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Math: Moving fwd vs staying put to solidify for a while


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Two of my kiddos still rely heavily on C-rods for math:
1) My youngest (K/1st) can look at a page with single-digit addition & subtraction problems, set up the C-rods, solve, and write the answer down completely independently, if she has C-rods in front of her. But without the C-rods, she seems to still lack enough understanding to even use her fingers to add or subtract (despite having used other manipulative and talked about and explained addition/subtraction concepts in a WIDE variety of ways).
2) My 4th grader still uses C-rods all the time, particularly to help with fractions work (3/5 of 40, etc.) and it's really starting to hamper her in bigger examples (3/5 of 160, for example, is much more difficult to "see" with C-rods, because they don't make rods that are 32 units long... lol).

My other two kiddos (6th and 2nd grades) just naturally transitioned from using C-rods to doing things in their heads. With the 4th grader, I assumed that it would "click" eventually, and she'd stop using them, so I continued to move forward because she did seem to be "getting things" easily enough with the C-rods. Now I'm wondering if I've done her wrong, and I'm particularly concerned about making the same mistake with the youngest (if it was, indeed, a mistake). Somebody please tell me if letting them continue to use C-rods to move forward is ok, or if I really need to put the brakes on and spend more time practicing/solidifying until they can also do the problems without C-rods? I feel like I'm totally out of my element here!!! lol. 

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19 minutes ago, square_25 said:

Also, are there strategies for adding you'd like her to use? I used counting on a lot, and that's easy to demonstrate with cards... there's also near doubles and making 10 and lots of other things. 

Yeah, we've tried counting on a *lot,* and it still doesn't seem to gel, even with just +2. And, despite being able to count up to 100 orally quite easily, and go backwards as well, she still can't do subtracting by counting down at all (not even -1 -- we tried today! lol) And we've drawn little number lines and talking about hopping up and down the number line, but no dice with that either. And we count beans and add and take away there. I feel like if I *tell her* to either put more beans in or take beans out, she'll do it, and get the answer, but she never seems to add/take the beans out correctly on her own. But if I say "You have 6 cookies, and I eat 1 (or 2, or 3), how many do you have?" She can usually think about it and answer 5 correctly (or 4, or 3). 

But, if I'm reading you right, you would definitely suggest "staying here" for a while until some of these other ideas gel, correct? 

ETA: Ok, now that I wrote all that out, it's obvious she shouldn't be moving forward.... I should face palm myself.

Edited by 4KookieKids
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12 hours ago, 4KookieKids said:

Two of my kiddos still rely heavily on C-rods for math:
1) My youngest (K/1st) can look at a page with single-digit addition & subtraction problems, set up the C-rods, solve, and write the answer down completely independently, if she has C-rods in front of her. But without the C-rods, she seems to still lack enough understanding to even use her fingers to add or subtract (despite having used other manipulative and talked about and explained addition/subtraction concepts in a WIDE variety of ways).
2) My 4th grader still uses C-rods all the time, particularly to help with fractions work (3/5 of 40, etc.) and it's really starting to hamper her in bigger examples (3/5 of 160, for example, is much more difficult to "see" with C-rods, because they don't make rods that are 32 units long... lol).

My other two kiddos (6th and 2nd grades) just naturally transitioned from using C-rods to doing things in their heads. With the 4th grader, I assumed that it would "click" eventually, and she'd stop using them, so I continued to move forward because she did seem to be "getting things" easily enough with the C-rods. Now I'm wondering if I've done her wrong, and I'm particularly concerned about making the same mistake with the youngest (if it was, indeed, a mistake). Somebody please tell me if letting them continue to use C-rods to move forward is ok, or if I really need to put the brakes on and spend more time practicing/solidifying until they can also do the problems without C-rods? I feel like I'm totally out of my element here!!! lol. 

Do you think this is due to a lack of understanding, or a lack of confidence? Can your kids articulate what they're doing with the rods? Like can your youngest explain how she's using them to solve an addition problem? Can your fourth grade explain what 3/5 means?

 

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13 hours ago, Little Green Leaves said:

Do you think this is due to a lack of understanding, or a lack of confidence? Can your kids articulate what they're doing with the rods? Like can your youngest explain how she's using them to solve an addition problem? Can your fourth grade explain what 3/5 means?

 

I will have to think about this, some to see if I can gauge their actual understanding. The short answer is that they cannot articulate what they are doing, and the older will even flat out tell me that she has no idea what's going on. But they are also both autistic, so articulation of thought processes is a special challenge for them in general, and not just with regards to mathematics..

ETA: the 4th grader can say 3/5 means to break into five equal pieces and take three of them, and she can draw a matching bar diagram or pizza picture where she correctly shades 3/5. But she's not able to articulate exactly how she's using it or what it means in a particular problem, if that makes sense?

Edited by 4KookieKids
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Work on subitizing right away like a PP said. Teach her how to "see" 6 items right away by mentally grouping them into two groups of three, for instance. You can draw dots on a paper or whiteboard in random patterns, then have her circle groupings to "see" the number without counting. 

Greg Tang's book Math for All Seasons is perfect for practicing this skill. 

If she doesn't get that or can't translate that skill into what she's doing in math, you could try switching to a different manipulative with the purpose of developing a visual frame of reference. She's already use to using the c rods like a calculator, so try an abacus or ten frames. I personally prefer a ten frame, then double ten frame, for this particular purpose.

So print a ten frame, have her practice putting disks like poker chips or colored paper cutouts. Do very simple addition problems using two different colored disks. Then ask her to close her eyes and imagine what the ten frame looks like in her head for the problems. A few days later, have her put just the first number of an addition problem on the ten frame but only imagine the second number. What does that look like in her mind? What is the answer to the addition problem?

Practice this for a few days, and gradually increase her ability to visualize the whole math problem by looking at a blank ten frame, then to no ten frame at all. You want to work up to using a double ten frame so she can visualize regrouping, then she can do this for any place value.

This is just based on what I do with my particular kids, but I hope it helps.

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6 hours ago, square_25 said:

That seems like a fine understanding to start with :-). Can she try to use that for bigger problems?

Also, mind telling me how you explain “counting on”? 

Bigger problems become problematic because she usually doesn't think to actually do division; she guesses a random number and then multiplies by 5 to see if it's right. She is dyslexic and has a difficult time with fact memorization (which is why I let her rely on C-rods for so long), so by the time she actually gets a number divided by 5, she has usually forgotten why she even did it. lol.

I have explained "counting on" by using C-rods by making our staircase and pointing to one and asking her to start there and keep counting (easy for her), counting up/bigger, one more each time, or going to the "next" (whole) number. We played games of what comes before and what comes after before trying to learn counting on. 

I really appreciate all of the thoughts all of you have shared with me on this thread. It's helpful to be able to talk these things out. Math was the one field I really thought I wouldn't need help teaching (as it's what I have my degree in)! lol Just goes to show!

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59 minutes ago, square_25 said:

So, what you're describing for counting on isn't actually an explanation for WHY it works, right? Like, why is that the same thing as what you get when you add 8 + 3? That may not be making sense in her head. What does she think 8 + 3 means, in words, if that's possible for her to explain? Would she understand she was doing 8 + 3 if she was playing Addition War and she had an 8 card and a 3 card? 

Hmmm, so we talked about "why" we count on when she is just counting everything together (for 8 + 3, we'd get 8 beans out in one pile, put 3 in another pile, and she'd count them to get 11), by talking about how we don't have to count the first 8 because we already know they're in the 8 pile, so we just have to start at 8 and keep counting up the extra 3 in the other pile. And then we practiced this a lot. Is that what you mean, or something else regarding the "why"?

She knows addition means "put together," and she can do it with dice just by counting all the dots on both dice combined. 

As I consider what happens when we sit down, I think that, when we do addition exclusively, and I give the example of how to use today's manipulative (counters, dice, 10-frames pictures, and c-rods are the main ones), she can repeat the process independently with whatever numbers I give her to get correct answers. And she can do the same with subtraction. But it's clear that she's not really "getting it," because the moment I introduce mixed practice (either mixing up the manipulative between problems, or mixing up addition and subtraction problems) she reverts to shouting out of random numbers rather than doing anything logical. And they're not even showing a concept of more/less, so she'll get 4+3 and shout out "9! 2! 5?..." (without even processing that adding 4 and 3 should certainly not give you 2, even in random guessing! lol)

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2 minutes ago, square_25 said:

It sounds like she's not generalizing the idea. This is something you might ask about on the Learning Challenges board, because I've heard people mention this issue with autism before. 

Ah, I hadn't thought of that. I have so much to learn! lol. I'll ask about it.

2 minutes ago, square_25 said:

 C-rods can feel a bit magical to me. 

I thought they were magical for a while, because it got these two kiddos *doing* math, when they were stuck (my other two kiddos have never really gotten stuck on something in math).  I think they just didn't actually make the leaps I thought they had, and instead, they only learned to use the manipulative correctly. lol. 

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44 minutes ago, square_25 said:

Well, I'm sure they did learn things along the way!! But yes, in my experience, kids will do things in the way requiring the least effort, on average ;-). If you want them to make a conceptual leap, you tend to need to give them questions that actually require that conceptual leap. If you just give them a tool, they will often use the tool without understanding, because it's, well, easier. And this applies very broadly to manipulatives, formulas, explanations, algorithms... anything that can be done without thinking about it will be :-P. 

I'm not even trying to be mean about this -- I'm like this, too! Good understanding takes real effort, and most people only have so much energy. So I no longer blame kids for this, or lecture them... I just try to make sure to provide situations in which understanding what you're doing is actually the method of least resistance. 

I don't take offense - no worries. I feel like I'm pretty good at teaching ideas/understanding in higher level math, primarily because I've always loathed formulas and algorithms. So I just started taking my oldest kiddo through AoPS algebra, and we are having a wonderful time and he wrote out a great proof that sqrt(2) is irrational and it's SO MUCH fun! lol. BUT, I have a lot more practice teaching some of that sort of stuff than elementary age stuff, and I just never struggled with understanding any part of arithmetic, personally. So I think I *thought* I was a pretty good teacher, but it turns out, I'm just a good teacher when my kids actually think just like me, and I need definite help when my kids don't process things the same way as I do! I think that my first and third kiddos went through these stages so seamlessly that I'm not even sure I *realized* that there was conceptual learning going on, if that makes any sense. Somehow, I just assumed it was all common sense and would happen naturally as we played with math and did life together. I've taught college courses, and even taught college courses for in-service middle and high school teachers, but the two kiddos I wrote this post about definitely have made me realize that I actually have NO real understanding of what actually happens (in their brains) when young children learn math. 😂

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At my volunteer gig, I worked with a kid who struggled for years to do addition and subtraction.  I know that he was held back one year, and I worked with him every chance that i got but it was such a struggle.  I told everybody that I could who worked there that he needed more than mom help - his behavior dysfunction was often frustration, and he needed somebody who knew what they were doing in addition to the normal 'homework help' that the kids got.  At some point they found a coach for him, because after not seeing him for a few weeks we sat down to do some work and he started with some addition - say, adding 8 + 3 in the ones column - and he looked at me and said 'first I put the 8 in my brain, and then I count 3 more' as his way of explaining 'counting on'.  I had coached him to count on, and sometimes he could, but that wording seemed to make it click with him.  I had taught him to start with the bigger number so there were fewer to count, so he had turned that into 'put the bigger number in your brain' when adding.  Whoever worked with him seemed to help him overcome some sort of mental block, because the last time that I saw him he was working on multiplication and said that he knew up through the 5s or 6s already...and he did.  He's still behind, but I figured if he got solid on arithmetic, he'd be in much better shape.

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1 hour ago, square_25 said:

DD7 was also very young when I was teaching her some of these things, so I wound up needing to connect a lot of dots for her. (I think I taught her to count on at age 4; having spent more time with more average kids, I can now attest to the fact that most 4 year olds cannot learn this at all, but I didn't realize that at the time.) 

Maybe that's why it has thrown me that two of my kiddos don't just absorb math; I think my other two kiddos are more similar to your dd, as am I. I'm still learning to process what is "normal"! Even my 4th grader who still uses c-rods, definitely understood counting on at age 4, as did my other two kiddos, so it honestly never even occurred to me that my youngest wouldn't have it figured out by shortly thereafter, and that that is actually ok/normal.

1 hour ago, square_25 said:

Let me know if my counting on explanation helps at all... I did find that it helped the kids in my classes, and I also found that a good number of kids had not figured out counting on for themselves by age 6 or 7, which is I think considerably older than the age during which they would have been able to understand the explanation. I had a kid whose mom said she had tried to teach counting on and he had never gotten it, but he really didn't seem to have trouble with the explanation... sometimes, someone guiding you over a hump is really helpful! 

I'm definitely going to try it as "counting really fast!" the next time we sit down. I can't recall if I said that exactly last time we tried (I know I *thought* it!), but I know that she looked at me like I had three heads when I said, "Hey! Since you just counted 8 beans out, you already know there are 8 here! Why don't we say we counted these 8, and then keep counting with this little pile of 3!" And, after looking at me silently for several moments, she very calmly and deliberately just started at "1....2...." and counted all 11 beans. 😂

 

1 hour ago, square_25 said:

Oh, the other thing to be mindful of is that ideas take as long as they take to absorb. I could never rush DD7 to the next concept without having some aspect of her understanding undermined. That doesn't mean that I didn't care whether she made progress... but for me, sufficient progress was "understanding the idea we're studying better and better." And we'd definitely study lots of ideas concurrently, so we wouldn't get bored staying put. 

Yes, this is why we've traditionally done more than one program (education unboxed + Singapore + base 10 blocks + counters/dice games, adding in LoF and then BA later, etc.) because my kids tend to hit a little wall, and then want to do something else for a while while those things "marinated" a bit. 🙂 We have so many manipulative at this point that I think I actually vary the manipulatives less now than earlier. Isn't that odd? I think I just forget about some of them. Maybe I should just make a plan like Monday: Counter chips (different colored sides so fun to play parts/whole games), Tuesday: Base 10 blocks, Wednesday: dominoes, Thursday: 10-frame, Friday: dice games.... Uh oh, I'm out of days and not out of manipulative yet... May have to go in 2-week intervals or double some of them up!

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9 hours ago, Patty Joanna said:

I mentioned this conversation to my son, and he (who has a pithy way of saying things) said, "My student is confused about arithmetic.  Let's keep moving ahead.  That will help."   He did not eye-roll, but I got his point.  

 

Ha ha. Yeah, I think that when I first posted, I thought that maybe there was just an over-reliance manipulatives, in which case, it does seem ok to continue forward, while we wean from the dependence on manipulatives. Through this conversation and upon further thought, it's now become clear that what we're actually talking about it a lack of understanding of some basic concepts, which merit figuring out before going any further. In hindsight, it was a dumb question initially, and of course moving forward when foundational pieces are missing is a bad idea. lol. But I've really appreciated the conversation and feel like it gives me some ideas for how to teach the missing pieces, because I don't expect that just "staying in place" and continuing to review/cover things in the same way as I had been would have been the right answer, either. 

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One way I encourage my kids to think about the big picture "does this answer make sense" in math is to make up silly scenarios regarding numbers changing and the kids have to think of what could have caused the numbers to change that way.

For instance...

A squirrel was hiding nuts for winter. She already had 9 nuts in her hiding place and she could carry two nuts at a time. After two trips she counted her nuts again. How many were there? Then when kiddo says something random that makes no sense, like 5, make up something silly that happened, like a ninja squirrel stealing the nuts. Kids could make up the scenarios themselves too.

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