Carrying and Borrowing in Math: Stay Put, Move On??

[Rebecca in KY]

My son turned 7 in October and is working through Singapore Math 2A. We recently hit a snag with carrying and borrowing in addition and subtraction, and I'm not sure what to do.

 

He can do all of one type of problem with very few errors. But when the problems are mixed together (addition and subtraction, some require carrying, some don't), he gets bogged down and makes too many errors.

 

We have already spent one week focusing on addition problems only, and another week on subtraction only. I pulled some worksheets off the internet and made up some of my own for those weeks. But we're still seeing too many errors when it's mixed together. The other day he missed 5 problems out of 15.

 

I've taught him to first check for what the problem is asking him to do (plus or minus), and he's good at that. But then...

 

Here's an example. Today he had 362 - 178. Instead of 2 - 8 giving him a heads up that he needs to borrow, he did 8 - 2. It's like he flipped it around in his head. It seems like he forgets the rules and gets them all jumbled up.

 

So, should I:

1. Stop moving forward and keep working on these skills until he can do mixed problem sets well?

2. Keep moving forward but give him a few (maybe 3-5) mixed problems daily?

3. Something I haven't even thought about yet?

 

Our next topic in the book is multiplication and then division. We won't hit these skills again for quite some time. This is the first time we've struggled with any math concepts. We began Singapore last year in 1B.

[Arcadia]

He would need the same skill for double digit multiplication e.g. 15 x 2

For your option 2, you could "drill" the 2, 3 and 5 times tables first. The 5 times table is useful for reading an analog clock.

[OneStepAtATime]

Have you tried working extensively with manipulatives and a lot of real world application math?  This can be developmental, too.  Some kids need more review, some kids need to approach the topic from multiple angles, some kids need more time for brain development, some kids may struggle with this for years.  And some kids need a lot of spiral review or whatever they do finally achieve gets lost again.  Hard to know what your child will need.  But I wouldn't rush forward to multiplication and division just yet.  I would slow down, try different approaches, see if you can find something to help him understand what he is doing.  That may require more than just more worksheets.

[Rebecca in KY]

We haven't really worked with manipulatives for this--good call.

 

I am concerned that this may be developmental, in which case we need to let him catch up. What I don't want to do is camp out on this one topic and beat it to death--DS hates that like poison, which is one of the reasons Singapore has worked thus far. So I need ways to make it interesting and variable enough that he works with me instead of fighting against me.

[deerforest]

Try these videos from Education Unboxed: subtraction and addition algorithm

[Brit29]

Following. 

 

We are a month behind you in age and sounds like a lesson or two behind in 2A. :) 

[MotherGoose]

Use manipulatives, especially place value discs. Do you have the Teacher's Manual/Home Instructor's Guide? They go into lots of detail on how to use them. I stayed on the topic for awhile, having dd work through them with place value discs, just a few problems a day. It will click, but she really needed the manipulatives, and I realized that I really understood the concept after using the discs too--it became more than an algorithim. You can make place value discs with paper or something if you don't want to but them--just do one color each for 1s, 10s, 100s, and 1000s. Make prob 20s, 20 10s, and 11 100s, and a chart with ones place, tens place, 100s place, 1000s place

[Kiara.I]

We use Rightstart and they do a TON with place value and trading, such that when it came up in written problems for addition it was a very easy thing to do.  So yes, I vote for manipulatives.  And place value work.

 

And, equally, RS leaves subtraction until quite a bit after it's been done in addition, so that it's not new there either, so you might want to make sure it's solid in addition before working on subtraction, and definitely not doing multiplication until the other two are solid.

 

One of my *favourite* things for teaching it has been the Base 10 Picture Cards.  There's a good deal of work on building the numbers with those cards.  Then building two numbers with those cards, and then adding them together.  And for subtraction, building the first number with those cards, then using those to build the second number, and seeing what you have left (which necessitates trading sometimes, in order to have enough cards to build the second number.)

[serendipitous journey]

>editing<

 

[serendipitous journey]

I'd suggest going with your option 2 -- doing a few mixed review problems, probably with you right with him in case of errors, each day.   You could do this at a separate, review math session and precede it by trying some of the teaching approaches offered above -- try a new approach to the concept, do some problems.  Or just problems. 

 

He's also getting some stuff right -- like the difference between 2 and 8.  Have you introduced the idea of negative numbers yet?  It helped my little guy when I would say that, for the carrying algorithm to work, "you can't go negative".  So since 2 - 8 leaves -6, we need to regroup a 10 over to the units to give 12 and keep from going negative. 

 

But that last might be much more confusing than it's worth. 

 

Also -- I'm so glad you are tackling this now!  The mixed-problem challenge is a real one for lots of children, and it is great you are noticing and working with it right away. 

[Rebecca in KY]

Yes, Serendipitous, we've introduced negatives because HE figured it out himself. I had to spend some time helping him understand that even though 2 - 8 is -6, we can't do that in the problems we are working through.

 

I appreciate all the feedback!

[lmrich]

Repeating what everyone else said - use manipulatives. Also just do a few problems at a time and for some reason use a white board - kid love those. From my expericence teaching 2nd grade this is a skill that takes time to master, but please use the manipulatives.

[Tsuga]

An abacus has been very useful for this concept and tens with my children.

[Farrar]

Another vote to turn to manipulatives - C-rods or an abacus or both.

 

I usually advocate for doing review and moving on slowly at the same time, but it does sound like it might be a bit developmental. I wouldn't just keep drilling it, because that's no fun, but I think I might keep reviewing it and take a short Singapore break to do some math enrichment for awhile - games, living math books, projects. I'd look at Soror's Relaxed Math thread for some ideas. Just for maybe a month or so. And then see where you stand.

[kitten18]

I would move forward and write out just 2-3 of these problems to do each day.

 

When my kids have gotten stuck on something I have never regretted moving forward and then coming back around to it.

[Ausmumof3]

We had a lot of trouble with this as well. We did a lot of manipulatives too. In an odd way when you are just talking about numbers it "seems" to make sense because you can flip the numbers around with addition to make the column easier to add. We just slowed down a bit and I said a million times here are two counters - now take away six. Oh you can't do it. So what do we do now?

 

One thing I haven't resolved with the Singapore method is how messy it gets when you have to trade in a ten and then trade in a hundred. There's so much crossing out etc.

 

The 1 up 1 down method is harder to explain for the kids but producing a much easier neater sheet of work.

[blondeviolin]

I'd just do a few with manipulatives as an opener for math and then continue on with the next lesson.

[bethben]

With subtraction and addition mixed in, I will have my child either circle all the addition problems or just do all of the same type of problem all at once.  Honestly, the speed at which Singapore went coupled with the fact that my 3rd child really needed more practice with things to cement them better in his brain is what made us ditch Singapore.  My 2nd child did well with it until fractions (addition/subtraction) and didn't get the practice he needed regularly, so he would forget whole concepts he had previously mastered.  It got to be too much to try to add spiral review into Singapore.  If someone wants to create a spiral method Singapore math book, I will be your first customer.

[Sahamamama]

My son turned 7 in October and is working through Singapore Math 2A. We recently hit a snag with carrying and borrowing in addition and subtraction, and I'm not sure what to do.

 

He can do all of one type of problem with very few errors. But when the problems are mixed together (addition and subtraction, some require carrying, some don't), he gets bogged down and makes too many errors.

 

We have already spent one week focusing on addition problems only, and another week on subtraction only. I pulled some worksheets off the internet and made up some of my own for those weeks. But we're still seeing too many errors when it's mixed together. The other day he missed 5 problems out of 15.

 

I've taught him to first check for what the problem is asking him to do (plus or minus), and he's good at that. But then...

 

Here's an example. Today he had 362 - 178. Instead of 2 - 8 giving him a heads up that he needs to borrow, he did 8 - 2. It's like he flipped it around in his head. It seems like he forgets the rules and gets them all jumbled up.

 

So, should I:

1. Stop moving forward and keep working on these skills until he can do mixed problem sets well?

2. Keep moving forward but give him a few (maybe 3-5) mixed problems daily?

3. Something I haven't even thought about yet?

 

Our next topic in the book is multiplication and then division. We won't hit these skills again for quite some time. This is the first time we've struggled with any math concepts. We began Singapore last year in 1B.

 

Try this:

 

Make a worksheet with addition & subtraction problems (mathfactcafe.com has a worksheet builder).

 

Give your son two light-colored highlighters, perhaps yellow and pink, or two light-colored pencils.

 

First, make a "key" at the top or bottom of the worksheet.

 

Pink = Addition (I may have to carry -- Memory Sentence: I may carry a pink pig home from the market)

Yellow = Subtraction (I may have to borrow -- Memory Sentence: I may borrow yellow butter from Betty Botter)

 

Next, have him focus only on the sign (+ or -) and color-code each problem according to its operation.

 

Then, have him do all the sutbtraction problems, then all the addition problems. Have him do the subtraction problems first. For some reason, I've noticed that kids usually have a harder time remembering to switch from addition to subtraction than the other way around. If he gets the subtraction problems done first, he can easily complete the addition.

 

Another time, do all these steps, but ask him to do all the YELLOW problems in one row, then all the PINK problems in that same row. Next row, the same way. And so on.

 

Work towards being able to color-code all the problems, then do them all in the order they appear on the worksheet.

 

Finally, color-code only the signs (+ or -) and solve the problems.

 

This will begin to teach him to focus on the sign, and to think about what operation he must do for each problem. Sometimes a visual cue is useful for flipping the switch from addition (the facts and the processes we use to do that operation) to subtraction (the facts and the processes we use to do that operation).

 

Also, continue to drill math facts with flash cards, but mix up the addition and subtraction packs. As he practices paying more attention to the signs on these simple facts, it will help him to do the same when he solves mixed addition & subtraction problems.

 

HTH.

[Cosmos]

Yes, Serendipitous, we've introduced negatives because HE figured it out himself. I had to spend some time helping him understand that even though 2 - 8 is -6, we can't do that in the problems we are working through.

 

Actually you can. Here's another subtraction method that might click with a kid who thinks in negative numbers.

 

Take the example you posted: 362 - 178

 

Start with the leftmost column. You have 300 - 100, which is 200. Write down 200.

Now go to next column. You have 60 - 70, which is -10. Write down -10 under the 200.

Now go to the final column. You have 2 - 8, which is -6. Write down -6 under the -10.

 

So your problem will look like this:

 

 362

-178

------

 200

  -10

    -6

 

Now add up the rows you have constructed. 200 plus -10 plus -6. This is the same as 200 - 10 - 6, which is 184.

[EKS]

I'd break out the manipulatives and have him do one problem each day with them, explaining to you what he's doing. Once he's good with the manipulatives have him do one problem each day on paper. And I'd also move on even if you need to shorten the new lessons to accommodate the review.

[rocassie]

My DD is 7 too, and borrowing for subtraction was difficult for her to grasp.  Addition no problem.  We took a step back, worked on basics, and used a lot of manipulatives. We did some games in which we started out with 100 pts and worked down to 0 pts with DD keeping track, and then increased the starting points when she had mastered 100 to 0pts.  

 

When DD struggles with a math concept, I usually take a break from the textbook and worksheets and find practical, fun applications using real life, games, and manipulatives. 

[Arboreal TJ]

Knowing And Teaching Elementary Mathematics by Liping Ma is an excellent resource for anyone teaching math.

 

I agree with the others, pull out the manipulatives. Also ditch the terms carry and borrow, compose and decompose are more descriptive. If the student is thinking in terms of place value math is easier. Example 24 is 2 tens and 4 units. Read Ma or check out the Education Unboxed videos for a better idea.

[JanetC]

How are you and DS doing emotionally?  If he's frustrated, it's time to rotate this topic with something else just to keep things fresh.  Otherwise, there's no reason not to stick with it (and try different ways of teaching it, such as the things others have suggested above, to come at it from a different angle).

 

Be aware that Singapore level 2 is a bump up in difficulty. Don't push yourself to finish 2A and 2B in one school year if your DS needs a little extra time to finish.  This curriculum is more advanced and some kids need a bit more time to get through it.

[Incognito]

We're in the same part of Singapore 2A (HI! :)).  Use manipulatives.  FWIW, I would say that based on the errors you are talking about, your child would really benefit from working on this conceptually.

 

What I have done is I start by getting my child the blocks for the top # in the problem.  Then the child tells me what they need to do and will trade with me (I run the block bank, I guess).  For addition we then find the right amount to add on, then group them together and get our totals.  The logic of borrowing becomes very evident when you actually have to trade in your 10 bar or 100 block to get 10 ones/tens to do the actual math.

 

I don't do this for all the problems - once my child has gotten going, I have my child do the trading themselves too (unless it is a distracted day, then I end up staying or math would take forever).  If my child gets the problem wrong, I only tell them to re-do that problem.  I don't tell them what they did wrong - they figure it out.  If they can't, we do those manipulatives together again like in the beginning.  

 

ETA:  Have you done all the practice problems for this already?  If not, use them and stay where you are. If you have, then make up your own.  

 

 

[Rebecca in KY]

How are you and DS doing emotionally?  If he's frustrated, it's time to rotate this topic with something else just to keep things fresh.  Otherwise, there's no reason not to stick with it (and try different ways of teaching it, such as the things others have suggested above, to come at it from a different angle).

 

Be aware that Singapore level 2 is a bump up in difficulty. Don't push yourself to finish 2A and 2B in one school year if your DS needs a little extra time to finish.  This curriculum is more advanced and some kids need a bit more time to get through it.

 

We're doing well in our relationship and with how we approach school. I've noticed that level 2 is a bump up. Math takes longer each day, and DS doesn't like that - so we've been breaking the sections down into smaller, more bite-sized pieces. I have to remember what you've said! I find myself trying to figure out how to get through all of level 2 this year, when I already know it's not necessary.

[Rebecca in KY]

Actually you can. Here's another subtraction method that might click with a kid who thinks in negative numbers.

 

Take the example you posted: 362 - 178

 

Start with the leftmost column. You have 300 - 100, which is 200. Write down 200.

Now go to next column. You have 60 - 70, which is -10. Write down -10 under the 200.

Now go to the final column. You have 2 - 8, which is -6. Write down -6 under the -10.

 

So your problem will look like this:

 

 362

-178

------

 200

  -10

    -6

 

Now add up the rows you have constructed. 200 plus -10 plus -6. This is the same as 200 - 10 - 6, which is 184.

 

This is totally amazing! As someone who always just plugged through math but never really understood it, seeing this example is hugely enlightening. Wow! I bet this will help DS. It frustrates him to be unable to work with negative numbers when he "sees" them on the page.

 

 

[Rebecca in KY]

With subtraction and addition mixed in, I will have my child either circle all the addition problems or just do all of the same type of problem all at once.  Honestly, the speed at which Singapore went coupled with the fact that my 3rd child really needed more practice with things to cement them better in his brain is what made us ditch Singapore.  My 2nd child did well with it until fractions (addition/subtraction) and didn't get the practice he needed regularly, so he would forget whole concepts he had previously mastered.  It got to be too much to try to add spiral review into Singapore.  If someone wants to create a spiral method Singapore math book, I will be your first customer.

 

This is beginning to concern me. We've done just fine with Singapore thus far, but now we've been stuck on this one concept for over a month. I'm afraid that once he "masters" it, if we move on, he'll forget how to do it and I'll have to re-teach it again. The longer we work through the books, the more I see how little review is incorporated into the program.

 

I've been looking through samples of CLE and really like the incremental review built into it. I'm thinking through switching for next year, or maybe even sooner than that. I don't like curriculum-hopping, especially in math with the varying scope-and-sequence challenges, but I'm beginning to think this is a situation in which it is called for.

[EKS]

This is totally amazing! As someone who always just plugged through math but never really understood it, seeing this example is hugely enlightening. Wow! I bet this will help DS. It frustrates him to be unable to work with negative numbers when he "sees" them on the page.

 

 

Be careful with this. He needs to understand that he is subtracting a number--one number--from another number. He is not subtracting three different columns of numbers. Before reteaching regrouping with manipulatives, teaching subtraction from left to right, first using manipulatives, will help. So for 254-175, 254-100 is 154, 154-70 is 84, and 84-5 is 79. Obviously you'd start with problems that are easier than that one and work your way up. Anyway, after he's good at that, use manipulatives going from right to left and then show the paper and pencil method.

[kiana]

Out of curiosity, how does he do with two-digit addition/subtraction involving this? It might be worthwhile to go back and keep practicing with that a bit more if he's struggling with 3 digits.

[Rebecca in KY]

He does fine with 2-digit adding an subtracting. Maybe it's a place-value thing, or maybe the bigger numbers are just too big right now.

 

We will just take it slow and steady.

[SparklyUnicorn]

You could move on and then each day do a couple of those types of problems.  It would be easy enough to just make up a few of your own problems.

 

 

[kiana]

If he does fine with two-digit and struggles with three-digit I'd suspect it's a 'too many steps' issue right now.

[letsplaymath]

He just turned 7? I would drop it entirely and focus on mental math and on other topics that encourage him to think about what the numbers mean and how they behave.

 

The point of the pencil-and-paper borrowing/carrying/renaming/whatever technique (called an algorithm) for calculating answers is that it can be done automatically, without any understanding, just by following programmed steps. It can be done by computers, and even by dollar-store calculators. In earlier days, it could be done by low-paid scribes in an accounting house (think "Christmas Carol"). All you have to do is turn off the mind and crank through the motions.

 

This is what Alfred North Whitehead meant when he said, "It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them."

 

BUT what we really want from our children at this age is that they THINK about the numbers, that they build up a foundational understanding of number properties and behavior. The algorithm works against this sort of understanding.

• We say numbers from left to right, big place value to smaller, and children naturally work with them that way. But the algorithm forces them to go against intuition and work from right to left.
   
• We want children to think about the meaning of numbers and their relationships to each other. But the algorithm treats each individual digit as a stand-alone entity, independent of its place in the number. That is, you use the same methods whether you are adding/subtracting a number in the millions or millionths.

This is one of the few places where I disagree with Singapore math (and most other curricula---and most of the people who have posted answers here, too). I think that teaching the algorithm at this age is developmentally unsound. I don't think it does quite as much damage in Asia, where most of the teachers understand math and where the style of teaching emphasizes discussing alternative methods and using mental math. In American classrooms, however, I think this topic is one major contributor to the math anxiety epidemic.

 

 

 

[deerforest]

Since I was one of the people to actually use the word "algorithm" in my response, I'll just point out that the videos I mentioned explicitly avoid just getting kids to regurgitate the algorithm. Instead, they use cuisenaire rods to explain the idea conceptually. As stated in the video's description:

 

This free math video shows the “WHY†behind the traditional subtraction algorithm normally taught in 2nd grade in the USA. Many children don’t understand what is happening in the algorithm that they are expected to memorize. Because of this, they can’t remember the steps and get confused and frustrated. Doing these types of problems with manipulative many times while also doing it on paper will help those kids who struggle as well as help those who don’t struggle with the memory task to actually understand what they are doing.

 

 

[Incognito]

 

 

BUT what we really want from our children at this age is that they THINK about the numbers, that they build up a foundational understanding of number properties and behavior. The algorithm works against this sort of understanding.

• We say numbers from left to right, big place value to smaller, and children naturally work with them that way. But the algorithm forces them to go against intuition and work from right to left.

   

• We want children to think about the meaning of numbers and their relationships to each other. But the algorithm treats each individual digit as a stand-alone entity, independent of its place in the number. That is, you use the same methods whether you are adding/subtracting a number in the millions or millionths.

This is one of the few places where I disagree with Singapore math (and most other curricula---and most of the people who have posted answers here, too). I think that teaching the algorithm at this age is developmentally unsound. I don't think it does quite as much damage in Asia, where most of the teachers understand math and where the style of teaching emphasizes discussing alternative methods and using mental math. In American classrooms, however, I think this topic is one major contributor to the math anxiety epidemic.

 

 

I agree with you that kids should be understanding the #'s and not the algorithm, but when I look at Singapore I think that is exactly what they are trying to do - it is all in the teaching of it, I think.  The idea is for kids to be playing with the #'s and understanding what they are really doing with them - not to just memorize borrowing and lining things up.  

 

Life of Fred (an oft maligned curriculum here) actually teaches right into the millions when starting place value (IIRC), doing just what you are talking about - helping the child to see the system of it, not just the one spot and what you do with that one value.  BUT, it is all in how the content is delivered - if there isn't scaffolding and a basic background of mathematical perspective, then it is easy for a person to not see outside of the exact problem being worked and to focus on just the problem instead of on the concept in the background.  So LoF is dubbed random and inconsistent and fluffy, because it assumes a background perspective of the teacher as someone who sees the world in a mathematical way.  AND I would put your description of Singapore as algorithm based as an unfair assessment too, because it is meant to be conceptual, but it is all in the hands of whoever is doing the teaching, how it will actually come to fruition.

[letsplaymath]

I agree with you that kids should be understanding the #'s and not the algorithm, but when I look at Singapore I think that is exactly what they are trying to do - it is all in the teaching of it, I think.  The idea is for kids to be playing with the #'s and understanding what they are really doing with them - not to just memorize borrowing and lining things up.  

 

Life of Fred (an oft maligned curriculum here) actually teaches right into the millions when starting place value (IIRC), doing just what you are talking about - helping the child to see the system of it, not just the one spot and what you do with that one value.  BUT, it is all in how the content is delivered - if there isn't scaffolding and a basic background of mathematical perspective, then it is easy for a person to not see outside of the exact problem being worked and to focus on just the problem instead of on the concept in the background.  So LoF is dubbed random and inconsistent and fluffy, because it assumes a background perspective of the teacher as someone who sees the world in a mathematical way.  AND I would put your description of Singapore as algorithm based as an unfair assessment too, because it is meant to be conceptual, but it is all in the hands of whoever is doing the teaching, how it will actually come to fruition.

 

You misunderstand my point. I'm not criticizing Singapore math for the way it teaches this, only for the timing. EVERY textbook that teaches the algorithm tries to explain it and tie it back to fundamental concepts. And almost every teacher tries to emphasize the meaning of the place value columns and build conceptual understanding.

 

A few teachers resort to meaningless mnemonics like "More on the floor? Go next door!" to teach borrowing, but by far most people really are trying to communicate to kids WHY the algorithm works.

 

The problem is in the algorithm itself.

• Standard algorithms unteach place value.

To use any algorithm efficiently, you have to follow the steps without stopping to think about them. If you try to think, you will probably forget what step you are on and make mistakes. Therefore, every time the student does a calculation, he or she is training the mind to skip out on math.

 

And because the algorithm goes against how most children intuitively think about numbers, he or she may also be training the mind to think, "Math is not about making sense. It is about following steps. And if I mess up the steps, I must not be good at math."

 

 

 

 

[Incognito]

You misunderstand my point. I'm not criticizing Singapore math for the way it teaches this, only for the timing. EVERY textbook that teaches the algorithm tries to explain it and tie it back to fundamental concepts. And almost every teacher tries to emphasize the meaning of the place value columns and build conceptual understanding.

 

A few teachers resort to meaningless mnemonics like "More on the floor? Go next door!" to teach borrowing, but by far most people really are trying to communicate to kids WHY the algorithm works.

 

The problem is in the algorithm itself.

• Standard algorithms unteach place value.

To use any algorithm efficiently, you have to follow the steps without stopping to think about them. If you try to think, you will probably forget what step you are on and make mistakes. Therefore, every time the student does a calculation, he or she is training the mind to skip out on math.

 

And because the algorithm goes against how most children intuitively think about numbers, he or she may also be training the mind to think, "Math is not about making sense. It is about following steps. And if I mess up the steps, I must not be good at math."

 

So, how old do you think a child should be when they learn to line up #'s on a page?

 

I think in a way we are still missing one another in the discussion.  I would say that if a child is messing up in their math problems like the OP's son, he just plain doesn't understand what he is doing.  So I suggested she use blocks so he can get his hands on the math and really understand what he is up to.  Then, if he understands what he is doing, I would think it should be just fine for him to write it on a page (use the algorithm).  

 

But it sounds to me like you just think he is too young to do this math and it is wrong to be teaching it to him.  I'd say a LOT of kids can handle writing the math and understanding it at 7.  Some sure can't, but many can.  Or perhaps you are asserting that he is too young to write it down?  I'd say that depends on the kid too, but many can successfully.

 

Or maybe I misunderstand the word algorithm.  I understand it as putting the math of it on paper - not mindlessly crossing things off and adding things on.  If the algorithm is thoughtless, then I wholeheartedly agree with you - terribly out of place for young kids.   

[letsplaymath]

So, how old do you think a child should be when they learn to line up #'s on a page?

 

I think in a way we are still missing one another in the discussion.  I would say that if a child is messing up in their math problems like the OP's son, he just plain doesn't understand what he is doing.  So I suggested she use blocks so he can get his hands on the math and really understand what he is up to.  Then, if he understands what he is doing, I would think it should be just fine for him to write it on a page (use the algorithm).  

 

But it sounds to me like you just think he is too young to do this math and it is wrong to be teaching it to him.  I'd say a LOT of kids can handle writing the math and understanding it at 7.  Some sure can't, but many can.  Or perhaps you are asserting that he is too young to write it down?  I'd say that depends on the kid too, but many can successfully.

 

Or maybe I misunderstand the word algorithm.  I understand it as putting the math of it on paper - not mindlessly crossing things off and adding things on.  If the algorithm is thoughtless, then I wholeheartedly agree with you - terribly out of place for young kids.   

 

Perhaps a better question is, how old do you think children should be when they learn to line up numbers on a page because we said so. If the child is lining up the numbers because it helps him express the thoughts that are in his mind, then he can do it whenever he wants. But if he is lining up numbers because an adult has told him it's the right thing to do, that is harmful. 

 

And it really doesn't matter how many times you try to demonstrate with blocks what is happening. It doesn't even matter if the child can repeat back to you a description that makes it sound like he understands place value and what the algorithm is doing. Children are very good at figuring out what makes an adult happy, but when you push a little harder, you find that what sounded like understanding was just a Clever Hans or Benny's Rules effect.

 

Seriously, watch the video at the second link above. It's amazing. As Ben Blum-Smith said, "Never underestimate your ability to fool yourself into believing your students understand something when really what they are doing is watching you."

 

Yes, children can be trained to do things that go against their intuition. They can be trained to line up numbers on a page to make Mom and Dad happy. They can be trained to remember and follow certain steps with those numbers. They can be trained to say certain words about those numbers, like "tens" and "ones." They can be trained to perform well on standardized tests -- or at least, many of them can.

 

But is this a worthwhile use of a 7-year-old boy's time? Only if it's HIS idea in the first place. Otherwise, there are plenty of deeper and more interesting things to play around with in math!

 

The boy in the original post showed excellent understanding of mental addition and subtraction until the nitpicky steps of the algorithm messed him up. If the parent focuses on the details of the algorithm now, she risks damaging his relationship with math as a whole. But if she allows him to use whatever intuitive method he wants on the numbers, and together they go on to explore new topics -- and there are some seriously interesting things to study in the rest of Singapore level 2 -- then she can easily come back to try the algorithm again next year. By then, he just may have figured it out on his own, or he will at least have some additional maturity to help his attention span.

 

P.S.:

An algorithm is just a set of procedures to get from point A to point B. Like a recipe. You have these ingredients (numbers), you follow these steps, you automatically get the result. No understanding required. A computer can do it.

[Incognito]

I think you are underestimating 7 yos.  I agree that if it isn't working and a child doesn't understand, then drop it and come back again later, but if the child hasn't even done it with blocks yet, I don't think we know this is beyond the child.  I also think that it is just fine for adults to expect kids to do things that they are capable of - and that writing #'s on paper isn't necessarily harmful.  So apparently we disagree in parenting style.

 

The OP's son does show insight with his use of negative #'s, and while I would also encourage using his understanding and insights, I don't think it is harmful to give him another way to see the problem if he is able to understand that other angle.  If the OP's son tries it with manipulatives and doesn't understand it, then yeah, move on.  Do something fun.  Or if he is frustrated and upset, of course, find something else to do.  But I don't hear that is a problem, I hear that the workbook pages are a problem right now and we don't know if he is or is not currently capable of understanding this.

 

FWIW, I have a 7yo who has just now started with written math work and is very capable of this stuff - before this we played with #'s and tagged along with sibling math and had interesting discussions, but in my mind it was time for us to sit down and get some numbers on paper, and SM2A has been where we started.  I don't believe it has been detrimental to do so.  My 7yo is better at mental math and at discussing the concepts, but I felt that learning to write what he understood would be helpful to him and would make moving forward in math easier for us in some ways, so we are doing it.  I think it is fundamentally important to have a child really understand #'s and use them, but I think it is just fine for me to expect my child to try to learn how to write it too.  

 

In the video, Cena didn't understand what the woman was asking for - you are right, she doesn't understand it.  But that doesn't mean she isn't old enough to learn it.  We don't know.  I'm also not sure she doesn't understand place value at all - I think she is trying to please the questioner, and is answering the way she thinks the questioner wants.  Then in the end, Cena does clearly show that she is wrong by her choice of what to do with the extra 9 tiles, but really  - she is 7 and either she has a deeply flawed sense of #s or she is uncomfortable with this questioner and is just trying to move on because she doesn't understand what the woman is getting at and/or she is being silly because she knows she has been caught making a mistake but she doesn't immediately see what the mistake was.  I always find those types of questions from people confusing - I understand place value very well, and have a great concept of math and no anxiety, but questions like that are hard to interpret from people  (For me, place value questions always made no sense - of course the 1 means 1 - it means either 1 or 1 ten, or 1 hundred or 1 whatever, but the 1 itself always means 1.  The place itself gives the other information, but the digit never changes its worth.  I learned to answer the way they expected in school - that the 1 in the tens column mean 10, but I actually disagreed with it and still do :)  ).

 

 

Cena did show some level of ability to understand it in the classroom setting, but perhaps didn't remember/hadn't really internalized the concept.  Or maybe she is like me and just thinks people asking those questions are being obtuse.  Kids do misinterpret or not quite get everything about a lot of things.  It doesn't mean it isn't worth it to expose them to the material.  It does mean we need to be mindful that we are exposing and to keep our expectations in line with what is appropriate for the specific child and circumstances, but it doesn't mean that writing on paper is wrong.

 

I appreciate from your website that you have designed a program for teaching the fundamental understanding of math that is based on a lot of play and fun - and I would say that is how we have generally approached math in our house as well.  I think your immersion in the issue makes your feelings about a distaste for expected written output in math a little overly strong. 

[letsplaymath]

I think you are underestimating 7 yos.  I agree that if it isn't working and a child doesn't understand, then drop it and come back again later, but if the child hasn't even done it with blocks yet, I don't think we know this is beyond the child.  I also think that it is just fine for adults to expect kids to do things that they are capable of - and that writing #'s on paper isn't necessarily harmful.  So apparently we disagree in parenting style.

 

... Kids do misinterpret or not quite get everything about a lot of things.  It doesn't mean it isn't worth it to expose them to the material.  It does mean we need to be mindful that we are exposing and to keep our expectations in line with what is appropriate for the specific child and circumstances, but it doesn't mean that writing on paper is wrong.

 

I appreciate from your website that you have designed a program for teaching the fundamental understanding of math that is based on a lot of play and fun - and I would say that is how we have generally approached math in our house as well.  I think your immersion in the issue makes your feelings about a distaste for expected written output in math a little overly strong. 

 

I agree that we are talking past each other. If what has come across in my posts is the idea that "writing numbers on paper is wrong," then I am clearly not communicating what I intended. That's frustrating, because from reading your posts I suspect we would agree on most aspects of teaching.

 

As for my "program" for teaching math, um, I have a blog. And I've expanded some of the more popular blog posts into an inexpensive ebook. Does that make it a program?

 

But you are right, my feelings on this topic are coming out overly strong -- not so much in reaction to this thread (which I will stop hijacking), but probably as a boil-over from other things I've been reading online. My sincere apologies!