Moving past counting blocks with addition/subtraction.

[minkajane]

DS (6yo) does great with addition and subtraction as long as he's able to use counting blocks. Like if it's 5+2, he'll count out 5 blocks, then 2 blocks, then count the whole thing together for the answer. This is great for introducing him to the concepts, but how do we help him move to the point where he doesn't need them? They're not really feasible for when he gets into numbers over 20 or so. Is it just something that will come with time? DS tends to get really frustrated if he doesn't get something the first time, so I'm concerned that he'll refuse to try not using them because he won't know the answer right away.

[AAK]

I think it will come with time.  But, you could help transition him by showing how to do it with pictures on paper.  So, for 5 + 2, he would draw 5 tallies (or x's or squares) and then draw two more.  Underneath it, I would have him write it out.  

     x x x x x             x x 

          5           +       2    = 7

If it seems that he is going to want to do bigger math before he is able to leave the counters behind, perhaps you could get an abacus and show him how to use that.  Or, switch to using unit cubes, ten sticks, etc.

Amy

[moominmamma]

I agree with Amy that it will likely come with time. One idea that you might introduce is that, in the 5 + 2 example, he has already counted the 5 blocks, so he only needs to keep counting up from five. So rather than moving the 5 blocks and the 2 blocks into one big group and then counting them all over again from 1, he should look at the group of five and think "Five..." and then count up from five: "Six, seven."

You could introduce this approach like a super-secret short-cut.

A similar way of introducing the same concept would be to show him how to use a number line to add. It's the same sort of "counting up."

Miranda

[pigpokey]

When he gets to 2 digit numbers do you not want to line them up vertically and have him do the 1s column first and then the 10's column etc?

We did sheets of one digit plus one digit for a while.  Then two digit.

[Daffodil]

Originally Posted by moominmamma 

I agree with Amy that it will likely come with time. One idea that you might introduce is that, in the 5 + 2 example, he has already counted the 5 blocks, so he only needs to keep counting up from five. So rather than moving the 5 blocks and the 2 blocks into one big group and then counting them all over again from 1, he should look at the group of five and think "Five..." and then count up from five: "Six, seven."

And then you can point out that he doesn't actually even have to count out the group of 5.  He already knows there will be 5 in that group, so he doesn't have to make the group.  He can just imagine it's there, say "five" and count up from there with the other 2 blocks.

[moominmamma]

Quote:

Originally Posted by pigpokey 

When he gets to 2 digit numbers do you not want to line them up vertically and have him do the 1s column first and then the 10's column etc?

This is one of those philosophical issues in math education, but I definitely think it's best NOT to do this at first. To a kid who doesn't necessarily have a robust understanding of place-value yet, this will seem like a bit of hocus-pocus rather than the result of "grouping units of like value together" and "regrouping to form units of higher place value." I wanted my kids to develop excellent facility constructing and deconstructing tens for mental math addition up to about 30, so that the grouping of tens seems entirely self-evident, before introducing the algorithm you describe. It doesn't sound to me like Mandy's ds is anywhere near ready for that yet.

Miranda 
 

[pigpokey]

I look at it as, the child will see that the algorithm for addition is a tool to solve the problem. 

IMO the current trend in abstraction for early elementary math is like making six year old kids sit down and study physics of bike-riding ("now children, explain to me why turning the wheel away from the fall drives you harder into the ground") at the expense of actual time on the bike.  What you will get is a lower percentage of kids who can ride a bike. 

Looking for understanding goes against everything I have read and was taught about this age.  I thought concrete operations started about age 7 and formal operations not until age 11.  I think it is not the best way to spend time.  That is not to say that my children don't learn place value but I do not expect them to be able to manipulate it outside of algorithms that are introduced and practiced in small steps.

I have tutored enough teenagers who flubbed, over and over, their calculations.  If you don't get the calculations right all your understanding is irrelevant.  Your bridge will still collapse or your chemical reaction fail.

[MomtoDandJ]

Yes, "counting on" is a complicated idea!  For us, it has worked to tell our son to "hold the bigger number in his head" and then he can use his fingers to add another number to it.  That obviously only works if one of the numbers is 10 or under, but it might be a way to help him transition from needing to recount both numbers with blocks.  So when my son is adding 35 and 7, just an example, we tell him to hold 35 in his head and then he'll stick out 7 fingers and begin counting them from 35.

[moominmamma]

Originally Posted by pigpokey 

Looking for understanding goes against everything I have read and was taught about this age. 

I think you're confusing concrete with prescriptive. I would encourage you to read Liping Ma's book "Knowing and Teaching Elementary Mathematics" which basically compares the Asian and American approaches to mathematics education. I think Ma would agree with you about the idea that concrete should come before formal / abstract. But she would disagree about whether the algorithms are concrete. Grouping objects in units of value (tens, ones, hundreds, etc.) can be learned very concretely, using manipulatives and symbols and starting out with numbers that are small enough to be visualized and to have a clear relationship to things the child can see and understand in the real world. The algorithms are prescriptive, but they're not concrete in the slightest; in fact they're much more abstract than the place-value learning Asian kids do in 1st grade. What does "adding 8 and 9 in this column and getting 17 and writing the 7 here, and then putting the 1 up at the top of next column to the left, and adding it in with the 1 and the 2 to get the other part of the answer" have in it that makes it a concrete way to add 18 to 29? 

Miranda

[moominmamma]

Originally Posted by MomtoDandJ 

Yes, "counting on" is a complicated idea!  For us, it has worked to tell our son to "hold the bigger number in his head" and then he can use his fingers to add another number to it.  That obviously only works if one of the numbers is 10 or under, but it might be a way to help him transition from needing to recount both numbers with blocks.  So when my son is adding 35 and 7, just an example, we tell him to hold 35 in his head and then he'll stick out 7 fingers and begin counting them from 35.

This is a great way to do it. I just realized that we're all assuming he can "count up" without starting from one. If he's merely remembering a sort of language-pattern that goes "one-two-three-four-five-six-seven... " etc., he may not be able to start at 6 and tell you what the the next biggest number is. That would be something to check out first. If you say "I'm thinking of a secret number. The clue is that it's one bigger than 8," can he figure out your number? If not, you'll need to backtrack a bit and work on that skill.

If he can "start at 5 and keep counting," you're good to go with developing the counting-up concept. My kids always enjoyed being introduced to this sort of thing as a "magic smartness trick." Here's one way to do it. You'll need the blocks, an opaque cup or two and some cards labelled with the numbers from 1 to 10 or 12.

If adding 3 + 4, place the requisite blocks in two groups on the table. Ask your child to check that there are the correct number in each group.

Now ask him if he remembers that there are 3 in the first group. If he nods, then hide the 3 under the cup. "Do you still remember how many there are, even though they're hidden?"

"Yeah, three," he'll say.

"Okay, let's put this card that says 'three' on it on top of the hat so we don't forget. Now here comes the magic part. Can you figure out how many blocks there are on the table, even though you can't count the three under the cup any more?"

With luck he'll get this at a first go. Remove the cup with a flourish ("abdracadabra!") and count every last block to prove that there are seven just like he said. "Wow, it's like you have xray vision!"

If he doesn't get it first go, you can say "Well, my magic trick works like this. I know there are three under there, so I can count them even though I can't see them. Onetwothree. And then I can keep counting these ones... four, five, six, seven. See? Three, then four, five, six, seven. Let's check ... abdracadabra!.... "

Then repeat it with different numbers of blocks under the cup, and different labelling cards on top to 'remember' how many the cup is covering. Eventually he'll get so confident with it that you can just sweep 6 blocks quickly under the cup and place a "6" label on it and ask him to add two more blocks to that amount. He'll trust you on the fact that there are six blocks under the cup. And then, you can start putting "invisible" blocks under the labelled cup. Now he's just adding numerals to blocks. He'll probably think it's hilarious that you're moving invisible blocks around, and that he's able to count and add them just fine. Then you can try it with two cups covering two different piles of blocks. And two cups covering two imaginary piles of blocks. Which is, of course, just adding number to number with no manipulatives. It might days or weeks or months to get through the whole progression, but the nice thing is that it is very gradual. There's lots of opportunity to feel success at every small step.

Have fun!

Miranda

[pigpokey]

Originally Posted by moominmamma 

... What does "adding 8 and 9 in this column and getting 17 and writing the 7 here, and then putting the 1 up at the top of next column to the left, and adding it in with the 1 and the 2 to get the other part of the answer" have in it that makes it a concrete way to add 18 to 29? 

Miranda

It is a set of steps that can be learned and applied, and the math involves relatively small numbers.  Success can be had at a lower developmental stage, yet it is a skill needed by adults.

As a side bonus, doing sets of problems every day cements math facts. 

[moominmamma]

Originally Posted by pigpokey 

It is a set of steps that can be learned and applied, and the math involves relatively small numbers.  Success can be had at a lower developmental stage, yet it is a skill needed by adults.

Okay, we clearly have different definitions of 'concrete.' We'll have to agree to disagree.

Miranda

[NightOwlwithowlet]

I used a number line to help my son transition from blocks to higher numbers.  I just wrote a number line 0 to 20 on the top of the page.  For some reason having the visual cue made the transition seamless.  Slightly off topic, my son grasped the concept of ones, tens and hundreds very easily, but regrouping was a nightmare.  I think I have an old thread here about it, I tried different web sites, several different explanations, I even had a friend who is a 2nd grade teacher try to show him. DH with a minor in math couldn't figure out regrouping, eventually I gave up for three months.  I showed him how to borrow and he grasped the concept in two days and mastered in two weeks.  Several friends with same age kids in public school have told me, their schools are dropping the regrouping method and returning to borrowing and carrying.

[moominmamma]

Originally Posted by NightOwlwithowlet  Slightly off topic, my son grasped the concept of ones, tens and hundreds very easily, but regrouping was a nightmare.  I think I have an old thread here about it, I tried different web sites, several different explanations, I even had a friend who is a 2nd grade teacher try to show him. DH with a minor in math couldn't figure out regrouping, eventually I gave up for three months.  I showed him how to borrow and he grasped the concept in two days and mastered in two weeks.  Several friends with same age kids in public school have told me, their schools are dropping the regrouping method and returning to borrowing and carrying.

Following you off-topic with interest. I think I must be missing something. What's the difference between regrouping and borrowing/carrying? For example, when adding 19 and 28, you "regroup" the 10 from the 17 into the tens column. When subtracting 19 from 28, you "regroup" one of the tens from the tens column into the ones column so that you have 18 from which to subtract the 9. I always thought regrouping was just using the more conceptually-based terminology of regrouping to or from units of higher place value, rather than calling it "carrying the 1" or "borrowing 1." 

Miranda
 

[Greenmama2]

Originally Posted by moominmamma 

Following you off-topic with interest. I think I must be missing something. What's the difference between regrouping and borrowing/carrying? For example, when adding 19 and 28, you "regroup" the 10 from the 17 into the tens column. When subtracting 19 from 28, you "regroup" one of the tens from the tens column into the ones column so that you have 18 from which to subtract the 9. I always thought regrouping was just using the more conceptually-based terminology of regrouping to or from units of higher place value, rather than calling it "carrying the 1" or "borrowing 1." 

Miranda
 

I'm interested too. Also not really sure what the difference is?

[NightOwlwithowlet]

Originally Posted by Greenmama2 

Originally Posted by moominmamma 

Following you off-topic with interest. I think I must be missing something. What's the difference between regrouping and borrowing/carrying? For example, when adding 19 and 28, you "regroup" the 10 from the 17 into the tens column. When subtracting 19 from 28, you "regroup" one of the tens from the tens column into the ones column so that you have 18 from which to subtract the 9. I always thought regrouping was just using the more conceptually-based terminology of regrouping to or from units of higher place value, rather than calling it "carrying the 1" or "borrowing 1." 

Miranda
 

I'm interested too. Also not really sure what the difference is?

Essentially, there is no difference, except my ability to teach it.  For some reason, the additional steps of regrouping tripped me up.  The more I tried to explain, the more confused my son became.  He clearly understood the concept of the value and placement of the numbers.  Regrouping just seems so fiddly to me, there is nothing instinctive about it.   I wish I could explain why it was so hard for him to grasp because had no problems with the borrowing/carrying method from there he was able to quickly make the leap to mental addition/subtracting of double and triple digit numbers.  I pretty sure his biggest issue was me.  

[TAKLinda]

To teach addition, use the points on the numbers to add on.  eg., 5+3 with his pencil he touches the 5 then counts up as he touches the points on the 3.  This works for +1 to + 4.  While driving in the car, teach the doubles, eg., 2+2=4, 3+3=6, 4+4=8, etc.  You can then use the doubles to teach the doubles plus one, eg., if 6+6=12 then 7+6 must be one more or 13.  Eventually, you will use the doubles to regroup, eg., 6+8=7+7.

[Plummeting]

This is something that just comes with time, IMO. If you have a smart phone, there are a few good games for practicing math facts. It's essentially just about practicing, practicing, practicing those 1-10 facts. Once you get those down, the over 10s can all be arrived at using strategies and that eventually leads to memorization. The 1-10 facts just require lots and lots of review, though. :)
 

Originally Posted by moominmamma 

I think you're confusing concrete with prescriptive. I would encourage you to read Liping Ma's book "Knowing and Teaching Elementary Mathematics" which basically compares the Asian and American approaches to mathematics education. I think Ma would agree with you about the idea that concrete should come before formal / abstract. But she would disagree about whether the algorithms are concrete. Grouping objects in units of value (tens, ones, hundreds, etc.) can be learned very concretely, using manipulatives and symbols and starting out with numbers that are small enough to be visualized and to have a clear relationship to things the child can see and understand in the real world. The algorithms are prescriptive, but they're not concrete in the slightest; in fact they're much more abstract than the place-value learning Asian kids do in 1st grade. What does "adding 8 and 9 in this column and getting 17 and writing the 7 here, and then putting the 1 up at the top of next column to the left, and adding it in with the 1 and the 2 to get the other part of the answer" have in it that makes it a concrete way to add 18 to 29? 

I agree and strongly second your recommendation of this book. One thing I simply cannot wrap my mind around is why Americans (and I am one, so everyone chill) will see our dismal math performance, compare it to the astounding performance of Asian students, and then insist that the cure for our poor performance is more of the same type of teaching. It doesn't work, and I'm pretty sure that Asian children are not genetically superior mathematicians, so if it works for them to learn the concepts first, then I believe it will work just as well for most American children. My 6yo (7 in May) can mentally add 2 digit numbers with regrouping both over 100 and over 10 in one problem (i.e. 58+67), and she doesn't make careless errors in calculation precisely because she understands what she's doing. It would be extremely difficult for her to do that (probably impossible, based on what I know about my own child) if she had to remember to "carry a one". It's not intuitive if you have no idea why you're doing it.

[Thandiwe]

How about an abacus?  My little guy, about 5.75 years old, is doing excellent with addition/subtraction by using an abacus (not the traditional Chinese, just a basic abacus).  It's easier than blocks, and maybe a step closer to independence?  I think I bought ours, Melissa and Doug, for $10.  The way I figure it, he will get it in his head on his own; as it was he figured out addition/subtraction on his own.

Good luck!

[minkajane]

I don't even know how to use an abacus! Is there a website or something with directions?