You could sneak in trades.

What I loved doing in class was playing a lot of blackjack with poker chips. There were constant trades, and we'd also constantly name the numbers everyone had. I think it helped kids internalize the trading, which is really the main IDEA of place value 

[PeterPan]

1 minute ago, Not_a_Number said:

 As in, you think he wouldn't be able to remember the name?

Or the completely clueless look that tells you he filed in somewhere else in his brain (things we do with round stuff, things I did for school, whatever) instead of "math concepts I can apply to other useful scenarios with trades". It's autism. It's like my ds knew 7+4 with one manipulative and you'd go to the next and start totally over. You have to do it so many different ways (they usually say 6) that eventually it "generalizes" or gets filed somewhere as a concept rather than a novelty.

[PeterPan]

Just now, Not_a_Number said:

What I loved doing in class was playing a lot of blackjack with poker chips. There were constant trades, and we'd also constantly name the numbers everyone had. I think it helped kids internalize the trading, which is really the main IDEA of place value 

Yeah, RB has a positive negative turnovers game, similar thing with trades. We were doing it with cards and equivalences. It has been a long time since I played the game, mercy. It's free in her Card Games ebook (free). But op needs an apple device to access it, grr. 

Maybe RB has a video of playing it? It was a good suggestion to try her FB and youtube. 

[Lecka]

https://www.amazon.com/Learning-Resources-Blocks-Childhood-Skills/dp/B00IJC7VLC/ref=mp_s_a_1_1_sspa?dchild=1&keywords=ones+and+tens+blocks&qid=1613789983&sprefix=ones+and+&sr=8-1-spons&psc=1&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUEzNlNGV0VYVVlVRTlOJmVuY3J5cHRlZElkPUEwNDYyMzcyMjc2TFZRWExHOU8wUiZlbmNyeXB0ZWRBZElkPUEwNzc1MTgwMlhOQkdGUVdaUURBRyZ3aWRnZXROYW1lPXNwX3Bob25lX3NlYXJjaF9hdGYmYWN0aW9uPWNsaWNrUmVkaXJlY3QmZG9Ob3RMb2dDbGljaz10cnVl
 

My son did better with something like this when learning numbers 11-20.  
 

All the same color.  
 

It is a lot simpler.  

 

Not that it was the only thing, but I wanted to use Cuisenaire rods and they were too much going on for him at that point.

I got them back out for him later, and he still played with them sometimes.

 

[Not_a_Number]

2 minutes ago, PeterPan said:

Or the completely clueless look that tells you he filed in somewhere else in his brain (things we do with round stuff, things I did for school, whatever) instead of "math concepts I can apply to other useful scenarios with trades". It's autism. It's like my ds knew 7+4 with one manipulative and you'd go to the next and start totally over. You have to do it so many different ways (they usually say 6) that eventually it "generalizes" or gets filed somewhere as a concept rather than a novelty.

No, I understand this would probably not generalize to other manipulatives. That's why I was suggesting working on singles and one-to-one correspondence concurrently with lots of objects. 

It's kind of the C-rod idea, isn't it? You restrict your attention to ONE representation of number, and work a lot with that representation, which frees up your brain to think about other stuff without worrying what number IS. And then the question is... how do you generalize from this one representation of number? Does a kid understand that because a 3 C-rod and a 4 C-rod make a 7 C-rod, then 3 sheep and 4 sheep make 7 sheep? And the answer is... it depends. 

So, just because a kid can learn that 2 yellow chips and a green chip MEANS the same thing as a pile of 21 chips, I know that it doesn't mean that it would make it easy to look at a 21 in other representations and name it. You'd have to scaffold that. But the trading idea would be absorbed in context, and then you could do number manipulations... in the model of number where number is "place value chips." And how do we get to other models of number? That I don't know exactly, but since they have one-to-one correspondence already, I was hoping that there was a way. 

[Clickie]

Just throwing this out there, as the OP mentioned that workbook pages could be of interest to her child.  Critical Thinking Company's Mathematical Reasoning Beginning 1 (numbers 1 to 5) and Beginning 2 (numbers 1 to 15) are robust early math workbooks.  They are both big 250-ish pages, colorful, pages aren't too busy.  There are sample pages online to look at.  My son did every single page of both of these books when he was 3 - 5 years old (I tear the pages out and get him to do about three pages a day. Discard.)  I felt it contributed to his early confidence in mathematics.

On the app side, I didn't catch whether or not apps work in your situation, but DragonBox Numbers (iPad, Android) is great for number sense (not DragonBox Big Numbers, that's regrouping). We loved both of those apps, and we did them way later, past the number sense stage  https://dragonbox.com/products/numbers 

The iPad app "Bugs and Buttons" is a lot of fun, too. Number sense. Anyways, I don't usually post, but thought I might mention the above, in the case that it could be of slight interest.

[Drama Llama]

9 minutes ago, Clickie said:

Just throwing this out there, as the OP mentioned that workbook pages could be of interest to her child.  Critical Thinking Company's Mathematical Reasoning Beginning 1 (numbers 1 to 5) and Beginning 2 (numbers 1 to 15) are robust early math workbooks.  They are both big 250-ish pages, colorful, pages aren't too busy.  There are sample pages online to look at.  My son did every single page of both of these books when he was 3 - 5 years old (I tear the pages out and get him to do about three pages a day. Discard.)  I felt it contributed to his early confidence in mathematics.

On the app side, I didn't catch whether or not apps work in your situation, but DragonBox Numbers (iPad, Android) is great for number sense (not DragonBox Big Numbers, that's regrouping). We loved both of those apps, and we did them way later, past the number sense stage  https://dragonbox.com/products/numbers 

The iPad app "Bugs and Buttons" is a lot of fun, too. Number sense. Anyways, I don't usually post, but thought I might mention the above, in the case that it could be of slight interest.

Actually, ironically my students who are at @Cake and Pi's level love Big Numbers.

It's supposed to teach regrouping, but a huge amount of it is picking apples, or stones or whatever, and then going and making groups and tracing the number that goes with the group.  Lots of practice for conservation of number, and subitizing, and composing and decomposing, and collecting things to make a set of an assigned size.  All the things he needs. 

If you get to the point in the app where it shows you numbers, and not dot patterns, just scribble for a little while and it will reset back.  

I would say that most of my kids like and are able to play Big Numbers and get those things from it, before they can play Numbers.  Then they go to numbers, and probably could come back and play it again.  

[Clickie]

My son and I each got totally addicted to Big Numbers -- it is a lot of fun.  It's interesting to hear you say that the skills in Big Numbers can be developed that feed into the skills for Numbers.  I would not have known that, thanks.

[Drama Llama]

Just now, Clickie said:

My son and I each got totally addicted to Big Numbers -- it is a lot of fun.  It's interesting to hear you say that the skills in Big Numbers can be developed that feed into the skills for Numbers.  I would not have known that, thanks.

It is addictive isn't it?   I've won twice.  The first time, I could justify as I was researching it on behalf of my students.  The second time?  Yeah, I have no excuse for wasting that much time.  

[Clickie]

lol, that's funny.  My son has won twice, me only the once.  But I went back and still did fishing, even though it was no longer advantageous to harvest that way. 🙂  I also liked harvesting apples just for fun in the days after.

[Cake and Pi]

50 minutes ago, Not_a_Number said:

OK, let me make sure I have this clear, because I'm not sure I'm understanding. As in, you think he wouldn't be able to remember the name? Because ultimately, I don't think there IS any meaning in place value other than the fact that you can trade one unit for another one. Place value is a convention that's not any more meaningful than the fact that you can trade chips. 

Maybe let me see if I can dig a bit deeper here 🙂 . What would you like his understanding of place value to look like? Would you like him to be able to see something like 

* * * * * * * * * * 

* * * * * * * * * * 

* * * 

on the abacus or in the C-rods or something and to be able to quickly say "twenty three"? Or would you like him to be able to take the written number 23 and measure this out on the abacus? What kind of proficiency are you looking for? 

If he looks at a 31, do you think he'd be able to get out 3 greens and 1 yellow out? Or is the issue that you don't think he'd connect the 3 greens and 1 yellow to any numbers in singles? 

If I give him two 10-chips and three 1-chips he cannot name, write, or point to the number/quantity represented, neither can he represent the quantity with any other manipulative (ie. on abacus, with base-ten blocks, etc). If I show him the written number "23" or say it or build 23 with abacus/base-10 manipulatives he cannot choose two 10-chips and three 1-chips to represent it. There is really zero understanding. He just knows that he can exchange ten 1-chips for a 10-chip (but again, not the other way around). Honestly, it means about as much as if I'd trained him to always exchange 7 tooth brushes for a can of spaghetti sauce.

Eventually I hope he'll have the understanding and fluency to translate back and forth between verbal, written, 1-to-1 manipulatives (like abacus or base-10 blocks), and abstract manipulatives (like place-value chips) for any number, but that is a quite a long ways off.

1 hour ago, PeterPan said:

Is he interested in playing store?

Yes! We have a play cash register and he loves to play with it. However, he's not learning *anything* other than you can trade plastic circles and rectangular papers for like ANYTHING you want to put in a store. Money is too abstract. He has a ton of fun with it, though, and I figure having positive associations with stuff that looks like money will help when I eventually try to teach him that money has specific values and such. He'll be more interested and it'll be easier to get his buy-in.

I suppose I could take the play money out of the drawer and replace it with the 1-chips... then we could work more on matching 1-chips to written numbers on play tags... maybe we could work up from that, eventually.

1 hour ago, Lecka said:

My son did better with something like this when learning numbers 11-20.  

We have base-10 blocks and use them a ton. Ours are yellow!

 

1 hour ago, Clickie said:

Just throwing this out there, as the OP mentioned that workbook pages could be of interest to her child.  Critical Thinking Company's Mathematical Reasoning Beginning 1 (numbers 1 to 5) and Beginning 2 (numbers 1 to 15) are robust early math workbooks.  They are both big 250-ish pages, colorful, pages aren't too busy.  There are sample pages online to look at.  My son did every single page of both of these books when he was 3 - 5 years old (I tear the pages out and get him to do about three pages a day. Discard.)  I felt it contributed to his early confidence in mathematics.

On the app side, I didn't catch whether or not apps work in your situation, but DragonBox Numbers (iPad, Android) is great for number sense (not DragonBox Big Numbers, that's regrouping). We loved both of those apps, and we did them way later, past the number sense stage  https://dragonbox.com/products/numbers 

Thanks! I will look into the Mathematical Reasoning books. I didn't even consider them because I guess I mostly think of the CTC as catering to test prep and resources aimed more toward gifted and high achieving kids.

We have the DragonBox Numbers app and DS 7 enjoys it. He's been playing with it for a couple of years now and does well with the puzzles but can't do the "running" game at all. Anyway, the Nooms are a big reason I made our c-rods myself. I wanted to match the Nooms' colors. I was afraid that after playing the app, changing the colors of the numbers would mess him up.

1 hour ago, BaseballandHockey said:

I would say that most of my kids like and are able to play Big Numbers and get those things from it, before they can play Numbers.  Then they go to numbers, and probably could come back and play it again.  

How interesting! I didn't even consider putting Big Numbers on his tablet because he can't do like half of the activities in the Numbers app. I'll try putting it on his tablet and see how he does.

 

[Drama Llama]

2 minutes ago, Cake and Pi said:

 

 

Those are fantastic!

 

[Clickie]

They are wonderful.  My son just walked by and immediately recognized them!  We haven't played that app in a few years.  They are perfect!

[Not_a_Number]

7 hours ago, Cake and Pi said:

If I give him two 10-chips and three 1-chips he cannot name, write, or point to the number/quantity represented, neither can he represent the quantity with any other manipulative (ie. on abacus, with base-ten blocks, etc). If I show him the written number "23" or say it or build 23 with abacus/base-10 manipulatives he cannot choose two 10-chips and three 1-chips to represent it. There is really zero understanding. He just knows that he can exchange ten 1-chips for a 10-chip (but again, not the other way around). Honestly, it means about as much as if I'd trained him to always exchange 7 tooth brushes for a can of spaghetti sauce.

Yes. I understand. But why do you expect all those understandings at once? My question was whether you could START with teaching him the names for numbers purely using place value chips. This could temporarily be his model for bigger numbers, while you worked on one-to-one correspondence, counting, and counting on with other items. As he keeps trading and playing games and naming the numbers, his understanding might increase. I understand that it won’t generalize to other manipulatives or to the concept of place value in general, but it’s a model. 

For what it’s worth, that’s how I teach place value. I tell a kid it’s made of 10s and 1s (with DD8, it was pictures of boxes and dots, with DD4 it was poker chips) and then we do LOTS of work with that. I don’t expect them to match the numbers in poker chips (or pictures) to any other manipulative. DD4 can now do lots of math with poker chips, but I doubt that she’d be able to figure out a number on the abacus, and I’m not even sure she knows why 31 is more than 29, except by rote counting up. I figure all those understandings are actually really tricky. In fact, they are trickier than adding or subtracting numbers in this form, which is why we start by adding and subtracting. I don’t remember how long it took us to connect this model to other manipulatives, but then I didn’t try very hard.

If you think about it, deciding on a place value chip model of number isn’t any weirder than deciding on a written model of number. Like, the number 23 is REALLY abstract!! It doesn’t contain its quantities in a helpful way. And yet it’s supposed to indicate 23 of lots of things.

Do you think having him associate a 23 with 2 yellows and 3 greens will hamper him when you work with other manipulatives? 

Edited February 20, 2021 by Not_a_Number

[Not_a_Number]

So... again, I could be totally wrong, because I've never tried it. It's just that this is where I'd start. But I think part of the difficult thing about place value is the idea that one thing represents 10 of another thing. I think having a single physical unit that's ten of another thing is kind of a bridge to that idea, instead of having a convention about the place of the digit representing ten units. 

I think that right now, you want him to LOOK at a ten (in a ten frame, in a hand, as a base-10 block, etc) and immediately grasp it as a single entity that ought to be categorized as such when you make up a number, and he's not ready for that. However, since he CAN trade, he could work on his association between 10 units and a single 10. The nice thing about a place value chip is that it's obviously a single entity and not 10 things. You don't get distracted into counting it, because there's nothing to count. So every time you trade up and down, you work on the idea that a 10 (at least in place value chips!) really is represented by a single thing. You work on it in a physical way with a specific object and you accept that having an easy model means that it won't be a widely applicable model. 

I've often had to make the trade of keeping my model accessible and easy to work with but less applicable at the beginning. And I've found that once the model is firm, it's easier to generalize from it to other things, because they no longer have to think about the model. For what it's worth, I don't actually find the chip model in any way fast to communicate. I've been working on arithmetic with DD4 for a few months now, and we've had to have the conversation "Did you change the number when you did that trade?" probably something like 20 times. The idea that one thing is genuinely worth another thing (not just trading spaghetti for toothpicks, but that there's a general RULE about it, and moreover, there are no OTHER trades you're allowed to make) takes quite a while to penetrate. 

Am I making any sense here at all? I think the difference in my perspective here is that I expect a model to do very limited work at the beginning. 

[Lecka]

I have been thinking about this — with the pizza, if there is an issue I think it could be same/different.

I really think he could just want his pizza in 3 pieces just because.

Have you done same/different?

If you have — you can line up 3 little pieces and 3 big pieces and ask him if he thinks they are the same or different.

Are you going down a list of these from speech therapy or something?

My some struggled a lot with part/whole relationships and had a lot of trouble with some/all.  He learned half before some, but some was very hard.  
 

I don’t remember same/different being particularly hard but he spent time on it.

Keep in mind — a lot of these have a 50/50 chance of getting the right answer!  So missing very many is probably a sign they aren’t solid.

It can be the language not solid, the concept not solid, or both.

And it can be very hard to go back and forth between A is more than B, and B is less than A.  Those have the same meaning but they are hard sentences, and it’s hard to get to:

A is less than B, so: (child says B is less than A).  This is a big causal relationship and a hard sentence.  
 

Making that kind of leap can be where expectations get too hard.  If that kind of thought process is difficult. 
 

If he can do same/different, working on sentences like “A and B are the same, but A and C are different” uses the word “but” which is another hard word to teach and also expresses a relationship. Or you could do “A and B are the same, and A and C are the same,” and practice with the word “and,” and have to choose to use the sentence frame with “and” or “but” depending on if you have a relationship comparing things that are the same or different.

 

These are really hard sentences and concepts.  
 

But it does start with just comparing two things and saying same or different, and that can be hard.  There are things too where they have a brown dog and a brown cat, the same size, and they are different.  But two totally different looking dogs are the same.  There can be a lot to it.

 

[Lecka]

I tried to look something up, and it said for same/different, add a new item.  So I guess for pizza you could have more pieces and see if they went with the big or little pieces.

What would really be wanted would be to say — “they are both 3 slices of pizza, but they are different sizes, so the bigger pieces are more pizza than the smaller pieces.”  Which is really complicated.

Have you done anything with sorting by feature, function, and class?  There is a lot out there for this with autism.

With pizza — feature could be pepperoni or sausage, function could be something you eat, and class could be food. Feature could also be size — large or small.

This is a tricky question for sorting:  let’s say there are pepperoni and sausage pizza slices, and some are big and some are little.

You can sort them by all pepperoni or by all sausage.  With different sizes going together.

Or you can sort them by size, with all the big and little slices going together, but mixed up between pepperoni or sausage.  
 

That kind of thing would be on the advanced side for feature/function/class.  
 

There is a lot out there for it and it’s supposed to help with abstract thought and language.  
 

It is supposed to help with forming categories and thinking about why things do or don’t belong to a category.  It’s supposed to help with mental organization also.  

[Kanin]

15 hours ago, Cake and Pi said:

Maybe! I have place value chips. Each place value has its own color, and on one side the units all have "1" the tens all have "10" etc. He understands trading. I've played games with him where he can trade me ten yellow 1 chips for a green 10 chip, and he gets that and thinks its hilarious for some reason. But I haven't gotten him to connect that back to physical representations of numbers, like base-ten blocks, the abacus, or c-rods, or to written (or place-value card) two-digit numbers. 

I wonder if you could play the same game, but with base-ten blocks instead of chips. I'd be wondering if he just thinks ten yellows equals a green, but just because it's a hilarious part of the game, not because it actually happens in real life math. 

[Kanin]

10 hours ago, Cake and Pi said:

Honestly, it means about as much as if I'd trained him to always exchange 7 tooth brushes for a can of spaghetti sauce.

I'm a bit late to the conversation, so I see you already beat me to it! Your phrasing is hilarious. 😅

Order up those Ronit Bird books for sure. The Dragonbox apps are amazing and well worth the $. You can get them on non-Apple devices.

[Kanin]

2 hours ago, Not_a_Number said:

The nice thing about a place value chip is that it's obviously a single entity and not 10 things. You don't get distracted into counting it, because there's nothing to count. So every time you trade up and down, you work on the idea that a 10 (at least in place value chips!) really is represented by a single thing.

This is why I love c-rods, and I think why Ronit Bird loves them, too. With my students, we'd find a c-rod, say, the number 8, then find other ways to make 8... 4+4, 2+3+3, etc. Then we'd pretend to glue the smaller pieces together to make the larger one. I'd explicitly guide them, over and over and over, to notice that they can't pull apart the 8 rod, but if we had a saw or something, we could saw it into smaller pieces. If we had glue, we could glue the smaller pieces together to make the bigger one. I think it really helped. After a while they'd say, "Ah! If I just had a saw I could chop this up!" Lol. 

[Lecka]

I kind-of think making trades is probably more advanced than needed at this point. 
 

Not that I would say don’t do it, don’t provide exposure.

 

But I think there are other things that are probably more foundational. 
 

I think for homeschool you might look at Math-U-See or Schiller. 

[Lecka]

I think there is a distinction to make:

To some extent, does the child already have good concepts of bigger/smaller and ordering by size in daily life?  
 

Or not so much?

 

If the first one — you are using existing concepts to teach a new concept.

 

If the second one — you are doing a math activity and using the activity to target these language/conceptual goals.  And, it is also exposure to math.  

 

The same activity can be done in a way that focuses on the first or second. 
 

And if something is hard — an easier game could just use one of the steps of a harder game.

 

[Not_a_Number]

17 minutes ago, Kanin said:

This is why I love c-rods, and I think why Ronit Bird loves them, too. With my students, we'd find a c-rod, say, the number 8, then find other ways to make 8... 4+4, 2+3+3, etc. Then we'd pretend to glue the smaller pieces together to make the larger one. I'd explicitly guide them, over and over and over, to notice that they can't pull apart the 8 rod, but if we had a saw or something, we could saw it into smaller pieces. If we had glue, we could glue the smaller pieces together to make the bigger one. I think it really helped. After a while they'd say, "Ah! If I just had a saw I could chop this up!" Lol. 

Whereas I kind of don't like C-rods, lol! But I understand what they are going for. It's just not my favorite model. 

 

16 minutes ago, Lecka said:

I kind-of think making trades is probably more advanced than needed at this point. 

It's possible. I'm not sure if it's the right thing or not -- as I said, I don't tend to work on that until counting on is solid. So I don't think I'd be doing it with this child yet. It just seemed like the easiest access point to place value ONCE he's ready. 

I'm just explaining how I would work on place value, once that goal is reached. But I'm also failing to explain the conceptual framework, I think. It took me many years to understand teaching within this framework, and I've had really good results with it, but I haven't been able to communicate to anyone what I mean. Well, except for maybe my husband, who's been listening to this for years now, and who's a math professor himself. 

[Not_a_Number]

38 minutes ago, Kanin said:

I wonder if you could play the same game, but with base-ten blocks instead of chips. I'd be wondering if he just thinks ten yellows equals a green, but just because it's a hilarious part of the game, not because it actually happens in real life math. 

 But I actually think chips are a better model. When you see a "2" in "25," it's not like you can cut the 2 up into twenty pieces. It's really just a 2, in the same way that 2 yellows is just 2 yellows and in no way is the same as 20 greens stuck together. 

I read an essay that said that poker chips are an equally abstract model to how we write numbers, it's just an easier one to pick up in an intuitive way, and I very much agree with that. The idea is that of "units" and of the fact that units can be decomposed into units of a different rank (if anyone's read Liping Ma, she uses something like that phrase.) In poker chips, the units are colorful poker chips. In writing numbers, the units are numbers are different places. It's really an identical model. 

However, this is precisely the idea I'm having zero luck communicating, so I think I'm going to give up for the time being 🙂 . I hope someone believes me that there's an idea there, lol! 

Edited February 20, 2021 by Not_a_Number

[Lecka]

With any abstract concept, to some extent, you do it concretely a bunch of times and then the concept generalizes to abstract.  

I think it makes sense!  

You can never make anything generalize, but you can provide clear, concrete explanations and examples.   

 

[Not_a_Number]

Just now, Lecka said:

With any abstract concept, to some extent, you do it concretely a bunch of times and then the concept generalizes to abstract.  

I think it makes sense!  

You can never make anything generalize, but you can provide clear, concrete explanations and examples.   

That's definitely the beginnings of the idea, but it's not the whole idea. I'm obviously having trouble communicating the whole idea, but I think it's me, not you -- I've just been immersed in this idea for so long that I'm taking lots of words and thoughts for granted, and really I'd need to write a long essay about how I came to think about this and what I now believe about mental models and their place in mathematical teaching. 

So, it's kind of a circular issue 😉 . My whole point is that I try to teach in a way that takes into account what's going on in someone's head to a very detailed extent, but the words I'm saying are not creating the models in your heads that I wish they were, and yet I'm acting like they are. Whereas I think it's ultimately the teacher's responsibility to make sure that the mental model created in a student's head is efficient and effective and matches what's in the teacher's head as much as possible. 

Anyway, this is all to say that what we've got here is a failure to communicate, but it's me, and not you 😉 . 

[Kanin]

27 minutes ago, Not_a_Number said:

 But I actually think chips are a better model. When you see a "2" in "25," it's not like you can cut the 2 up into twenty pieces. It's really just a 2, in the same way that 2 yellows is just 2 yellows and in no way is the same as 20 greens stuck together. 

I read an essay that said that poker chips are an equally abstract model to how we write numbers, it's just an easier one to pick up in an intuitive way, and I very much agree with that. The idea is that of "units" and of the fact that units can be decomposed into units of a different rank (if anyone's read Liping Ma, she uses something like that phrase.) In poker chips, the units are colorful poker chips. In writing numbers, the units are numbers are different places. It's really an identical model. 

However, this is precisely the idea I'm having zero luck communicating, so I think I'm going to give up for the time being 🙂 . I hope someone believes me that there's an idea there, lol! 

I think I do understand what you're saying - maybe? - but I'm not sure your way would be as effective for teaching a student with a language disability who is very visual. You said that the chips are equally abstract to how we write numbers, but what we're trying to do here is make the abstract part of numbering as concrete as possible. At least, I think that's what we're trying to do. 

When I think of the "2" in "25," or at least when I do it with students, we build it so the 2 is two physical blocks that are ten units long, and the 5 is 5 unit blocks. So they're not picturing cutting up any numbers into pieces. Maybe this would ruin the abstract understanding of the chips, but I'd probably build numbers with both base ten blocks and chips, side by side. 

[Not_a_Number]

6 minutes ago, Kanin said:

I think I do understand what you're saying - maybe? - but I'm not sure your way would be as effective for teaching a student with a language disability who is very visual. You said that the chips are equally abstract to how we write numbers, but what we're trying to do here is make the abstract part of numbering as concrete as possible. At least, I think that's what we're trying to do. 

I'm actually not trying to make the abstract part of numbering as concrete as possible. What I'm trying to do is create an equivalent model that is possible to appreciate using lots of hands-on experimentation. And generally, I don't expect this model to be internalized in days or even in weeks. I expect it to be slowly internalized over the course of months. 

I haven't worked with students with a language disability, so I really can't claim that I have data that I do not. I'm really just chiming in because I know that the way I conceive of teaching place value isn't much like what I've seen anyone else do. But since I'm not finding myself able to explain why it works or how it's different, I probably should stop. 

 

6 minutes ago, Kanin said:

When I think of the "2" in "25," or at least when I do it with students, we build it so the 2 is two physical blocks that are ten units long, and the 5 is 5 unit blocks. So they're not picturing cutting up any numbers into pieces. Maybe this would ruin the abstract understanding of the chips, but I'd probably build numbers with both base ten blocks and chips, side by side. 

Yeah, I know that's the standard thing to do. That's just not how I conceive of place value, though. But as I've said, I'm completely failing to communicate how I conceive of it.

[Not_a_Number]

I think part of the issue is that we're so used to our usual notation, that we don't appreciate that there's nothing natural or logical about how we write numbers. The way we write numbers is purely a convention. There's nothing to understand in this convention. We've DECIDED that the second unit is worth 10. We could have just easily decided that the green unit is worth 10. None of us would be able to do mental arithmetic in the land of Colorfulia, where they write numbers like this: 

4 4 7

You might THINK this number means 4 hundreds, 4 tens, and 7 ones, but in Colorfulia, it does not! Because in Colorfulia, the order the number is written it doesn't matter in the least. All it matters are the colors. Actually, this number is 7 thousands, 4 hundreds, and 4. 

And if that feels confusing to you (it does to me!), I submit that there's for this is that we're all very used to the fact that later units are worth more than earlier units. In fact, we're used to them being worth 10 times more than the previous unit... although of course, really they could be worth twice more or 5 times more, and that would also be a valid system. 

So... I think a part of learning how to use place value is getting experience with the convention. And I think that getting a true feel for the convention in poker chip colors can actually be a lot easier for a visual kid than remembering the convention in writing. And just like a kid being able to parrot "a 21 is 2 tens and 1 one" doesn't actually show much understanding or much feeling for numbers and is rather simply memorization, being able to pull out two yellows and a green when someone says "twenty one" (or perhaps writes a 21, whatever is easier) doesn't actually mean that a kid has the kind of understanding one could use for other numbers. The thing I like about poker chip models is that this model is flexible enough to allow a kid to get their own understanding of it in a hands-on way, whereas written numbers don't really give you this opportunity. 

But then I conceive of understanding as a slowly developing photograph. I never expect understanding as a result of anything I say; only as the result of experience with the model I give them. And I expect that process to be really arduous. 

[Clickie]

Just throwing another app out there, both iPad and Android, is Todo Math.  It has a lot of number sense, place value mini-games.  It's a full pre-K to Grade 2 app.  Very fun, well done, we did it for a long time.    https://www.todomath.com/curriculum/kinder

[PeterPan]

Just throwing this out there, but at first I really balked about the c-rods. I was so into the abacus, having repped for RightStart and really liking the curriculum, that I thought moveable was better, blah blah. And I kinda gave Ronit Bird a hard time, even writing her and saying hey you're cracked, kids with language disabilities don't need this extra hurdle of memorizing names and connecting them to these stupid fixed colored rods. 

And she wrote me back so graciously and she's like GIVE IT A CHANCE. And I'm like no, fixed rods are sin, you're memorizing names and it's language and it means nothing to our kids.

But what I finally realized, when it finally clicked, was that that was the point. We had to make the words mean something. There is a *one* inside of every larger number, and until *one* means something and he has a name and you know him and he's your friend and he's there every time, it's really hard to find him, kwim? And you make friends with *two* and learn his name and see the *one* inside him. And you play her inane games where you keep seeing numbers inside numbers and make friends and learn their names. And then you generalizing those names so that one isn't one white cube but it's the MATH ONE where it could be one m&m or one button or one tiny turtle or one finger or one centimeter or one spoon of pudding or one anything.

I think it's the most fundamental thing to get down, and it's hard because it's language. And it's counter intuitive because we don't want to pin language to it. But it's a pretty big step, when the language and the math start to connect. I don't think it's necessary to rush it to go forward.

After we did Dots, we played positive/negative turnovers to extend it to subtraction. Since we were writing equations anyway, we went ahead and started doing it with positive/negative numbers. So at that point he had all that. Then, since he had single digits, I was like hey let's throw inhibition to the wind and try FRACTIONS! So we turned over two cards each for War, formed fractions where we very tediously talked about how many parts were in our whole, how many of those parts we had, and found the fraction on the RightStart fractions puzzle. (which I LOVE, LOVE, LOVE, LOVE, LOVE) And we did that for a LONG TIME, like 1-3 months, and then he was like OH and started regrouping and being able to compare. We spent months just with that fractions board and those single digit cards. 

So a simple thing, completely understood, opened up other things. I think place value is maybe the least useful thing, kwim? It's kinda funky and advanced. There are harder things that can be simpler that are also terribly useful. Fractions are crazy useful!!! Fractions make measuring easier. So to me, why horse around with place value when you can do something else, kwim? Place value you sorta work on, come back, work, come back. 

Total, total aside, but integrating retained reflexes will often result in a language bump. I THINK my ds' sense of place value for the math came in after the language bump he got when we integrated his retained reflexes. So again, why work on place value? Work on retained reflexes and see if his math improves.

[Not_a_Number]

2 minutes ago, PeterPan said:

And you play her inane games where you keep seeing numbers inside numbers and make friends and learn their names. And then you generalizing those names so that one isn't one white cube but it's the MATH ONE where it could be one m&m or one button or one tiny turtle or one finger or one centimeter or one spoon of pudding or one anything.

I think it's the most fundamental thing to get down, and it's hard because it's language. And it's counter intuitive because we don't want to pin language to it. But it's a pretty big step, when the language and the math start to connect. I don't think it's necessary to rush it to go forward.

Well... for what it's worth, I like the dice games a lot better than the C-rods. I'm all for seeing numbers inside other numbers. And I don't really care about movability -- my older kiddo didn't use anything movable at any point in her math education, period. But for me, the C-rod model is funky. 

So, I totally get the idea of making friends with "one" and with "two" and whatnot. I'm just not convinced that C-rods are the fastest route there. Do you think the C-rods were really key to communicating what a "1" is and what a number is in general? Or were the dice games and the repetition more helpful? 

It's not like I tried C-rods and they didn't work and that's therefore my opinion. I have an uninformed prejudice here, because C-rods don't fit into my own mental framework very well. 

 

8 minutes ago, PeterPan said:

So a simple thing, completely understood, opened up other things. I think place value is maybe the least useful thing, kwim? It's kinda funky and advanced.

Yeah, place value is definitely tricky and advanced. That's actually why I like starting it early, to be honest -- like, if a kid only understands trading in one context for a while... that's fine by me. But there shouldn't be any pressure to really use it "properly," and I agree that there are lots of other things you could introduce. Like... multiplication. Or division. Or equality. Or fractions. It's all totally unrelated to place value, and you can do them concurrently. And you can see what clicks first. 

[ieta_cassiopeia]

Alternative solution for the Ronit Bird issue: consider downloading (or downgrading) to version 12.6.5.3 of iTunes, which is the last version of iTunes to allow management of books. That should allow you to run the Ronit Bird eBooks on your computer. (But whatever you do, don't let iTunes update itself, and keep a backup of the installer file in case).

 

Old versions of iBooks can still get books, so in principle it should be possible for older versions of iTunes to do so (since it appears books are still classified as such at Apple's end, in a way that is backwards-compatible).

Edited February 20, 2021 by ieta_cassiopeia

[Kanin]

5 hours ago, Not_a_Number said:

You might THINK this number means 4 hundreds, 4 tens, and 7 ones, but in Colorfulia, it does not! Because in Colorfulia, the order the number is written it doesn't matter in the least. All it matters are the colors. Actually, this number is 7 thousands, 4 hundreds, and 4. 

And if that feels confusing to you (it does to me!), I submit that there's for this is that we're all very used to the fact that later units are worth more than earlier units. In fact, we're used to them being worth 10 times more than the previous unit... although of course, really they could be worth twice more or 5 times more, and that would also be a valid system. 

Well, I don't think that's confusing, and I know that base ten is just a choice that people made oh so long ago. I agree with you about understanding taking a long time. For the OP, I wouldn't worry that he doesn't understand yet. 

[Not_a_Number]

13 minutes ago, Kanin said:

Well, I don't think that's confusing, and I know that base ten is just a choice that people made oh so long ago. I agree with you about understanding taking a long time. For the OP, I wouldn't worry that he doesn't understand yet. 

I don't know about you, but I'd have a harder time adding in the new color system in which the order of the digits didn't matter. Just like I'd have a harder time using the letters a, b, c, d, e, f, g, h, i, j for digits... Like, what's c + g? Apparently, it's i, but I didn't figure out that all that quickly, lol. I'm not used to using these letters for numbers. They don't FEEL meaningful. 

So, what I'm really trying to say is that a poker chip model isn't any more or less "reasonable" than the written number model. However, the difference with the chip model is that it allows kid to get a feel for what a 25 means... or, more concretely, that it allows one to have a real feel for what it might mean to have one thing be worth more than another thing. Like, truly worth, not just in the "this is a weird rule" way. 

I would also say that I've found that most kids (learning disabilities or not) haven't truly integrated place value into their understanding of numbers. Most kids can name 10s and 1s but have very little hands-on experience trading up and down in a practical way -- for games, for addition, for subtraction. And that tends to bite kids when they need to learn later operations as well as algorithms. I saw this a ton in my homeschool math classes.  

Anyway, place value is just hard 🙂 . So I agree that it's not anything to worry about. 

[Drama Llama]

4 hours ago, Clickie said:

Just throwing another app out there, both iPad and Android, is Todo Math.  It has a lot of number sense, place value mini-games.  It's a full pre-K to Grade 2 app.  Very fun, well done, we did it for a long time.    https://www.todomath.com/curriculum/kinder

I love Todo Math!

[Lecka]

Not_a_number ----- are your poker chips different from having pennies and dimes?  I think pennies and dimes are very helpful.  

Is it teaching one color poker chip is worth ten of the other color?  

I am picturing this as -- the same as using pennies and dimes to teach place value -- not sure if there are some differences there.  

[Not_a_Number]

6 minutes ago, Lecka said:

Not_a_number ----- are your poker chips different from having pennies and dimes?  I think pennies and dimes are very helpful.  

Is it teaching one color poker chip is worth ten of the other color?  

I am picturing this as -- the same as using pennies and dimes to teach place value -- not sure if there are some differences there.  

Hah, not very different! They are easier to stack and they don't make your hands smell? 😉 But yes, that's the exact same idea, if you did all your arithmetic using pennies and dimes for a good long while and spent a lot of time talking about trades that change numbers and trades that don't. 

[Cake and Pi]

21 hours ago, Not_a_Number said:

Super interesting! So you genuinely have a length-based model. Fascinating. 

Okay. So just for fun, I decided to ask the numerate members of my immediate family about their mental models for numbers. 

DH:
He says he sees just the digits of a number, the way one would write the number, in his mind with an implicit understanding that the digits are multiplied by powers of ten, and that's it unless he is given context. With context he switches to visualizing quantities, quite accurately, so it sounds like he uses a quantity model. He gave me some examples that I'm not going to remember correctly. Something like one cubic foot of concrete is a cube *this* big (holds out hands). Ten cubic feet of concrete is a traffic barrier. Three hundred cubic feet of concrete is the foundation of our home, etc. He said if the context were stuffed animals, "one" would be a kid holding a stuffy. "Six" or "ten" might be a shelf with stuffies lined up on it for sale in a store. "Fifty" might be a claw machine full of stuffies, etc. 

For reference, he is a chemical engineer who does mostly process and controls engineering in mining, mineral processing, and water treatment. He also handles the budgets and cost estimates for enormous large-scale projects. He's sort of a Jack of all trades in the engineering world. He routinely works with very large numbers that represent large *physical* quantities, usually in the millions, of um, STUFF, lol. He doesn't remember early elementary or the process of learning place value.

DS 13:
Numbers were not automatically associated with any mental image, just sounds. He has great trouble visualizing in general. However, when pressed to "picture" a number in his head, he arrived at a length model. He said 27 is a vertical stack of 27 balls next to a measuring tape with the height of the stack equivalent to the quantity in the stack. When I asked about bigger numbers, like 1,123, he said he envisioned basically block letters of "1123" rotated 90 degrees counterclockwise and stretched until the height of a measuring tape at 1,123.

Background: DS 13 is 4ish years accelerated in math, ASD, PG, and NOT a visual thinker (unusual for ASD, but there you go). Compared to my other 2e kids, he had a relatively difficult time understanding place value. I think we worked on the idea of place value for a solid month when he was 4y 11m to 5y 0m old before it clicked. 

DS 11:
For small numbers, under 10, he envisioned groups of cakes, lol. For slightly bigger numbers, his mental model was cookies lined up horizontally with groupings of 5. He described 27 as essentially OOOOO  OOOOO  OOOOO  OOOOO  OOOOO OO. 1,123 was a big pile of 1,000 cookies with a smaller pile of 100 cookies to the right of that and a line (again grouped in 5s) of 23 more cookies to the right of that.

Context: DS 11 is maybe 2 years accelerated in math, PG with a small army of other Es, and suspected ASD. He picked up on place value immediately the first time I introduced it at 4y 3m. As in, he was immediately able to do *all* of the skills I described in an earlier response to you that I hope to get DS#4 doing eventually. As soon as I explained that 1,000 was 10 hundreds, he could say that there must be 100 tens in 1,000, could build numbers with any of our manipulatives from a number said aloud or written down, could write and say a number based on a physical model of it, etc. After all the trouble DS#1 had, I was armed and ready to hunker down and focus on place value for a while, but he seriously got it the very first day I introduced it to him.

DS 9: 
DS 9's response basically blew my mind. I obviously don't operate on the same wavelength as him. I asked him what he saw in his mind when he thought of 27 and his response was "three cubed." Okay, new tactic needed. I figured he'd have some other model for non-square, non-cube numbers but figured composite numbers might elicit a similar response, and so I asked him what he envisioned with the first prime number that popped in my head, 31.
"Two to the fifth minus one."
Oooookaaaaaaaay...... "But what do you SEE in your mind for 31?" I asked.
"I see zeros and ones," he answered.
"Seventy two. What do you see for seventy-two?" 
"A seven and a two with a one and a two under them."
"Whaaaaat? What are the one and the two?"
"They're places for the hoppers. You can change the orders of the numbers and hoppers go with them so they're the same number."
Then he described putting apples into his "inventory" and using "block commands" and "burning up" 9 apples to "transfigure" 10 regular apples into a golden apple and having separate hoppers to put regular apples and golden apples in... but then he said his hoppers hold multiples of 64. Oh, but he doesn't always use hoppers. He also has buckets. The buckets can be empty or full of water, and those are all zeros and ones (binary). Or he can stack two buckets and add water to the inner bucket and have all zeros, ones, and twos (base 3, I guess). So I essentially have no idea what, if any, mental model he has for numbers, place value, anything. There seems to be an evolving complicated something going on in his head all the time.

Background: DS 9 is like 9+ years accelerated in math, ASD, PG. I went through the first 24 lessons of RS A (1st edition) with him very informally here and there between 3y 8m and 4y 1m -- as in, I never pulled out the book. I was just familiar enough with the beginning concepts that I was able to introduce the idea of subitizing and grouping numbers into five-and-somethings, counting out snacks by 2s and patterning in rows with toys and food, etc. to work it into everyday interactions. He played extensively with math manipulatives (mostly base-ten blocks, place-value cards, the abacus, math balance, geometric solids, Magnatiles, pattern blocks) independently and, as much as it pains me to admit it, was largely neglected by me because he was content to occupy himself quietly while I focused enormous energies into DS#4's early intervention therapies while trying to keep up with regular household tasks and homeschool lessons for DS#1 and DS#2, who were in early elementary at the time. He went to a play-based preschool for half the day to keep him gainfully occupied (he had a tendency toward destructive curiosity). One day I had 830+177 written on the white board as a review problem for then-6yo DS#2. DS#3 looked up and announced the sum was 1007, then returned to his play. I didn't even know he could read 3-digit numbers at the time, so I was pretty floored. My only explanation is that he was paying attention to my conversations with older siblings as he played quietly in the same room 🤷‍♀️. After that I quickly went through all the place value exercises I'd uses with the older two to make sure he was really solid on the idea and there were no holes in his understanding, but I'm not sure it was really necessary. He was able to do anything I asked the first time.  

11 hours ago, Not_a_Number said:

I've often had to make the trade of keeping my model accessible and easy to work with but less applicable at the beginning. And I've found that once the model is firm, it's easier to generalize from it to other things, because they no longer have to think about the model. For what it's worth, I don't actually find the chip model in any way fast to communicate. I've been working on arithmetic with DD4 for a few months now, and we've had to have the conversation "Did you change the number when you did that trade?" probably something like 20 times. The idea that one thing is genuinely worth another thing (not just trading spaghetti for toothpicks, but that there's a general RULE about it, and moreover, there are no OTHER trades you're allowed to make) takes quite a while to penetrate. 

Am I making any sense here at all? I think the difference in my perspective here is that I expect a model to do very limited work at the beginning.

It's making sense, but it's also not lining up with my personal experiences of young kids learning place value.

7 hours ago, Not_a_Number said:

But then I conceive of understanding as a slowly developing photograph. I never expect understanding as a result of anything I say; only as the result of experience with the model I give them. And I expect that process to be really arduous. 

Maybe my experiences don't really count though, since none of my kids are neurotypical? I did not have to focus on a single model of place value before introducing another. I always taught them all concurrently. And, for my older kids at least, I'd not think of the process as a slowly developing photograph but more of a lightbulb suddenly being switched on. 

I taught trading as soon as place value was well understood with the RS abacus. There were beads that represented units (ones, but we do like MUS and call that place value units so as not to confuse one 100 or one 1,000 and so on), beads for tens, beads for hundreds and thousands. There's room for 20 unit beads, but once you hit 10 units you're supposed to trade them for a 10 bead because you can't write fifteen or twenty in the unit's place.

I *never* taught counting on to the older three boys, at all, ever -- so they obviously mastered place value without first having that skill.

[Lecka]

I don't think my son could have understood pennies and dimes without already understanding something like the ones and tens cubes/logs where he could "see" that ten of the cubes was the same length as one log.  

He already knew about ten ones being the same as one ten, before he started pennies and dimes.  

And I included "hto charts" with pennies and dimes.  Not the only way we used them, but I did use them and thought there were helpful.  

And hundreds charts.  

I think he already knew how to count by tens, could count dimes by tens, could point at and count by tens on a hundreds chart, or count those numbers individually, etc.  

I went back and forth between a hto chart and abacus also, doing the same problem with the abacus and pennies and dimes on the hto chart. 

But also just -- lots of doing things with pennies and dimes. 

He was at this stage for so long, there was plenty of time to do all kinds of things a lot.  Whereas most kids probably just do not spend as much time and then do not do as many activities in as many ways, just because they are moving on.  

I do not think there is any reason to use extra things that are not needed or helpful to kids.  But if some kids can benefit from more -- I think that is fine, too.  

The other thing was my son would have goals for counting money and skip-counting with money -- so that was happening anyway, and he learned to skip-count with nickels and dimes and compare amounts with those coins.  He also liked coins.  They would also be considered good for fine motor!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!  Which -- he did not have fine motor goals like that at the time, but they could do the same activities with kids who had fine motor goals and it never hurts to do more fine motor, either.  

The school my son went to at the time used Math in Focus and they liked "hto charts" a lot.  

They could do tally marks in the columns, or pennies/dimes, or they had chips that would have 1 and 10 written on them, or they could use base ten blocks, etc.  

I also think he was helped by doing "number of the day," which he did for probably 3 years with small numbers, where you have a number and write one more than that number, one less than that number, ten more than that number, ten less than that number, and then they would have a couple more activities that might be coloring in circles or writing an addition or subtraction problem using the number.  That is something he did at school that I liked, and really did help with one more and one less, which was not something he picked up really easily.  

    

[Not_a_Number]

1 minute ago, Cake and Pi said:

Maybe my experiences don't really count though, since none of my kids are neurotypical? I did not have to focus on a single model of place value before introducing another. I always taught them all concurrently. And, for my older kids at least, I'd not think of the process as a slowly developing photograph but more of a lightbulb suddenly being switched on. 

You're right that some concepts can absolutely be taught as lightbulbs being suddenly switched on. That generally works best with concepts that kids are naturally exposed to over and over again, and it also works best for naturally mathy kids. So, mathy kids have spent time reading numbers, and thinking about groups, and maybe using money, and maybe listening to you talk to older kids... so they are already primed for the concept. 

However, I teach LOTS of things most kids aren't in any way primed for. I teach kids vectors, and I teach kids variables, and I teach kids trig, and I teach kids parametrizations, and even advanced kids tend to need to have time with the concepts to have them really sink it. Usually, the time they need is measured in months and not days. And if you don't give the kids that time, they pretty much never figure out the concept, no matter how many times we try to USE the concept. Using it isn't the thing that gets kids fluent. It's being exposed to the model that makes them fluent. This was a serious epiphany for me, and it has vastly improved my teaching. 

So for me, the idea that place value would also require lots of time is just a natural outgrowth of other things I teach. I also teach my own kids quite young, which means I'm often teaching at the bottom end of when it's developmentally reasonable. That also extends the required exposure time (but also makes them fluent sooner, of course.) 

 

4 minutes ago, Cake and Pi said:

I *never* taught counting on to the older three boys, at all, ever -- so they obviously mastered place value without first having that skill.

Oh, obviously you don't need to count on to do place value -- they are disjoint skills. But "counting on" can show awareness of a number as a concept, particularly if you use it in conjunction with subitising. And that makes it likelier that a kid will be able to accept that a differently-colored chip might genuinely mean a different quantity.

[Lecka]

Just for the record -- I consider place value to be a 2nd grade skill, needed for regrouping with addition and subtraction.  

I think a lot of kids are still developing this skill and then putting it into practice in 2nd grade.  

Do I think my other two kids picked all this up easily way before 2nd grade?  Yes, yes I do.  

[Not_a_Number]

1 minute ago, Lecka said:

I don't think my son could have understood pennies and dimes without already understanding something like the ones and tens cubes/logs where he could "see" that ten of the cubes was the same length as one log.  

He already knew about ten ones being the same as one ten, before he started pennies and dimes.  

I am not sure you're right about that 🙂 . There isn't anything to understand, you see -- you just tell them that a dime is "worth" 10 pennies, and then you practice a LOT. And at first, it isn't very intuitive, and then it gets a lot better. But it takes a while. 

You don't need to visualize an actual 25 pennies to understand a 25 as 2 dimes and 5 pennies. You really don't. 

I can't comment on whether this would have definitely worked, but it has worked with all the kids I've worked with so far.