but one is for comprehension of concepts... one is for working in reality.

sP

than on the way that subtraction was presented to many of us in the wayback, and the way subtraction was often used to find answers on pencils out in school book store or to keep track of boards in the lumber yard.

In the common sense world of people now grandparents, when we think of the value arrived at when we do the operation 32 minus 12 we think about what's left over.

In our day the story problem for this would be set up as 32 boards - 12 boards = ? boards left.

In math speak, the result of a subtraction is called difference (it isn't a remainder--a word that actually sounds like what's left over...'remainder' is the word goes with what's left over when integers don't divide evenly it's also called modulus).

If you read 32 - 12 as "what's the difference between 12 and 32 rather than what's left when 12 is taken away from 32 (which absolutely -IS- the difference); then the approach presented is a matter of adding up in steps the 'difference' between 12 and 32.

It seems weird to do this, because, well it's just not the way we conceptually oriented to subtraction.

Because if the problem is

"Tom has 32 2x4s. He sells a dozen to Bill. What's the difference???" The answer from some smart ass DUer will be, "it makes no freakin difference to me at all as long as Bill paid a fair price for the boards, and capitalist Tom pays his fair share of taxes on the profit while paying the guy out in the lumber shed a living wage."

Both concepts work.

9846 minus 2938 (typed at random)

2938 needs 62 to reach 3000

3000 to 9000 is easy... 6000

so 62 plus 846 = 908 + 6000 = 6908

It's the intuitive way of doing such problems in your head.

2938 needs 62 to reach 3000
It seems this method only works if you already have answers

You know the number combinations that get you to 100, and go from there.

You need 2 to get to the next multiple of ten (2940)
You need 10 to get to the next multiple of 50 (2950)
You need 50 to get to the next multiple of 100 (3000)

2+10+50 = 62.

Or, thinking about it the way we used to make change

$29.38 + 2 pennies makes $29.40 + 1 dime makes $29.50 + 2 quarters makes $30.00

So to get to $30.00 (in the context of what at least older people will remember), you need to give the customer back 2 quarters, a dime, and 2 pennies (or 68 cents).

Consider 63-17

Add 6 to 17 to get 23

Subtract: 63-23 = 40

Add : 6+40=46

That is the way you would do it in your head
Don't stop at 5 or count up by 10.

142-37

Add 5 to 37 to get 42

Subtract: 142-42=100

Add: 5+100= 105

142-37.

I see 37 is 5 away from 142, and I add 100. 105.

Isn't that closer to doing it the old way?

It doesn't matter if there's a one in front of 42. Just add that in with the final result.

63 is 3 more than 60
17 is 3 under 20
20 needs 40 to get to 60.
40 + 3 + 3 = 46

142 is 2 more than 140
37 is 3 under 40
40 needs 100 to get to 140
100 + 2 + 3 = 105

I think the point of the exercise is to demonstrate that there's more than one way to arrive at a solution plus it also demonstrates how subtraction and addition are related.

The algorithmic approach working from the least significant digit toward the more significant digits and borrowing as needed from the next column is just an algorithm but not the way most people would do it in their head.

I am a CPA and have always loved working with numbers. My wife, not so much. Our son had some homework that made ZERO sense to our son, my wife and myself. After the two of them spending a couple hours, I sat down with my son. I looked at the problem and explained 2 or 3 different ways to get to the correct answer and one them really stuck him. We then spent time explaining how and why it works. We then showed how it was similar to the other methods. By the end, he understood all the different methods, except the funky one. He nailed them all, and was given an incomplete because he did not do it the "right" way.

i.e., who's in control right now. They don't want the right answer, they want the right process

"Think the way we do, or you fail."

There is a new one every 15 years or so. The only truly harmful one I remember was the whole language only fad of 20-25 years ago.

A good teacher has more than one technique in their quiver and can change to help a child. Thankfully, most teachers are good.

But, if they're teaching Common Core, aren't they restricted to that format only? Otherwise the school is going to come down on them hard until they teach the way that particular system expects. I haven't heard of much leeway in that system or the current one which teaches to the tests.

(I think he would have liked this as it's a little warped...)

Consider 63-17:

add 1 to 17 to get 18
add 1 to 18 to get 19
add 1 to 19 to get 20
add 1 to 20 to get 21
add 1 to 21 to get 22
add 1 to 22 to get 23
add 1 to 23 to get 24
add 1 to 24 to get 25
add 1 to 25 to get 26
add 1 to 26 to get 27
add 1 to 27 to get 28
add 1 to 28 to get 29
add 1 to 29 to get 30
add 1 to 30 to get 31
add 1 to 31 to get 32
add 1 to 32 to get 33
add 1 to 33 to get 34
add 1 to 34 to get 35
add 1 to 35 to get 36
add 1 to 36 to get 37
add 1 to 37 to get 38
add 1 to 38 to get 39
add 1 to 39 to get 40
add 1 to 40 to get 41
add 1 to 41 to get 42
add 1 to 42 to get 43
add 1 to 43 to get 44
add 1 to 44 to get 45
add 1 to 45 to get 46
add 1 to 46 to get 47
add 1 to 47 to get 48
add 1 to 48 to get 49
add 1 to 49 to get 50
add 1 to 50 to get 51
add 1 to 51 to get 52
add 1 to 52 to get 53
add 1 to 53 to get 54
add 1 to 54 to get 55
add 1 to 55 to get 56
add 1 to 56 to get 57
add 1 to 57 to get 58
add 1 to 58 to get 59
add 1 to 59 to get 60
add 1 to 60 to get 61
add 1 to 61 to get 62
add 1 to 62 to get 63

Add up all the 'ones' to get 46

63 - 20 and add 3 to the result. I think the point is that they should stick with teaching math and let people develop their own strategies.

The new way is just confusing to me...and I teach math to adults!
Hahahaha

Yes, I do change in my head the other way.

Doesn't matter how I look at it it makes little sense. The 2s cancel out each other leaving 30 - 10. What could be easier? I get what they were trying to show. They failed, however.

The way you say is easier is the traditional way. First subtract the ones place: 2-2=0 (as you point out, they cancel), then subtract the 10s place 3-1=2.

abut I often use 'indirect' math like that.

For example, when adding 8 to something, it's easier for me to add 10, then subtract 2. Same for numbers like 98 or 95, for example. Add 100 (very easy) then subtract the difference. It's using associative and commutative properties of addition.

My child is not a throw away just because he cannot keep up with elite intellectuals. My child has determination, grit, creativity, and is funny, bright, and fun to be around. He struggles with math but has a passion for astronomy, chemistry and ice cream. I can't wait to see what he does with his life.

It is a blessing and a curse. I tend to do math in my head and don't like to wrote out answers because my hands cramp. I used to have terrible math grades because teachers thought I was cheating. Anything in school that required a lot of writing was hard and I was considered a slow student. Until I started out performing the better students with right answers when those around me were wrong.

This is reminiscent of the "new math" of the 60's. It may be how one person thinks and it can be justified, but it is not how all people think. expecting all people to use the same method instead of teaching choices, is stifling. I remember coming home from college and looking at my younger brothers grammar school math and seeing vector theory. I taught him another way of doing it, but told him his teacher wanted to see this method, so first get the answer then work backwards to give them this answer. I think a lot of people get messed up in math in those days, my brother did very well though.

That it's good because instead of abandoning kids it brings them all up.

Special education is suppose to go by the Individual Education Plans or IEPs. But now because of Common Core they are all forced to keep up with their general education peers whether they are capable and ready for it or not. It sucks. The theory that increasing accountability without placing the proper funding and support behind it is ridiculous. It is also ridiculous to think that all children learn the exact same things at the exact same rate. Some kids learn at a different pace. Some learn in different ways. Some have learning disabilities. Some have other issues such as being foster kids or living in poverty. What happens to these kids you may wonder? Does Common Core bring them all up? No. It leaves them behind and the school simply passes them to the next grade whether they know the material or not. All through middle school my son was pushed along from 6th grade math, to 7th grade math and then 8th grade math. We moved right before he hit 9th grade. The new school district we are in placed him in a class where they are going back and teaching him the basics of adding, subtracting, multiplying, and dividing because he never really learned those skills even though he technically passed all his math classes. That literally is what they do too. My son has been struggling in his general education computer classroom. You know what they told me? If things don't improve they can change the grading to a pass/fail grade instead of a letter grade. Same thing happened to my husband when he went to school. He was legally blind. Instead of accommodating his disability they simply passed him from one grade to the next until he hit high school. It was then he really started falling further and further behind until it came down to him going to an alternative high school where he did finally find the support he needed and really passed and graduated. I'm starting to wonder if my son should be going to an alternative high school. Maybe he would actually get the support he needs there instead of being treated like cattle where he is.

he and his cal tech buddies would test each other with tough problems to solve without writing anything down.
so feynman learned a lot of ways to get the approximate answer first and then refine the answer in his head.

the above trick reminds me of such an intuitive, approximation approach, essentially, subtracting multiples of 5 or 10 is easier to do, then you can adjust for how different the actual numbers are from the easier numbers.

anyway, feynman thought he had all the tricks to beat the reigning champ, but when he challenged him, the champ quickly shot back with "what's the tangent of 10 to the 100th power?" and feynman was toast. intuitively, you first would have to figure out the remainder of 10^100 when divided by pi/2, and how on earth to you even come close to approximating that mentally?


i read that book ages ago, hope i'm getting the details right....

"So Paul (Olum) is walking past the lunch place and these guys are all excited. "Hey, Paul!" they call out. "Feynman's terrific! We give him a problem that can be stated in ten seconds, and in a minute he gets the answer to 10 percent. Why don't you give him one?" Without hardly stopping, he says, "The tangent of 10 to the 100th." I was sunk: you have to divide by pi to 100 decimal places! It was hopeless."

Just saying: 32-10=22, 22-2=20.

would be even better to show how the new way actually is easier to conceptualize how to treat math operations on 4 digit or higher type of numbers.

you have no idea how long I looked at my daughters homework with an example like this on it while thinking WTF is this.

Why the hell is she adding when she should be subtracting? Now it starts to make sense.

Conservatives have trouble with ambiguity and with independent thinking.
The first way is an algorithm. Everyone does it exactly the same way to arrive at the answer.
The second way is a strategy for thinking about it but it can be implemented in many ways, such as:
12+8 = 20 , 20+10 = 30, 30+2 = 32 and so 8+10+2 = 20
-----
A Conservative's objection could be based on the fact that there is no iron-clad 'right way' to get to the solution and doing things the 'right way' is equally important to conservatives as is getting to the right answer. For example, conservatives are more likely to side with police or other authority if they follow the rules, no matter what the outcome is. A good health care system that is socialized is unacceptable but a 'free market' system is best, even if it is more expensive and performs poorly on many measures.

In the conservative world, learning is a discipline where you are expected not only to come to the proper conclusion but to have arrived at it in the right way. Conservatives think it is more important that everyone in the class should speak the same language than it is that everyone in the class should learn something.

Disclaimer: I do not speak for conservatives but I do wonder what they are thinking.

Both methods are algorithms. In no way does the second method (the "new way&quot reflect liberal values or fail to reflect conservative values. Preferring the second method because of dubious claims to it representing liberal aesthetics is so absurd it might as well be parody.

Exploring the neurobiology of politics, scientists have found that liberals tolerate ambiguity and conflict better than conservatives because of how their brains work.

In a simple experiment reported today in the journal Nature Neuroscience, scientists at New York University and UCLA show that political orientation is related to differences in how the brain processes information.

You're right, though. The two debated subtraction algorithms are both unambiguously ideological and furthermore their ideologies map into the common understandings of the American political left and right. I didn't see this at first because I'm mired in the morass of close-minded conservative rule-based thinking.

Love your reply.

that's a load of bullshit dressed in academic garb. I know bullshit when I see it...there's a W.C. Fields quote on my desk that reads:



(Not my desk.)

You seem to be a good person to ask …

There are many kinds of algorithms and perhaps both methods are. I've looked around and haven't found the answer to a question I have.

To my understanding, two runs of an algorithm should take the same number of steps to reach a conclusion if the input parameters are the same each time.
The 'new method' does not do this since the number of steps depends on an arbitrary decision made during runtime.
Is the presence - requirement - of an arbitrary decision in the process consistent with an algorithm? You can't flowchart that.

Thanks ..

You're on the safe side as long as you know you only have to take finite steps.

I'm interested …

Non-deterministic algorithm. The kind you're thinking of is called a deterministic algorithm.

It's quite often much easier to express an algorithm in a non-deterministic way. The most important characteristic of the concept of algorithms is "finite steps" not "always the same".

I think I should point out that the 2nd subtraction method gives no hint that it's actually non-deterministic. For all you and I know, they may be teaching people to add up to 5 first and then 0 in exactly 2 steps for digits less than 5. And the way the algorithm makes the most sense for people that can add (i.e. adults) is to deterministically go to 0 in one step.

Because I don't get that "new" math at all.

How do they get those numbers to work with? Where did they get the 12+3?

That is so ridiculously convoluted. Old way is soooo much easier.

Maybe I should read the article since I'm seeing posts saying it's fine. But just looking at the pic it makes absolutely no sense and I was a straight A math student.

you know and can manipulate very easily. Instead of solving one problem you might mess up, you break it down into smaller, easier steps where an error might be less likely to occur (or more easily noticed if it does). So, instead of starting with 32 and subtracting 12, you start with 12 and work up to the number 32, keeping track of how much you have to add to 12 to get there. It's easy to count up by fives and tens, so the first middle column is how much do you have to had to count up to the nearest 5. After that, you keep bumping up by 5s or 10s until you reach the round number nearest your target, and then add the final 2 to get 32.

It's a lengthy process, but it works if you've got enough paper. Since most people are more likely to make a mistake subtracting than they are adding, and addition mistakes are more likely when you have to carry, the problem is broken down and approached in a way to avoid that.

My daughter was taught this method years ago, and used it on homework when required. She was also taught the method you're likely most familiar with, and prefers it. After a brief time (less than a year), she was allowed to use whatever method she wanted.

So what is the controversy? I guess if it works it's fine, but to me it seems like a really long, roundabout way to get to the answer.

And what do they do to help them add that column they end up with? Do the same thing with subtraction? They will be in an endless loop!

Yeah, it is a longer process, but especially for beginners, it can be less error-prone. For that last column they add up, they add in the traditional way, but that column will always have lots of zeroes in it, so the addition should be pretty simple.

I'd object if kids were forced to do things this way all the time, for years on end, but I don't see a problem with teaching this approach with others, and eventually letting the kid do whatever makes sense to them.

The explanation given in the article cited isn't parsimonious.

My explanation uses a number line as a math concept and can be stated in one sentence

Along a number line with positive numbers increasing to the right, the answer to the problem is located 12 units to the left of 32.

The concept of difference along a number line also anticipates additional math concepts a student will need to know after grades 1-5, such as distance, and even absolute value of an error in a measurement.

In an age when calculations are done by machine (calculator, spreadsheet, etc) I can't see how yours or the explanation given in the article cited actually constructs foundational understanding to more advanced concepts.

I'm not advocating that method, merely explaining how it was explained to me when my kid had to learn it.

Last edited Sat Mar 8, 2014, 12:36 PM - Edit history (1)

Additionally, for kids with dyscalculia every manipulation is an opportunity for an additional error.

The vertical columns method for 32-12 = ? requires 2 subtractions to get the answer.

The approach used in the article uses 5...increasing the likelihood of a visual processing error and it doesn't lay any foundation.

Our daughter learned it somewhere, we didn't teach it to her, but in 4th grade she was chastised for not using the traditional method. From that point on her math skills have sucked. She has an IQ of 145.

it should be whatever works best for the person doing it. It's not affecting anyone else. The point is to learn.

doesn't show their work, exactly as demonstrated in the example, the child will NOT get credit even for a correct answer. So your daughter's experience will now be replicated by ALL kids across the country.

The common core sucks.

Old fashioned. Not old fashion.

Remember, the two problems are two different things.

The first is a math problem to be solved.

The second is an illustration of how math works. Totally different thing.

One thing that's telling is that the problem in NOT 32-7.Simply writing

32
-7
--
25

Explains nothing. The problem would require you to use "borrowing" to solve it, and would therefore require you to understand something that's not self-explanatory, even if you understand basic subtraction.

The key to actually being comfortable at math, even at the most rudimentary arithmetic levels, if to understand its inner workings (that's also the key to understanding its beauty). I have no dog in the hunt, but if that's what Common Core is going for, it's doing something right.

They don't just put the numbers in front of you and expect you to figure it out intuitively.

To be clear, I think people should do whatever works for them, but just saying that the original method was taught, not just put out there with no explanation.

that "borrowing" itself, if your truly understand it and don't just memorize it as a way of doing things, actually teaches you some rudimentary mathematical concepts about base 10. But that would have messed up conservatives' pretty picture of "obvious" math.

You're not borrowing "1" you're borrowing "10".

I seriously doubt understanding is taught with either method. My impression of basic math instruction is that the solution methods are presented and the students are told to solve a ton of problems with the hope that eventually this will lead to understanding. This isn't actually a bad approach for the students that do eventually understand. It's a terrible approach for the rest of the students.

You may be right that both methods will be taught without emphasizing understanding, and that's a shame -- but then it doesn't make common core any better or worse than any other curriculum. I also doubt that teh second method in the image is actually being taught as a method of problem-solving, but as an illustration of numerical relationships, which is something a good math class should teach. But if teachers don't understand teh concept any better than the students -- then you're not accomplishing much.

It's the first thing that they teach you in teacher ed or clinical psych.

The way that you solve problems, or learned to solve problems, doesn't work for everyone.
You, as a teacher or therapist, have to join with each individual and help them understand the problem in a way
that works for *them*, not what works for you.
You need a lot of tools in your toolbag.

Which is why the factory approach to education ( all children are identical, and respond identically
to the same treatments ) doesn't work.

To turn a better writer than I to illustrate the point:

“The most thoroughly and relentlessly damned, banned, excluded, condemned, forbidden, ostracized, ignored, suppressed, repressed, robbed, brutalized and defamed of all 'Damned Things' is the individual human being. The social engineers, statisticians, psychologists, sociologists, market researchers, landlords, bureaucrats, captains of industry, bankers, governors, commissars, kings and presidents are perpetually forcing this 'Damned Thing' into carefully prepared blueprints and perpetually irritated that the 'Damned Thing' will not fit into the slot assigned it. The theologians call it a sinner and try to reform it. The governor calls it a criminal and tries to punish it. The psychologist calls it a neurotic and tries to cure it. Still, the 'Damned Thing' will not fit into their slots.”

-- Robert Anton Wilson

way. The second example she gives shows that most of us use similar techniques, but forcing someone to learn with a specific technique is a bad idea because it stifles creativity. For example, in her second example:

Suppose you buy coffee and it costs $4.30 but all you have is a $20 bill. How much change should the barista give you back? (Assume for a second the register is broken.)

You sure as hell aren’t going to get out a sheet of paper and do this:



Instead, you’d just figure it out this way: It’d take 70 cents to get to $5… and another $15 to get to $20… so you should get back $15.70.

In my experience, school doesn't work that way.

For example, back in the late 1970s, when I was taking Geometry and we were on proofs. You were taught the basics, the ways of going about a proof in order to get to a certain answer. Okay, I did that. However, I did not put my proofs together in the expected way, so therefore, I failed. (I even had to take that level of math over in summer school because I didn't think the way the school expected me to think.)

I'd love it if they did teach the basics and let kids reach the expected answers in their own way. But they want us to think in their way only, not our own. Creativity is frowned upon, or downright squelched.

that we forget what place value means and why zeros are needed. The above problem is easy in any number base because no "borrowing" is required. Change it to 32 - 13 and change the base to, say 5, and see how many people will get the correct answer: 14

Using the common core algorithm:

13 + 2 = 20
20 + 10 = 30
30 + 2 = 32

adding the addends you get 14

If they knew that the zero was not invented by white Christians, conservatives would probably go back to Roman numerals.

Going through the school system at this point.

Though do not mind paying the taxes that fund it, as politicized as it's become.

The answer is instantly obvious, so I don't use either method.

But, I understand both methods. Still, for two-digit numbers, I simply look and know the answer. The second method gets less and less useful, though, the more digits there are in the numbers being calculated. The first method doesn't care how many digits there are. It works the same for numbers of any length. So, I prefer the "old-fashion" {sic} method if I have to subtract numbers I can't do in my head.

It's good to know multiple calculation methods, though. By learning them, you're better able to work in other bases than 10 more easily.

the 20 different ways they would do this problem. That's good. Each of them is using the way that is easiest for their brain. Someone was allowed to teach them all different ways, and they picked the one that made sense.

The problem with the common core as illustrated in the example is that there is ONLY ONE way that can be used to solve the problem on the Almighty Pearson-Funded-By-Gates test. I personally find that way to be ridiculous and unnecessarily difficult. If the child does it another way, and comes up with the correct answer on the test, the child DOESN'T get credit, because he has to show his work, and the work HAS to be as illustrated in this example.

I'm sorry, but that is completely asinine.

"Just write down the answer" isn't an educational technique at all, it's as much use as standing at the front of a class cracking a whip and yelling "Subtract, I say, subtract for your lives, you maggots!". In fairness, it might be an illustration of a sane approach to long subtraction, but there's no evidence that it is.

"Turn your subtraction into a series of small additions" works, but gives you many more opportunities to make a mistake, is grossly inefficient, and doesn't give you any insight into what's going on.

You can't illustrate the Right way (and I think that it probably is the one Right way) to learn subtraction on 32-12. If you want to illustrate it, you need a sum with some carries, like 312 - 25. Here's an attempt to illustrate it in ascii (I hate ascii).

/3/ 2 ^1 /1/0 ^1 2
2 5
2 8 7

I hope that what I mean is deducible from that...

How would you even know what numbers to put in the box? I guess it doesn't matter; there is more than one way to skin that cat. Still, I think simpler is better, easier to remember.

I'm hoping this is one of those fads that will quickly disappear.

Then add up those, and add up the differences. If you look at the grid, the lowest number in the problem is in the upper left box, and the largest number is in the lower right box. The number in the upper right box is the nearest 5-increment to the lowest number. Ignore the center column until you're ready to do the calculations. Just get all the nearest 5s and 10s in place on the outside columns, and then start subtracting the differences between them.

Yes, it's convoluted, but if you've read the whole thread, there are many solutions that rely on the same nearest 5s and 10s method of finding the answer. This one is just the most complex of all of them. Hopefully kids will learn the better methods later, either on their own or from smart parents

especially with larger numbers. It's hard to keep a sum in your head as done on top but the bottom shows how you could think about it. Interestingly the old way just shows you a mechanism that works without necessarily showing what goes on underlying the exercise.

I love the fact that there are conservatives who are simply too ignorant and dumb to understand how simple mathematical concepts work. Both methods on this example work are simple to understand if you have a clue about math.