Common Core “compensation” mathematics a metaphor for Obamacare
Guess wrong, figure it out later.
Compensation means how much you get paid, right? Or payment for a loss? Or any of the other common usages, right?
Reader Bronwyn was not happy when she learned that in Common Core mathematics, compensation means encouraging students to guess the wrong but easier answer, then teaching them how to compensate for the wrong answer to get to the right result:
My 4th grade daughter attends a Christian school here in Orange Co., Ca.
I do not like their choice in a common core math book at all, but I have been particularly amused by the use of the term “compensation.”
The teachers actually had to send home an email because none of the parents had any idea what the term meant. I attached a copy of the definition in the book because it just seems so fitting during this Obama Administration.
“compensation: you choose numbers close to the numbers in the problem to make the computation easier and then adjust the answer for the numbers chosen.”
All this under the lesson- Using mental math to multiply. This must be Obama’s math!
Here we parents thought it was how we got paid?
Here’s the question posed to the students to which the featured image was the answer:
In related news, HealthCare.gov will meet deadline for fixes, White House officials say:
Administration officials are preparing to announce Sunday that they have met their Saturday deadline for improving HealthCare.gov, according to government officials, in part by expanding the site’s capacity so that it can handle 50,000 users at once. But they have yet to meet all their internal goals for repairing the federal health-care site, and it will not become clear how many consumers it can accommodate until more people try to use it….
Nov. 30 was not originally intended to be a key date for the online enrollment system, but it took on outsize political and public importance when administration officials announced five weeks ago that the “vast majority of users” would be able to sign up for insurance through the site by that day. A combination of federal employees, outside contractors and a handful of technical and management experts have worked at breakneck speed for five weeks to improve the Web site’s performance as the White House has come under withering criticism from its political opponents and some consumers.
Update: H/t to commenter genes for this video — what possibly could go wrong with “compensation” mathematics?
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Common Core Math
What is the big deal about calling it “compensation” math? Obviously part of the procedure for coming up with the correct answer using this method is that you must “compensate” (or adjust) the approximate answer you first calculated to the correct answer using another simple calculation.
There are plenty of patently bad things in Common Core, there is no need to go hunting for invisible boogie-men.
This used to be called “estimating”. But teaching “estimating”, which helps facilitate speed, is for those who already understand what they are doing and why. Otherwise, it has no value in and of itself to facilitate comprehension. This is a really and truly stupid approach to teaching math.
I want to give you (3*18) thumbs up.
No, it is not estimating. At the end of the process you arrive at the correct (not estimate) answer. And no, this is not a stupid way to teach math. Read below where myself and other commentators explain that we have used this process for years to perform mental arithmetic.
You are simply wrong.
It IS a stupid way to teach math because it ignores that this kind of method does not make sense if you don’t know your multiplication tables in the first place.
But you’re right when you say that it’s not estimation.
I’ve used this method for years as well, and taught it to my children. It can be faster to get 95 + 95 = 190 by 100 + 100 – 10. This particular example isn’t as bad as some of the examples I’ve seen for doing some of these problems, like these: http://twitchy.com/2013/10/04/must-see-common-core-math-problems-of-the-day-pics/
15-7 is 15-7, 15-7=8.
I’ve seen similar material in our kids’ public school math books.
My Japanese wife, volunteering in our kids’ elementary classrooms, was blown away with the bizarre approaches to teaching math, all resulting in poor results.
“The kids can’t add and subtract and multiply in their heads. What is this crap?” Her English really hit stride there.
So, our kids go to a local Kumon-style math school run by Japanese where most of the students are the kids of Japanese businessmen over here temporarily on assignment. It’s meant to keep the kids up to speed for when they go back to Japan. There, they go over increasingly difficult drills, drills, drills, mercilessly.
And our kids always stand out in math back at their public school. Imagine that. And, ironically, they love math.
Don’t worry, university education departments have to justify their existence, so they will fix things even better as time goes on!!!! You can’t spell progressive, without progress.
Kumon is excellent. I heartily endorse it, for those who are not familiar. If you have little kids who have to be in public school for one reason or another, it’s well worth it.
There’s no shortage of huge problems with Common Core.
This is not one of them.
I figured out how to do math as described above, on my own, decades before Common Core. Still do it, regularly. The bigger the numbers, the more likely it is that you can multiply/divide in your head using this method. Thermodynamics, statistics, whatever, I rarely need either a pencil or a calculator.
I taught myself how to do this AFTER I had multiplication tables memorized. The tables are straight-forward, one-step, and uniform — these processes are the kind of things that vary from person to person.
What’s the problem? This is basically how I do math in my head (when I have to). It’s also a good way to check answers for ‘reasonableness’.
I agree. As a pilot I’m always having to determine a reciprocal heading by adding (or subtracting) 180 degrees. I always do the mental math by adding 200 and subtracting 20 (or the other way around). I’m sorry, Professor, but this is non-issue.
I agree that Common Core is not good, but there is nothing wrong with performing arithmetic this way. Like a couple posters above I taught myself how to do this a long time ago and it greatly simplifies doing arithmetic problems mentally.
Unlike the clueless parent seems to think, doing simple mental calculations correctly is the exact opposite of “Obama’s math.”
Gotta agree with the comments asking, “What’s the problem?”.
I knew this as “estimation”, which is a very valuable tool in quickly coming to a close approximation of a correct number.
This gives you something very close to the right answer, and an immediate knowledge of the magnitude of the computed number. You then can “back into” the correct number in just the manner shown.
I’ve actually used that method (which I was never taught as a “method”) for numbers with three or more digits ending in 99 or 98, e.g. It can be an efficient way to handle largish computations mentally, and it really is not “guessing”; it’s “rounding” followed by a precise adjustment.
For 18 x 3, I would have gone a more traditional route of adding 8 x 3 to 10 x 3, which is essentially as you would multiply on paper. But the round-and-adjust lesson for mental math is not the weirdest example of Common Core math that I’ve seen.
So instead of going 18×3 = 8*3 + 10*3 this method is suggesting 18*3 = (20 – 2)*3 = 60 – 6 = 54.
Different ways of skinning the same cat. Just as in teaching reading having a mixture of phonics and whole language is reasonable, so too does there need a mixture in mathematics. Drilling for automaticity* is good, but so too is understanding the structure of numbers and what it means to do multiplication. Sometimes I use this method. Works for me, when appropriate.
*”automaticity” is the term from the National Council of Teachers of Mathematics for answering a question, such as a math fact, quickly and without thinking. Or thoughtlessl
No, actually. 18*3 would equal (20*3) – (2*3). Because you’re always working with multiples of 5 or 10 and/or one digit numbers, it’s extremely easy to reach the correct answer (not an estimation) to complex problems in your head.
This method is not a replacement, obviously, for knowing math facts.
I also do not see the problem with it. As a programmer, I have to frequently ‘estimate for reasonability’ in my head with large datasets, and do ‘on the fly’ mental cross checks of reports, other people’s math, etc. IMO it’s a valid technique.
Yes, I do, too. But I was never taught to do that. It’s just using numerical common sense.
Nothing so uncommon as common sense. This method is akin to Machiavelli’s The Prince. If you already know how to do this instinctively, having it pointed out to you is redundant. If you don’t “get it”, having it pointed out to you is just more confusion.
I bet the IRS will continue to use traditional hardcore arithmetic when they audit tax returns. Especially returns from conservatives or suspected conservatives.
I’m no fan of common core, but I agree there is nothing wrong with this method. It does not involve “guessing” or “estimation”. It allows you to compute the correct answer by breaking the problem into easier chunks, allowing you to mentally solve a problem that normally would require paper or a calculator. Drills only promote regurgitation without understanding. This method promotes full understanding of numerical values and computation.
“Drills only promote regurgitation without understanding.”
Like most of the other commenters, I would agree that estimation techniques in and of themselves are not out of line. The main problem occurs when they are introduced too early, before kids get a solid grasp on the basics. Contrary to popular belief, as illustrated in the comment above, drills are a necessary element in producing facility (and therefore comprehension) with numbers in children. Such practice often gets derided as “drill and kill,” but the reality is that we humans generally need repetition and overlearning to become adept at a task or activity. Funny how some people have no problem acknowledging that concept when it comes to music lessons or sports practice, but these same folks usually bad-mouth it when it comes to arithmetic.
Mental arithmetic has its place – after kids have learned their “timeses” and “guzintas.” Otherwise, it’s putting the cart before the horse ~
Exactly. The language arts analogy is the disaster that was the “whole language” approach to learning reading in which phonics was just so passé.
I learned my timses and guzintas. But the multiplication tables only go up to 10 x 10. How does this method not use timses and guzintas?
“Drills only promote regurgitation without understanding.”
If that’s what YOU got out of them, you were doing them wrong.
Yeah, I’m with everybody else. I’ve been an electrical engineer and software designer for decades. Use this method all the time for mental arithmetic. Although I’ve never heard it called compensation before, seems like a reasonable term to me.
I agree with Aarradin … students need to learn the basics, but learning some “tricks” is also useful.
I don’t see how the term “compensation” is being used to introduce some liberal “theology”. I am not a teacher, but the vague concept should be explained at some point in normal math terms, like:
3 X 18 = ?
3 X 18 = 3 X (20 – 2) = (3 X 20) – (3 X 2)
I’m thinkin they need to learn basics first,
then some associative properties,
THEN think in terms of groovy “compensation” (20-2=18) shortcuts.
Common core could offer various possibilities from the top down, but when they start mandating national testing, teachers will teach to the test. And as we saw, certain ordained companies will sell all the books, and insert their dogma.
If we need the federal level education department at all, they should be a resource, not a dictator. Many online resources could be accessed by any school, even home schoolers, since we ALL pay the taxes. There could even be conservative OPTIONS that might teach the constitution and real free market economics, instead of Keynesian voodoo, and the evils of slave holder white men.
Sorry, by ALL pay taxes, I mean 50% pay income taxes, of the 50% that actually file, to support the federal bureaucracy and all their quid pro quo deals with special interests and communist groups. That would be a good compensation problem.
If 42% of 320 million file taxes,
and 47% of them pay no income tax,
and Yellin is shoveling a trillion a year of coal into the steam engine, which is outlawed by the EPA
and the bridge is out and the brakes don’t work
forget it … let’s go to music class
and sing praises to Obama … yes we can can can
and it seems there is an open italic html tag up there somewhere
A bigger concern on my part are people who are so wed to electronic calculators that they cannot tell when the number that pops up on the screen can have no basis in reality.
You can thank education departments for that little disastrous experiment, too. Relying on calculators in school was finally abandoned here in California.
But the damage was done to a certain set of people who were unfortunate enough to be learning math in public schools at the time of that educational fad.
I dropped out of my college courses in part because I could not get my set theory answers to match up to the answers provided. When I discussed it with my academic adviser, the calculator question came up. That’s when I told her I was using the graphics calculator to check my answers, otherwise it felt like cheating. I think she’s still laughing. Of course, I’m still out of college as well, but the free courses are 1) much cheaper, needless to say and 2) actually give me real world applications for the employment I want.
This method is valid and correct, but not for the reasons you guys think. And calling it “estimation” is probably wrong(er) than calling it “compensation.”
This method is a simple application of the distributive property of multiplication over addition. This is how it works:
3*18 = 3*(20-2) = 3*20 – 3*2 = 60-6 = 54
The problems with teaching this method to fourth graders are:
1- They’re not skill-wise ready for it. Like others said before, you use this method when you’re already proficient in basic arithmetic. And guess what, the students are not even required to learn their multiplication tables.
2- This is just one of many methods that are being taught. As a result, our kids are being exposed to a large number of techniques, but not taught to master any.
distributive is what I meant, not associative.
But I’ll feel better about myself if you grade it as correct for showing my work. 😉
We can thank ‘Everyday Math’, used by so many school districts in this country, for the sad decline in elementary school math.
And that’s why we need to do away with the Department of (mis)Education.
Somebody makes a mistake(s), and that(those) mistake(s) is spread down to every child in America.
But I was taught “New” math well before the Department of Education was created. From the Wikipedia entry on New Math:
“In the Algebra preface of his book Precalculus Mathematics in a Nutshell, Professor George F. Simmons wrote that the New Math produced students who had “heard of the commutative law, but did not know the multiplication table.”
In 1965, physicist Richard Feynman wrote in the essay New Textbooks for the “New” mathematics:
“If we would like to, we can and do say, ‘The answer is a whole number less than 9 and bigger than 6,’ but we do not have to say, ‘The answer is a member of the set which is the intersection of the set of those numbers which is larger than 6 and the set of numbers which are smaller than 9’ … In the ‘new’ mathematics, then, first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don’t think it is worth while teaching such material.”
In 1973, Morris Kline published his critical book Why Johnny Can’t Add: the Failure of the New Math. It explains the desire to be relevant with mathematics representing something more modern than traditional topics. He says certain advocates of the new topics “ignored completely the fact that mathematics is a cumulative development and that it is practically impossible to learn the newer creations if one does not know the older ones” (p. 17). Furthermore, noting the trend to abstraction in New Math, Kline says “abstraction is not the first stage but the last stage in a mathematical development” (p. 98).”
The department of education existed before, only that it was not a cabinet-level department.
Interestingly, it was part of…the Department of Health and Human Services!!
I am from the right. But please don’t do the knee jerk stridency thing that is so damaging to our brand.
The second picture clearly shows the title of this being mental math. Can the author of this post multiply 3 x 18 mentally without a technique like this or the other ways shown in the comments?
Deserving derision is the use of the term compensate. It shows you who came up with common core. And I am adamantly against common core and the type of people who created it.
“Can the author of this post multiply 3 x 18 mentally without a technique like this or the other ways shown in the comments?”
Teach the process, have them memorize the tables, and then teach some short-cuts.
Did no one else learn calculus? We learned the theory behind integration, and how to apply, and had to show we COULD do it, before we learned the shortcuts.
A person that has developed a proficiency in arithmetic would easily solve that “problem” using different methods:
This is the traditional method, only that as kids we were taught to stack the numbers:
3*18 = 3*(10+8) = 3*10 + 3*8 = 30+24 = 54
This other one is easy(er) and fast(er), when you know your multiplication tables, of course:
3*18 = 3*3*6 = (3*3)*6 = 9*6 = 54
3*18 = 3*2*9 = (3*2)*9 = 6*9 = 54
I do this all the time to figure quickly something in my head when I don’t have paper/pencil.
Why they call it this is the mystery. It confuses the issue and is being seized on to make it look like something it’s not.
This is a basic rule in math that if you divide up the problem, you can make it easier and faster to figure that doing it direct.
You’re not guessing nor do you come up with the wrong answer. You make the problem easier to solve without paper and then adjust your answer to remove any amounts you added(or substracted) to do so.
then you have the final and correct answer.
What we are doing here is what my aunts used to call “agreeing at the top of our lungs.”
What we are discussing is a legitimate computation tool, which I was taught long ago under the rubric “estimation” and later “distributive property.”
The unease expressed in these comments with the term “compensation” is also legitimate, because although the term is technically descriptive, people recognize it as a politically charged analogy that will be used in other contexts. The change of the term from “estimation” or “math trick” to “compensation arithmetic” was likely a political decision that had nothing to do with teaching math, and everything to do with a desire to have that be a familiar term for school children when they get a little older.
slide rules and decimal points come to mind. Compensation/estimation/approximation are all available to those who are taught the basics.
Dang! If I hadn’t gone out and finished the work on the front gate I would have beat you to this one MSO.
We didn’t have the fancy name for it, but you needed to diddle around with the round numbers to make sure you got the decimal in the correct place. Mess up that one and you have a genuine mess… I still have three of them critters in the 3rd desk drawer… but the problem is now with the eyes, not the decimal.
“Teenager in the 70s”, hah! Whippersnapper! 🙂
Yes, you can use that method to get where you need to go. But it strikes me as extra complicated, when I can do it by parts like so:
Sorry to pile on, but I went to MIT in the ’80s, and while I’ve never heard of a formal name for it before today, this is exactly how most if not almost all of us did a lot of “mental math”, which is after all exactly what this section of the book is claiming to teach. A very good thing, I believe, but of course not a substitution but augmentation of learning your math tables out to the 9s or 12s.
You used that method while at MIT, not exactly close to being a fourth grader who was never required to learn the mult. tables in the first place.
Don’t you think?
Exiliado: You used that method while at MIT, not exactly close to being a fourth grader who was never required to learn the mult. tables in the first place.
You can’t use compensation unless you know your basic multiplication table. Consider it a mental exercise of those tables.
This is billed as “mental math” … how does that work if you have to whip out your calculator to do the individual steps? Sort of implies to me this is intended to be part of a curriculum where the tables are already mastered.
It’s standard mental arithmetic. It has the advantage of giving a close answer, which can help avoid misplaced decimal points, then a more precise answer, depending on the number of digits involved.
3 x 18 = a bit less than 60
3 x 18 = 60 – (3 x 2) = 54
32 x 18 = about 30 x 20 = 600
30 x 18 = 540, so more than 540
2 x 18 = 36
32 x 18 = 36 more than 540 = 576
32.2 x 18.3 = about 30 x 20 = 600
32 x 18 = 576 method above
a bit less than a third of 30 + a fifth of 18 = roughly 13
or you can do .3 x 32 + .2 x 18 = about 13
or even simpler .3 x 30 + .2 * 20 = 13
576 + 13 = 589
We can continue compensating to get an exact answer, but the polynomial is a bit cumbersome for most people. The key is to solve from the larger to the smaller parts, remember what is left to be solved, and whether you are high or low to the answer.
Depending on the number of digits, you can zero in on the answer. In engineering, you rarely need more than a handful of digits, but you need to avoid being off by orders of magnitude, even if the digits are correct. Most people use a calculator for exact answers, but use compensation to determine a rough value to make sure they are within reason.
32.214 x 18.28
From above, it’s about 589. You rarely need the rest of the digits. If you do, a calculator will give you 588.87192
I’ve also used this mental math method for a looong time, since my teens in the early 70s at least.
I also use it for spelling but that hasn’t worked as god.
“I also use it for spelling but that hasn’t worked as god.”
You certainly have high expectations. There are at least two possible typos here, but I thought, what the heck…
Do you play poker? If so, be at my house tomorrow night at 7.
The issue here is that this is going on in fourth and fifth grade, not with high school students who already (allegedly) have a grasp of basic multiplication. The short-cuts are being taught before or instead of the multiplication facts.
The reliance on calculators is crippling. Twelve years ago I taught at a high school where they had developed a lower-level, freshman-year math curriculum that relied entirely on computers and calculators. In no time at all the kids figured out how to game the system. They did not know any basic math facts, nor did they care to know any. They knew what needed to be typed into the computer to get an answer and that was all they did.
When my son came home in second grade with a math homework page that included an algebraic word problem—and the instruction to “guess and check,” well that was the beginning of the end. I met with the teacher, and then with the teacher and the principal, after asking to see the entire curriculum. They wouldn’t show it to me: not for math nor reading nor anything else. We’ve been homeschooling ever since.
Fifteen minutes of your time will give you a clear idea of the many ways math is being taught—this video is considered a classic.
Elementary math instruction has been in the toilet for at least 20 years.
OT: Is anyone else seeing everything in this thread in italics? It’s making me correctively tilt to my left, where I definitely do NOT want to go.
I’m a gay Democrat who thinks Obamacare is a disaster (yes, one of exists), but I’m having a laugh at how mental math is supposedly part of the conspiracy.
Silly wingnuts, don’t you know that WE invented mental math in San Francisco in 1966 to keep track of, ah, things? I was always multiplying 3 times 18, which everyone knows is 30 plus 24.
Obamacare? Booooooring! Couldn’t these people ask Grindr how to handle more than 50,000 at once?
RandomCrank: dissecting your inane posting… I do hope you were being sarcastic.
1)Using the pejorative,”wingnuts”,will not win you friends and immediately identifies you as someone who is a divider. I use “divider” here as a pejorative.
2) That you are gay and opposed to Obamacare is not pertinent to the discussion in any way. Your lifestyle has no importance or bearing on the discussion. None.
3) Using “Boooooring” demonstrates a laziness in using the English language.
4) Mental math invented in 1966? In San Francisco? Really? I do hope this posting is sarcastic, otherwise…
5) The discussion here is principally about the appropriateness of teaching mental shortcut math to 3rd-4th graders before they are fully grounded in addition/ subtraction and multiplication/division. It is not about a conspiracy. One of my major complaints with education in the US in last 30-40 years is the idea that the old tried and true methods are wrong and that those who know better, the progressives, are right. This is hubris and elitism at its worse. As noted in other comments, dismissal of phonics and allowing calculator use are perfect examples of this kind of hubris/elitism.
I cheated on my 7 and 8 times tables in third grade and it has dogged me to this day fifty years later. The 10 year old mind has an amazing capacity for imprinting – imprinting that is not lost in a lifetime. However, the imprinting process should not be lost in a fog of superfluous concepts that only serve to confuse what is important – learning to add and subtract, multiply and divide. What else is important? Imprinting these processes as hard as possible, for they are the foundation of all math processes. I coached this age group for years in soccer and anything outside the basic fundamentals just confused the kids and reduced their performance.
Like any of us with decent math skills, we learned the tricks later once we had the foundation. In many respects, geometry, algebra, calculus, et al are tricks of the mind and math that we learn once we master the basics. Is there a conspiracy in Common Core? To the extent that all these progressives/intellectuals who devised it have a group mind set that they know what is best, then the answer is yes. Is it a liberal or socialistic conspiracy? I kind of doubt it. I just think it is hubris. As a coach of this age group I found that each child is different and no single approach worked, but sticking to the basics always worked.
Off topic but fun, or disconcerting, to discuss: if you are a hermaphrodite and have sex with someone, are you straight or gay? I’ve gotten some hilarious reactions from folks on this question recently.
What do you mean by “WE“?
People who think o-care is a disaster?
So YOU guys, whoever that is, invented mental math in 1966. Were you awarded the Nobel Mathematics Prize for that accomplishment?
Estimation was part of my high school Arithmetic book. To do it right, you must know the basic multiplication tables. If you can’t add, subtract, multiply in your head, then you can’t estimate. That is the point of what everybody is saying on the board. So, first teach the children the basics.