Thursday, October 06, 2011


Fifth Grade Math Homework

(Congratulations to anyone who actually read this post after seeing the title!)

Many years ago, MAD magazine had a piece about joke courses for college football players. The one I remember was called “Subtraction: Addition’s Tricky Pal.”

The Boy is discovering that there’s some truth to that.

He’s struggling with “borrowing.” Say you want to subtract 235 from 700. Turning the last zero into a ten requires borrowing from the digit before, but that’s a zero, so...

Oh, the humanity.

I learned “borrowing” sometime during the Carter administration, so I’m a little fuzzy on the step-by-step of teaching it. I suggested thinking of 700 as seventy ‘tens.’ To borrow a ten for the last column would leave you with 69 tens. Then, in order from right to left, ten minus five is five, nine minus three is six, and six minus two is four. 465, done and done.

That doesn’t seem to take, though. The idea of “seventy tens” seems a little too abstract.

He’s fine when just borrowing from a single digit. 82 minus 28 is fine, since borrowing from the 8 still leaves enough to subtract the 2. But borrowing across multiple digits remains mysterious.

Trying to convey ‘borrowing’ to a ten year old really brings home the difference between knowing how to do something and knowing how to teach it. I’ve walked him through problems step by step, narrating each step as I go, and he seems to follow. Then I have him try a few problems, and he does great. But it’s gone by the next day. It doesn’t stick.

His frustration is palpable and insidious. He’ll get back worksheets with middling grades, but the middling grade really just reflects a single mistake repeated over and over again. He honestly wants to get it right, but just can’t seem to hold the concept for very long.

I’m okay with him learning to struggle a bit, since that’s a valuable life skill. (Sometimes I think the only way in which my social science grad school training prepared me for my current job was in teaching me how to take a punch.) But I don’t want him to get so discouraged that he starts to doubt his own abilities. I don’t want him to become the kid who writes off math as something you’re either born with or not, like wiggling your ears.

He can’t be the first bright, motivated kid to have trouble holding on to a mathematical concept overnight. So I’m sending out this message in a bottle -- okay, in a blog -- hoping that someone has found a way to help that kind of concept stick in the mind of a ten year old.

Wise and worldly readers, have you found a way to explain this sort of math that’s both simple enough for a ten year old, and sticky enough to make it to the next day? TB and I would be terribly grateful for anything that works.

(Program note: We’re trekking across several states over the next few days, so the next post will be on Tuesday the 11th.)

As an elementary school teacher, I completely understand the frustration here. Borrowing is complex and really just an algorithm to remember for most kids because they don't fully understand what they are doing. They just follow the steps. That makes it easy to make mistakes.

Here's a youtube video that shows a fairly traditional algorithm after showing partial differences, a strategy that builds off the place value. Maybe it will help.
Pennies, dimes and dollars are the ones, tens, and hundreds places. Work out the problem each time in coins, and exchange the dollars for an extra 10 dimes, etc. Physically doing it helps a lot. This helps cement the concept.

Watch the khan academy videos as well, doing problems alongside the video a few times. This should help with the algorithm.
How about (you or him) actually writing out the algorithm?

Most of our teaching of algorithms is done via examples and "talking" the algorithm. The talk vanishes pretty quickly. Afterwards you're left with a bunch of worked examples that are written in this highly coded symbolic system.

Divide a page in half, label the left column: General description of the steps

Label the right half: example of the steps

Then, write out the steps without reference to a specific example in the left-hand column.

If he does it he might remember the steps longer, and if you do it he'll at least have a reference to go back to when he's forgotten the steps and there's not an adult around.

I also support having kids change the algorithm, like Jenny suggested, using partial differences or Wendy suggested using money.

Tangent alert:
Your comment about getting a middling grade for making the same mistake over and over is provocative. It asks the question, "what should a (final) grade indicate?" For most students it indicates whether or not they mastered the content at the rate determined by the teacher. Isn't "did you learn it?" a better thing to assess than, "did you learn it by the day I arbitrarily decided would be 'test day'?

Standards-based assessment is cool...
Wendy's idea is similar to mine, and hers is probably better if you're traveling. Mine was to use either manipulatives (though ten is probably a bit old for this, and given the type of problem you'd need a LOT) or graphical representations. 700 = seven squares of a hundred tiny boxes. To take away 235 tiny boxes requires taking away two of the large squares and then ... breaking up one of the remaining five. Have him work through several problems in this way, just with the paper representation of hundreds (squares), tens (rods), and ones (tiny squares). Then ask him how HE'd represent it when just working with the numbers. It may take him a while to see that borrowing/regrouping is the same thing, but it'll probably have a chance of sticking well, and the draw-a-manipulative can ALWAYS be used, if he has time.
There's (at least) two ways to do this, both of which are in the comments already.

1. Gives a full understanding of what is happening. This method uses manipulatives. I like pixel blocks, but Wendy is 100% correct that pennies, dimes, and dollars can be used instead. Teachers will often have specially made manipulatives for exactly this concept. Starfall and other math games also have games that illustrate this concept.

2. Only gives an algorithmic way of doing them. The student can do the subtraction but doesn't know why he or she is doing it that way. This is what Jenny and Tim are suggesting-- break the problem down into steps and have him do each step. It's ok that he doesn't have a full understanding because many kids will pick up understanding *after* they've done several problems and mastered the algorithm.

In our house we did a combination of the two. But my kid still needs to memorize his addition and subtraction facts before he actually enjoys doing borrowing. Right now we've switched over to Singapore math at home which has a completely different focus at his level.
Pennies, dimes, and dollar coins. The best way to visualize it, if you ask me. Wendy gets my +1 :)
Does he have to borrow?

If the task is to get the subtraction correct, then counting up is often more accurate and quicker.

So: starting at 235, add 5 to get 240, add 60 to get to 300, add 400 to get to 700. In total you've added 5 + 60 +400 = 465, which is the difference between 235 and 700. (For additional clarity, write this out on a number line).

I teach 16-18 year olds who have previously failed their GCSE (standardised national assessment at age 16), and many of them have no difficulty using this method but struggled with abstract algorithms.

I would say that a student who understands what they are doing, and can reliably get the right answer, has successfully understood subtraction; knowing a particular algorithm doesn't strike me as an issue.

On the other hand, if the teacher is assessing the method rather than the outcome, then previous commenters have offered good advice.
I'd suggest using some addition problems, like 6999 + 1, to look at how the carries work when doing the reverse problem.

Carefully articulate each step (in writing, as one person suggested), then repeat for the borrowing needed to do 7000 - 1.

I'm also thinking about the "gone by the next day" issue. Is that next evening or next morning?

BTW, there is also the method of addition to the number down below rather than borrowing from the one up above (what Tom Lehrer referred to as the "catholic school method" in his New Math song). Using 7000-1, there you never change the number on top. You subtract 1 from (1)0, then subtract the (1) from (1)0, then subtract a (1) from (1)0, then subtract a (1) from 7. This has a lot in common with the add-back method of making change in the olden days.

PS - They are teaching this in the fifth grade? What did they do in third and fourth grade?
Calculator. Yep, that's all I've got. Math was always a struggle for me, making the same mistakes. I could do it at home but bombed the tests. Give me a paper to write or history and I'm a rockstar.

Make sure he's taking his time. I found that I often rushed through tests. That plus probably a little dsylexia made things works.
This doesn't help with borrowing in general, but in the specific case where the larger number has a lot of zeros, you can just show him how subtracting 1 from each number will give you the same answer. For example, 8 minus 3 is the same as 7 minus 2. So similarly, 700-235 = 699-234. The latter problem is much easier to solve. I still use this method as an adult sometimes :)
I must be a "loonn" based on what I just got as a word verification.

I should have sent you to the homeschool math entry on this topic. It includes the use of manipulatives and a review of the meaning of "place" in our number system. (His problem might be that he doesn't understand placeholders.)

Finally, people trying to compensate for schools using the "Everyday Math" curriculum with teachers who don't know any math also swear by the Singapore math books. You might want to get a set, starting in 2nd grade on an as-needed basis until he can do long division.
Everyone needs to start using the FAR SUPERIOR method of Austrian subtraction:

I posted about it a while back:

It was a total REVELATION to me in my grade school days!
Well, the way I learned it obviously worked for me, since I remember it 30 years later.

You borrow from your neighbor, not anyone else. If he doesn't have anything to lend you, then he goes and borrows from HIS neighbor, and THEN lends to you.

We wrote this out, with the middle 0 in 700 becoming a 10, then crossed out and replaced with 9 and the last 0 becoming a 10.
Is he allowed to not use borrowing?

To do mental math, I'm far more likely to use a handy cognitive trick than the way I was taught. Usually I'll 'count up and add'. i.e. 235+5 = 240. 240+60 = 300. 300 + 400 = 700. 765. It sounds cumbersome when I write it out, and it may not be very efficient as a cognitive strategy, but it's what I actually do. I have a good short term memory though.
It may be time to get the Feynman book and have him read about having a "mental box of tools".

Clancy- I just checked that out, and it does seem far superior. At least for those of us who always hated scratching out the written numbers (I don't know why I hated that, but I did)
The homeschool method CCPhysicist quotes is very similar to Singapore math way.

Also with CCPhysicist, I'm curious that he's doing this in 5th grade. If he's struggling with this in 5th, I'd classify it as a hole in his understanding, and make me wonder if he's got other holes. I've got a kindergartener and 4th grader: Everyday Math does this in the 3rd grade, and cements it in the 4th. (I think, we're a bit disconnected from the sequence in my house.)

If you suspect other gaps, you might want to give him Singapore Placement tests (free) to find and plug them.

@JohnInCambridgeUK ~ Welcome to NCLB, where to get credit on state tests, you not only have to get the answer right, but you also have to do it their way. (Personally, I subtract in my head by totaling positive and negatives from each place value, so that 312-28 would be +300-10-6=290-6=284, a method that requires neither pencil and paper nor calculator, but would get me half points on the Ohio third grade test.)
Heh. I had a similar "procedural" mental block with long division. One step of the process just would not stick in my head. Drove me nuts. Took ages to get past it.

So it's certainly in the blood.

Back in the Ye Olden Tymes, DD himself engaged in a bitter death-battle with geometry. I was certain that year would end with him stabbing somebody in the neck with a protractor. (He didn't, but it was a near thing.)
I'd start with Tom Lehrer's song "New Math," which does a lot of what's suggested above, but is also howlingly funny:
I like the count up and add trick. Subtraction is just the inverse of addition, and it makes a problem easier.

I've realized that I generally work a problem from the left, rather than the right. So, 700-235 reduces to 500-35 reduces to 470-5 reduces to 465.

Reduction is another key. 700-200-30-5 can be done in any order, so long as the problem is broken down. One hard problem can be broken into three easy problems.
Use money.

Play Monopoly or Farming Game or something that requires him to make change. Making change is the same thing as borrowing.

(If you don't know about The Farming Game, check it out. It's an old family classic, and 10 years old is exactly when it starts to get fun for kids.)
If you can, acquire or build an abacus; it's the easiest manipulative to explain place value, and generalizes beautifully.
The way I learned it, and the way I still think about it, is by writing out the problem vertically and looking at the actual numbers. (I'm very, very visual.)

For 700-238, I look at the ones column: 0 over 8. Whenever I see 0, I think "10", and look left to steal a "1". If there's no "1" to the immediate left, I put my ones column on hold and go looking for my "1", as far left as I need to go. So I cross out my "7", write a "6" above it, and mentally steal that "1" for the tens column "0", making it a "10". I literally write the "1" in there, making the "10" visible. Then I go back to the ones column, and now I see a "10" to the left, so I can steal a "1" and make it "9". So I cross out the "10" I just wrote there in the tens column, and write "9", and steal the "1" for my ones column. Now I have "10" over "8" in my ones column, "9" over "3" in my tens column, and "6" over "2" in my hundreds column.

When it's done, it looks messy, but my brain has never captured the concept any other way as successfully.
I turned math problems into a narrative stories. I seem to recall an early teacher explaining things that way, especially subtraction. If The Boy is more of a Reader it might help to reframe the arithmetic into word/story problems.

I started typing out a full narrative for 700-235, but it became a novella filled with intrigue, OCD, secret clubhouses, and sugar cookies. Here, then, is a treatment of my unpublished work, The Story of 700-235: A Reduction of Life on the Streets in Subtractionville":

The Zero Neighbors in 700 literally "borrow" from each other and Seven until they have enough to loan (or pay off, depending on how noir you are) their across-the-street neighbors, Two, Three, and Five who are short on materials needed for them to complete a job. The final answer is the completed task.

Production rights are in negotiation stages.
I have no wisdom on borrowing -- I just do it automatically, and often do the mental arithmetic -- but in 5th grade I vividly remember not being able to understand how rectangles with the same perimeter did not enclose the same area. I think my teacher thought there was something wrong with me, because I just wouldn't believe it. I got the math, the formulas, etc, but it just seemed WRONG. (Still does, frankly, though I know it's true.)
I don't have any suggestions, but I can tell you something that *might* be encouraging.

When I was learning long division it took ages for it to stick. I would learn it one day, get comfy with it, and then the next I was back to square one. It was unbelievably frustrating. I can't remember how long it took, but eventually it did stick. I'm sure that'll also happen with The Boy.

Sometimes it just takes a little while for our brains to be ready to properly process and then hold info.

Good luck to The Boy! I know he can do it. :)
Hehe, I am Austrian, and it is true that I did not understand where this particular problem with borrowing could come from. The way we learned it in third grade, borrowing from the hundreds is not more difficult than borrowing from the tens.
I can't answer your question, but this article about teaching elementary school math showed up in my mailbox last week, and I think you might find it interesting.
I second Sarita. I think we sometimes equate smarts with speed to comprehension. But they're not the same. Your son may just need to understand that for math, he will need to practice more than other people do to get some concepts. Some brains just work that way. Getting through a year of Calculus required me to do 3-4 times more problems than were assigned because it was the only way for me to get fast enough to pass my exams. I had to do something similar with Chemistry (thermodynamics! NMR spectroscopy! Kill me now!) and stats. Sometimes, I would just do the same problem over and over just to mentally rehearse all the steps. The benefit for me in being "slow" is that once I finally get something, it tends to stick. Perhaps I have to use a different kind of memory to get this stuff? Whatever the reason, your son may just have to learn that patient persistence will pay off for him. Getting through the frustrating part is one of those life lessons that will pay off in other areas as well – if he learns to move from “I hate this and I suck” to “oh – this is one of the things I have to study repeatedly to get.” Practice practice practice!

How’s it going with guitar?
I would say that you need to make that last "0" a "10" so you take from its neighbor. But you can't take from "0". So first you have to make that "0" a "10". You do this by borrowing from "7" (which he understands) Now you can borrow from the new "10" to make the last "0" a "10". So you have "6","9","10"
I would say that you need to make that last "0" a "10" so you take from its neighbor. But you can't take from "0". So first you have to make that "0" a "10". You do this by borrowing from "7" (which he understands) Now you can borrow from the new "10" to make the last "0" a "10". So you have "6","9","10"
Am I the only one whose head exploded half way through the comments? I barely understood the question, let alone the answers. But as a mother of a bright ten year old who tends to rush tests, I think you're probably already doing all the right stuff.

I enjoyed the moment in the Steve Jobs 2005 Stanford commencement speech that the whole world is rewatching today where he mentions that we need can only see how the dots have connected in the past, and we need to trust that the dots will connect in the future. So perhaps just take the advice you've given your readers so often: make it safe to fail. My slight thought is that our ten year olds are still as impressed by our supportive reactions to their efforts as by formal 'middling' feedback.
Nothing new to say here but I vote for the money. My 10 year old is a visual learner and the hands on factor made it more interesting. Turning the hundred into 10 tens and then taking one ten and changing that to 10 ones seemed to make it 'click'. We used monopoly money.

Like other commenters I am curious about the math concepts he has been working on. My son learned borrowing from two zeroes away sometime in the first semester of 4th grade. They are on fractions and exponents now. And, generally, I consider my state (MO) to be one of the ones "behind" the norm.
Adding to all the comments above - I use the following virtual manipulative to illustrate teaching subtraction to future ele. school teachers.

Click on the instruction button in the applet to see how it works. "Borrowing" is no longer magic!!
The way I learned it was, if you add a carry mark on top, you have to add the same amount to the bottom so it balances.

So you'd go from 700-235 to 7 0 *0 - 2 3* 5 (or 70[10] - 245) to 7 *0 *0 - 2* 3* 5 (or 7[10][10] - 345) to something you can subtract properly.

Don't know if that's weird, but it worked for me, and I still do it that way. It also doesn't require crossing out - just make a little dot by the numbers.
A +1 to the pennies, dimes, and dollars idea, but make sure to emphasize the place-value system while you're doing it.

What kind of learner is he? Does he tend to do ok with memorizing algorithms in general (so the problem is that the subtraction algorithm doesn't stick overnight for some reason), or does he tend never to remember algorithms until he understands why they work?

Also, as John asked, is he being graded on the method or the answer? If the answer, then it could be worth exploring Austrian subtraction or something similar. Even if it's the method being tested, though, you could teach him an alternate method and use it as a means of checking his answer. Another reason this is good is that I've also found that sometimes kids learn algorithm A better once they've already learned algorithm B.

Another thing that can be successful (but doesn't work with everyone) is to give him mental subtraction exercises. Start simple, with problems he can do easily... it's a game; toss one out when you're in the car, and another when waiting at the supermarket. This can work well as a confidence-building exercise, or as a way to get his competitive juices flowing.

Finally, instead of seven hundred being 'seventy tens', try thinking of it as '6 hundreds and ten tens'. Now he can borrow from the tens! (This is also close to the dollars, dimes, pennies approach.)
I'd tell him that he is learning an algorithm that makes doing subtraction quicker but doing the algorithm doesn't help you understand what is going on.

Then I'd go on to say but I can expand out the algorithm by doing the subtraction on each piece of the two numbers i.e. the hundreds, tens and ones. It starts making more sense when you can see it expanded out.

Try and easy one first
675 - 243
= 600 + 70 + 5 - 200 - 40 - 3
= 600 - 200 + 70 - 40 + 5 - 3 i.e. regroup the 100's, 10's and 1's
= 400 + 30 + 2
= 432

and then try a difficult one
700 - 235

= 700 - 200 - 30 - 5

= (700 - 200) + (0 - 30) + (0 - 5)

= (600 - 200) + (100 - 30) + (0 - 5) * borrow 100 from 700
(and you could add in an extra step where you add in 700-100+100)
= (600 - 200) + (90 - 30) + (10 - 5) * borrow 10 from 90

= 400 + 60 + 5

= 465

Although in this way I've worked left to right whereas his problem is with working right to left. But I think once he sees there is a bit of logic to it in might make things easier.

Id' leave off the brackets when I'm teaching him and just make physical space between bits otherwise the brackets might hinder the process.

At the end say something about how much work that was writing it all out that way. And isn't it great there is short-hand method for doing it.
Here's how I thought about it:

If the guy on top dips below zero, then the guy to his left spots him a ten.
I'm with JohninCambridge and Becca: borrowing is a useless algorithm that has come down to us by historical accident. It is always easier to add up to a round number. For example:
964 - 789
First add 1 to 790, then 10 to 800, (so 11 so far), then add 164 to 11.
Many teachers enforce a specific algorithm though.
It just makes me so happy to see all these educators chiming in to help your kid with his subtraction issue.

And then it makes me sad -- if only we could do the same for every kid, that kind of individualized attention combined with flexibility of practice and thought.

My own kids are still small, 2 and 4, but I'm pretty confident that as the kids of two professors, if they run into educational stumbling blocks, we'll have access to a ton of help for them. Every kid should have the same.
My kids went to a Waldorf School. They learned borrowing and carrying in first grade by collecting sticks and putting them in bundles of tens.

To do 23-6, you need to break up a bundle of 10 and put that 10 with the 3 you already have. Then you have enough singles to subtract.

You could do it with groups of 10 legos too.

Also - why do we like 10 as the digits for our number system? Why not 8 or 20?

Well, some cultures used 20 and they didn't wear shoes.....

Yup, it's from counting on your fingers. 10 just makes sense to us.
Plenty of good advice here -- whatever techniques you try, just don't get him addicted to using the calculators! I am hearing reports from middle school teachers that some kids can longer do addition, let alone subtraction.
My son just learned this. He's using the EPGY program, and I liked the approach they used. (It's been a while, but I think singapore math does something similar.) Basically, they start with teaching regrouping: 25 is the same as 2 tens and 5 ones. Then you have them regroup by borrowing: this is also the same as 1 ten and 15 ones. Have him do some regrouping like that. Then you can show him the shorthand for it. It's a pretty quick way, and I think they catch onto it fairly quickly.
"I don’t want him to become the kid who writes off math as something you’re either born with or not, like wiggling your ears. " - Not letting your child setting aside Math just like that is a great aspiration. It's like getting defeated over something that will make one a better person. Math is a beautiful thing to learn. There is beauty in numbers.- Layce of
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