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Think back to your elementary school math classes. Were you told to think of a greater-than sign as Pac-Man or to cross-multiply when dividing fractions? You weren’t alone. Tricks to help kids get the right answers to difficult problems have long been a staple of American math education.

But if Common Core supporters have their way, shortcuts like these will soon disappear from the nation’s classrooms.

In the age of Common Core, getting the right answer to a math problem is only step one. The Common Core math standards, which are in place in more than 40 states, say that it is just as important for students to understand the mathematical principles at work in a problem.

Related: What makes a good Common Core math question?

This emphasis on principles poses a problem for popular techniques like Please Excuse My Dear Aunt Sally, a mnemonic device for remembering the order of operations that teachers complain is imprecise, and the butterfly method for adding and subtracting fractions. If correctly applied, the tricks always result in the correct answer, but math experts say they allow students to skip the sort of conceptual thinking the standards are trying to encourage in students.

Stacey Jacobson-Francis works on math homework with her 6 year old daughter Luci Wednesday, May 14, 2014, at their home in Berkeley, Calif. As schools around the U.S. implement national Common Core learning standards, parents trying to help their kids with math homework say that adding, subtracting, multiplying and dividing has become as complicated as calculus. Stacey Jacobson-Francis, 41, of Berkeley, California, said her daughter’s homework requires her to know four different ways to add. “That is way too much to ask of a first grader. She can’t remember them all, and I don’t know them all, so we just do the best that we can,” she said. (AP Photo) Credit: AP photo

Linda Gojak, president of the National Council of Teachers of Mathematics, is waging a war against the old advice that students should cross off zeros when dividing, for example. Using this technique students can quickly solve a problem like 4000 divided by 100 by eliminating two zeroes from each number and simplifying the problem to 40 divided by 1.

“I get teachers that get mad when I tell them they should stop,” said Gojak. “But I envision students dragging in a big bag of tricks into standardized tests and not really thinking about the questions.”

“It is your justification that makes your answer right or wrong,” Gojak added.

Related: Are math specialists the answer to teaching better math?

Critics, including parents who remember the way they learned math in school, worry the standards are throwing out proven computational techniques in favor of overly complex methods. They say new, convoluted approaches are turning kids off of math.

But Phil Daro, one of the lead writers of Common Core math, says math tricks have already tarnished the math brand for countless students.

“Take the butterfly method. It doesn’t articulate any mathematics,” said Daro at a conference of the Association of Mathematics Teachers of New Jersey last month. “Nothing in school is perceived to be useful by the kids, but in math they are going farther and saying, ‘why are we even doing this?’”

Related: What happens when a robotics class starts the year with no robots?

Steve Leinwand, principal researcher at the American Institutes for Research’s education program, also argues that America’s math teachers should embrace the shift away from right answers.

“Common Core has the audacity to use the word understand 218 times,” said Leinwand.

Daro does see some limited room for shortcuts in math.

“Now students have to arrive at a grade level way of thinking about the problem,” said Daro. “You can spend the first two-thirds of a lesson letting kids use the varied ways of thinking but for the last one-third we need to get them to the standards’ way of thinking.”

As for the tricks, Daro says, “I’d only settle for something like [the butterfly method], some days for some kids.”

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Emmanuel Felton is a former staff writer. Prior to joining The Hechinger Report, he covered education, juvenile justice and child services for the New York World. He received a bachelor’s degree from...

Letters to the Editor

149 Letters

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  1. First, no finger pointing, i.e., liberals or conservatives…it’s immature and without basis. Math should be made as simple as possible for students to understand. Maybe some learners need to use these methods, but certainly not all. It can be frustrating and overwhelming for many students.

  2. I fell in love with maths with a high school teacher who related it from the atom to the same precision of the universe. I went on for a MS in Sciences. Regretfully, we teach common denominators and not the tangents. We have become common, which Tomas Jefferson fought against when he established a public education for everyone.

  3. @JOECROUSE you can not divide by zero and when you “cancel out” a^2-ab you are really dividing by zero. The people teaching common core should know that this error has occurred and it can’t be correct.

  4. Unless SAT and ACT are giving out tests using common core type of exams and colleges/universities are going to accept those test score, common core is totally not working. Pretty soon, the rest of the world is going to laugh at American kids and ask why the fuck it takes so long for our kids to do a simple math problem.

  5. Does ANYONE like the Common Core methods, other than the people who designed and created it? I certainly find them obtuse and confusing.

  6. The key word/phrase in this comes in the quote by Kojak : “into standardized tests”. This is what “education” has become; the ability to pass a test created by some DEd/PhD who has spent ZERO hours in a classroom with actual human students.


  7. JOECROUSE: in the last step of your proof that 2=1 you divided by zero. Not sure if that was intentional or an error on your part. And no, Common Core does not teach that division by zero is acceptable.

    In fact, Common Core math is about students understanding math, not just doing math. If a student were truly taught using Common Core as a guide he/she would be able to critique your reasoning (that’s an actual standard) and point out your error.

    It surprises me that so many of the comments hear seem to be in support of kids learning tricks to get answers instead of developing mathematical reasoning. Tricks like the butterfly method do nothing to help students understand anything about math and that’s what we should want for our children? Really?

  8. The only comment I even begin to agree with is the one from Dave Eckstrom. I am a middle school math teacher who actually understands the math behind what I teach. No shortcuts. I formerly taught 4th grade, and I understood the math behind what I taught then too. No shortcuts then, either. Elementary grade teachers need to do a better job with teaching the math behind the math! That way the kids can come home and do their OWN homework without their parents. Who cares about standardized tests? Talk about companies and people trying to earn money from US education!! THOSE are the people you should be mad at. THOSE are the people profiting from our kids. Kids should be taught how to think, not get right answers. Standardized tests celebrate right answers, not understanding or thought processes. Flexible thinking and reasoning skills are what the world needs. That’s where ideas come from!

  9. I’m sorry, but I do agree with this article in principle. I teach 6th grade math. Too many students know shortcuts but don’t have true understanding of of math. We have to teach students to be critical thinkers- that is why teaching the conceptual understanding is so important. Common Core is not the problem. Teaching by rote is. Common Core takes being able to teach by rote out of the realm of possibility. How can anyone argue that is a bad things? One day these students will be making government, Medicare and Social Security decisions that will impact me. I want them to be critical thinkers. But that’s just me.

  10. This mathematician says that students shouldn’t be using the shortcut of crossing out zeroes to reduce the fraction. Perhaps he wasn’t in the classroom when the teacher was teaching reducing fractions, as 4000/100 REDUCES to 40/1 by virtue of dividing both the numerator and denominator by 100, the greatest common factor. It’s not a shortcut, that’s good math. You are supposed to reduce a fraction before working with it.

  11. forget that kids are not all the same!!! And not all kids can learn the same way. This is setting them up to be failures when their brains can figure out what it just can’t do!! I have a private school and is slowly filling with children with learning disability or can get the common core and they are failing and not frustrated. Our teacher teach MATH they are excelling and learning.

  12. Common Core creators are like the people who created “No Child Left Behind.” “No Child Left Behind” actually stood for “No Child Gets Ahead – everyone kept back and slowed down so that the unable can appear to be ‘normal'” and Common Core is supposed to help do that by insisting that getting the right (and USEFUL) answer is unimportant compared to ‘method.’ Right. Here’s the future scenario for these kids:
    (Rocket trajectory/Satellite orbit calculating time)
    Director: So, what’s the correct trajectory and best orbit?
    Worker: Um just a minute . . . I can’t be sure the answer figures are correct, but here’s the method for finding them if you can figure THAT out.
    Director: All right, then, if you know the method, the correctness of the numbers is irrelevant. LAUNCH!
    TV Announcer: NASA today launched a satellite and placed it into orbit around the Empire State Building. GOOD SHOW! The Common Core approach – for when answers don’t really matter.

  13. Reminds me of the new math I learned in 4th grade where I was supposed to learn long division. Spent all year trying to learn it, where to put the side numbers, etc. The night before the final test I still didn’t get what to do. My mom taught me how to divide the way she’d learned and then fake the side numbers. I was the only kid in the class who passed the test. The teacher was so proud that I’d ‘got it’ until my mom informed her of how I did the math. The teacher yelled at my mom that she’d ‘ruined’ a whole year of teaching…my mom said back, ‘you had a whole year to teach it and you didn’t, I taught her how to do it in one evening.” No more discussion.

  14. [ [ but in math they are going farther and saying, ‘why are we even doing this?’” ] ]

    that should be the first thing the teachers, teach to give a good motive to learn math, science, english .

  15. [ “Nothing in school is perceived to be useful by the kids, but in math they are going farther and saying, ‘why are we even doing this?’” ]
    is that not part of the parents, teachers job to give good reasons and examples why it is important to learn math, science english

  16. I think the article, or the quotes from people in it, are misleading, and that’s what’s got some people angry. The point that the article failed to emphasize is that it’s OK to teach shortcuts as long as it’s AFTER understanding of the concepts is achieved, and not INSTEAD of understanding. Kids should understand WHY the shortcut works. I tutor math, and what I’ve seen trip kids up is applying tricks in the wrong situations. If understanding is achieved first, then kids will know when to apply a shortcut, and when not to.

    As in most things, the answer is somewhere between the extremes. Teaching only tricks & shortcuts is bad, and teaching only understanding is bad. We need both. Common Core moves us in a direction we’ve needed to go, towards more understanding, but some of those implementing Common Core take it too far by eliminating useful tricks and learning techniques.

  17. I do not want a surgeon operating on me whose education taught him or her not to focus on getting the right answer. I was a teacher specializing in Math for 30 years. I have always taght critical thinking, but being able to get the right answer is absolutely the most important part of solving a Math problem. There are different strategies to use which require critical thinking skills – teach different strategies to students, but do not teach students that it is not imperative to get the correct answer. I wish that these so-called experts in education would quit trying to fix things that are not broken. Consider me not a proponent of “Common Core”.

  18. We have a cultural problem in this country- we want everything to be easy. So we resist anything that takes time. The processes outlined in the common core (NOT the curricula some use to teach the core- granted, some of it is goofy) take time because they are building number sense and conceptual understanding. There is a learning curve here; results will not happen overnight, and change is hard. There are many many people out there who claim to not understand and hate math. These are the same ones complaining that they want their children taught the same way they were taught. You can’t have it both ways, people. Math is important. Understanding why the math works is important. Shortcuts are fine IF you understand why they work. Learning different ways a problem can be solved increases conceptual understanding and critical thinking. Of course the kids are going to do it the fast way once they understand that it works. And different kids will pick different methods. But just telling them to memorize some algorithm with no basis is the easy way out- with no lasting understanding.

  19. Someone mentioned that elementary teachers don’t have the knowledge of math they need and said elementary schools need math specialists just like we have reading specialists. I TOTALLY AGREE!! I think.elementary school should be taught like high school. Why should one teacher be teaching all these subjects they are not specialized in? If math teachers taught math and reading teachers taught reading from the very beginning there wouldn’t be any misunderstandings.

  20. Best line from the experts: ‘If correctly applied, the tricks always result in the correct answer’ …LOL couldn’t of said it better myself. Those ‘experts’ are paid by Bill Gates and Bill Gates is afraid of students in public schools doing well.

  21. I am most disturbed by the premise supporting these radical, government enforced changes in educational methods, that American children do not do as well on standardized tests. First, this is apples and oranges. Kids in Asia, particularly in China, are taught ONLY to the test, to the exclusion of any conceptual or practical understanding. They can get test scores — an art in itself which their school people believe is the ONLY outcome of schooling — understanding and ability to apply, to assimilate, to infer have absolutely no place in their vision, and those are schooled out of children very early in life. Second, this kind of testing involves recognition and memory, the lowest level of cognitive participation.
    In addition, if the CCS is attempting to promulgate conceptual thinking in six and seven year olds, as its proponents claim it is, it is ignoring utterly generations of research and practice in child development. This is a reinvented wheel that remains blissfully ignorant of what was already known before it.

  22. Have the people who made this policy ever been in education as a teacher? Mnominic (which is what the pac-man analogy for the < symbol is called) is a technique used by people with the greatest memories and ability to recall information! In fact, I used Mnominic to find the word mnominic! I remember the Wikipedia article on photographic memory that recall seeing the word used in, even though I didn't remember the word itself.

    What mnominic does is store the information in different parts of the brain. It's like backing up your hard drive. Because of my teacher's use of mnominics I was able to get a good grade on my college placement scores even if I hadn't taken a math class in years.

  23. This is an interesting read, but I have to disagree with the author. When I comes to math, my philosophy has always been, 1) “There’s more than one way to skin a cat.” and 2) It doesn’t need to be any harder than necessary”. When I look at the way CC standards are implemented to teach basic math, it looks very convoluted. A simple addition question takes half a page Those math “tricks” help me even today to to do mental math quickly, and you know math was not my strong point. Try calculating percentages in a grocery store, balancing a checkbook, doing your taxes or doing a budget this way (since CC also requires real-word application.)
    I also call into question using these methods on a timed standardized test (like ACT, SAT, etc.) All these tests measure is the correctness of your answer, not your methodology! Besides, if a student only has 60 minutes to do 60 questions, and he takes three minutes to do one question, then how is he supposed to finish? He will only be able to do 20 questions on the test. (I know that because 6/3 is 2, and if I add the zero back that gives me 20 questions. Common Core considers that to be a “hack”. I consider it to be a right answer.)

  24. You know, most teachers I have talked to have either seen no major change to their curriculum or seen a negative change. No teacher is ever talking about a positive change coming from common core. That means, for the students, common core is at best useless or at worse damaging.

    Even if you still support the philosophy of common core, no one is talking about what’s going on behind the scenes with common core. Are they now tracking your students? If so, what do they want to do with that information? Who will have access to it?

  25. yes! The math tricks do not allow the student to learn and understand the concept!! We want students to be able to talk about, and write about, math so that they understand why the problem works (or doesn’t work). Talking about (or writing about) the shortcut that you taught them does not qualify!! The shortcuts and tricks were taught so that they would do well on the test, we are moving away from that. We want kids to understand the math NOT just get a correct answer! Please stop teaching them those “tricks”…

  26. The move to take away shortcuts is wrong and shortsighted. When dealing with a financial transaction, such as leaving tips in a restaurant or figuring out how much a 9% sales tax will add to a purchase in a store to be charged to a credit card, individuals need to be able to do this off the top of their heads in a very limited amount of time, without calculators, or pencil and paper. Shortcuts help with these matters, so that the individual leaves an appropriate tip, or ensures that the amount being charged to a credit card, will not be harmful to a budget. Repeal common core and give the kids the tools they need to survive in the future.

  27. I will continue to teach patterns in math. We discovery why crossing off those zeros works. I believe knowing why we do what we do is part of the common core. As far as fractions go we need to examine those old methods. How many children believe reducing a fraction makes it smaller. We simplify fractions because we know our prime numbers and understand the construction of numbers.

  28. @steve just reading your example proves you have no idea what you’re talking about. If a=b then a^2=ab so (a^2)-(ab) sums to zero.

    I’m a huge supporter of common core math, though I think it definitely needs tweaked. I’m currently an electrical engineering student (top of my class with insane mathematical skills) and it’s because I have a level of understanding of mathematics that many students weren’t taught in school. I taught myself the ideas and concepts behind algebra and calculus rather than memorizing worthless dumbed down shortcuts. PEMDAS and other trucks are just ways to circumvent teaching the ideas behind mathematical operations. This is why so many students struggle with word problems and math heavy sciences.

  29. As a math tutor, I am all for understanding math, but “understanding” is a progression and should be age appropriate. As an engineer, I know that efficient methods resulting in correct answers are important too. The real question here is, who is really benefiting from all of this new curriculum? Follow the money.

  30. The Common Core is not based on research, and over 500 childhood development experts have condoned these Standards as being age inappropriate, especially at the primary level. Not one teacher was invited to the table to write them, but hedge fund managers were. What do they know about education? Nothing! Bil Gates put millions behind them to implement without vetting or public debate. Why should teachers pay attention to these fatally flawed and unscientific nonsense?

  31. First, I had NOTHING to do with the writing of the CCmath, but I ABSOLUTELY agree with the methods some wrongly call “cc math” (these methods and ideas long predate CC). Secondly, I work about 1 hour each day trying to help get rid of all things Common Core.
    Where to start? Common Core has only been implemented for the last few years, close to 4 in some states, less than 1 in other states. Hardly enough time to see any real trends in international scores or in any longterm national tests (the new tests cannot be used to establish any trends because, well, they are new and have cut scores that are constantly changing).
    Next, people make comments about Chinese education based on assumptions, not facts. I encourage them to read the book called Knowing and Teaching Elementary Mathematics by Liping Ma. In it she interviews two sets of elementary teachers, all rated anywhere from highly skilled to low skilled, one group from China and one group from the US. She gives them 4 very basic elementary math problems (62-49, 123×645, 1 3/4 divided by 1/2, and how area changes as perimeter changes) and has them solve them, and then listens to them discuss how they would go about teaching these topics to their students and why they would make the choices they do. People will be surprised to learn (spoiler alert) that it is the Chinese teachers who use what some call the “common core craziness”…that they get all their students to see many different ways to solve a problem, that they work very hard to get their students to understand the why (the “common core” stuff) first before they teach their students the “how to get the right answer” stuff (the “good old” traditional way). It is also disheartening to see how the US teachers (who did not go to elementary school during the Common Core craziness) could ONLY solve and discuss these 4 topics in terms of the algorithms.
    Could go on, but don’t have time right now. “Talk amongst yourselves” for awhile….

  32. Why wouldn’t you cross off the zeros? You mean to tell me u rather multiply 400 x 200 the long algorithm-way instead of saying it is just 8 with 4 zeros? Cmon man.

  33. “It is your justification that makes your answer right or wrong,” Gojak added. Ummm, not in math. A correct answer is absolute.

    You can justify a wrong answer. Maybe these intellectuals didn’t watch enough TV when they were kids.

  34. I am an elementary teacher and I love common core math. It gives me the freedom to teach my class in a way that makes sense to them. My third graders understand basic math concepts better than I did until I was an adult. The problem with common core is that it is new and that many teachers don’t quite understand it yet because it so very different from the previous standards. It has taken me three years to feel comfortable with it and finally feel like I really understand it. Give it time, people. I honestly believe it is a huge step in the right direction. The problem with math tricks that we learned as kids is that while they may get us to the right answer quickly, they show no true understanding of the concepts. There is a time and a place for these tricks, AFTER the child has the basic understanding of why the tricks work the way they do.

  35. I am working on my PhD in education and prior to that taught math and science for 13 years. I know that there are education researchers who have not been in an actual classroom, but I am not one of those.

    First of all a fact that is missed by so many people is that Common Core is standards not curriculum. If you are outside the field of education this might not seem like an important difference, but it is. Standards are what we expect students to know and be able to do and curriculum is a tool to get there. So if you don’t like the way schools are trying to get students to meet the standards then attack the curriculum not the standards.

    However, if you don’t believe that students should have conceptual understanding of mathematics then please attack the standards all you want.

    Secondly, as someone who has taught middle school math (Pre-Algebra, Algebra, and Geometry) and now works with preservice teachers learning to teach elementary school I can tell you that using shortcuts without the conceptual understanding that makes the shortcuts work is a recipe for disaster. Every semester I hear from preservice elementary teachers who say they never understood the mathematics in a particular kind of problem until they got to college because they were taught shortcuts to get to the right answer and they were successful “plugging and chugging” to get the right answer. I don’t know about you, but I want my children to understand why a mathematical solution to a problem works in addition to being able to solve it.

  36. I use a lot of the Common Core math methods. Most of them I had already developed on my own during years of doing math. I have been using similar techniques with the students I have been tutoring for 12 years. Now they are finally emerging as mainstream. I do not like many of the techniques that are not based on fundamental properties of how math works; these mindless Simon Says methods cause students to eventually become frustrated and quit math, usually at some point during high school.

    The explanations of some of the Common Core methods may seem long, but once you understand them, you can then do problems faster and more accurately. You are also better prepared to keep advancing in math and science over the long haul.

  37. SULLBHIT – you nailed it! This is totally for national testing and NOT for improving student learning. Period.

  38. This is all about teaching to the test. My kids are not given enough time or repetition of the math basic before the school starts heaping convoluted, complex methodolgies on them. In what world does solving a problem in 10 steps better than doing it in 5?

    Stop now before a whole generation of kids are so put off by math that they can’t think at all.

  39. More and more teachers pass students you do not deserve it.

    They do it to please the students, administration, to get goo evaluations.

    The questions are either leaked to the students, or they get a “study guide” which often identical.

    Many students can’t figure out what 0/3 is ! They need a calculator.

  40. It’s a toss-up as to who comes off as more ignorant: Linda Gojak and Phil Daro, both of whom should know better than to say some of the insupportable things they say in this article (their point about tricks is valid: the examples they give are so poorly chosen as to be counter-productive), or some of the remarkably ridiculous commenters here who attack things when they have neither an historical perspective on mathematics education nor any perspective on mathematics, mathematicians, or mathematics educators and researchers. So many of these comments are knee-jerk reactions made out of ignorance and/or pure stupidity. Not that this sort of thing is peculiar to this subject, but this is what I do professionally, and it’s depressing to see how many people who know little or nothing are willing to shoot their mouths off when they don’t have a clue. Lots of luck, American kids for another couple of decades, when it comes to actually learning mathematics rather than doing more donkey arithmetic, just like your parents, grandparents, et al., as if there has been NOTHING learned in the last 100 or so years. Awful.

  41. Math has not changed in 100 years. Right is right and wrong is wrong and if you want to justify a wrong answer please don’t do it while building an airplane or a house or calculating my grandmothers medication.

  42. Most of the posters miss the point COMPLETELY. Common Core is trying to get kids to understand the underlying process in what they do. That is VERY important. Kids who can solve a problem in multiple ways will develop a better understanding and achieve a greater comfort level with math.

    It does NOT matter if it takes 10 minutes to solve a problem. It doesn’t matter if if takes 20 minutes to solve a problem. BOTH the result AND the process are important. When the kids are grown up, and working in the real world, they’re going to use calculators anyway. But they’ll have a better understanding of numbers and be able to SET UP the problem in the first place.

    The traditional methods don’t work. The truth is that most kids aren’t able to adapt to the Common Core way because they are NOT as smart as their parents think they are…

    For those who can adapt, let them continue with Common Core.

  43. The liberals are bent on destroying everything good about this country. They have a warped sense that humans are equal in everything. They never take into consideration that humans have different capabilities. One child might be great in math, but not so good in English or vice versa. When will these people ever stop destroying the USA?

  44. 4000/100 being the same as 40/1 can easily be explained by any teacher. 40 $100 bills divided by 1 $100 bill. 40 bags of 100 balls divided by 1 bag of 100 balls. How much simpler does that get? There is some real understanding. Does Gojak really not see that? I am so tired of math being about estimates and justifying why your answer should be right, but isn’t. If the math isn’t accurate, the robot won’t work right, no matter how you try to justify your answer. Period. No wonder math is so messed up.

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