When a child can't count on their fingers...

[Kanin]

What would you think? Say, for something like 9+3. Child puts out 9 fingers, then gets stuck because there aren't 3 more fingers to count. When told to "put 9 in your head," and then count 3 more fingers.... blank stare.

Clearly, this child doesn't understand "counting on." It's just a developmental stage that eventually comes with enough number sense, right? There's no rushing it, in my experience. This child does well with number lines.

[Arctic Bunny]

If they can use a different method, why use fingers at all? 

[EKS]

Why do you need to use fingers at all?

I wouldn't say that the child doesn't understand counting on.  I would say that the child doesn't understand reusing fingers.  Have you tried using something else, like beans, where you don't run out?

[kbutton]

I never understood how to count on fingers (beyond ten) until my son was doing it (self-taught), lol! So, yeah, I don't think your explanation would've helped me at all. 

[Lecka]

Can you model starting at 9 and then counting on, with 3 fingers, while counting on.

Like -- maybe first show on the number line.  

Then say "we can count on without a number line.  we can start with 9 and then count on 3 numbers."  Then model it.  Could the child copy an exact model vs. doing it from you giving an explanation.

Possibly make a point of counting "1, 2, 3" with the jumps on the number line.  Possibly draw a hand with one finger, two fingers, three, fingers, on the number line (or on a number line) to make the connection of the skips to the fingers.  

I would wonder if it is a coordination issue and offer using counters (but giving 3 counters and modeling counting on from 9 and encouraging using the counters to count on) if there is any chance it is a coordination issue.  Do they seem confused about how to hold their fingers if they copy you?  Really some kids don't have that automatic awareness of their fingers where it is an awesome strategy because of "feeling" their fingers.  

Using a ten frame might be a way to go.

I think you have a lot of options!  I think do try the most explicit modeling you can to start 🙂 And try to link it to the number line as explicitly as possible 🙂

And watch out for subtraction where to subtract you "count on" like subtracting years to find out how many years have gone by and things like that.  I have seen things that talk about 3 kinds of subtraction, and this one was the hardest for my son and it effected his understanding of some word problems.  And some mental math.  

I think it is worth working on but not necessarily with using fingers.  Maybe going from a number line to practicing verbally or mental math.  I don't know if it's worthwhile to focus on using fingers if it is not a natural fit.  I could see it depending on the kid.  You could try ten frames or unifix cubes etc to see if there is a manipulative that clicks better, and might want to use a variety of manipulatives for the same concept.  

Good luck!  My son has made a lot of progress on this!!!!!!  But is has taken so long, sigh.  

 

[PeterPan]

Geodob had a method where one hand is tens, the other ones. So you make 1-9 with your right hand and move the tens over to the left hand.

Have you looked at Ronit Bird? Maybe some more strategies? 

[Melissa in Australia]

Math is one of those things that fits very nicely into Piaget's theory of cognitive development.

I would say, just only by the information that you have provided , that it is too abstract for that child to think  number 9 then count on 3 more.

 much more work with manipulatives of all sorts may be needed

 

[Ausmumof3]

16 hours ago, Mainer said:

What would you think? Say, for something like 9+3. Child puts out 9 fingers, then gets stuck because there aren't 3 more fingers to count. When told to "put 9 in your head," and then count 3 more fingers.... blank stare.

Clearly, this child doesn't understand "counting on." It's just a developmental stage that eventually comes with enough number sense, right? There's no rushing it, in my experience. This child does well with number lines.

Can you make up a ten frame and use counters instead?  I’m not sure what you’re doing for math but singapore does a tonne of work with making tens.

[geodob]

The problem that the child probably has?  Is that they are using their fingers to remember the order of numbers.
So that numbers are learned and recalled, in a similar way to Alphabet.
But the problem they likely have, is that they don't have a sense of each number.  As representing a different sized group?
Rather, that 9 is the finger next to the finger named 8.

What they need to develop, is a sense of different sized groups. Being represented by different numbers.

[Kanin]

Hi all, thanks for the ideas. This child is 8. We're doing lots of Ronit Bird, and he does well with Cuisenaire rods. We will continue to use manipulatives as long as needed for counting and adding. He is pretty good at visualizing a number line and counting up in his head, but when it's a lot of counting up, like 8 + 7 or something, he makes mistakes. He can count up mentally +3 or +4 with no problem. 

It's just surprising to me because I've only met one other child who hasn't been able to count up by using fingers. I definitely think it's a matter of not really having a sense of quantities, like geodob said above... fingers are not connected to actual quantities when counting, it's just a recitation. 

[PeterPan]

2 hours ago, geodob said:

The problem that the child probably has?  Is that they are using their fingers to remember the order of numbers.
So that numbers are learned and recalled, in a similar way to Alphabet.
But the problem they likely have, is that they don't have a sense of each number.  As representing a different sized group?
Rather, that 9 is the finger next to the finger named 8.

What they need to develop, is a sense of different sized groups. Being represented by different numbers.

Bingo. That's why the Ronit Bird stuff would be good. She has some games in her C-Rods ebook that help them work on bridging. (=crossing over the 10 while adding/subtracting). I would probably back up and skim through Dots, just to make sure they really understanding smaller quantities and addition within 10. They need to be solid subitizing the smaller quantities before they're ready to bridge. If they're not seeing the smaller quantities in the larger quantities, they can't bridge. At that point, you're asking whether Fred + Ernie = Big Bird, ie. memorizing names instead of thinking in terms of quantity.

The method of representing the smaller quantities in the larger quantities is irrelevant, and one manip does not make it better than another. So abacus, dominoes, fingers, beads, tally marks, whatever. The real issue is seeing the smaller quantities inside, and that's the real reason the kid can't bridge, because they aren't thinking that way. So once you realize that's the goal, you work on the subitizing (at which Ronit Bird is most brilliant) and do it across manipulatives, lots of ways. So I did every manipulative I could find. Ante poker cards, dominoes, candies on a table, tally marks, different kinds of abacus (RightStart, Soroban, etc.), 100 charts, money, fractions, decimals, etc. 

It's the biggest win in all of math, and everything else flows from it.

[PeterPan]

1 hour ago, Mainer said:

Hi all, thanks for the ideas. This child is 8. We're doing lots of Ronit Bird, and he does well with Cuisenaire rods. We will continue to use manipulatives as long as needed for counting and adding. He is pretty good at visualizing a number line and counting up in his head, but when it's a lot of counting up, like 8 + 7 or something, he makes mistakes. He can count up mentally +3 or +4 with no problem. 

It's just surprising to me because I've only met one other child who hasn't been able to count up by using fingers. I definitely think it's a matter of not really having a sense of quantities, like geodob said above... fingers are not connected to actual quantities when counting, it's just a recitation. 

Does he need help to generalize? My ds was like that. He could do it with one tool and TOTALLY NOT KNOW IT with the next. Page numbers in a book, haha. Yes the number lines are good, but change the direction to vertical. Change to thermometers. Go back to the manips you used in Dots. Every skill as many ways as possible. I think they say the rule of thumb with generalization is to go for 6. So whatever concept you're doing, park and apply it lots of ways.

-money

-measuring

-time

-graphing (coordinate planes or data/bar)

-100 board--I have a whole book of games to play on a 100 board and RB has games

-smaller charts like RB has you make with 20 or 30 or cutting out to make custom frames. I cut them out of craft foam with prints so they're cute.

-fractions

-thermometers

-word problems

At least for my ds, that's what I've had to do, because he can do a concept one way or seem to have it and not have it in the next setting, like a total blank. 

You might also look at Tang Math as another way to get him processing and connecting the math. Right now I'm particularly liking Pack 3, which has Shape Up, Pictarithms, TangaRow. You can take the grade level down to where it's not too hard, and it will really make him think. At least that is good for my ds.

And you know, none of this is stuff you don't know. It's more just the shock of realizing how extensive the disability is and what might be necessary. When starting any new manip, you may need to go all the way back to Dots level understanding, make sure he gets it, then go forward. It just won't take as long because he has figured it out before with another manip.

[Kanin]

23 hours ago, PeterPan said:

And you know, none of this is stuff you don't know. It's more just the shock of realizing how extensive the disability is and what might be necessary. When starting any new manip, you may need to go all the way back to Dots level understanding, make sure he gets it, then go forward. It just won't take as long because he has figured it out before with another manip.

Thanks for reminding me about generalization. I will find out how he does with our math concepts in other contexts. Also, gonna mix up the manips!

[Janeway]

I never teach using fingers. I think it makes it too easy to go back to fingers later. It is like the difference between sucking a thumb and a pacifier. Forget the fingers, they are a bad habit. Just use blocks, beads, even beans or anything you can find around your house that can be counted...cheerios, anything. And don't worry about ever teaching to do any of it on fingers.

[PeterPan]

If you want to use fingers, you can make cards or games that interweave the fingers and other manips. So you could play a memory game or Clumsy Thief type game where the goal is to make target numbers (13, whatever) but use cards with pictures of fingers, pictures of tally marks, pictures of dot patterns, digits, etc. interchangeably. We had them with RightStart, so it's just something I have lying around. 

Have you played Positive/Negative Turnovers? It's one of RB's games from her free games ebook. Runs in my mind we used bridging for that but I so cannot remember. I know the cards are ante poker cards. They're a really handy shape. 

Here's that 100 board book I mentioned earlier

Carson-Dellosa › Shop AllHundred Number Board Activities Resource Book Grade 2-3 ...

Turns out the book we have is marked gr4-5

Carson-Dellosa › Shop AllHundred Number Board Activities Resource Book Grade 4-5 ...

[lewelma]

On 11/28/2019 at 1:46 AM, Mainer said:

It's just surprising to me because I've only met one other child who hasn't been able to count up by using fingers. I definitely think it's a matter of not really having a sense of quantities, like geodob said above... fingers are not connected to actual quantities when counting, it's just a recitation. 

I'm a tutor, and I just wanted to let you know that I have taught a kid who at the age of 17 could not subtract 9-6.  When I asked her to do it, it took her 2 full minutes with a tally chart to get 4. She had been born at 24 weeks and had obviously been diagnosed with dyscalculia.

I got her through 11th and 12th grade statistics with the use of a calculator. 

[PeterPan]

8 hours ago, lewelma said:

I got her through 11th and 12th grade statistics with the use of a calculator. 

Yup, right now I have ds learning to use "math machines" because I'm hoping he'll see patterns. But very soon I'm just gonna hand him a calculator and plow forward. The writing is so on the wall for him. 

[Kanin]

14 hours ago, Janeway said:

I never teach using fingers. I think it makes it too easy to go back to fingers later. It is like the difference between sucking a thumb and a pacifier. Forget the fingers, they are a bad habit. Just use blocks, beads, even beans or anything you can find around your house that can be counted...cheerios, anything. And don't worry about ever teaching to do any of it on fingers.

I'm not trying to teach him to count on his fingers, I'm just surprised that he CAN'T, because for most kids it's so natural. I'm trying to figure out what his holdup is with "counting on"... maybe he's just not ready to move to a more abstract way of thinking about math. 

[Kanin]

36 minutes ago, PeterPan said:

Yup, right now I have ds learning to use "math machines" because I'm hoping he'll see patterns. But very soon I'm just gonna hand him a calculator and plow forward. The writing is so on the wall for him. 

Could be the same for this one! As long as he understands what he's doing, I'm fine with it... 

[Kanin]

8 hours ago, lewelma said:

I'm a tutor, and I just wanted to let you know that I have taught a kid who at the age of 17 could not subtract 9-6.  When I asked her to do it, it took her 2 full minutes with a tally chart to get 4. She had been born at 24 weeks and had obviously been diagnosed with dyscalculia.

I got her through 11th and 12th grade statistics with the use of a calculator. 

Wow. That's awesome that she got through such difficult math 🙂 Did she ever talk about how she felt about her math skills?

[lewelma]

1 hour ago, Mainer said:

Wow. That's awesome that she got through such difficult math 🙂 Did she ever talk about how she felt about her math skills?

Yes, she did. When I first got her, she told me that using a calculator made her feel slow and stupid.  And it took me many many months to convince her that the goal was to get the question right in a reasonable amount of time, and that the calculator was a tool she could use to accomplish this.  We focused on interpreting statistics using a qualitative program (although she had to do some mathematical statistics as these was the NZ national exams she was doing). I kept talking to her over and over about how: yes, there are people who *do* the math, and then there are people who *use* the math.  She could be in the latter group and math could still be her forte. Understanding errors and assumptions in surveys, experiments, observations, inferences, etc, this would serve her well in political science (the field she was interested in). She did not have to DO the math for it to still be a useful skill. I convinced her that she wanted to do what computers couldn't - design and interpret statistics; and that she shouldn't try to be a human calculator because computation is a low level skill.  Statistics was never easy for her, but she pushed through because I convinced her that 1) she could do it as long as she had her calculator and 2) she was learning valuable skills that she would actually use in the future. 

Edited November 29, 2019 by lewelma

[sweet2ndchance]

Something I don't think I saw mentioned (sorry if I missed it somewhere) is using touch points.

It is a similar strategy to what TouchMath uses but you don't need to use that curriculum to use the idea of touch points.

Basically, you learn where on the numbers to touch and in what order. You do it until it becomes automatic. Some numbers like six and nine have double touch points where you touch twice to get enough points to make the correct quantity.

One of my kids despised using manipulatives but still needed something concrete. This was a great way to give him something concrete to work with but not manipulatives or fingers. Since each number has exactly the number of touch points it needs, you never "run out" of manipulatives or fingers to count. ;-)

Here is a link with lots of different takes on touch points.

[lewelma]

24 minutes ago, square_25 said:

The key here is that it's obvious to a child that they are still counting all the things together, just in a more efficient way. 

Over the years I have tutored, I have been absolutely fascinated to see how kids think about things.  I am so very appreciative when a kid asks me a 'stupid' question because it clarifies deep misunderstandings, and can be so helpful in teaching them. I also taught Chemistry to the girl I described above, and she could not tell me the number of electrons in the outer-most shell because she could not count up in the way you are talking about. For example, you have 12 electrons to put in shells: 2 for the first shell, 8 for the second shell, and 2 for the final shell. She would have to draw all three shells around the nucleus and added in the electrons one by one counting up to finally tell me that there were 2 electrons in the outer shell.  I tried many times to convince her that you could add 2+8 to get 10 and then subtract 12-10.  That if the number of total electrons was more than 10, that all you had to do was subtract off 10 to get the electrons in the outer shell. She could not understand this, and felt *very* strongly that it was almost sleight-of-hand and a trick.  I tried place value, I tried blocks, I tried drill, it was just not to be.  She was 17. And yes, I did get her to pass 11th and 12th grade chemistry. haha. I taught her to love her calculator.  🙂 

Edited November 29, 2019 by lewelma

[lewelma]

yes, she is an unusual case.  But what helped me more than anything to better understand the profound lack of understanding was that she considered 12-10=2 to be sleight of hand.  It really felt like magic to her, a trick that I could implement to amaze an audience. Here is a kid who can read and interpret complex probability problems and code them correctly into a calculator, but then thinks subtracting 2 is an advanced skill. Oh to understand the human brain! But this situation taught me a few things. 1) there comes a point in time where you should abandon basic numeracy and switch to a calculator and more advanced work. It is hard to know exactly where that point is, but the decision must be made. 2) Many kids who are not as articulate as this girl, may have deep misunderstandings and simulate numeracy through memorizing tricks.  I must always be vigilant to root out these types of computational-skills-without-understanding to insure kids can move forward. I have learned to ask very basic questions about how a student does something that they seem very confident and capable of doing, and at least 50% of the time, they have no idea. And I am talking very basic stuff for 16/17 year olds taking pre-calculus. Stuff like why is 0.1+0.01=0.11. or why is 0.95 = 95%. Shocking how many kids are using sleight of hand and don't recognized it as such like this girl did. Her metacognition was excellent.

Edited November 29, 2019 by lewelma

[PeterPan]

3 hours ago, Mainer said:

I'm not trying to teach him to count on his fingers, I'm just surprised that he CAN'T, because for most kids it's so natural. I'm trying to figure out what his holdup is with "counting on"... maybe he's just not ready to move to a more abstract way of thinking about math. 

Abstract like what? For my ds, easy things are hard and hard things are easy. And no he has never used his fingers for math. 

So jumping to fractions is easy. Applying to other things, easy. +/-, easy. 9+3, not easy.

There's a really sharp book from Didax on using geometry to teach fractions. The number sense is in one part of the brain (per what I've read) and math concepts in another. He might be fine with some really advanced things even while being bunk at this. 

Are you playing the games Ronit Bird has for bridging? She has quite a few.

I think sometimes you can tell a dc how to get there but until he discovers it for himself, like really in the moment goes OH, it's not his. It's just what you said to do, filed away in this side folder. 

[PeterPan]

24 minutes ago, square_25 said:

You know, I've generally found that kids don't connect "counting up" and "subtracting" all that naturally.

Yeah and it's not really something Ronit Bird encourages. I need to review, but counting is not her gig. It's a totally different thing in the brain, encouraging sequences rather than quantity based thinking. The question is how many to here, how many to the next step, not counting.

[lewelma]

19 minutes ago, square_25 said:

 

Quote

Yeah, I've had this experience many, many times. Most kids I've taught didn't think there WAS a why, never mind what the why might be. That's why I tend to be suspicious of procedural teaching, unless it's important to get a kid through their next test (I believe we don't tend to agree on that one.)

Ah, yes, we had a great conversation about that one. 🙂 Sometimes I just find kids are so caught up in learning the procedural skill that they won't actually put brain power to the conceptual understanding until they master the procedural skill.  So I drill in the skill without understanding, then back up when they are confident and work on the conceptual understanding. Not all kids, but there are definitely some. Partly, I think it is because they have been taught for so many years is that math is only procedural understanding. So they focus on that and simply can't focus on anything else until they can manipulate as required.  Sad but true. 

Quote

I think the worst one I've seen was fractions. I would guess that about 80% of my first year calculus kids had no clue at all where the fraction rules came from. If you asked them to show to you that 1/x + 1/y is not 2/(x+y), they really would have no idea where to start. More of them could do 1/2 + 1/3 = 5/6, but they wouldn't really be able to explain this one either, except for procedurally. 

Oh my, yes. Fractions. Sigh.  I teach their conceptual understanding almost every single lesson to every single student all the way up to calculus.  I am *constantly* reminding them of the concepts behind fractional manipulation. Over and over and over. The only kid I have whose fractional knowledge is rock solid is the one who did MUS fractions.  I really hate MUS and I hate the Fraction book because it is just drill and kill, but this kid gets it. So I may have to swallow my opinions. 🙂 

Edited November 29, 2019 by lewelma

[lewelma]

Oh my, I have totally munted the quoting feature! yikes. Off to make my response in blue as I can't seem to fix it!

[lewelma]

11 minutes ago, square_25 said:

@lewelmaI've done purely conceptual fractions for my 7 year old so far, with really excellent results. It's taking her a while to get the procedural stuff down, but she's super solid, she can think about it, and she jumped into decimals without any trouble whatsoever ("They are just fractions, right?"). Of course, her place value was completely solid by the time we got to fractions, so that helped with decimals as well. 

From my AoPS experience, once kids have learned to engage with something procedurally, it's a really uphill battle to get them to even THINK about the concept as something that ought to have an idea or a picture behind it. Somehow, they not only internalize the procedure, they internalize that they aren't supposed to understand it! And that's with AoPS instruction, where we DO introduce it conceptually at first. But somehow, not having a chance to actually engage conceptually before learning the procedure would mess up all but my one or two strongest kids. If we took a few weeks to engage conceptually, I'd instead have 20-30 kids who were engaging conceptually. It was really remarkable. 

Isn't it great to be able to teach your dd how you wish all kids could be taught! What a great gift! 

My older boy has always said that math is his most creative subject.  I think that 99.99% of the population would have absolutely NO idea how this could be the case.  Mathematics instruction is awful in both the USA and NZ. I was a product of the American procedural math focus, and when I got to university and was asked to think conceptually, I failed. It has been a slow uphill battle for me to work conceptually because this is not how I was trained for a decade. 

I am working with a kid who I could barely squeeze in so the mom got a second math tutor for him.  I just about had a heart attack when the kid showed me the 'upside-down picnic table method' for adding fractions with different denominators. Multiply this horizontally, cross multiply that, add it up.  OMG! Horrible horrible horrible.  This kid quite happily drew the picnic table crosses on the two fractions and added them up with no idea why it worked or, like you said, no idea that there was even a why to understand. 

Edited November 29, 2019 by lewelma

[lewelma]

2 minutes ago, square_25 said:

(Why is fraction multiplication so much easier than fraction addition? Who knows....) 

LOL!  Just about spit out my tea!

[lewelma]

But come on, with a name like the "upside-down picnic table method" who could possibly forget it!?!?!?!?! Why bother to understand when it is so much easier to just drag out the picnic table upside-down.