# Math Educators: Is there a name for this?

December 1, 2015 7:35 PM Subscribe

My eight year old has devised his own way of doing subtraction with re-grouping and it makes me wonder if this is a known "wrong" way of doing things that is part of the "normal" learning process, such as the way that many children write letters backwards and it doesn't necessarily mean anything, or if it is more unusual. For what it's worth, his teacher, who has a masters degree in mathematics herself, has not seen a child do things this way before.

If you ask my son what 3 minus 5 is, he will tell you 2. His method for solving problems that involve regrouping, such as 23 minus 15, is not to turn the problem into 13 minus 5, but rather to turn it into 10 minus 2. He subtracts the lower number from the higher one in each column, regardless of which one is on top or bottom, then, if the number is a negative number (although he doesn't refer to it that way), he borrows 10 and subtracts his number from it. He can do and understands doing subtraction with regrouping the "normal" way, but devised this way for himself and this is his preferred way of doing it.

His math education comes mainly from Singapore Math and Khan Academy. My math education came mainly from "this is how you do this, memorize it, and do it", so the idea that someone might do basic subtraction their own way was a bit of a surprise to me. Is it actually surprising to anyone who teaches elementary mathematics? Does it possibly indicate anything about my son in terms of learning challenges he may have?

I hope this doesn't sound ungracious, but I'm looking for answers from people who know something about how children acquire math skills, rather than anecdotes from people about how they learned math themselves.

If you ask my son what 3 minus 5 is, he will tell you 2. His method for solving problems that involve regrouping, such as 23 minus 15, is not to turn the problem into 13 minus 5, but rather to turn it into 10 minus 2. He subtracts the lower number from the higher one in each column, regardless of which one is on top or bottom, then, if the number is a negative number (although he doesn't refer to it that way), he borrows 10 and subtracts his number from it. He can do and understands doing subtraction with regrouping the "normal" way, but devised this way for himself and this is his preferred way of doing it.

His math education comes mainly from Singapore Math and Khan Academy. My math education came mainly from "this is how you do this, memorize it, and do it", so the idea that someone might do basic subtraction their own way was a bit of a surprise to me. Is it actually surprising to anyone who teaches elementary mathematics? Does it possibly indicate anything about my son in terms of learning challenges he may have?

I hope this doesn't sound ungracious, but I'm looking for answers from people who know something about how children acquire math skills, rather than anecdotes from people about how they learned math themselves.

I did an early childhood education degree recently though haven't gone so far as to be a regular classroom teacher.

It doesn't surprise me that his teacher hasn't seen a child do things this way. There is a real shift over the last few years into how mathematics is being taught and if the teacher's degree was in maths rather than pedagogy they may not have been aware of some of the changes. This article discusses it a bit.

I don't have the terminology for you but that he is creating his own method and understands his reasoning is awesome and will give him so much better ownership of the learning process. In all of my maths education classes we had so many 'ah ha!' moments but in the classroom I found it tended to see be more old school (like how you describe your experiences). But your son is doing what we were hoping to get our students to achieve.

I just turned up this article which might be interesting to you too (I'm book marking it to read).

posted by kitten magic at 7:51 PM on December 1, 2015 [2 favorites]

It doesn't surprise me that his teacher hasn't seen a child do things this way. There is a real shift over the last few years into how mathematics is being taught and if the teacher's degree was in maths rather than pedagogy they may not have been aware of some of the changes. This article discusses it a bit.

I don't have the terminology for you but that he is creating his own method and understands his reasoning is awesome and will give him so much better ownership of the learning process. In all of my maths education classes we had so many 'ah ha!' moments but in the classroom I found it tended to see be more old school (like how you describe your experiences). But your son is doing what we were hoping to get our students to achieve.

I just turned up this article which might be interesting to you too (I'm book marking it to read).

posted by kitten magic at 7:51 PM on December 1, 2015 [2 favorites]

This reminds me of the partial differences method (there is also partial sums for addition, partial products for multiplication, etc.) It's how math is being taught these days -- understanding rather than blindly following algorithms. The general idea is that you can deal with each column separately and turn a hard problem into a series of easy problems as long as you have a good grip on place values. It's not exactly the same but it is in the same spirit as the approach used in Everyday Math which I think is based pretty heavily on Singapore math. The spirit is, that there are many ways to get to the right answer and if you play with the numbers and devise your own way then you will have true understanding. I'm not a math educator but I do have a 9yr old and have helped with a lot of math homework. I found this article on new math teaching methods very interesting.

posted by selfmedicating at 8:47 PM on December 1, 2015 [3 favorites]

posted by selfmedicating at 8:47 PM on December 1, 2015 [3 favorites]

I can't speak to early childhood development, and while I note your gracious aside, it's worth mentioning that for anyone who deals with money (cashiers in particular), this is incredibly common, as it's generally faster than doing the subtractions in a more traditional way. Is your kid dealing with money? Bus fares?

posted by klangklangston at 9:15 PM on December 1, 2015 [6 favorites]

posted by klangklangston at 9:15 PM on December 1, 2015 [6 favorites]

I think selfmedicating is onto something with the partial differences method. I had a hard time understanding your description of his method, but if I have it right, he does something like this:

23-15

(20-10)- ( 5-3)

10-2

8

My own 8-year-old is doing Khan Academy math and has learned a couple of methods that are very similar to this, as well as learning similar techniques through the Everyday Math curriculum used by our public school district.

It's very common in contemporary elementary math curricula to teach these kinds of methods, and to teach multiple methods, including the "traditional" way of borrowing that you and I learned. kitten magic may be right that, depending on the teacher's background, she may not be familiar with methods like this but they're very common in elementary math curricula in the US these days, and what your son is doing is, at most, a variation on techniques being taught in schools all over the US.

posted by not that girl at 9:36 PM on December 1, 2015 [2 favorites]

23-15

(20-10)- ( 5-3)

10-2

8

My own 8-year-old is doing Khan Academy math and has learned a couple of methods that are very similar to this, as well as learning similar techniques through the Everyday Math curriculum used by our public school district.

It's very common in contemporary elementary math curricula to teach these kinds of methods, and to teach multiple methods, including the "traditional" way of borrowing that you and I learned. kitten magic may be right that, depending on the teacher's background, she may not be familiar with methods like this but they're very common in elementary math curricula in the US these days, and what your son is doing is, at most, a variation on techniques being taught in schools all over the US.

posted by not that girl at 9:36 PM on December 1, 2015 [2 favorites]

If NotThatGirl has correctly defined the modality your son uses, then he is using a method which my son was taught in primary school when he was your son's age. I've also seen similar methods on Khan. Which isn't to take away from him grasping it, internalizing it, and being able to do it. That's awesome.

posted by SecretAgentSockpuppet at 9:56 PM on December 1, 2015 [1 favorite]

posted by SecretAgentSockpuppet at 9:56 PM on December 1, 2015 [1 favorite]

Did you make a typo? Because if your son is saying 3 minus 5 = 2, that's a problem.

But interpreting (23 - 15) = (20 + 3) - (10+5) = (20 - 10) + (3-5) =20+(-2) = 20-2 just shows a good grasp of place value.

My only worry would be that he may miss-generalize, and add when he should subtract or something.

posted by leahwrenn at 10:01 PM on December 1, 2015 [3 favorites]

But interpreting (23 - 15) = (20 + 3) - (10+5) = (20 - 10) + (3-5) =20+(-2) = 20-2 just shows a good grasp of place value.

My only worry would be that he may miss-generalize, and add when he should subtract or something.

posted by leahwrenn at 10:01 PM on December 1, 2015 [3 favorites]

I would also call this the opposite of a learning challenge; his method shows a very good understanding of what's going on, provided he can explain it. The main obstacle he may face later on is a teacher who doesn't understand math well enough to figure out when his unorthodox methods are correct or not, and may penalize him for not following the methods from class.

Speaking as someone with a PhD in math who does math enrichment with elementary school kids, kids are constantly surprising me with new ideas they have discovered or learned. It's an amazing, beautiful thing. This is not something I would try to squash. However, I would push your kid more towards making sure he can explain his ideas. This is one of the biggest challenges I see from bright kids -- often their method seems so clear to them, it's hard for them to understand how someone else

posted by ktkt at 10:34 PM on December 1, 2015 [11 favorites]

Speaking as someone with a PhD in math who does math enrichment with elementary school kids, kids are constantly surprising me with new ideas they have discovered or learned. It's an amazing, beautiful thing. This is not something I would try to squash. However, I would push your kid more towards making sure he can explain his ideas. This is one of the biggest challenges I see from bright kids -- often their method seems so clear to them, it's hard for them to understand how someone else

**wouldn't**get it. (And frankly, sometimes the kids are dead wrong about some of their creative methods. But the exploring and trying and testing is what's important, so definitely keep encouraging that.)posted by ktkt at 10:34 PM on December 1, 2015 [11 favorites]

Yeah, as math PhD dropout, if I understand your description of what he's doing, it sounds like he's basically just switching the order of a couple of steps from the "standard" algorithm.

Agreeing with those above that this doesn't seem like a conceptual problem, but also that if he's someone with a tendency towards this variety of auto-didacticism, making sure he is able to articulate what he's doing to others is an important skill to develop - especially the pieces around not assuming other people know or think what he does, and erring towards over-explaining as opposed to skipping things he thinks are obvious. Save a lot of frustration down the road.

posted by PMdixon at 10:48 PM on December 1, 2015 [1 favorite]

Agreeing with those above that this doesn't seem like a conceptual problem, but also that if he's someone with a tendency towards this variety of auto-didacticism, making sure he is able to articulate what he's doing to others is an important skill to develop - especially the pieces around not assuming other people know or think what he does, and erring towards over-explaining as opposed to skipping things he thinks are obvious. Save a lot of frustration down the road.

posted by PMdixon at 10:48 PM on December 1, 2015 [1 favorite]

No, he's doing 5 minus 3, and getting 2. Since he knows it's not actually 2, but negative 2, subtracts 2 from 10 (or adds -2 to 10) to get 8.

If the top number is bigger, he doesn't put it through the whole subtracting-from-10 process.

I have never heard of this but it seems great and I'll probably use it from now on.

posted by internet fraud detective squad, station number 9 at 10:58 PM on December 1, 2015

If the top number is bigger, he doesn't put it through the whole subtracting-from-10 process.

I have never heard of this but it seems great and I'll probably use it from now on.

posted by internet fraud detective squad, station number 9 at 10:58 PM on December 1, 2015

(although I should note that it's basically the same as borrowing ten, he's just doing the steps in a different order. I see no real reason why the order of the steps matters.)

posted by internet fraud detective squad, station number 9 at 11:00 PM on December 1, 2015

posted by internet fraud detective squad, station number 9 at 11:00 PM on December 1, 2015

*His method for solving problems that involve regrouping, such as 23 minus 15, is not to turn the problem into 13 minus 5, but rather to turn it into 10 minus 2. He subtracts the lower number from the higher one in each column, regardless of which one is on top or bottom, then, if the number is a negative number (although he doesn't refer to it that way), he borrows 10 and subtracts his number from it.*

I do this too, having also worked it out in third grade.

The method's main advantage is reducing the size and complexity of the subtraction tables you need to memorise. All you need is the smaller digit from larger single digit results, along with five "friendly number" pairs that sum to 10, and you're good to go.

For people who still don't quite see how it works: say you're working on

314159 -271828 -------Nine take eight is one; write it down.

Five take two is three; write it down.

Eight take one is seven, that was upside down so write down seven's friend which is three.

Used a friend, so four becomes three, take one is two, write it down.

Seven take one is six, that was upside down so write down six's friend which is four.

Used a friend, so three becomes two, take two is nothing. Done.

314159 -271828 ------- 42331If your son likes that kind of thing, he'll probably also enjoy the nines complement method:

314159 -271828 -------First make sure that the bottom number is indeed smaller than the top one. Now replace all digits with their nines complement and change the minus to a plus:

314159There will always be a 1 carried out of the leftmost pair with this method, and instead of writing that down we're going to wrap it around and carry it back in on the right. So:~~-271828~~+728171 -------

One and nine and one is eleven, write one carry one.

One and five and seven is thirteen, write three carry one.

One and one and one is three, write it.

Four and eight is twelve, write two carry one.

One and one and one is four, write it.

Three and seven is ten, write zero and we already carried the one.

314159It may interest your son to know that this nines-complement method was how early mechanical cash registers, with adders built out of gear wheels, did subtraction.~~-271828~~+728171 ------- 042331

posted by flabdablet at 1:09 AM on December 2, 2015 [21 favorites]

It reminds me a bit of how Singapore Math teaches the concept of number bonds. For example, 5 has the bonds of 2 and 3. Kids memorize these bonds and when doing subtraction will remember that if there's a 2 and a 5 in a problem, the other bond is 3. This could be what he's doing. This could also just be super creative.

When kids are younger, some will have an affinity to take numbers and look at them in ways adults haven't considered. Honestly, get a bunch of youngsters and some will tell you that 3 is a happy number and 7 is always red and everyone knows that.

posted by kinetic at 2:43 AM on December 2, 2015 [1 favorite]

When kids are younger, some will have an affinity to take numbers and look at them in ways adults haven't considered. Honestly, get a bunch of youngsters and some will tell you that 3 is a happy number and 7 is always red and everyone knows that.

posted by kinetic at 2:43 AM on December 2, 2015 [1 favorite]

Husbunny is ABD for a mathematics Ph.D. and he's an actuary, he says the way we're taught to do math in the US is terrible. I've heard things about common core, but I lived through New Math so...

If you're concerned, have a conversation with his teacher about it. To me, it sounds like your child has an affinity for math and is a creative problem solver.

The only thing that might be an issue is when you get to higher mathematics is that the order of operations becomes important.

posted by Ruthless Bunny at 5:20 AM on December 2, 2015

If you're concerned, have a conversation with his teacher about it. To me, it sounds like your child has an affinity for math and is a creative problem solver.

The only thing that might be an issue is when you get to higher mathematics is that the order of operations becomes important.

posted by Ruthless Bunny at 5:20 AM on December 2, 2015

I am a former K-1-2 teacher and trained to teach math in the mid-90s, when the new methods spoken of above were first coming in (Everyday Math was mine). Today, teachers encourage experimentation and discovery with numbers and this is the kind of method a kid with good number sense would discover.These are sometimes called "invented algorithms," analogous to what is called "invented writing" in language learning.

posted by Miko at 5:27 AM on December 2, 2015 [2 favorites]

Once these background skills are in place, and before students are taught standard algorithms, they are encouraged to invent and share their own ways for doing operations. This approach requires students to focus on the meaning of the operation. They learn to think and use their common sense, as well as new skills and knowledge — building on the conceptual understanding they were first taught. Quite often, students come up with their own unique algorithms that prove that they “get it.”It's not indicative of a challenge, as you probably suspect, but of doing

*well*with math. He's got good number sense and has invented (or found) a method. If he can explain his logic, break down his operations, and get an accurate result, he's doing math right.posted by Miko at 5:27 AM on December 2, 2015 [2 favorites]

Is there a chance that he picked up how to borrow ten from someone older, and he's just doing it this way because it's different and seems more logical to him? (This method is alluded to in Mair from 1772, and I can't keep myself from chanting “…

posted by scruss at 5:42 AM on December 2, 2015 [1 favorite]

*five from three I cannot, so borrowing a unit, five from thirteen is eight*, …”)posted by scruss at 5:42 AM on December 2, 2015 [1 favorite]

FWIW, The way your kid is doing subtraction is how I, personally, have always done it, as a kid and even now. And math was always my easiest subject in the past and today.

posted by TinWhistle at 6:12 AM on December 2, 2015

posted by TinWhistle at 6:12 AM on December 2, 2015

All through high school I would do math problems the "wrong" way, but get the correct answer. What was actually going on is that I intuitively understood the math problem, so I could solve it my own way, which meant that I didn't bother memorizing the way we were taught to solve the problems. I ended up majoring in math in college, and really enjoyed it, though I now do something totally unrelated.

It sounds like your son is doing the same kind of thing - he understands how numbers are put together, so he uses his own method to solve the problem. I think it's a good sign!

posted by insectosaurus at 8:01 AM on December 2, 2015

It sounds like your son is doing the same kind of thing - he understands how numbers are put together, so he uses his own method to solve the problem. I think it's a good sign!

posted by insectosaurus at 8:01 AM on December 2, 2015

kinetic - somehow I had forgotten the endless pages of number bonds! That does sound exactly why we've ended up with him solving 3-5=2. Yesterday we messed around with some physical objects rather than numbers on paper, which got him thinking outside of memorized number bonds and reassured me that no, he doesn't actually think you can take 5 plastic hamster toys out of a pile of 3 plastic hamster toys....and end up with 2 plastic hamster toys.

It sounds as if the rest of his way of doing things has likely come from Khan Academy, which he works through on his own. I didn't realize the site taught subtraction this way.

Thanks much for all the help, there are some interesting resources here, and some new ways for us to play around with numbers!

posted by toodles at 8:40 AM on December 2, 2015

It sounds as if the rest of his way of doing things has likely come from Khan Academy, which he works through on his own. I didn't realize the site taught subtraction this way.

Thanks much for all the help, there are some interesting resources here, and some new ways for us to play around with numbers!

posted by toodles at 8:40 AM on December 2, 2015

Having invented a nifty algorithm, it can be instructive to try to find edge cases that stress it. Have him apply his technique to problems like

posted by flabdablet at 7:15 AM on December 3, 2015

90036 -19952 ------and see whether he can come up with a reliable method for propagating his borrows across multiple columns.

posted by flabdablet at 7:15 AM on December 3, 2015

Another non-school-curriculum method I ran across when I was about his age was lattice multiplication. I found it in a book (can't remember which one), thought it was totally cool, and have been using it ever since.

Khan covers it.

posted by flabdablet at 7:31 AM on December 3, 2015

Khan covers it.

posted by flabdablet at 7:31 AM on December 3, 2015

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