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

[Cake and Pi]

I'm trying to decide what direction to go with my 7.5yo (2nd grade age) SN learner and would love your opinions and input.

I'm using a mashup of Right Start 1st edition level A, ST Math kindergarten level, MUS Primer, and my own inventiveness with a variety of other materials and manipulatives. I have a strong math background and already successfully taught my three older kids all of this stuff already, but he's his own animal. Nothing I thought I knew applies.

Altogether it took him 6-7 months to master (90% accuracy or so) subitizing and counting with 1-to-1 correspondence 0-10 along with numeral recognition and matching language, but he eventually got it. He's able to trace numbers 0-10 and can independently write maybe half of them (with a high rate of reversals).

I've now been trying to teach him numbers bigger than 10 for about the last four months and made only very modest progress. He cannot read or comprehend numbers bigger than 10. Fifteen and fifty-five are exactly the same to him because they both have 5s, as are twelve and twenty because they both have 2s. Using RS's special naming system almost seems to make it *worse* because he'll take a number like 25, which RS would call "2-ten 5" and say "two, ten, five," put them all together to make 17, and think it's 25. I've tried using the abacus, base-ten blocks, home-made, Noom-colored c-rods, place-value cards, place-value chips, tally sticks and tally marks, ten-frames, beads and pipe cleaners, MUS's Decimal Street, using fingers from multiple family members, making bundles of rods, basically EVERYTHING at this point. If a number is bigger than 10, he doesn't get it.

Should I keep trying? It took half a year to get 0-10; maybe we just haven't sat with this concept long enough. Perhaps I'm being impatient. 

Should I step away from numbers >10 for a few months and focus more heavily on other math concepts? He can't get past the 25-30% mark in any of these kindergarten curricula we're using because they all incorporate place value to some extent from there on out. Should I ditch the curricula and work on teaching *whatever* he can understand? He seems to get basic addition and subtraction within 10 with manipulatives. Maybe I should try to continue along that line and try to teach him addition and subtraction with symbols? He's pretty good with visual patterns; we could do more patterning. He's starting to show a better grasp of bigger/smaller, greater/less, side/middle (but not left/right). He spontaneously builds some pretty complex things out of blocks, magnatiles, and math manipulatives -- see picture below.

 

Background in case you need it: Recently diagnosed ASD 2 with language impairment/disorder; conversationally verbal with lots of echolalia/scripting... except when he's not and just cries/grunts/moans; encephalopathy; global delays but average IQ; basement-low verbal memory and general memory funkyness; dx deferred, but still strongly suspected SLDs in reading, writing, and math. He left public school 11 months ago, and I've been homeschooling him continuously ever since. We did not take a summer break. (He left public school originally due to covid school closure, but I am a seasoned homeschooler in my 9th year with one or more of my three other kids.)

Edited March 6, 2021 by Cake and Pi

[PeterPan]

6 minutes ago, Cake and Pi said:

encephalopathy

This is the wild card to me. Just in general, I would say you need Ronit Bird. It's a bit more super powered than RightStart so it might give you a few more tools. Where is his IQ, if you don't mind us asking? I just know diddley about encephalopathy. 

I think you teach them where they are and what happens happens. It doesn't mean you're doing a bad job. You're going to focus the math on where it gets APPLIED. So with my ds (exceptionally similar to yours except maybe IQ, dunno), I would do something (like this is the number 5, we spent a month on the number 5) and then APPLY it. Your word of the day is GENERALIZATION. So 5 in cereal, 5 in meausring, 5 in time, 5 with a zillion times of manips. Cut it, measure it, pour it. 

In other words, you know how there are like 12 chapters in the math book that are the fun stuff? (a regular math book) That's all the application for generalization. So instead of teaching first, generalizing later, you flip that and do a little, a little.

I also thikn it's fine to go with a more pragmatic, scripted approach. I follow a really detailed blog (Epiphany) that just had a whole post on math that challenged me. https://epiphanyasd.blogspot.com/2021/02/core-vs-extended-maths-unexpected.html  Now this man's ds has a type of ASD that responds to a medication, an alzheimer's medication. He mentions it in the article. With the med the chemistry was there for his ds to learn with enough specialized instruction. And dopamine plays a part. But that's not everyone's reality.

I think you're just going to do what you can do, see what clicks.

Maybe read the article and see what you think. It left my head swirling, lol. In the midst of SO MANY THINGS we are supposed to be doing, math kinda slides. My ds isn't writing numbers yet, because we've been booking it on vmi=visual motor integration. I think he'll be able to now, but I can't swear he'll actualy be able to write enough to write his math. And one bit of STRESS and the WHOLE THING SHUTS DOWN. 

So I'm not sure any perfect method would get you farther. You're doing what you can. See if anything makes sense to you, like the Ronit Bird (for conceptual) or say Saxon (for scripted). We did some Saxon K5 (which is different in the early years) and it was adorable. It's just you could tell my ds wasn't getting things. You're describing dyscalculia, yes, so that's why I'm suggesting looking at Ronit Bird, seeing how far you can get. Maybe a blend. There are some other theories and curricula, but yeah I'd be looking for dyscalculia specific materials definitely.

[PeterPan]

How does he do with workbooks? some of the publishers like Teacher Created Materials and Evan-Moor have wonderful workbooks. There's a cut and paste math. There's graphing math art where they color squares (you could use a bingo blotter to make holiday designs. 

Word problems are another thing where you can back up, hit say a K5 level, and get something that lets him apply the math diversely. I've done a lot of that over the years. There was a series Using the Standards sold by I forget whom (one of the big ebook curriculum places) that had books applying to measuring, etc. They were a lot of fun because we'd measure, do nonstandard units, etc. 

Lawrence/GEMS has units tha tmight work for him. He might enjoy the Frog and Toad unit. That was my first oh my lands this is not just SLDs, because it was hitting things like sorting.

Timberdoodle sells a ton of logic games. If he enjoys them, you know back up to that kind of preschool/k5 level and see what clicks. Games are so good for working memory, sequencing, coordinates, lots of early skills. 

We had some early level games we played a LOT. Go Nuts has this squirrel them. Any of your elementary yahtzee games. 

[PeterPan]

18 minutes ago, Cake and Pi said:

He can't get past the 25-30% mark in any of these kindergarten curricula

Bingo. You're going to have to ditch them.

[PeterPan]

19 minutes ago, Cake and Pi said:

It took half a year to get 0-10

Sounds fine to me. Like I said, my ds has a gifted IQ plus the dyscalculia with the ASD2, and he was pretty mch one month per number. Ronit Bird saved my sanity on that. Her Dots ebook is all the 0-10 and how to apply them to basic math. Then she has C-rods where she starts going up. 

I just don't know what part the enceph. plays. For me the hardest thing in that was being PATIENT and finding materials to let him tread water in a concept when he needed to tread water. You're happy for him to tread water, apply, etc., you just need to be connected with more resources. So what I was doing was going to every online curriculum publisher that had pdf ebooks and looking through EVERYTHING.

https://www.teachercreated.com/books/math?level=preschool

https://www.teachercreated.com/books/math?level=kindergarten

https://www.teachercreated.com/products/cut-and-paste-math-3708

Here, there's teacher created resources and teacher created materials. I've used them both, but the above are what I'm saying for now. 

 

[PeterPan]

http://www.ronitbird.com

[PeterPan]

https://www.carsondellosa.com/0742418200-eb--using-the-standards-problem-solving-workbook-grade-k-ebook-0742418200-eb/  Here's one of the books from the Using the Standards series I was talking about. We were doing gr2 and up, but see what you can find. I think they used to have more. Maybe you can find them in print on ebay or amazon or something if you find they're a good fit. 

Remember too, you probably know this, but there's Vygotsky and his sorry I forget the term. There's a theory of instrction and you can read about it. Sorry, brain blip. But it's very american to say what do you know, like it's a test. But what you can do is say what can they do with support, what can they walk up to doing, what are they on the cusp of doing indepenendently, kwim? Zone of proximal instruction. 

So part of what you're feeling with him and don't maybe have words for is that idea that he can do things together that he can't yet do independently. And then you can look at your math sesssions that way. What is independent level, what is together, and start to have a MIX, kwim? So you can set him up STATIONS where he can work INDEPENDENTLY. Might mean he's matching things with file folder games. Christine Reeve is the bomb for this and she has a whole website, FB, TPT store, etc. You will LOVE her stuff. 

When I toured a small autism school locally, they had that kind of balance. Three math session components, one independent, one with the teacher at instructional level, and one with an aide that was semi-independent. So then you're doing more but still fitting him. 

[domestic_engineer]

I know you’re well versed in rightstart.  So I’m just brainstorming with you here .....  

What about doing the time, money, and Roman numeral lessons with him while you wait for it to click.   If it goes well, maybe you can use those concepts to get back to your current struggles.   (You could also jump ahead to the solids/geometry lessons.).
 

FWIW - my struggling learner will still have days where 25 is 2+10+5. I usually have to get out the cuisenaire rods (and pray for patience).  Maybe try saying the 2-ten part loudly and the 5 part softly?
 

one other idea to help the place value struggle:  James Tanton’s Exploding Dots materials. When my youngers tagged along on viewing these videos, the RS lessons were a breeze when we got to that point. 

[PeterPan]

C-rods are superior to the abacus for dyscalculia, because they're fixed quantities. Once you get the crods down you go BACK to the abacus.

[Kanin]

I like all of the ideas above. Ronit Bird is the bomb. I love her stuff.

 

Can he count objects 1-1 above 10? You could just work on doing that, but not attaching any numerals to the amounts. 

 

How about math stories, like if 4 cats are on the bed and 3 more join them, how many cats are on the bed now?

 

 

 

[Kanin]

He is SO cute, by the way! 

[Cake and Pi]

13 hours ago, PeterPan said:

Where is his IQ, if you don't mind us asking?

Average as of last testing, but it keeps going up as he gets older. This could be related to his brain slowly remodeling after perinatal injury, or maybe they just haven't been able to accurately measure his IQ all along because of his communication (esp. low receptive) difficulties. 

12 hours ago, PeterPan said:

I just don't know what part the enceph. plays.

I don't either! It seems to affect every part of him, though. He technically has super mild CP (the rare, non-spastic low-tone kind), growth problems, learning and processing struggles, and I presume his unusual memory stuff is also a direct result. We have a strong family history of giftedness and autism, so I presume that his profile is exactly that of an autistic, gifted individual with a hypoxic ischemic brain injury, who therefore may no longer be gifted, but perhaps has remnants of that wiring.

The biggest struggle is that he has regressions. He can learn something, KNOW it, and then loose it, gone, and have to slowly, painfully relearn it.

13 hours ago, PeterPan said:

How does he do with workbooks? some of the publishers like Teacher Created Materials and Evan-Moor have wonderful workbooks. There's a cut and paste math. There's graphing math art where they color squares (you could use a bingo blotter to make holiday designs. 

He likes workbooks. He can't necessarily write his responses, but he can say or point to what he wants on a choice board and I write it in a light color so he can trace it after. We've also had some success with letting him use stamps... if/when he doesn't get distracted by the stamps and start stacking or lining them up. He LOVES dab-and-dot markers and cut-and-paste activities and can do those kinds of activities for hours at a time. Thanks for all the links and suggestions. I'll order some and we'll see how it goes.

12 hours ago, PeterPan said:

Zone of proximal instruction.

I think my mental struggle is that he's not able to do this stuff with numbers over 10 even *with* me right there coaxing him along. It's just beyond him. We have days where he seems to be on the cusp of getting it, he says or builds or points to something that's right, then it's gone. We had about a week where he was able to say or pull out the matching place -value card for multiples of 10 presented to him on the abacus, but not with base-10 blocks or c-rods or anything else, and then we we took the weekend off and he couldn't do it anymore the following Monday. It's been a couple more weeks and the skill hasn't come back yet. He's frustrated. I'm heartbroken. It's too hard.

12 hours ago, domestic_engineer said:

What about doing the time, money, and Roman numeral lessons with him while you wait for it to click.   If it goes well, maybe you can use those concepts to get back to your current struggles.

I think I'm going to try introducing the whole-part circle sets next week and do some partitioning with 5. He's not yet grasping even/odd or counting by 2s. He's struggling with the "comes after" game. He can't say what number comes after another unless he starts at 1 and counts up, but he often forgets what number he was counting up to, so he just doesn't ever get there. We could really just camp out on these first 20-25 lessons in RS for a while long, I think. He doesn't know the days of the week. We could work on recognize numbers on the dot cards (he only subitizes in particular patterns, and that's not one of them). Clocks are really out of reach, but we can focus on parts of the day like morning, afternoon, evening, earlier, now, later, etc. He gets time words confused and is super stuck on the concept of "tomorrow."

I'll check into that Exploding Dots resource you suggested and try altering the way I'm saying RS number names. Thank you.

11 hours ago, Kanin said:

Can he count objects 1-1 above 10? You could just work on doing that, but not attaching any numerals to the amounts. 

 

How about math stories, like if 4 cats are on the bed and 3 more join them, how many cats are on the bed now?

I think it's probably a good idea to work on counting with 1-to-1 above 10. Thanks for the idea. Math stories are waaaay hard for him. His narrative language is pretty spotty. You can give him 4 bear counters and tell him to get 3 more. Then he can count out the 3 more and count the entire 7 (starting from 1). If you remind him or strategically arrange the 7 into groups of 5 (arranged like the 5s on dice) and 2, he may say there are 7 without counting.

Edited February 19, 2021 by Cake and Pi

[PeterPan]

4 hours ago, Cake and Pi said:

I'll order some and we'll see how it goes.

Just a disclaimer. I bought the books as ebooks and printed so I could give him one page at a time. I don't know which things are consumable if bought as print or how that works. 

4 hours ago, Cake and Pi said:

I think my mental struggle is that he's not able to do this stuff with numbers over 10 even *with* me right there coaxing him along. It's just beyond him. We have days where he seems to be on the cusp of getting it, he says or builds or points to something that's right, then it's gone. We had about a week where he was able to say or pull out the matching place -value card for multiples of 10 presented to him on the abacus, but not with base-10 blocks or c-rods or anything else, and then we we took the weekend off and he couldn't do it anymore the following Monday. It's been a couple more weeks and the skill hasn't come back yet. He's frustrated. I'm heartbroken. It's too hard.

Well check out Ronit Bird. She breaks it down more than RS. If not, then maybe kinda go with the anything that works approach. Like you may need to google for materials for ID for math (if dyscalculia materials won't get him there) and use their strategies. That CP and the early challenges is rough because you get affected parts of the brain. Has he had a neuropsych eval to pinpoint these things? Oh, you said tons of psych testing. But I think try the Ronit Bird, because it *may* get you there. And if it doesn't, then Touch Math or something that is identified as appropriate for ID. People can be contradictory and multiple things at once. It's ok to use what works. 

 

4 hours ago, Cake and Pi said:

I think I'm going to try introducing the whole-part circle sets next week and do some partitioning with 5. He's not yet grasping even/odd or counting by 2s. He's struggling with the "comes after" game.

Yeah, maybe go back and do the Dots ebook from Ronit Bird. It sounds like that's where he needs to be. And it's $10, so you're not out a ton if it doesn't it. That's what I was saying about RS always being too hard, no matter what you did.

It's not a surprise that he doesn't get "comes after". Have you tried any sequencing games? Sequencing is hard with autism, and you've got the math disability on top of it. Lakeshore sells some adorable sequencing puzzles for numbers, but you can just find them online and print, make them yourself, whatever you want. 

https://www.lakeshorelearning.com/products/math/numbers-counting/sequencing-numbers-span-stylewhite-space-nowrap1-10-span-puzzles-setof3/p/LL567  Here's an example. They start small and go more complex. You can doubtless find printable files on TPT or free googling to be similar. 

4 hours ago, Cake and Pi said:

If you remind him or strategically arrange the 7 into groups of 5 (arranged like the 5s on dice) and 2, he may say there are 7 without counting.

Yes, Ronit Bird's Dots ebook would help clear this up. RightStart doesn't go as far as they need to for dyscalculia. RB backs it up, building a sense of ALL the numbers inside the number. So we play games (yes, games plural) for seeing numbers inside of other numbers. They're actually pretty witty! And you can change up the manipulatives, extend it, whatever. You'll see. Get the ebook, start at the beginning, work forward.

If you do RB's Dots book and REALLY DO IT (and if it fits him obviously, can't guarantee that part), then by the end he should have internalized and nailed all the addition facts. At that point you take the logic and extend it to another of her games ( Positive/Negative Turnovers) and then he's doing subtraction. So at that point you've nailed all the single digit addition and subtraction. 

I wouldn't try to go forward. Actually nail the concepts in Dots, then go forward. 

4 hours ago, Cake and Pi said:

Math stories are waaaay hard for him.

Just checking, is he getting SLP services? I know you have a lot going on, but it sounds like he has some needs there.

[PeterPan]

I love that his scores are going up btw. I'm jealous of you getting a nonverbal IQ test. It always kinda irks me that they don't run this on ds, even though we know his language is funky and drops when stressed. 

[Not_a_Number]

7 hours ago, Cake and Pi said:

I've now been trying to teach him numbers bigger than 10 for about the last four months and made only very modest progress. He cannot read or comprehend numbers bigger than 10. Fifteen and fifty-five are exactly the same to him because they both have 5s, as are twelve and twenty because they both have 2s. Using RS's special naming system almost seems to make it *worse* because he'll take a number like 25, which RS would call "2-ten 5" and say "two, ten, five," put them all together to make 17, and think it's 25. I've tried using the abacus, base-ten blocks, home-made, Noom-colored c-rods, place-value cards, place-value chips, tally sticks and tally marks, ten-frames, beads and pipe cleaners, MUS's Decimal Street, using fingers from multiple family members, making bundles of rods, basically EVERYTHING at this point. If a number is bigger than 10, he doesn't get it.

OK, I know absolutely nothing about this kind of profile. Absolutely nothing. So tell me if I'm off-base... 

... but is it possible you're using TOO MANY things at this point? And also, is it possible that writing the numbers down is actually confusing matters for him? Perhaps having a single mental model that you use over and over and over and over again (perhaps minus the written component) would help him a lot more? 

Edited February 19, 2021 by Not_a_Number

[Terabith]

I think you need to talk to @BaseballandHockey.  This is her wheelhouse.

[Drama Llama]

8 hours ago, Cake and Pi said:

I'm trying to decide what direction to go with my 7.5yo (2nd grade age) SN learner and would love your opinions and input.

I'm using a mashup of Right Start 1st edition level A, ST Math kindergarten level, MUS Primer, and my own inventiveness with a variety of other materials and manipulatives. I have a strong math background and already successfully taught my three older kids all of this stuff already, but he's his own animal. Nothing I thought I knew applies.

Altogether it took him 6-7 months to master (90% accuracy or so) subitizing and counting with 1-to-1 correspondence 0-10 along with numeral recognition and matching language, but he eventually got it. He's able to trace numbers 0-10 and can independently write maybe half of them (with a high rate of reversals).

I've now been trying to teach him numbers bigger than 10 for about the last four months and made only very modest progress. He cannot read or comprehend numbers bigger than 10. Fifteen and fifty-five are exactly the same to him because they both have 5s, as are twelve and twenty because they both have 2s. Using RS's special naming system almost seems to make it *worse* because he'll take a number like 25, which RS would call "2-ten 5" and say "two, ten, five," put them all together to make 17, and think it's 25. I've tried using the abacus, base-ten blocks, home-made, Noom-colored c-rods, place-value cards, place-value chips, tally sticks and tally marks, ten-frames, beads and pipe cleaners, MUS's Decimal Street, using fingers from multiple family members, making bundles of rods, basically EVERYTHING at this point. If a number is bigger than 10, he doesn't get it.

Should I keep trying? It took half a year to get 0-10; maybe we just haven't sat with this concept long enough. Perhaps I'm being impatient. 

Should I step away from numbers >10 for a few months and focus more heavily on other math concepts? He can't get past the 25-30% mark in any of these kindergarten curricula we're using because they all incorporate place value to some extent from there on out. Should I ditch the curricula and work on teaching *whatever* he can understand? He seems to get basic addition and subtraction within 10 with manipulatives. Maybe I should try to continue along that line and try to teach him addition and subtraction with symbols? He's pretty good with visual patterns; we could do more patterning. He's starting to show a better grasp of bigger/smaller, greater/less, side/middle (but not left/right). He spontaneously builds some pretty complex things out of blocks, magnatiles, and math manipulatives -- see picture below.

 

Background in case you need it: Recently diagnosed ASD 2 with language impairment/disorder; conversationally verbal with lots of echolalia/scripting... except when he's not and just cries/grunts/moans; encephalopathy; global delays but average IQ; basement-low verbal memory and general memory funkyness; dx deferred, but still strongly suspected SLDs in reading, writing, and math. He left public school 11 months ago, and I've been homeschooling him continuously ever since. We did not take a summer break. (He left public school originally due to covid school closure, but I am a seasoned homeschooler in my 9th year with one or more of my three other kids.)

So, first of all, what an adorable picture.

I'd start by saying that I have zero experience with Right Start Math, so I can't comment at all on that.  Also I only read the first post and @Terabith's so I might be repeating what has been said.  

I seem to remember something from you about pulling him out of school, maybe due to covid?  Are you crisis schooling like me, and  planning on sending him back to school soon?  If so, what kind of setting?  

If you're homeschooling, my instinct would be to  stick with numbers 0 - 10 or even 0 - 5 until he's showing  lots of fluency with quantities he can subitize and visualize.  There is so much math you can do with those small numbers.  

Estimation and measurement with objects (e.g. how many inch cubes long is this pencil)

Comparing

Picture graphs

Composing and decomposing dice patterns, and then connecting those to addition and subtraction. 

Odd/Even games like in the Ronit Bird Dice patterns book, and using this book https://apps.apple.com/us/app/domino-addition/id912574293

Odds and Evens games with fingers (lots of games with fingers)

Counting on as a strategy for addition (I do a lot of dice games with one dice modified to be 0, 1, and 2) 

Matching games with different sets (e.g. matching a  photo of him holding up 3 fingers, to a dice pattern, to a written numeral)

Addition and subtraction word problems with very small numbers.

Work with 5 frames and then 10 frames.

Number lines

Counting pennies

Counting combinations of 1 nickel and pennies 


When I saw fluency with those kinds of things, a solid understanding of concepts like 1 more, counting on, or of 3 and 2 being a "name" for 5, visualizing addition and subtraction, etc . . .  Then I'd start on about place value to 20.  And then I'd stick within 20 until he's solid there. 

Place value is really developmental, I wouldn't rush it.  

[Not_a_Number]

Nm.

Edited February 19, 2021 by Not_a_Number

[Cake and Pi]

6 hours ago, PeterPan said:

Well check out Ronit Bird.

I cannot figure out how to buy or view her ebooks. We only have Android devices, and I can't find an Apple book reader for my phone or PC. 😞 

6 hours ago, PeterPan said:

Just checking, is he getting SLP services?

Not at the moment; we're trying to get speech up and going again. Other than the four-month gap in services we're currently in, he's had 1-4 hours of speech therapy per week pretty much continuously since 16mo, though. Hopefully we'll get through the wait list here soon.

6 hours ago, PeterPan said:

Have you tried any sequencing games? Sequencing is hard with autism

He sequences pictures like a pro, just not words. He can't tell you (and doesn't understand) a short verbally delivered story. But if you give him pictures of an event that doesn't require him to have heard and understood a story, he can sequence great. For example, he can sequence pictures of a snowman melting or pictures of someone eating a meal. 

He can put the numerals in order when I give him cards with numbers on them. He can put c-rods in order. He can also sequence pictures of hands holding up 1-10 fingers. We've worked on this a lot. But if you ask him, "What comes after 5?" he can't tell you without counting up from 1. He's also just as likely to decide that 4 is the number after 5 as he is to correctly arrive at 6, which seems like it would take some serious extra mental power -- counting 1, 2, 3, 4, 5, recognizing that 4 was adjacent to 5 in that sequence, and remembering it even after passing it. I can put 5 on the abacus to have in front of him when I ask what comes after it, and he STILL (usually) wants to count up from 1.

4 hours ago, BaseballandHockey said:

I seem to remember something from you about pulling him out of school, maybe due to covid?  Are you crisis schooling like me, and  planning on sending him back to school soon?  If so, what kind of setting?

Yes, I pulled him because remote learning was utterly ineffective for him. Six months ago I would have said that our plans were to send him back asap, but now that I'm (sort of) getting the hang of teaching him and balancing him with my other three, and now that I've seen him make so much better progress than he was at school, I don't know. His anxiety is also significantly improved being home. If we can smooth out the sibling dynamics a bit, I think I may be homeschooling him longer term. In any event, I have no intention of sending him back while there's a reasonably high risk of covid closures and necessitated remote learning for any part of the school year.

4 hours ago, BaseballandHockey said:

When I saw fluency with those kinds of things, a solid understanding of concepts like 1 more, counting on, or of 3 and 2 being a "name" for 5, visualizing addition and subtraction, etc . . .  Then I'd start on about place value to 20.  And then I'd stick within 20 until he's solid there.

Thank you for the list of ideas! This is a good place for us to start, and I think you're right that we do need to sit with a smaller set of numbers for now. I think my biggest problem is that I feel like I need a guide book to tell me what all to cover. I need a check list of skills, iykwim. 

5 hours ago, Not_a_Number said:

... but is it possible you're using TOO MANY things at this point? And also, is it possible that writing the numbers down is actually confusing matters for him? Perhaps having a single mental model that you use over and over and over and over again (perhaps minus the written component) would help him a lot more?

Using too many things? No, probably not because we don't use them all at once, and I've done acrobatics with the lesson sequences so that we're covering the same main topics no matter what curricula or other resources we're using. We *must* use different kinds of manipulatives or he doesn't generalize. He knew that 5 fingers was "five" an entire year before he could identify or compose 5 of literally anything else (and before he knew 2, 3, or 4 🤷‍♀️). Similarly, he could distinguish between 4 fingers and 5 fingers for many months before he could do the same with blocks, pencils, crackers, etc. Other kids learn a skill and apply it wherever it fits. He has to be taught the same skill in each separate scenario.

Trying to learn too many different things at once? Yes. Definitely yes. RS has too many unrelated activities in each lesson. One of the reasons I bought MUS is because I thought a mastery approach would work better. MUS would probably work great if it had triple or quadruple the practice pages for each lesson and wasn't trying to introduce place value 30% of the way into kindergarten and then incorporating that place value into almost every lesson for the rest of the book! Sigh. We're just going to have to ditch curricula for the time being. (Aaaaah! I feel like a ship without a rudder!)

With my other kids you can introduce them to a topic and let it sort of marinate while you move on to other things. Then when you come back to that original topic they have a deeper understanding and are better able to tackle it. It's not like that with DS 7. If you introduce a topic and don't revisit it every day for weeks or months, it's like you never visited it at all... except when he randomly remembers exactly how a lilyturf flower cluster looks after a walk and draws it nearly perfectly, or he recounts in detail that time when Grammy took him trick-or-treating four years ago (before he was speaking), or he spouts off entire sentences someone said weeks or months ago with that person's exact tone of voice and posturing. Very funky memory stuff.

The writing is a conundrum. Even though he is presumed to have a SLD in writing and had writing goals in his school IEP, writing and fine motor are actually relative strengths for him. He enjoys coloring, drawing, and writing for short stints on good days. His hypotonia limits his writing stamina, and he struggles to form letters and numbers from memory... except when he doesn't, lol.

Edited February 19, 2021 by Cake and Pi

[Not_a_Number]

4 hours ago, Cake and Pi said:

I cannot figure out how to buy or view her ebooks. We only have Android devices, and I can't find an Apple book reader for my phone or PC. 😞 

Not at the moment; we're trying to get speech up and going again. Other than the four-month gap in services we're currently in, he's had 1-4 hours of speech therapy per week pretty much continuously since 16mo, though. Hopefully we'll get through the wait list here soon.

He sequences pictures like a pro, just not words. He can't tell you (and doesn't understand) a short verbally delivered story. But if you give him pictures of an event that doesn't require him to have heard and understood a story, he can sequence great. For example, he can sequence pictures of a snowman melting or pictures of someone eating a meal. 

He can put the numerals in order when I give him cards with numbers on them. He can put c-rods in order. He can also sequence pictures of hands holding up 1-10 fingers. We've worked on this a lot. But if you ask him, "What comes after 5?" he can't tell you without counting up from 1. He's also just as likely to decide that 4 is the number after 5 as he is to correctly arrive at 6, which seems like it would take some serious extra mental power -- counting 1, 2, 3, 4, 5, recognizing that 4 was adjacent to 5 in that sequence, and remembering it even after passing it. I can put 5 on the abacus to have in front of him when I ask what comes after it, and he STILL (usually) wants to count up from 1.

Yes, I pulled him because remote learning was utterly ineffective for him. Six months ago I would have said that our plans were to send him back asap, but now that I'm (sort of) getting the hang of teaching him and balancing him with my other three, and now that I've seen him make so much better progress than he was at school, I don't know. His anxiety is also significantly improved being home. If we can smooth out the sibling dynamics a bit, I think I may be homeschooling him longer term. In any event, I have no intention of sending him back while there's a reasonably high risk of covid closures and necessitated remote learning for any part of the school year.

Thank you for the list of ideas! This is a good place for us to start, and I think you're right that we do need to sit with a smaller set of numbers for now. I think my biggest problem is that I feel like I need a guide book to tell me what all to cover. I need a check list of skills, iykwim. 

Using too many things? No, probably not because we don't use them all at once, and I've done acrobatics with the lesson sequences so that we're covering the same main topics no matter what curricula or other resources we're using. We *must* use different kinds of manipulatives or he doesn't generalize. He knew that 5 fingers was "five" an entire year before he could identify or compose 5 of literally anything else (and before he knew 2, 3, or 4 🤷‍♀️). Similarly, he could distinguish between 4 fingers and 5 fingers for many months before he could do the same with blocks, pencils, crackers, etc. Other kids learn a skill and apply it wherever it fits. He has to be taught the same skill in each separate scenario.

Trying to learn too many different things at once? Yes. Definitely yes. RS has too many unrelated activities in each lesson. One of the reasons I bought MUS is because I thought a mastery approach would work better. MUS would probably work great if it had triple or quadruple the practice pages for each lesson and wasn't trying to introduce place value 30% of the way into kindergarten and then incorporating that place value into almost every lesson for the rest of the book! Sigh. We're just going to have to ditch curricula for the time being. (Aaaaah! I feel like a ship without a rudder!)

With my other kids you can introduce them to a topic and let it sort of marinate while you move on to other things. Then when you come back to that original topic they have a deeper understanding and are better able to tackle it. It's not like that with DS 7. If you introduce a topic and don't revisit it every day for weeks or months, it's like you never visited it at all... except when he randomly remembers exactly how a lilyturf flower cluster looks after a walk and draws it nearly perfectly, or he recounts in detail that time when Grammy took him trick-or-treating four years ago (before he was speaking), or he spouts off entire sentences someone said weeks or months ago with that person's exact tone of voice and posturing. Very funky memory stuff.

The writing is a conundrum. Even though he is presumed to have a SLD in writing and had writing goals in his school IEP, writing and fine motor are actually relative strengths for him. He enjoys coloring, drawing, and writing for short stints on good days. His hypotonia limits his writing stamina, and he struggles to form letters and numbers from memory... except when he doesn't, lol.

Yeah, from this description, I’d guess that waiting on place value is a good idea. 

I know that when I was doing the homeschooling classes, it felt like a good idea to wait for “counting on” to do place value. Not that they were technically related, but somehow it felt like if kids weren’t able to do that, place value wouldn’t make any sense. And it sounds like he’s not there.

Place value is quite abstract. I know that the specific things we do won’t work for your kiddo due to the fact that he doesn’t generalize, but we work on it with differently-colored poker chips, and having spent lots of time doing it — the idea of different things that look very similar being worth different amounts is just HARD. I wonder if playing games where you can use a single poker chip for a 10 would at least be a signal for him being ready? Because poker chips are at least easier than numbers in writing, and yet they are still hard and there’s still something THERE. 

But I think @BaseballandHockey’s list is great. I remember working on subitizing with my homeschool classes using Tiny Polka Dot — some kids really did need it. And he probably needs more than that, including finger stuff.

[Kanin]

5 hours ago, Cake and Pi said:

But if you ask him, "What comes after 5?" he can't tell you without counting up from 1. He's also just as likely to decide that 4 is the number after 5 as he is to correctly arrive at 6

I have an older student who does this. Turns out she doesn't understand the meaning of the words "before" and "after" in other contexts, either. 

[Kanin]

16 hours ago, Cake and Pi said:

I think my mental struggle is that he's not able to do this stuff with numbers over 10 even *with* me right there coaxing him along. It's just beyond him. We have days where he seems to be on the cusp of getting it, he says or builds or points to something that's right, then it's gone. We had about a week where he was able to say or pull out the matching place -value card for multiples of 10 presented to him on the abacus, but not with base-10 blocks or c-rods or anything else, and then we we took the weekend off and he couldn't do it anymore the following Monday. It's been a couple more weeks and the skill hasn't come back yet. He's frustrated. I'm heartbroken. It's too hard.

I hear you. It's probably time to give yourself a mental break and just go with what he's able to do right now. I know the pressure of wanting to - needing to! - move forward, but ultimately, we can't control that. 

I had a student a long time ago who had no understanding of numbers over 10. Place value? No way. We did Ronit Bird's dots book, and used her other materials. We built numbers 1-20, on a big table, over and over and over again, with cuisenaire rods, with dot cards, with base ten blocks, etc. We did this for a LONG LONG TIME. Like, almost a year. Finally, one day, he just seemed to... understand! He just needed a really long time to sit with the idea and play with it and explore. After the fundamental idea of place value was there, numbers higher than 20 were no problem. 

[PeterPan]

5 hours ago, Cake and Pi said:

I cannot figure out how to buy or view her ebooks. We only have Android devices, and I can't find an Apple book reader for my phone or PC. 😞 

Can you *borrow* an apple device? Yes, the ebooks are only apple. She has print books, so you'd be looking at Dyscalculia Toolkit. It contains the equivalent of the first three ebooks and more. I just really like the ebook because it has embedded videos and clear instruction sequences that worked well for ds. And it's a great starter price ($10). So if you could borrow even for a while or like buy it for the apple device and go visit that person once a week and go prep the next couple sections, that could work.

5 hours ago, Cake and Pi said:

Not at the moment; we're trying to get speech up and going again. Other than the four-month gap in services we're currently in, he's had 1-4 hours of speech therapy per week pretty much continuously since 16mo, though. Hopefully we'll get through the wait list here soon.

Having btdt, I finally gave up on hoping SLPs would do everything that needs to be done. You might try ProEdInc and Super Duper and pick out some things to do with him yourself. Has he had ABA? They do a heavy language emphasis in the early years and it sounds like he missed some of that instruction and those skills. So any of the checklists they use would also get you going forward. Northern Speech Services has a kit for early language ABA stuff. I use ProEdInc/Linguisystems a lot. I'm also keen on Therapro but for speech they're usually reselling ProEd stuff. Super Duper has some good things and they're more on the visual/fun/colorful side. Their Grammar Processing Program is terrific. 

5 hours ago, Cake and Pi said:

He sequences pictures like a pro, just not words. He can't tell you (and doesn't understand) a short verbally delivered story. But if you give him pictures of an event that doesn't require him to have heard and understood a story, he can sequence great. For example, he can sequence pictures of a snowman melting or pictures of someone eating a meal. 

Have they tried him on AAC? LAMP will go on sale pretty soon. If could go on the least expensive ipad and then you'd have a device for the Ronit Bird ebooks too.

5 hours ago, Cake and Pi said:

remote learning was utterly ineffective for him.

Yes, that's pretty much what I'm hearing locally for kids with significant disabilities. My ds will do short teletherapy sessions, but the minute they go high demand he's outta there. 

5 hours ago, Cake and Pi said:

with a smaller set of numbers for now. I think my biggest problem is that I feel like I need a guide book to tell me what all to cover. I need a check list of skills, iykwim. 

Ronit Bird's Dots ebook (or her print Toolkit book) will do that for you. 

5 hours ago, Cake and Pi said:

We *must* use different kinds of manipulatives or he doesn't generalize.

ABSOLUTELY. You are so right on this. 

5 hours ago, Cake and Pi said:

MUS would probably work great if it had triple or quadruple the practice pages for each lesson and wasn't trying to introduce place value

It's more than that. The steps are too big. Ronit Bird breaks everything down into smaller steps cognitively so you can spend more time on each one. You don't need more time on a BIG step. You need more time on SMALLER steps.

5 hours ago, Cake and Pi said:

If you introduce a topic and don't revisit it every day for weeks or months, it's like you never visited it at all... except when he randomly remembers exactly how a lilyturf flower cluster looks after a walk and draws it nearly perfectly, or he recounts in detail that time when Grammy took him trick-or-treating four years ago (before he was speaking), or he spouts off entire sentences someone said weeks or months ago with that person's exact tone of voice and posturing. Very funky memory stuff.

So I'm totally with you on the mindblindness, gestalt vs. details thing. He's showing you what his brain is noticing in situations, and he's missing the big picture. But I also think that what you're seeing specifically with the math was that the steps were so big that rather than being comprehended they went into SOME OTHER FILE FOLDER in his brain. Like they went into the "things I know about dominoes" folder or the "animals do this" folder or whatever. 

Do you have dominoes? My ds received a set as a gift and they are terrific! You can pair them with the Dots lessons to give you another manipulative. So we'd do the lesson one day with say colored stones. The next day with a set of tiny math counters that were animals. The next day with m&ms. The next day with dominoes. The next day with playing cards that showed fingers. And so on. All the same concept, but lots of different ways like you're saying. Till that teeny tiny step generalizes into his brain and gets put in the MATH folder, not the animals folder or the things I do after lunch folder or the things that are shiny folder or the *language* folder. 

You can look into this, but dyscalculia occurs because the brain actually does math on TWO sides of the brain, not just one. And it's foggy for me now, but one side is sort of conceptual, and on that side my ds is considered gifted. (And he actually is sort of surprisingly math intuitive!) And on the other side of the brain is where all that language goes and how it all does the numeracy and connects to language. I have no clue, look it up. But when you see the structures and that break, it makes sense why your dc can be sort of BOTH and why he's having language troubles and math troubles at the same time.

In other words, it matters what part of the brain we're hitting with our lesson and where things are getting filed.

Ok, I'm going to throw out for your consideration Rapid Prompting. And people are like oh that's horrible. Fine, I've seen the double blind studies, I know it's a form of facilitated communication that has serious issues. I also know people doing it who have loving, kind interactions with their extremely challenged children at that the youths are HAPPY and interacted with.

So anyways, with that huge disclaimer (that I agree there are issues), I would like to suggest you look into it anyway just to see more teaching ideas. For instance, she (the lady who wrote the book, who is in the videos, Soma something, I forget) shows methods using words on slips. Simplifying responses. Looking for the pathway that is OPEN. This flexibility, focusing on the response, is really brilliant. To me it's like communication by any means, kwim? If they can point, we point. And then how you could modify your math lesson to decrease language and increase COMMUNICATION.

You still need the Ronit Bird lessons to get the steps smaller. But I'm thinking the Rapid Prompting and just that idea of communication by all means, that maybe it could get you some *ideas* of how to improve communication in the math lessons. If he can sequence pictures but not do the language, it just means more is inside.

Has he ever had someone consider PROMPT for him? Now this is not Rapid Prompting. PROMPT is a motor planning of speech therapy and what my ds got. It can be good for CP, because CP involves motor planning. Might be something to look into. (As in I'm a huge, huge fan, think it's amazing.) Disclaimer, done by anyone less than certified and it is usually not done properly/completely. So if you've had it and they weren't certified, try again with someone certified. 

6 hours ago, Cake and Pi said:

he struggles to form letters and numbers from memory... except when he doesn't, lol.

Is he having some kind of seizures? I have no clue. I'm just with you that that's interesting that it varies. My ds does not VARY, kwim? Now his output can go *downhill* but it's pretty consistent. Varying would concern me and make me wonder why.

6 hours ago, Cake and Pi said:

The writing is a conundrum.

Have they done an assistive tech eval on him? I'm with you that the big challenge with a person with both speech and physical writing disabilities is you're trying to figure out how to help them get out what is inside. And it's tricky to tell whether it's in there and not coming out because of motor planning of speech problems or if the language is flat not there. I *assumed* the language was there with my ds and it wasn't. We saw that clearly once we got LAMP, because he couldn't even get out of the gate with it. It's normal to teach/introduce *some* but it was obvious he was not processing language at the individual word level and hence could not use tech that wanted you to input word by word. That was an ugly moment.

6 hours ago, Cake and Pi said:

I've seen him make so much better progress than he was at school

I LOVE this. Even with my ds home, I spent a lot of time trying to get "professionals" to help us. Sometimes they were good, but honestly we have probably wasted a lot of time with professionals. There are many reasons why it happens, but I'm just saying sometimes they don't get done what really needs to be done. And it's aggravating. 

So no guilt, just take steps forward. You're figuring out this stuff. These are the hardest kids, the most complex, and it really is rocket science. It's CUSTOM. And I think that's the problem. It takes so much time to sort it out and then make it happen, that kids can just slide on through with some sorta service happening but never the magic and breakthroughs. I just love that you're making progress now. That must feel so good even though it's also frustrating seeing more that needs to be done.

[PeterPan]

14 minutes ago, Kanin said:

just go with what he's able to do right now.

Total, total aside, but when audiologists do APD evals and screenings they generate an ear advantage score. It reflects maturation in the brain, and the numbers I've been given on my kids matched other numbers we had. The Vineland that they do for ASD screening will kick out a number I think. 

I know I get in this rut of thinking my ds is 12, he ought to do such and such, but by literal brain maturation and neurological readiness, he's working more like 9/10. And for a 7 yo with this many issues, he could be working at 4/5. We wouldn't expect a 4/5 yo to be nailing some of this stuff yet or even be ready to. 

Op can ask her people for that number of how much delayed he is. If he's been diagnosed ASD, they have a sense and a number they're working with. 

[Kanin]

Here are some photos of the early subitizing/place value work we did.

I'm not sure what we were doing with the c-rods to 30... I think we ran out of tens but he really wanted to get to 30 so we improvised.

 

[PeterPan]

A couple years ago we were through the Ronit Bird Dots book, he could do add/subtraction for basic facts and a bit more (within 100 probably), and some worksheet wanted him to find *pages in a book*. Nope, couldn't do it. 

So even when you think you win, you go to the next thing or the language changes and it's there, sigh. I'm not sure if he could do it now. In theory he could, but it's not *easy* for him. We've been doing some tracking pages for the alphabet and that's hard for him. I think tracking pages could be another way to go at the number sequencing. 

http://www.cdfieldtrips.com/02KOTshow/pages/pg13kot.html  Here's an example of what I'm talking about. I tried to buy the CDs of the books from this vendor and the order never went through. Eventually got a refund from Paypal. I made screenshots to print of the pages. They've been (appropriately) challenging for ds. I'm not sure really what that means, because per the VT (vision therapy) people his visual perception is fine. The only thing he had trouble with on their testing was VMI=visual motor integration.

So somewhere in there, where vision and math and sequencing and all this stuff merge, there are some glitches that tracking pages seem to catch. Ann Arbor has some books for tracking too. Therapro and High Noon sell them. But I don't know exactly why this is needful or challenging for ds, sigh. The alphabet tracking is WICKED HARD for him. 

[PeterPan]

1 hour ago, Kanin said:

but he really wanted to get to 30 so we improvised.

I love this! And what a great room to work in!! I'm totally warped from working with ds, because I look at pictures like that and immediately notice how he's sitting, his escape path if it gets overwhelming, the beanbag chair for a break, the light level and bouncing noise from the floor, etc. LOL 

[Kanin]

13 minutes ago, PeterPan said:

I love this! And what a great room to work in!! I'm totally warped from working with ds, because I look at pictures like that and immediately notice how he's sitting, his escape path if it gets overwhelming, the beanbag chair for a break, the light level and bouncing noise from the floor, etc. LOL 

Ohhh, it's a deceiving picture. The room is actually a big room split into 4 quadrants with no real walls in between, with one teacher and up to 4 kids in each quadrant. It's loud and chaotic and the complete opposite of what you would want for a kid with learning challenges. Ugh!

[PeterPan]

7 minutes ago, Kanin said:

Ohhh, it's a deceiving picture. The room is actually a big room split into 4 quadrants with no real walls in between, with one teacher and up to 4 kids in each quadrant. It's loud and chaotic and the complete opposite of what you would want for a kid with learning challenges. Ugh!

Yeah, I kinda figured with that commercial flooring. 

[Not_a_Number]

So... again, I have no experience with kids specifically like this, so all of my comments come from experience with kids with a different profile. So this should all be taken with a BOULDER of salt. 

However, I'll note I see failure to generalize at higher levels all the time. We often teach kids as if they should be able to extract the main idea of what we're teaching them despite all the noise, and maybe 50% of the time, it works, at least in the lower grades. But the rest of the time, kids never actually intuit what it is that we're trying to teach them in our muddled way, and that leaves them with giant gaps. I see this with variables. I see this with (typically developing) kids with place value. I see this with functions and graphs. 

So... my question to you, @Cake and Pi, would be what CONCEPT you're trying to communicate to him, and how to present a mental model that unites this concept as one. I've gotten my best results with picking one MODEL for a concept, but then applying it as broadly as possible. (That goes for lots of concepts. Operations, equals signs, fractions, vectors, whatever. They've all been situations where kids often fail to generalize from how we normally teach.) 

To me, it sounds like he's still struggling with the concept of number somewhat, which is where @BaseballandHockey's ideas are simply brilliant. But I'd also maybe spend some time thinking about what the uniting theme with the concept of number is. I've actually explicitly thought about this to some extent, and I've always thought the key idea with the concept of number is that of one-to-one correspondence, which I generally communicate by counting and also by subitizing and rearranging and other practice that comes sufficiently easy to my kids that I don't think about it a lot. But I definitely do stick to the idea of one-to-one correspondence very firmly with early math, which is where C-rods and measurement would personally give me pause -- they seem more abstract than that idea to me. (I know Ronit Bird uses them, so I may be off-base. But I just tend to wonder how they fit into the idea of one-to-one correspondence, which to me is the uniting concept it seems like you're working on.) 

I don't know if any of this is useful 🙂 . I just know that figuring out what the overarching concept for the thing I'm trying to teach is has always been fruitful to me. And since I haven't had to think much about this specific concept, it's possible my thoughts aren't much use. But this way of thinking about the problem has been helpful to me. 

[Drama Llama]

I think one to one correspondence is a great thing for a kid to have, but I also think over relying on counting as a strategy, can keep kids from really understanding the concept of quantity and a permanent attribute of a set, that can only be changed by changing the composition of the set, not by changing the location of the set, for example.   That is to say, that counting is one way we answer the question "How many are here?", but many children think it's the way we answer the question "What number will I say when I touch the last one?" 

Having that understanding, as well as the understanding that numbers can be composed of smaller number, and that that is also immutable is a pretty core understanding that you need to move on to place value, and I would keep working on it, until it appeared.  Another, simpler skill that requires those two understandings is counting on.  I wouldn't say that a kid needs instruction in counting on before instruction in place value, but I would say that a kid who is ready to understand and use place value, will often show us that by using counting on strategies fluently.

12 hours ago, Cake and Pi said:

I cannot figure out how to buy or view her ebooks. We only have Android devices, and I can't find an Apple book reader for my phone or PC. 😞 

I would buy her book, Dyscalculia Toolkit, and make use of the handful of videos that go with it on Youtube.  I'll link down below to the ones I'd use now. 

Having said that, her techniques are really awesome.  The books, at least the early ones, aren't really written for kids to use themselves.  They're written for adults to read and watch to learn how to demonstrate concepts for kids.  I'd consider reaching out to my circle to see if there's a half working device out there that you could get for free or almost free.  For example, I had an old iphone that wouldn't hold a charge.  It does me no good whatsoever as a phone, but I could absolutely sit next to an outlet and watch Ronit Bird videos on it using wifi without paying for a service.  

Maybe put out a request on FB or Freecycle, and see if you can get something? The products are that good. 

Here's a great video that lays out some of her big ideas.  I particularly like what she has to say about counting. 

Having said that, I wouldn't assume that he's got dyscalculia at this age, especially given that he's also got a significant language disability.  The ways we teach math, and the way most math curriculum is written in this country is very language intensive.  It might be that when you move to very visual methods, you will find that he makes more progress. 

Regardless of whether he has dyscalculia, however, I'd still suggest Ronit Bird.

12 hours ago, Cake and Pi said:

Not at the moment; we're trying to get speech up and going again. Other than the four-month gap in services we're currently in, he's had 1-4 hours of speech therapy per week pretty much continuously since 16mo, though. Hopefully we'll get through the wait list here soon.

He sequences pictures like a pro, just not words. He can't tell you (and doesn't understand) a short verbally delivered story. But if you give him pictures of an event that doesn't require him to have heard and understood a story, he can sequence great. For example, he can sequence pictures of a snowman melting or pictures of someone eating a meal. 

He can put the numerals in order when I give him cards with numbers on them. He can put c-rods in order. He can also sequence pictures of hands holding up 1-10 fingers. We've worked on this a lot. But if you ask him, "What comes after 5?" he can't tell you without counting up from 1. He's also just as likely to decide that 4 is the number after 5 as he is to correctly arrive at 6, which seems like it would take some serious extra mental power -- counting 1, 2, 3, 4, 5, recognizing that 4 was adjacent to 5 in that sequence, and remembering it even after passing it. I can put 5 on the abacus to have in front of him when I ask what comes after it, and he STILL (usually) wants to count up from 1.

Yes, I pulled him because remote learning was utterly ineffective for him. Six months ago I would have said that our plans were to send him back asap, but now that I'm (sort of) getting the hang of teaching him and balancing him with my other three, and now that I've seen him make so much better progress than he was at school, I don't know. His anxiety is also significantly improved being home. If we can smooth out the sibling dynamics a bit, I think I may be homeschooling him longer term. In any event, I have no intention of sending him back while there's a reasonably high risk of covid closures and necessitated remote learning for any part of the school year.

If you were going to send him back, I'd suggest some differences, because I think that aligning the sequence of what you're doing with the classroom curriculum is important. But it sounds like you're planning on keeping home for a while, so I'll offer suggestions for that. 

12 hours ago, Cake and Pi said:

Thank you for the list of ideas! This is a good place for us to start, and I think you're right that we do need to sit with a smaller set of numbers for now. I think my biggest problem is that I feel like I need a guide book to tell me what all to cover. I need a check list of skills, iykwim. 

 

Well since you asked.

I'm going to write up a checklist of skills and understandings I'd prioritize, as well as some tools.  If you like the first couple I'll send you more.  I'm going to put them in separate posts so I can write one, come back, write another one, etc . . . 

 

[Lecka]

Ime (I am sure it varies) you would be looking at Touch Math at public school.

 

There are pros and cons.  
 

Pros are it helps a lot of kids.

 

Cons are it can overly promote counting as a strategy.

 

My son was helped by it and then still moved on from counting, but some kids do not.  

 

Hundreds charts really helped my son.  

He did well with Cuisenaire rods later.

 

When he was at the point of struggling with numbers 11-20 ———- he did well with just those linking cubes that come in ones and tens.  They also come in hundreds flats.  He did well with just the ones and the tens, though.  It was simpler for him.  
 

It took him about a year to get numbers 11-20, and then from there he was faster to get up to 50, and then he went fairly quickly after that point.

 

Numbers 1-5, then 5-10, then 11-20, were the very slowest.  Then he did go more quickly once he was good with numbers through 20.

 

 

[Lecka]

There are so many ways to work on language through math.

Ime language gains helped math gains, too.  
 

My son also did well with having a number line available and making “jumps” with the number line.  
 

I am going to point out — there are language concepts involved in counting up or down.  If he is confusing more and less, it’s easy to see how more or less than a number could see to be one way or the other.  More or less is a more abstract concept (in its way) and can take endless concrete examples.
 

You might look at sentence frames.  This can be:  ______ is one more than ______.  ______ is one less than ______.

 

One more/one less and ten more/ten less can both be done with a hundreds chart.  
 

Good luck!  

[Drama Llama]

Understanding # 1:  Number is a permanent attribute of a set.  Or, in simpler terms, 4 is 4 is 4. 

Activity #1:  Introducing numbers like in Ronit Bird.

One change I'd make, is that instead of the Slavonic track she shows at the end (10 spaces, the first 5 of which are shaded), I'd use a 10's frame.  It's not that I think that 10's frames are better or worse, but they are pretty common in schools and curriculum, and so familiarity there is good.

I'd start out this way.  

Get a small whiteboard, and a set of 6 magnetic counters.  Use a Vis a Vis pen (wet erase pen) to draw a 10's frame at the top.  (Note:  You could d this on just a sheet of paper.  I really like the way the magnets don't slide around so much).

Take a dice, set it with the number up.  Tell him to make a pattern that looks like the pattern on the dice with the counters.  

Once he makes the pattern for 1.  Ask him how many.  He'll probably know.  Slide the counter up into the top right hand corner of the 10's frame.  Comment how it's still 1.  Then bring it back down.  Still 1.

Repeat with 2, 3, 4, 5 and 6.

Once you've done it that way several times, then switch it so that you roll the dice, name the number, build it, and then slide it on the track.  I'd do this as a warm up at the beginning of math class every day.  

Once you'd done it with rolling the numbers, then have one person roll the number, and tell the other person what number they rolled without showing it.  Have the other person make the number, move it up, and move it back. 

Activity # 2:  Matching games.

Dominos are a great game for this.

Make a set of cards that have the numbers 1 - 6 shown in various ways.  I usually do:

a) 1 - 6 in dice patterns

b) 1 - 6 shown on a 10's frame

c) 1 - 6 in line drawings or photos of hands with that many fingers held up.

d) 1 - 6 written as numerals, with Touchpoints added on:  https://www2.touchmath.com/. (Note: I think that Touchmath's curriculum falls really hard into that over relying on counting problem that Ronit Bird described in the video I linked in my last post.  I would not use it as a full curriculum.  I do like the Touchpoints as a way to visualize numbers.)

Use those cards to play 

Memory

Go Fish

A game like Crazy 8's, where you can either match the kind of card (e.g. fingers go on fingers) or the number

Snap 

Activity #3: Graphing

I think you have 4 children, is that right?  And 2 parents?  Assuming that's your whole family, you're in the perfect place to do lots of graphing.

Make a set of cards (or better yet magnets) representing each of the six of you.  If he consistently recognizes your names, then just use those.  Otherwise print out pictures.  

Think of a silly questions with 2 or 3 answers that you can ask.  "Which do you like better chocolate or vanilla ice cream?" or "Would you rather have a pet penguin or an elephant?"  Set up a board, with 2 pictures to symbolize the two answers.  Then go ask people, and come back and put the cards in the right place. Arrange the cards on the two sides in dot patterns.  So on the left you've got 4 pictures in the pattern of a 4, and on the right you've got the other 2 in a pattern of a 2.  Talk about how many are in each group.  Talk about which group has more. 

Get those round solid color stickers that you can buy for price tags for yard sales.   Cut them apart so that you've got one little square of paper with a sticker on it.  

Make sets of stickers in 2 different colors that look like the sets of names.  (e.g. if you've got 4 pictures of people, make a set of 4 stickers).  Originally you might need to make that set on top of the other one, literally, later you can probably copy it under).  

Talk about how it's still 4.  Count to prove it's still 4.  Don't count to find out of it's still 4.  It is.  You don't want to introduce the idea that it might not be.  

Then take those sets of stickers and make a graph.  

Activity # 4: Make sets for practical purposes

Do activities that require him to copy subitized groups, for practical purposes.  

For example:

You're making lunch for him and his siblings.   Name the number of people who are coming to lunch, while you put up your fingers.  John, Paul, George, Ringo, Mommy  -- Look at your fingers, oh it's 5 (don't count your fingers.  If he seems unsure, have him count to confirm after you've told him the number).  

Put out 5 plates.  Arrange them like a dot pattern. 

Tell him that today each person is going to have 4 crackers.  Show him how to make a set of 4 crackers on the first plate.  Have him recreate that set on the other plates.  

Make a matching set on top with pieces of cheese.  Still 4!

Serve

 

Work towards the point where he is automatically using that dot pattern strategy, or the putting up fingers while he lists things and then looking at the set instead of counting.  

 

I'll post Objective # 2 later. 

 

[Lecka]

Other little-kid sentence frames are things like:

_____ is bigger than _____
 

_____ is smaller than _____
 

______ is shorter than ______
 

______ is longer than ______
 

Heavier/lighter, greater than/less than etc all these can be reversed and they are all kind of abstract and require not just looking at word order.  
 

Lots of time acting out addition and subtraction sentences and using sentence frames is another thing that I saw a big focus on and I worked on it also.  
 

Addition and subtraction sentences use words like more and less, and be aware many kids get confused when the sentences are worded differently.  Sentence frames help and so do acting things out.  It’s easy to do with snacks or as part of a lot of games.  

Remember more and less are abstract words — they are abstract concepts.  They are not easy.  But they are pretty needed words/concepts for math!  It can take a long, long time, but then when they are solid it goes a long way.  
 

If you see any pronoun issues — you can use a picture or object with sentence frames.  Aka — a picture of him and a sibling, or a horse figurine and Lightning McQueen ———- with a sentence frame — can keep you away from he, she, him, her, me, you ———— which can introduce confusion.  Or include those too for pronoun practice with pictures/objects. B

[Lecka]

My son spent probably a year always tapping touch points with touch math.  But then he did move on.  
 

I always tried to do different things and I don’t think they overly pushed touch math at school.

 

But I don’t know why he moved on from always tapping touch points to not having to do this.  I would say to him “you know you don’t have to tap every number” but he was very into it past the point I wanted him to be doing it.

 

He had a point where he had gotten good at touch points and would add up dice and get up to numbers over 30, by tapping the numbers of the dice.  Aka — to add 6, tap times while counting up, to add 5, tap 5 times while saying “7,8,9,10,11.”  He did this for a long time, months, before he would just use a number line or ones and tens blocks or just trust he could use another strategy besides counting on.  He spent a lot of time doing this plus another strategy (whatever we were working on) because he was going to do this anyway — and then he could see he got the same answer.  

It is really known for kids getting “stuck” like this but also known for being helpful.  

[PeterPan]

Ronit Bird has the printables for the games from the Toolkit book on the resources section of her website.

[Lecka]

Overall though I would say there is a lot of language in math.  It’s easy to think math can move ahead without pesky language, and there are stories like that out there.

For many kids there is an abstract level with language and with math.  Language supports abstract thought for many kids.  In this way language supports math.

There is a lot of language just in:

A is more than B.

B is more than A.

It is easy to get the relationship confused if there is not solid understanding of words more and less, and they ARE abstract.

I keep mentioning this but it was something I had to have explained to me several times!

It was not obvious to me those would be abstract words.  
 

But really the number 10 can be more, or it can be less, it has to be compared to another number and have a relationship with another number.  

[Drama Llama]

1 minute ago, Lecka said:

My son spent probably a year always tapping touch points with touch math.  But then he did move on.  
 

I always tried to do different things and I don’t think they overly pushed touch math at school.

 

But I don’t know why he moved on from always tapping touch points to not having to do this.  I would say to him “you know you don’t have to tap every number” but he was very into it past the point I wanted him to be doing it.

 

He had a point where he had gotten good at touch points and would add up dice and get up to numbers over 30, by tapping the numbers of the dice.  Aka — to add 6, tap times while counting up, to add 5, tap 5 times while saying “7,8,9,10,11.”  He did this for a long time, months, before he would just use a number line or ones and tens blocks or just trust he could use another strategy besides counting on.  He spent a lot of time doing this plus another strategy (whatever we were working on) because he was going to do this anyway — and then he could see he got the same answer.  

It is really known for kids getting “stuck” like this but also known for being helpful.  

I think Touchmath can be helpful, if it's introduced at the right time, and in the right way.  I also think that some kids whose issue is really more language disability than dyscalculia can do well with it even if it isn't introduced in the right time and way. 

But I do think that getting stuck on counting is a real risk.  I think this happens when we use it to teach adding and subtracting to kids who conceptually aren't ready for adding and subtracting.  I'd wait to introduce their algorithms for adding and subtracting after the kid is pretty consistently composing and decomposing numbers they can subitize.

 

[Lecka]

What happens sometimes is — kids learn 10 is more because it’s always the most out of numbers 1-10.  
 

All of a sudden you have 11 and 10 is less than 11.

 

You can check using snacks if one kid has more, can they identify that, if you add more to one kid, can they change and say “now _____ has more and ____ is less.”

 

That is hard because it changes, it doesn’t stay the same.  

[Drama Llama]

2 minutes ago, Lecka said:

Overall though I would say there is a lot of language in math.  It’s easy to think math can move ahead without pesky language, and there are stories like that out there.

For many kids there is an abstract level with language and with math.  Language supports abstract thought for many kids.  In this way language supports math.

There is a lot of language just in:

A is more than B.

B is more than A.

It is easy to get the relationship confused if there is not solid understanding of words more and less, and they ARE abstract.

I keep mentioning this but it was something I had to have explained to me several times!

It was not obvious to me those would be abstract words.  
 

But really the number 10 can be more, or it can be less, it has to be compared to another number and have a relationship with another number.  

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.  

 

[Lecka]

I’m going to add — if you work on bigger/smaller etc with the same items, kids can memorize answers. (Edit — or really they can learn with a set and do well with that set.)  This is fine and good in that memorizing examples can be a step towards generalizing.

But I have had it happen before that skills are shown with one set of materials and then don’t generalize to another set of materials.  
 

Or a new item is introduced Thats bigger or smaller and it’s hard to fit it into tHe materials that have been practiced together. 
 

I was told once someone often saw kids introduced to 8 sets of materials before generalizing to a new set, and didn’t think 8 was a high number.  

Edited February 19, 2021 by Lecka

[Lecka]

So something about my son — bc we used reward charts to get to 5 or 10, he could answer a question “how many more until you get to ten” and he could do this for 5 and 10.  He could know if he had 7 circles filled in, in 3 more circles he would have all the circles filled in.

This did not make numbers 11-20 any easier.  

It also didn’t mean he could take that same language and use it in a different context.

 

Edited February 19, 2021 by Lecka

[Not_a_Number]

1 hour ago, BaseballandHockey said:

I think one to one correspondence is a great thing for a kid to have, but I also think over relying on counting as a strategy, can keep kids from really understanding the concept of quantity and a permanent attribute of a set, that can only be changed by changing the composition of the set, not by changing the location of the set, for example.   That is to say, that counting is one way we answer the question "How many are here?", but many children think it's the way we answer the question "What number will I say when I touch the last one?"

Right. I'm not arguing that counting is the only way to think about cardinality. I think this is a good example of a "concept" vs. "how to gain that concept." For SOME kids, counting will communicate what a number means, and for others it won't. However, the IDEA is still one-to-one correspondence. 

 

1 hour ago, BaseballandHockey said:

Having that understanding, as well as the understanding that numbers can be composed of smaller number, and that that is also immutable is a pretty core understanding that you need to move on to place value, and I would keep working on it, until it appeared.  Another, simpler skill that requires those two understandings is counting on.  I wouldn't say that a kid needs instruction in counting on before instruction in place value, but I would say that a kid who is ready to understand and use place value, will often show us that by using counting on strategies fluently.

Yeah, that's exactly what I felt. It's not that one seemed to require the other. It's that my kids seemed REALLY not ready to treat a unit as a "10" until they were able to treat a group of a certain size as that number, without counting it each time. 

Have you ever seen the Tiny Polka Dot cards, by the way? Here's a link: 

https://www.amazon.com/Math-Love-Tiny-Polka-Dot/dp/B01N1UUHP4/

 

I used them for the younger kids in my class for matching games and "make 10" type games, as well as using normal playing cards. I thought they were great as a way to communicate the concept of number. 

 

[Not_a_Number]

1 hour ago, BaseballandHockey said:

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've actually found that to be true for many kids without language disabilities as well. The less cluttered you make your presentation of an idea, the more kids will follow it, and that seems to include advanced kids. 

[Cake and Pi]

10 hours ago, Not_a_Number said:

I wonder if playing games where you can use a single poker chip for a 10 would at least be a signal for him being ready?

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. 

And we have and play Tiny Poka Dot!

9 hours ago, PeterPan said:

Can you *borrow* an apple device? Yes, the ebooks are only apple. She has print books, so you'd be looking at Dyscalculia Toolkit.

No, I don't think we can get our hands on an apple device. How would you proceed through the Dyscalculia Toolkit book? Just start at the beginning?

9 hours ago, PeterPan said:

You might try ProEdInc and Super Duper and pick out some things to do with him yourself. Has he had ABA? They do a heavy language emphasis in the early years and it sounds like he missed some of that instruction and those skills. So any of the checklists they use would also get you going forward. Northern Speech Services has a kit for early language ABA stuff.

Can you recommend anything in particular from ProEdInc and/or Super Duper? I've looked around on the websites and there is waaay too much to choose from. It's overwhelming. As far as speech/language goes, he has a moderate articulation disorder and just generally disordered language. He frequently gets pronouns reversed and speaks with long, complex sentences that usually make sense but often have really funky grammar and syntax. 

He has not had any direct ABA, as he was only just diagnosed ASD at the end of December. His ADOS at 3.5yo was "inconclusive" and they decided that his very low social skills (10mo level at the time) were just fine and expected since they were right in line with his receptive language abilities (11mo level at the time) and that low number they estimated his IQ at. Professionals have been telling us all along that he doesn't "seem" autistic. He's too social, makes great eye contact, and shows too much empathy, yet his repeat ADOS in December had moderate-high evidence for ASD, he stims all day long, he has the classic ASD echolalic/ scripted speech, etc.

Two of my older autistic kids have had extensive in-home ABA and BCBA consulting. DH and I "graduated" from parent ABA training. I'm currently doing a three-month-long weekly dual diagnosis (ASD + other dx) parenting  intensive with a clinical psychologist certified in ABA, which is turning out to be *the* most helpful, well-balanced, patient-centered approach to addressing behavioral concerns I've ever encountered.

I'll look into the ABA methods you listed and see if I think it's something worth trying. We've only ever used ABA to address dangerous and maladaptive behaviors.

9 hours ago, PeterPan said:

Have they tried him on AAC?

Nope. He can speak in long, complex sentences much of the time. He has word-finding/recall issues and occasionally goes mute. We use PEC/ choice board and some signs when he's not speaking, but it's the exception rather than the norm. Usually he speaks, and usually when he's struggling for a word he can just take my hand and lead me to what he wants or I have a good enough guess about what he's hunting for that I can figure it out without him speaking.

And I like your brain file folder analogy! 

9 hours ago, PeterPan said:

Is he having some kind of seizures?

Nope. He just had another 24-hour EEG a few months ago. His EEGs are abnormal, but he is not having seizures and has no evidence of epileptic discharges.

9 hours ago, PeterPan said:

So no guilt, just take steps forward. You're figuring out this stuff. These are the hardest kids, the most complex, and it really is rocket science. It's CUSTOM. And I think that's the problem. It takes so much time to sort it out and then make it happen, that kids can just slide on through with some sorta service happening but never the magic and breakthroughs. I just love that you're making progress now. That must feel so good even though it's also frustrating seeing more that needs to be done.

Thank you, and yes, as frustrating as the process can sometimes seem, it feels amazing to see the progress he's making. I'm proud of us both! 

9 hours ago, PeterPan said:

Op can ask her people for that number of how much delayed he is. If he's been diagnosed ASD, they have a sense and a number they're working with. 

He's functionally about 4yo but with lower receptive language and much higher splinter skills that are age-appropriate. His Vineland adaptive standard score was in the 1st percentile.

7 hours ago, Not_a_Number said:

But I definitely do stick to the idea of one-to-one correspondence very firmly with early math, which is where C-rods and measurement would personally give me pause -- they seem more abstract than that idea to me. (I know Ronit Bird uses them, so I may be off-base. But I just tend to wonder how they fit into the idea of one-to-one correspondence, which to me is the uniting concept it seems like you're working on.) 

I don't know if any of this is useful 🙂 . I just know that figuring out what the overarching concept for the thing I'm trying to teach is has always been fruitful to me. And since I haven't had to think much about this specific concept, it's possible my thoughts aren't much use. But this way of thinking about the problem has been helpful to me. 

I think with c-rods you get quantity represented by size and each number as a whole thing in and of itself. But by working with the c-rods you learn that numbers can be combined to be equivalent to other numbers. I'm not really sure about the concepts behind the pedagogy though. Our c-rods are home-made out of linking centimeter cubes, so they're like MUS blocks in that you can count the pieces in each rod. I rotated the cubes after 5 when I was gluing them, so they also have numbers pre-subitized into 5-and-whatever groupings. I was aiming for the best of all worlds when I made them, but I can't tell if they're actually working any better than classic c-rods.

I don't know if I have an answer for your question though. I'm not sure I understand what you mean by presenting a mental model that unifies the concept....? I know what *my* mental model is, but it's not necessarily the right or best model to teach him, I guess. Small numbers are lengths in my mind, extending from a mental "bottom" up a kind-of vertical number line to the height of the number... but that's only for the first 100 or so numbers, then my mental model shifts to a kind of line-square-cube thingy, pretty much exactly like base-ten blocks, except extrapolated. So 10,000 is a line of ten thousand cubes, just alike a giant ten rod; 100,000 is like a giant 100 flat composed of thousand cubes instead of unit cubes; 1,000,000 is a cube like a typical thousand cube but with each block itself a thousand cube instead of a unit; and so on. The bars, flats, and cubes just get bigger and bigger and bigger in my mind with cubes inside cubes inside cubes.

2 hours ago, BaseballandHockey said:

I'm going to write up a checklist of skills and understandings I'd prioritize, as well as some tools

Thank you!!!! Your lists are wonderful and super helpful!

1 hour ago, Lecka said:

It is easy to get the relationship confused if there is not solid understanding of words more and less, and they ARE abstract.

I keep mentioning this but it was something I had to have explained to me several times!

 

Yes, I think quantity, size, and position words tend to get all muddled in his head. For example, he will only eat one full slice of pizza, but his brothers will each eat three. He wants to have the same number that they have and will cry in outrage that it isn't fair that they get more (even though he will not eat the extra two slices if I put them on his plate!). But if I take his single slice and cut it into thirds, he is totally content. He has three. They have three. It's the same in his mind even though he obviously (or, obviously to everyone else) has significantly less pizza on his plate. Things in lines cause similar confusion -- 1st, 2nd, 3rd end up being called bigger even though they are the same size, just in a different position.

I will try your sentence frames for a while and see if we can make more progress through more consistent verbiage.

Edited February 19, 2021 by Cake and Pi

[Lecka]

If you are lining things up smallest to largest, make sure you go largest to smallest also.  At a certain point anyway.

Kids can learn that the biggest is to the right and be going off of the position and not the size.

It can be really hard to make sure kids are noticing the thing you want them to notice, and easy to use another pattern at the same time, and have it turn out the child was noticing the coincidental pattern and not the intended pattern.  This makes it hard to teach a lot of language concepts.  

[Not_a_Number]

1 minute 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. 

So... this is actually kind of what I mean about using a lot of representations maybe being confusing, and again, I might be totally off-base. What if you kept working on the place value chips in games, and possibly naming those quantities in chips, and constantly trading up and down... and you kept everything else just in singles and play the games that @BaseballandHockey suggests? That way, he could both work on "things being worth different amounts depending on the properties of the thing" and he could also keep working on the fundamentals of quantity. 

Do you think he'd be able to associate a "twenty one" (said out loud) with two green chips and a single chip? I can tell you some poker chip games we played with my homeschooling classes that require constantly trading up and down. And I can tell you that for some of my weaker kids, even ones without dyscalculia, it's still quite tricky to remember which trades are allowed and which ones change the number. Even for my advanced kiddos, a really thorough understanding of place value via trading has generally taking quite a while. 

Or do you think if he got the idea with place value chips, it wouldn't translate to understanding in other contexts? Perhaps you could sort of meet in the middle by working with 1s working up to higher numbers and also concurrently ONLY working with place value chips so that this is the only time he's really required go "group," which would keep grouping predictable? 

 

4 minutes ago, Cake and Pi said:

But by working with the c-rods you learn that numbers can be combined to be equivalent to other numbers.

I have to say, I've never had trouble communicating that concept without C-rods. I don't actually think you need numbers represented as lengths to communicate that idea. And if you represent numbers both as lengths and as counting, you're splintering your model, and that's always harder for kids to integrate than a model that's consistent (but widely applicable.) 

 

13 minutes ago, Cake and Pi said:

I don't know if I have an answer for your question though. I'm not sure I understand what you mean by presenting a mental model that unifies the concept....? I know what *my* mental model is, but it's not necessarily the right or best model to teach him, I guess. Small numbers are lengths in my mind, extending from a mental "bottom" up a kind-of vertical number line to the height of the number... but that's only for the first 100 or so numbers, then my mental model shifts to a kind of line-square-cube thingy, pretty much exactly like base-ten blocks, except extrapolated. So 10,000 is a line of ten thousand cubes, just alike a giant ten rod; 100,000 is like a giant 100 flat composed of thousand cubes instead of unit cubes; 1,000,000 is a cube like a typical thousand cube but with each block itself a thousand cube instead of a unit; and so on. The bars, flats, and cubes just get bigger and bigger and bigger in my mind with cubes inside cubes inside cubes.

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

So, what I'd probably do if I were you is I'd pick ONE model that you focus on. So, for example, if you want the model to be "length," I'd bring everything back to "length." I wouldn't expect him to count anything, ever. I'd only use C-rods and other ways to measure things, and I'd work on that really thoroughly. Any understanding I'd want him to work on, I'd bring back to length. So, for example, adding then becomes "putting lengths next to each other," and you could do that with C-rods and with rulers and with tape and with number lines and with all sorts of things. And you'd observe what happens and figure out patterns, etc.  

Now... that's not my favorite model because it's not widely applicable to things in real life -- it's easy to use and define, but it's hard to extend to countable problems. But it's certainly a model and you really could restrict your attention to it. And I do find that finding ONE model that kids can always refer back to clears kids' brains, because the model becomes much more automated, as opposed to having many competing models. 

If you used this model, I wouldn't expect him to be able to answer questions about countable things for a good long while. And I wouldn't feel like that's a problem, either. As you're assimilating a model, you don't tend to be able to apply it. 

Am I making any sense here? Working with much more advanced kids, I've found that having MULTIPLE models just clogged kids' brains up. Some kids can assimilate many models at once, but I've found that most can't. What they can do is assimilate ONE model that is clear in their head, and then they can slowly spin off from that model to its applications. So, an application of the length model turns out to be counting, but that's not at all obvious at first, and if a kid doesn't see that because 3 + 5 = 8 in the length model, it's also the case that 3 sheep and 5 sheep make 8 sheep... that's OK. What you need to do is to line up the sheep along the C-rods and talk about how that's the same thing because we have one sheep per unit of C-rod, and not to expect a kid to understand that length addition is the same thing as "counting" addition. They really aren't obviously the same thing! They are obvious to us, because we've assimilated both models. But if you have one model firmly in mind, it takes work to extrapolate from it. 

Am I making any sense here? I'm definitely sharing observation from a different class of kids, but it has held up across a really wide range of models that I've experimented with. 

[Not_a_Number]

11 minutes ago, Lecka said:

It can be really hard to make sure kids are noticing the thing you want them to notice, and easy to use another pattern at the same time, and have it turn out the child was noticing the coincidental pattern and not the intended pattern.  This makes it hard to teach a lot of language concepts.  

Yeah, I feel like that's the trickiest thing about teaching. Some kids naturally pay attention to the thing you want them to, and some really do not. And you have to figure out what it is they are focusing on. 

[Not_a_Number]

36 minutes ago, Cake and Pi said:

But if I take his single slice and cut it into thirds, he is totally content. He has three. They have three. 

So, at least to me, it sounds like his own natural model is not length-based. So then I'd probably try to spin off from the "quantity" model, but I wouldn't expect it to generalize to things he can't count or separate, like area or C-rods. But I'd probably work on arranging things into shapes that allow him to quickly recognize number, which it sounds like you do already.