I read it as procedural and declarative verbal memory may *appear* impaired in kids with DLD, but they're actually consistent with nonverbal IQ and working memory, which were presumably lower in this group than the TD control group.

Reading your other post, your DD sounds a lot like my youngest DS in terms of strengths and challenges (totally different diagnoses, though -- my kid is autistic and has encephalopathy + mild CP from his birth history). My DS#4 has relative strengths in visual processing and visual memory along with his basement-level verbal memory and a significant language disorder. He transitioned from EI to an inclusion SPED preschool at 3yo, did two years of prek and then two years of public kindergarten (mainstreamed with an IEP and lots of pull-outs) before switching to homeschool. Anyway, we started Right Start 1st edition level A about 15 months ago (and schooled through summer) at 6.5yo and it was soooooo slow going. He took about 3 months to get through the first 7 lessons, which were designed to take a typical 4-5yo less than 3 weeks to complete. We eeked out a few more lessons along side other math programs in the next 7ish months. Then he hit a solid brick wall and could not move forward no matter what I did. The hands-on component was excellent, but Right Start is surprisingly verbal! It also includes quite a few activities that require relatively high (aka at least low-average-ish) working memory. My son also needed waaaaay more practice with concepts than was built into RS, and the jumping around between topics that is standard in RS was problematic for him. It confused him to begin a new topic before the previous one was completely solid. There's a RS for struggling learners FB group that I mostly lurk on, and RS seems to work for many children with learning challenges; it just didn't work for mine. If you already have the materials or your budget has room for some experimentation, I say go for it. Just be ready to adjust things or ditch RS entirely if it doesn't work out. Also, in all fairness, my DS#4 also tried and got stuck in MUS Primer (at lesson 9, the same concept that he stalled on in RS) and ST Math (again, same concept). *NO* regular math curriculum was going to work for this concept and my kid at that time. He just wasn't developmentally ready. We took a couple of months off around surgery and now have spent the last 2.5 months in the Ronit Bird Dots book and doing other basic number sense activities. He's still struggling, but it feels less hopeless. I don't know if that's because of the change in approach or if at almost 8yo we're seeing the result of time and increasing mental maturity. I plan to switch him to a tier-3 public school intervention math program soon, SRA Connecting Math Concepts, because he's doing so well with SRA Reading Mastery Signature Edition, the reading and language program that uses the same DISTAR/Direct Instruction approach. Would it have worked when he was 5? Maybe, but I doubt it. I think his brain needed more time to grow and develop. Alternatives to RS to consider: ST Math - Online complete math curriculum that is 100% language-free until upper elementary. Also happens to be free this year. https://www.stmath.com/homeschool-math Ronit Bird dyscalculia materials. The Apple iBooks are a bazillion and one times better than the paperback books you can buy on Amazon. http://www.ronitbird.com/ebooks-for-learners-with-dyscalculia/

We're calling next year 2nd grade for my globally delayed kiddo, but it'll effectively be a lot like a fourth year of kindergarten. I'm planning: Blossom & Root Early Years Vol. 1 Let's-Read-and-Find-Out-Science level 1 -- second pass through Heggerty Bridge the Gap Intervention Lessons (phonological awareness) SRA Reading Mastery grade K, Reading Strand SRA Reading Mastery grade K, Oral Language Strand SRA Connecting Math Concepts grade K KiwiCo Kiwi Crate (if it's not a good fit we'll switch back to the Koala Crate) Keyboarding Without Tears, finish K and then... repeat K? Begin 1st? Speech therapy, social skills group, OT, and PT/swimming (depending on if the indoor pool is open this fall and winter) We've been working through Reading Mastery since early April, and I'm hopeful that we'll be to at least lesson 36 by August and can get through the remaining 124 lessons over the course of the year. The DISTAR/Direct Instruction approach is working so well for him for reading and language that we're switching to SRA Connecting Math Concepts as well. If it doesn't work out as planned, my fallback is going to be the final 2/3s of MUS Primer with RS and Ronit Bird games and perhaps to try ST Math again.

In the last 3.5 years my DS#2 just... outgrew it 🤷‍♀️. He very rarely does it anymore, anyway, and he never did actual speech therapy (his speech eval was for entrance to a social skills group). My DS#3 still does it, and it's still variable based on stress/excitement. Newest SLP is calling it "cluttering" or "cluttered speech" but officially diagnosed "childhood onset fluency disorder F80.81," which, it seems, is a shared diagnosis code for both cluttering and stuttering. Both do seem to be treated about the same way. Anyway, it's a bit better now at 9 than back at 6yo, but not all that much. We've focused much more heavily on social communication than on his articulation or fluency over the intervening years because he *wants* to improve his social communication but he doesn't yet care one tiny bit about the state of his articulation or speech fluency (and hence rarely cooperates in that aspect of therapy).

Once upon a time I had just a 5yo and an 8yo homeschooling. We did literature and social studies together, but I worked with them each separately for all the other subjects, alternating between them. The one I wasn't working with would play independently or participate in whatever therapy they were at (we did a ton of waiting-room-schooling). We use curricula, though, so things were maybe easier for me? At the time DS#1 was doing Beast Academy 5 and DS#3 was doing Right Start C/D. Each kid got the individual attention they needed. Now with four homeschooling there's no way I could do almost all the subjects with each individually. But basically, while I work with one on a subject that they need one-on-one attention for, the others either work independently or play. Right now I have two that need my undivided or nearly undivided attention and two that can do most things independently, just needing me to grade, give feedback and dictations, that kind of thing. It works out.

From talking with other parents I've gotten the impression that 10-14 is typical. For another set of data points, my oldest started working semi-independently (just needed supervision but not much interaction unless he got stuck) at 11 and was fully independent and reliably did assignments without reminders beyond a daily check list at 12. Second kid is 11.5 now and *just* in the last couple of months hit semi-independence, working independently as long as I'm nearby making sure he isn't goofing off on his computer (he's doing an AoPS class). Kid #3 was semi-independent at 7-8, but has recently regressed without ADHD meds and now at 9 isn't doing much of anything without *constant* redirection. He'll literally be mid-sentence explaining his reasoning on a problem and just stop and stare off into space or switch to talking about antimatter or Minecraft the fact that zippers are actually just tiny wedges and examples of simple machines. Its feeling like independence is a very long way out.

I'm using OUP The World in Ancient Times series with an outlier 3rd grader (plus 5th and 7th graders) this year. He thinks it's great. It is solidly middle school level, no more difficult than Hakim's History of US. I like to read (or have my kids read to themselves) the corresponding chapter in HQ as an overview before we dive deep in OUP. I also use the HQ guide a bit. Usually I give my kids Writing Revolution style assignments in history, but when we need a lighter day or I don't feel like coming up with a writing assignment I'll pull out the longer dictation or comprehension questions from HQ. We also do the map work. If we weren't so busy I know my 3rd grader would appreciate many of the hands-on activities in the guide as well.

How did you list these on the transcript and divvy out credits per year?

My oldest will be in 8th grade next year. Math: WTMA AoPS Precalculus class Science: Online G3 non-traditional physics and astronomy classes. History (combined with DS#2 & DS#3): The Medieval & Early Modern World (Oxford University Press) as a spine plus a half-dozen History Unboxed crates, a coordinating middle-grades literature list, and maybe-possibly-probably the History Quest Middle Times narrative. Writing Revolution style assignments worked in. English Language Arts: MCT 5 Lens I level with the lit trilogy, Fix It 4, Online G3 Essay Essentials and Shakespeare Tragedies & Sonnets classes. Other: Athena's Teen Goal-Setting class Digital art and graphic design using Adobe Creative Cloud Physical fitness training w/ Dad Family art crate from Outside the Box Creations Poetry Teatimes BW style

Adding to the topic... Bright Not Broken https://www.amazon.com/Bright-Not-Broken-Gifted-Autism-ebook/dp/B005HFBSHW Differently Wired https://www.amazon.com/Differently-Wired-Aspergers-Giftedness-Disabilities/dp/1523506318/ref=sr_1_1?keywords=differently+wired&qid=1616347865&sr=8-1

Here's an example of his self-generated speech on a really good day recently. As I've said, his performance varies *dramatically.* On not-so-good days he just says single words or short phrases or scripts, and on bad days he stops talking altogether. He also has days where he doesn't speak much English, but he jabbers at length in jargon. I video recorded this and then transcribed so I got it exactly as he said, translated for pronunciation difficulties. "You know, unicorn are actually horses they just magical sparkly one that fly in the air with no wings. I need to tell you something. When unicorns get faster, the rainbow gets bigger with more colors. When the unicorn gets very fast, so if someone saw a unicorn in real life and it getting faster, then would see that the rainbow they creating looks like it's getting bigger with more colors appearing, which makes it looks that it's getting wider and more bigger with more color. But some more color appearing when the unicorn stop the color slowly starts to disappear and get smaller and smaller until it's done until the color vanishes!" So, he's got some really big ideas, and there are some strong vocabulary words in there: actually, creating, appearing, disappear, vanishes. Now, if I wanted to convey the same information to him in language he would understand, I'd say something like, "Unicorns are like horses. They are pretend. Pretend, magical horses. They have no wings, but they can fly. They make rainbows when they fly. The rainbows get bigger when they go fast. Fast unicorn, big rainbow. (long pause) The rainbows get smaller then they go slow. Slow unicorn, small rainbow." And then I'd repeat that in three different ways using my hands to show the unicorn flying, pulling my hands apart to show bigger and pushing them together to show smaller. I'd show him a picture of a unicorn and a horse. Then at the end he might or might not understand what I said depending on the day, and if I wanted him to remember any of it I'd better tell him every day for several weeks (or longer, if he wasn't interested).

Slightly complicated answer. He has age appropriate (25th% or so) understanding of *individual* words, or did 1.5 years ago when we last officially checked, but when words are strung together, his comprehension breaks down. Understanding of spoken sentences was <1st%. We were only ever given percentiles, so the ~3yo comparison is my own estimate based on the target age of picture books he understands when being read aloud to. He does best when spoken to in phrases and very short sentences. (He actually speaks in much longer sentences -- like I said, receptive lags behind expressive by quite a bit.) He is better able to recall/recite a string of random words without meaning than he's able to recall/retell details in a typical short story aimed at kindergarten to 1st grade aged children. Do you mind linking to the specific resources you were thinking would help? I'm interested in checking them out. I've asked about CAPD several times, but audiology basically told us that's woooo and not an evidence-based diagnosis, that he has *language* processing problems associated with his language disorder, not *auditory* processing problems.

Sequencing is one that surprisingly did NOT come naturally to my DS 9. Actually, we still have micro visual schedules up to remind him about the steps for how to get dressed, how to sweep the dining room, how to brush his teeth, etc. and he can't reliably do those things without a step-by-step check list or someone standing there telling him what to do after he completes each step. And yet, he is incredibly good at math. Thank you for the reminder! I went back through my records and we definitely went over this in March of last year when started RS A. However, we only did the rearranging and renaming activities with numbers 1-3 because got stuck for a long time learning to differentiate 4 and 5. Perhaps he was actually *only* subitizing and last year and not recognizing that the quantity would stay the same even when rearranged. So either he used to understand and has since forgotten, or I mistakenly thought his ability to name 1-3 in any arrangement meant that he knew moving things didn't affect how many there were. I know better than to assume anything now, but I was new to teaching him back then. No sure I know what you mean? He's not reading yet. We're working on the alphabet, mostly. Got it! Thanks! Here seems fine. It's easier than re-explaining his whole back story on another thread. I can update the thread title to reflect the multi-part topic so that it can be searched if someone with a similar kid comes looking for ideas in the future. He has always had global delays and is overall functionally like a 4yo -- but with receptive language closer to 3yo and some nice strong 6-7yo splinter skills. Recent ABAS-II was <1st%. BUT he has 25th% overall IQ and his nonverbal IQ was 50th%. According to his school testing two years ago he was in the 3rd-5th% academically in all subjects. I won't have more current achievement data until this fall when he'll see private neuropsychology. I don't feel like he's had an average amount of growth over that time, though, so the percentiles may be lower now... except that my normal meter is totally broken, and maybe he *has* had a normal amount of growth for someone on his trajectory. I don't know how to judge, which is why I plan to let neuropsych do it for me, lol. So... I don't think he qualifies as having a discrepancy between achievement and *overall* IQ, but there's a 2+SD spread between his nonverbal ability and his achievement as well as between adaptive behavior and overall IQ. His profile is full of peeks and valleys. Which is to say, he may have SLD in all subjects, or maybe the lagging academic skills are just a result of his other, more foundational sill deficits and language issues. No, surprisingly. He was evaluated before his last language and literacy therapy group at the beginning of the school year, but I just went back through the documentation for his pre-assessment, and it doesn't have any actual data about his levels for phonemic awareness, though I know they tested him for that. It lists mixed receptive-expressive language disorder, phonological disorder, and delayed phonological awareness under background. His goals for the almost 4-month-long group look pretty vague to me... "associate targeted sounds with targeted letters," complete targeted phonemic awareness activities "isolating individual sounds, count words in a sentence, recognize rhymes, segment phonemes, blend phonemes, delete phonemes," and complete "language activities including: describing objects, grouping objects into categories, answering questions about a verbally presented story, and retelling stories with salient details" all with 85% accuracy, of course. I know he didn't meet a bunch of those goals, though. He did level 1 of this group the year before, so they automatically moved him into level 2, and by the end of the group they were saying he needed to go back to level 1 for his next language an literacy group. Oooooh, I used TYCTR in 100 EZ Lessons with two of my older kids. I'm very familiar with DI. I bet I could find a bunch of Reading Mastery practice pages and activities on TPT and extend the 100 EZ lesson enough that they might work for him if he responds to the approach. I hadn't even considered it since everyone keeps telling me he needs a strong O-G program. We've been doing O-G for years and it hasn't been working, so there really isn't anything to loose by trying. Thanks for sharing your son's experience! No problem at all! I'm getting over a many-day migraine and just made it back myself! I'm glad you came back to share more, even though I hadn't responded to previous comments. 🙂 Just requested it from my library! Thanks! I'm in the middle of Sousa's How the Special Needs Brain Learns right now and was just wondering what to read next. Yes, he has deficits in all of that and more. Yes, I read him the same 1-2 books multiple times every day. We spend 1-2 weeks on a book before I feel comfortable dropping it for a new one, and I intend to go though the whole sequence of books we started last May all over again this coming year. The more repetition, the better. This is a great idea! I think he would love doing something like this, too. Woot! We're already sort of doing this. I feel like I'm doing something right, lol. He's learning to type (Keyboarding Without Tears K) and has had an alphabet chart to point to since I brought him home. However, I the entire alphabet is too large a field. He often gets overwhelmed trying to find a letter. Someone suggested that I teach him the ASL alphabet recently, but I figured that would be about as easy as anything else, so I bought these AWESOME, big, chunky alphabet tiles for him to play with. They have the uppercase letter inscribed with a slightly roughened texture for finger tracing and the lowercase version (just stamped on) on the one side. On the other side is the hand sign for the letter. He is doing surprisingly amazing with these. I think we've had them maybe two weeks? And he's learned at least four signed letters already, which is incredibly fast for him. I don't want to get too hopeful, but I'm pretty excited. Thanks for all the links!

I laid out 5 tally sticks, 4 parallel to each other and the 5th across and on top of the other 4. He's learned that this is 5. So I asked him how many there were and he answered "five." I took the top stick off and moved it a few inches to the side as he was watching, then asked again how many there were, but this time he said "six." I put it back and he said there were 5 again. I told him I was just moving the stick, directed him to watch closely, and put it off to the side again, and again he said there were 6. I asked him to count them, which he did and then told me there were 5. I tried various other arrangements of the 5 sticks (a pentagon, group of 2 and a group of 3, a zig zag pattern) and he came up with different answers each time before I had him count them. Of course, in between each trial I reminded him that the sticks were all the same and I wasn't going to give or take any. He was able to parrot back to me that there were fives sticks, but then when I moved them it was like he was totally guessing about how many there might be.

... It's 7 big cubes, 7 square flat thingys, 4 bars, and 5 tiny cubes. 😄 Obviously that doesn't count as a formal definition mathematically speaking, but that's how the number exists in my mind in it's most basic form -- no trading involved. Now, trading absolutely starts happening when I try to manipulate 7745 on a larger scale, by, say adding or subtracting 2153. But if I'm just adding or subtracting a 1- or 2-digit number I'll be zoomed into that vertical number line jumping up or down it, and there's no trading there either. Yep. I was pretty floored. I sincerely thought he had this already. Maybe he used to. It's definitely time to back up and camp out, though.

I think so? Yes? I think our mental models are reversed, though. It goes the opposite direction in my head with trading flowing from place value instead of the other way around. Both models accomplish the same thing and seem to contain the same components. Maybe it's just a difference in how we each first made sense of quantity in our early days? Except... I go straight to trading when working in other bases... which again may be because that's the way I first understood other bases. Different mental models in different bases. Very inconsistent, lol. It sounds like you are working with kids who see "12," know it is called "twelve," can count to twelve, but don't realize that twelve is a 10 and two 1s until you teach that to them, which you do only after they're comfortable with working until 20. I was perhaps trying to go about this in the opposite direction. I was thinking that place value might make numbers 11-20 accessible to DS 7 since he's had such a hard time conceptualizing them. Learning to count and name numbers and quantities 11-20 was one of his IEP goals last year, but he never got beyond rote counting. DS 7 sees "12" and says "twenty" or "one two." He sees "20" and also says "twenty" or "two zero" or just "two," and has been at this point for at least the last year and a half. They didn't do any counting with addition. We did a bunch of work with the abacus and with visualizing, so they wouldn't need to count on. They'd just visualize a quantity and then some more and mentally subitize. With my kid who has trouble visualizing I focused more on +1 being "the next number" and +2 being the "next even" or "next odd" number. Perhaps this is actually a kind of counting and we're really talking a difference in semantics? It's good that you began at the beginning with these lists of activities. I started trying stuff out with DS 7 last week. I *thought* he knew that quantity didn't change when it was rearranged, but apparently I was mistaken! In general, do you you find it's better to run through all this with 0-5, then start over with 0-10, or is it better to focus on one task at a time first with 0-5 and then with 0-10 before moving on to the next task? I'm veering off topic here, but do you happen to have recommendations for beginning reading instruction as well...? Do you know of a curriculum or program for teaching reading to kids with significant language disabilities? DS 7 has had two years of sped preschool, two years of public kindergarten with 4 hours per week in the resource room with the O-G trained Learning Specialist, two summers of public summer school, and about 6 months of private language and literacy (O-G) targeted therapy spread out over the last year and a half. At home we've gone through the entire Explode the Code primer set (Get Ready, Get Set, and Go for the Code) TWICE and done a letter of the week type program. And, he still doesn't know the entire alphabet. Teaching him feels like trying to fill a cracked bucket with water. Thanks again for typing out all those activities to work on, the apps, and the videos. I also got the RB Dyscalculia Toolkit book and am reading though it now.

All of this 100% fits my observations with DS 7. Both, they're interrelated. Just google HIE. CP is a pretty misleading term for his condition, honestly. It's probably better to just stick with encephalopathy. Whole exome sequencing was clear. He has ASD and ADHD and probably SLDs, but so do some or all of the older three. The only difference between them and him is the encephalopathy. No, he's not really there yet. He's close, though. He can count on with with smaller quantities in context. Like he knows when his brothers have 3 slices of pizza and he only has 1 and can even tell you that he needs 2 more. Same with if he has 3 pieces of chocolate and thinks he should have 4. He can tell you he needs 1 more. He can add on his fingers up to 5 or so when it suits him. But he can't tell you how old he'll be on his next birthday, and when he gets tired basically anything bigger than 7 is "nine" or "twenty." It's coming though. We could use counting on within 5. Yes! He usually has 1-to-1 correspondence when counting to about 12 when he's fresh and interested. His performance fluctuates dramatically though. On a really good day, he can get to 17 or 18. On a bad day he looses it around 8 or 9. Maybe it's an attention thing? It seems like 1-to-1 should be a concept that you either understand or you don't. It doesn't seem like it should be something you can do and then not do all in the same few minutes. He can rote count to 23, perhaps missing or mixing up a number or two in the teens but just as often he can do it correctly. Why 23? Why not 24? I don't know. I've never heard him accurately count beyond 23. He knows number names, though, and will pull out counting words like twenty-nine, sixty-five, one hundred, a million, even infinity. Sorry, I wasn't being clear. The older ones are absolutely able to count on, they just don't. It's not a strategy they've ever used. I can do arithmetic in other bases, but my mental model in other bases is always the odometer, even though I've physically modeled other bases with linked centimeter cubes arranged into place value chunks analogous to base-10 blocks. I'm pretty sure the odometer model is my primary model. It's what I revert to when I'm tired. It's the first thing I remember using as a child. Counting on is *my* default, btw. I taught myself addition with my own personally contrived version of the touch points in Touch Math (a bit different of course, since I was making it up myself as a kid). I was stellar with complex math concepts, but I was pretty horrible with arithmetic. I relied pretty heavily on my calculator. I didn't master addition or multiplication facts until I'd drilled three kids to fluency. What is your mental model for other bases? It was super duper extremely helpful! Yes, please!

My older three kids never did counting on at all. They also never did the pre-skill to counting-on that counting-on replaces and is more effeciciant than: counting up from 1. So, when I gave them an expression like 4+3 or a group of 4 and 3 more objects, they never counted up from one like "I have 1, 2, 3, 4... 5, 6, 7" to get the sum and they also never counted on like "I already have 4, so 5, 6, 7." I taught them to subitize 0-10 as five-and-something first, absolutely no counting. Then they automatically regrouped with 5s instead of counting. So, 4+3 was "Move 1 from the 3 to the 4. Now I have 5 and 3, that's 8." They would do this with their fingers at first, but later just did it in their heads. I straight up never taught them to count objects. We used counting to mark time. Like, we counted to 20 when washing hands. I made them count the number of seconds for their timeouts, so at 2yo they were counting to 50-120 (as they were able), at 3yo they counted to 100-200 (depending on how serious the crime, lol), and so on. We used counting to keep track of quantities that were not easy to see, such as cups of flour for a big batch of homemade bread or drops of a water clarifying solution put into the fish tank. I suppose they picked up 1-to-1 correspondence that way? I don't know. It was just never NOT there. As soon as they started saying and reading numbers, the numbers were associated with the corresponding quantities. It seemed automatic. When determining number of objects, I taught them to line things up into groups of 10 made of two 5s, which they could do without any counting. So I guess place value was built into everything we did anyway. In kindergarten my DS#2 was asked to count in a group activity and he told the teacher no, announcing loudly that "counting is slow and inefficient and often inaccurate" and that he could figure out the number better by grouping. He got sent to the office and I got a phone call because the teacher felt like he was undermining her authority in front of the class. She insisted that he NEEDED to count and that if he didn't he would never understand numbers. She sent me studies showing that kids who could count to 20 when they entered kindergarten did better on math assessments in 4th or 6th grade or something than kids who couldn't count. I felt like she missed the point of those studies. I generally don't do any trading in my mental arithmetic. For smaller numbers, I'm just going up the number line. There are hops up my number line going on in my head. I used to have something like an old-time car odometer in head. In fact, I'm pretty sure I learned to count by watching the way the separate wheels would spin on odometer of my mom's car. The next wheel to the left would catch and spin with the 9, bringing the wheel to the left up 1 while the wheel on the right came back to 0. But that was when I used counting to add. When my strategies got more efficient, my mental model shifted to a vertical number line.

Unfortunately, this did not work. 😞 It was a great idea! I tried getting the sample on this version of iTunes and got "This book sample requires an iPad with the latest version of iOS and Apple Books installed and Automatic Downloads enabled." Our charter funds roll over in April, after which I'll have a fresh $1,800 at my disposal for technology and curriculum for DS 7. I may end up just buying an iPad at that point. This is not the first time I've felt like we really needed one for something.

Oh, I don't. I would think that if he was going to get it any time soon, though, he would understand *some* portion of all of that, not all of it. I was just illustrating that that he doesn't have any understanding at all. None of the pieces are there. I think @Kanin might be right about DS 7 thinking the trading game is hilarious precisely because it makes no sense to him and he thinks it's funny that such an obviously unfair trade (to him) is being promoted by me. No idea, really. Right now I can't even get him to associate 23 with 2 yellows and 3 greens, though, and I'm not sure I really want to put energy into trying to teach it to him to him anymore. I've really spent the last 4 months on this and the only progress we made was that he temporarily was able to name (or when he wasn't speaking, point out cards with the numbers written on them) some multiples of 10 represented on the abacus. That skill is gone now, so we're basically exactly where we were four months ago. I agree. I think I will ditch the place-value chips though, in favor of using dimes and pennies since that doubles as an applicable life skill, and right now we're just hoping that he'll be able to live independently as an adult some day. I think he hasn't made the leap that total amount does not necessarily relate directly to number, that pieces with different sizes represent different quantities or that you can break something into pieces and still have the same amount. For example, he has a medication he takes each night. His dosage is 3.5 squares, which ends up being four pieces (three wholes plus a half square). I break the pieces off of a bigger block, and if I leave two squares joined, he asks for more because he only has three separate pieces. He'll say that I didn't give him enough. We've done lots work with same/different. He likes worksheets where he is supposed to find the one picture in a group that is slightly different from all the others. It's just not translating to math or quantities. To use @PeterPan's analogy, I think it's getting filed in some other, non-math, non-quantity related folder. We need to work on it more. No for the first, but we probably should be. Yes for the second, for years. It's obviously in need of more work, though! Thanks pointing this out. When we get private speech up and running again, I'll be sure to include this his goals. Yes, I completely agree. We'll revisit trading, probably with dimes and pennies, in a while -- possibly a long while. I'm going to switch over to the Ronit Bird stuff for now. I think my DS 7 *needs* everything to be as concrete as possible. Honestly I'm not convinced that having such a model, other than in the context of money, is a gainful use of his time and limited mental energies. He's not going to just experiment with a model and discover things. He must be taught every. single. step. He must be pulled to every. single. conclusion. He has to be taught how to play with every. single. new toy. I like your idea for kids in general, maybe just not for kids like my DS 7.

Okay. So just for fun, I decided to ask the numerate members of my immediate family about their mental models for numbers. DH: He says he sees just the digits of a number, the way one would write the number, in his mind with an implicit understanding that the digits are multiplied by powers of ten, and that's it unless he is given context. With context he switches to visualizing quantities, quite accurately, so it sounds like he uses a quantity model. He gave me some examples that I'm not going to remember correctly. Something like one cubic foot of concrete is a cube *this* big (holds out hands). Ten cubic feet of concrete is a traffic barrier. Three hundred cubic feet of concrete is the foundation of our home, etc. He said if the context were stuffed animals, "one" would be a kid holding a stuffy. "Six" or "ten" might be a shelf with stuffies lined up on it for sale in a store. "Fifty" might be a claw machine full of stuffies, etc. For reference, he is a chemical engineer who does mostly process and controls engineering in mining, mineral processing, and water treatment. He also handles the budgets and cost estimates for enormous large-scale projects. He's sort of a Jack of all trades in the engineering world. He routinely works with very large numbers that represent large *physical* quantities, usually in the millions, of um, STUFF, lol. He doesn't remember early elementary or the process of learning place value. DS 13: Numbers were not automatically associated with any mental image, just sounds. He has great trouble visualizing in general. However, when pressed to "picture" a number in his head, he arrived at a length model. He said 27 is a vertical stack of 27 balls next to a measuring tape with the height of the stack equivalent to the quantity in the stack. When I asked about bigger numbers, like 1,123, he said he envisioned basically block letters of "1123" rotated 90 degrees counterclockwise and stretched until the height of a measuring tape at 1,123. Background: DS 13 is 4ish years accelerated in math, ASD, PG, and NOT a visual thinker (unusual for ASD, but there you go). Compared to my other 2e kids, he had a relatively difficult time understanding place value. I think we worked on the idea of place value for a solid month when he was 4y 11m to 5y 0m old before it clicked. DS 11: For small numbers, under 10, he envisioned groups of cakes, lol. For slightly bigger numbers, his mental model was cookies lined up horizontally with groupings of 5. He described 27 as essentially OOOOO OOOOO OOOOO OOOOO OOOOO OO. 1,123 was a big pile of 1,000 cookies with a smaller pile of 100 cookies to the right of that and a line (again grouped in 5s) of 23 more cookies to the right of that. Context: DS 11 is maybe 2 years accelerated in math, PG with a small army of other Es, and suspected ASD. He picked up on place value immediately the first time I introduced it at 4y 3m. As in, he was immediately able to do *all* of the skills I described in an earlier response to you that I hope to get DS#4 doing eventually. As soon as I explained that 1,000 was 10 hundreds, he could say that there must be 100 tens in 1,000, could build numbers with any of our manipulatives from a number said aloud or written down, could write and say a number based on a physical model of it, etc. After all the trouble DS#1 had, I was armed and ready to hunker down and focus on place value for a while, but he seriously got it the very first day I introduced it to him. DS 9: DS 9's response basically blew my mind. I obviously don't operate on the same wavelength as him. I asked him what he saw in his mind when he thought of 27 and his response was "three cubed." Okay, new tactic needed. I figured he'd have some other model for non-square, non-cube numbers but figured composite numbers might elicit a similar response, and so I asked him what he envisioned with the first prime number that popped in my head, 31. "Two to the fifth minus one." Oooookaaaaaaaay...... "But what do you SEE in your mind for 31?" I asked. "I see zeros and ones," he answered. "Seventy two. What do you see for seventy-two?" "A seven and a two with a one and a two under them." "Whaaaaat? What are the one and the two?" "They're places for the hoppers. You can change the orders of the numbers and hoppers go with them so they're the same number." Then he described putting apples into his "inventory" and using "block commands" and "burning up" 9 apples to "transfigure" 10 regular apples into a golden apple and having separate hoppers to put regular apples and golden apples in... but then he said his hoppers hold multiples of 64. Oh, but he doesn't always use hoppers. He also has buckets. The buckets can be empty or full of water, and those are all zeros and ones (binary). Or he can stack two buckets and add water to the inner bucket and have all zeros, ones, and twos (base 3, I guess). So I essentially have no idea what, if any, mental model he has for numbers, place value, anything. There seems to be an evolving complicated something going on in his head all the time. Background: DS 9 is like 9+ years accelerated in math, ASD, PG. I went through the first 24 lessons of RS A (1st edition) with him very informally here and there between 3y 8m and 4y 1m -- as in, I never pulled out the book. I was just familiar enough with the beginning concepts that I was able to introduce the idea of subitizing and grouping numbers into five-and-somethings, counting out snacks by 2s and patterning in rows with toys and food, etc. to work it into everyday interactions. He played extensively with math manipulatives (mostly base-ten blocks, place-value cards, the abacus, math balance, geometric solids, Magnatiles, pattern blocks) independently and, as much as it pains me to admit it, was largely neglected by me because he was content to occupy himself quietly while I focused enormous energies into DS#4's early intervention therapies while trying to keep up with regular household tasks and homeschool lessons for DS#1 and DS#2, who were in early elementary at the time. He went to a play-based preschool for half the day to keep him gainfully occupied (he had a tendency toward destructive curiosity). One day I had 830+177 written on the white board as a review problem for then-6yo DS#2. DS#3 looked up and announced the sum was 1007, then returned to his play. I didn't even know he could read 3-digit numbers at the time, so I was pretty floored. My only explanation is that he was paying attention to my conversations with older siblings as he played quietly in the same room 🤷‍♀️. After that I quickly went through all the place value exercises I'd uses with the older two to make sure he was really solid on the idea and there were no holes in his understanding, but I'm not sure it was really necessary. He was able to do anything I asked the first time. It's making sense, but it's also not lining up with my personal experiences of young kids learning place value. Maybe my experiences don't really count though, since none of my kids are neurotypical? I did not have to focus on a single model of place value before introducing another. I always taught them all concurrently. And, for my older kids at least, I'd not think of the process as a slowly developing photograph but more of a lightbulb suddenly being switched on. I taught trading as soon as place value was well understood with the RS abacus. There were beads that represented units (ones, but we do like MUS and call that place value units so as not to confuse one 100 or one 1,000 and so on), beads for tens, beads for hundreds and thousands. There's room for 20 unit beads, but once you hit 10 units you're supposed to trade them for a 10 bead because you can't write fifteen or twenty in the unit's place. I *never* taught counting on to the older three boys, at all, ever -- so they obviously mastered place value without first having that skill.

If I give him two 10-chips and three 1-chips he cannot name, write, or point to the number/quantity represented, neither can he represent the quantity with any other manipulative (ie. on abacus, with base-ten blocks, etc). If I show him the written number "23" or say it or build 23 with abacus/base-10 manipulatives he cannot choose two 10-chips and three 1-chips to represent it. There is really zero understanding. He just knows that he can exchange ten 1-chips for a 10-chip (but again, not the other way around). Honestly, it means about as much as if I'd trained him to always exchange 7 tooth brushes for a can of spaghetti sauce. Eventually I hope he'll have the understanding and fluency to translate back and forth between verbal, written, 1-to-1 manipulatives (like abacus or base-10 blocks), and abstract manipulatives (like place-value chips) for any number, but that is a quite a long ways off. Yes! We have a play cash register and he loves to play with it. However, he's not learning *anything* other than you can trade plastic circles and rectangular papers for like ANYTHING you want to put in a store. Money is too abstract. He has a ton of fun with it, though, and I figure having positive associations with stuff that looks like money will help when I eventually try to teach him that money has specific values and such. He'll be more interested and it'll be easier to get his buy-in. I suppose I could take the play money out of the drawer and replace it with the 1-chips... then we could work more on matching 1-chips to written numbers on play tags... maybe we could work up from that, eventually. We have base-10 blocks and use them a ton. Ours are yellow! Thanks! I will look into the Mathematical Reasoning books. I didn't even consider them because I guess I mostly think of the CTC as catering to test prep and resources aimed more toward gifted and high achieving kids. We have the DragonBox Numbers app and DS 7 enjoys it. He's been playing with it for a couple of years now and does well with the puzzles but can't do the "running" game at all. Anyway, the Nooms are a big reason I made our c-rods myself. I wanted to match the Nooms' colors. I was afraid that after playing the app, changing the colors of the numbers would mess him up. How interesting! I didn't even consider putting Big Numbers on his tablet because he can't do like half of the activities in the Numbers app. I'll try putting it on his tablet and see how he does.

Nope. The chip trading is just a memorized rule. He's memorized that he can trade the ten 1-chips for a 10-chip (but not vice versa) and that's exactly where the meaning ends. What would that look like? How would you present such a model? Would my home-made c-rods (that essentially show a quantity of glued-together cm cubes) fit in that model, or do you think they are more likely to cause confusion in a quantity-based model?

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

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.