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What is the largest number you can represent with 3 digits?

Posted by $ WilliamShipley 4 years, 11 months ago to Education
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No, it's not 999. This is a father's story on how he battled the entire educational system over his daughter's answer.
SOURCE URL: https://www.linkedin.com/pulse/whats-largest-number-you-can-represent-3-digits-nope-its-ed-trice


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  • Posted by $ Thoritsu 4 years, 11 months ago
    So foolish. Although I see value in standardized tests, wrong is wrong. Just admit it and move on. One of my favorite teachers, Dr Armand Dilpare at FIT, told me that there are three stages of learning:
    1. Learning a thing
    2. Doing a thing
    3. Teaching a thing to others
    ...and you learn more about the thing in each stage. Teachers that know it all are both poor teachers and are missing out.

    The story also reminds me of an incident as an undergraduate senior in macroeconomics as an elective. We were talking about elasticity of supply and demand by calculating the slope of the supply/demand curve. The graduate student teacher asserted that a forward difference was more accurate than a backwards or central difference for estimating the slope. After the class, I went to see her and noted that depending on the data, a central difference may likely be more accurate. She became hostile, and after raising her voice and reasserting her brilliance, she said, "In my country (India) students do not question the word of teachers". By that time I was irritated too, and reminded her she is not in her country, and in this country one has to be correct to be correct. One does not get to be right because on one's position. She promptly threw me out of her class, and told me not to come back. I had to go see the head of the business school department (the tail on the engineering dog at Dr Brenner's FIT back in the day) to get back in. He agreed to proctor the final and give me that grade for the class (3 of my last 6 credits). I got an A, and that was the last argument I had with a business major about math. However, one of my buddies with a PhD in EE, went and got his MBA at Chicago. He said the correlation mathematics he did there were harder than anything he did in graduate school.
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    • Posted by ewv 4 years, 11 months ago
      Voodoo economics uses very sophisticated numerology.

      As for your Indian Chief's pronouncement, central differences are 2nd order accuracy (truncation error proportional to the square of the spacing between the tabular points), and both forward and backward differences are first order. But that only works if the function is smooth enough and with arithmetic of sufficient precision. For discrete data samples numerical differentiation is notoriously unstable. Roundoff error alone propagates proportional to the inverse of the spacing size of the data. That is why it is generally best to smooth the data first with something like least squares and then differentiate analytically. Indian Chief should have been thrown out of class and sent back.
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      • Posted by $ Thoritsu 4 years, 11 months ago
        Agree with your assertions; however, the example in question was computing the slope at a point for which there is no data. I'd have to go think about it a bit, but I believe a weighted average (depending on the proximity of the points is better than using the point on either side). This type of data is very sparse, and rarely non-monotonic. As Dr. Brenner points out, derivatives are noisy. I seriously doubt with all the other factors not represented in the data (location, time of year, time of day, type of people) that a little averaging will not help better represent the actual information being sought. This is why it is macroeconomics, not physics.
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        • Posted by ewv 4 years, 11 months ago
          Computing at a point for which there is no data requires interpolation based on some number of points for which there is data. However you do whatever interpolation is required, numerical differentiation is still better analytically with central differences than either forward or backward single step differences because the truncation error is smaller in the underlying Taylor series, but none of them are reliable with ill conditioned calculations due to limited finite precision arithmetic and errors in the data. That is why smoothing is generally used whatever the source of the data.
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    • Posted by $ jbrenner 4 years, 11 months ago
      Interesting story, Thoritsu. ewv's analysis on this one is correct. To further ewv's point, it is pretty rare that the derivative of a data set is smooth enough to reliably differentiate again. Whenever possible, collect data sets as integrals. Congratulations on considering yourself your highest authority.
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  • Posted by $ jbrenner 4 years, 11 months ago
    This reminds me of an incident I had in first grade when I looked over at my best friend. The teacher had made three errors of nine + nine or less in the last couple of days. I asked my friend, "Should I volunteer to teach the class, or do you want to?" He said, "Go ahead." Knowing that I was going to get the paddle, I raised my hand politely, and when called upon, said, "Maam. This is the third mistake you have made in basic addition in the last couple of days. Do you mind if I get up and teach the class?" As was the student in the story, I was sent to the principal for being "disruptive".
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    • Posted by $ MichaelAarethun 4 years, 11 months ago
      An Army experience. the eventual Advanced and Senior Courses were based on a course previously set up to cross train our team members in a second skill set. But we had to attend both. The instructor was a flake and did not know the subject especially for that level.

      One of our group did much the same but since we outranked the instructor did not ask for permission. He sat in a student chair and we divided up the syllabus and turned it into a course for training senior people to be instructors. Then at the last moment turned the class back over and applauded our 'best student' who had accepted what was not an insult but a helping hand. Guy was professional enough and asked how we knew all that?

      Simple....we wrote the original course.

      Many would have called the principal except in this case the principal was an officer from that former group who wrote the syllabus.

      The rest of the class went back to their home units not only with extra skills but the ability to teach others.

      All in all it wasn't the waste of time to punch a ticket twice and we all benefited. For once so did the taxpayer.
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    • Posted by plusaf 4 years, 11 months ago
      That, of course, reminds me of a work story...
      As a relatively new-hired marketing engineer, I was still very shy about 'raising my hand' in situations like this, but well... that's another story, too.

      Every month, our division would have a mass meeting where various division managers would summarize the happenings in their departments.

      The controller put up a slide that showed the quarter-to-quarter Sales Growth Percentages for the whole division. It was gorgeous! A nearly perfect sawtooth 'curve' varying between nearly identical upper and lower 'limits'.

      And then he put up his forecast for the next quarter... the sawtooth was extended into the next quarter with a perfectly horizontal straight line.

      I leaned over to my closest friend and whispered, "How much do we pay that asshole to do shit like that?"
      He laughed softly and suggested that I not raise that question in public.

      I had MANY more opportunities and temptations to ask similar questions over the next 23 or so years with the company, but in the end resigned myself to the fact that there were too few people who had any 'input devices' to assimilate such feedback, let alone understand, 'grok' it or act on it.
      Then they offered me a generous retirement package.
      As some of you will recall, I snapped it up as fast as I could!

      Cheers! and yes, Critical Thinking is Dead.
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  • Posted by BradA 4 years, 11 months ago
    Fun brain food while I'm having morning coffee.
    Assuming you're limited to 3 numerical digits but not mathematical operators/notation, I started with the daughter's triple exponential 9^9^9 and after a bit of googling resulted in ~ 2.95e94. But that is obviously less than 9e99. Then I found a factorial calculator which says that 9!9!9 is way bigger than 9e99. Anyone know of any other obscure operators that arrive at a larger answer?
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  • Posted by $ CBJ 4 years, 11 months ago
    Before reading the article, I came up with the same answer his daughter did. I’m sure many others outside the education system did also. The fact that no other student or teacher in the entire country came up with this answer (or if they did, they kept quiet) is sufficient evidence that our education system is in need of a complete overhaul. A good start would be 100% privatization.
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  • Posted by $ MikeMarotta 4 years, 11 months ago
    Before I dove in here, I went, well "999… oh, well, 99^9 ah! 9^9^9…" You just have to stop and think. See Thinking Fast and Slow by Daniel Kahneman. Even people who work with statistics professionally show poor intuition with statistical problems from everyday life.

    One time in a physics class, we were going over the homework in the usual way: "How do you do Number 5?... What is the answer for Number 7, I got close but not the same one in the back?… What formula do you use for Number 4?" Then the prof stopped. He said, "You kids go out in the back yard and shoot hoops for 45 minutes and not make a single shot and still claim that you had a good time. How long did you spend on Number 5 before you gave up? How many methods did you try??"
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  • Posted by $ jlc 4 years, 11 months ago
    I ended up teaching 'the new math' to my 8th grade class because the doddering old nun we had as a teacher was totally adrift - and Wm can affirm how bad I am at math! If the 'bar' is really low (old nun and disinterested students), sometimes it is easy to excel.

    I note that the daughter is now majoring in Bio/BioMed. This puzzles me, because she obviously had a talent for math...and Bio is where you head if you love science and are not good at math (did that myself). I suspect that 'the system got to her' eventually.

    I thought it was superb that the father took the matter to national level and insisted that all the other answers be marked wrong because many of those students and their parents will now be upset at Common Core!

    Jan
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    • Posted by plusaf 4 years, 11 months ago
      Jan, I'd like to gently point out that it's not the Math ability she was showing, it was, if anything, the Lateral Thinking that can help bring her success and rewards in whatever field she finds to be the most fun.
      I've got an EE degree with about 48 years of rust and dust on it, but 'the way MY mind is wired' (I don't know how else to describe it...) there are a variety of fields in which I'm a whiz at finding solutions!
      As y'all have seen here, I love the Socratic Method of inquiry because it can lead me to answers in a wide range of subjects.
      There have been lots of books written about "Mavericks" or whatevers who can solve difficult problems that the 'regular professionals can't' and, virtually To A Person, they're shunned by 'regular professionals.'
      To the detriment of too many organizations.
      So, why ain't I rich? I'm not a good IMPLEMENTER of solutions. Maybe next lifetime...
      Cheers, all!
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  • Posted by CTYankee 4 years, 11 months ago
    A(9, A(9,9)) Or A(A(9,9), 9) or some such arrangement of Ackermann's function; whichever one goes up faster! Not certain if that's bigger than Graham's number, but it's gotta be a lot closer than 9^9^9 or pitiful little 999.

    Some might argue those are functions, but exponentiation is a function too. or at least it's a shorthand representation, but when I think about it so is decimal place notation. 9100+910+9 they're all higher functions, so my 1st answer stands.
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  • Posted by term2 4 years, 11 months ago
    Public school is a bad idea in the first place and should be abolished. Common core is just a logical extension of enforced public education. I call enforced public schools. ",government indoctrination centers". They might as well teach the earth is flat
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  • Posted by $ HeroWorship 4 years, 11 months ago
    Fun. inspiring tenacity and standing for truth. And the "you had your chance" bit that forces everyone else to get graded down because of an inflexible bureaucracy - an Asshole move. What is just in one context is not necessarily just in another.
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    • Posted by $ MichaelAarethun 4 years, 11 months ago
      Why shouldn't incorrect answers be graded as incorrect. Same with incorrect systems and incorrect teachers. The correct answer would have been to remove the question from the uncommonly cored list change the grades to reflect the question didn't exist and fire the teacher and program administrator.
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      • Posted by $ MichaelAarethun 4 years, 11 months ago
        given the absence of plus signs or any other indications the correct answer was probably 333. If you can assume a plus though you can assume a ! and E and N or an x. i would have gone with interrabang.
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  • Posted by Fish 4 years, 11 months ago
    Amazing story of how a whole establishment doesn't want to recognize an error that was considered correct for so long.

    Anyway, the answer is very difficult to give because 9^9^9 is the representation of the number assuming some operations related to the position of digits, but it is not the number itself.
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  • Posted by LibertyBelle 4 years, 11 months ago
    Hallelujah and kudos to you for winning. However,
    I do have the caveat that the others' answer (999)
    should not be counted against them, as they were
    not responsible for not having yet been taught a-
    bout exponents.
    I was not anywhere near being brilliant like
    your daughter in school, especially in math. But
    I couldn't stand my 8th grade math teacher. I
    would try to explain why my answer was right
    (though it probably wasn't), but the only reason
    I cared is because I wanted to know where the
    mistake was, so that I wouldn't make it again.
    And the old b---h would interrupt with, "There are
    some things you have to take on faith."
    The teacher I had in 5th grade had a science
    diagram put on the blackboard, with a "lever" and a "fulcrum". However, the "lever" in this pic-
    ture was not functioning as such; there that
    lever was on the board, with the triangle (ful-
    crum) sitting flat on top of it. When I objected
    to this on the ground that it didn't make sense,
    she just said that that was the way it was in
    the book (not one of ours, I think it was a sort of
    teacher's manual), and that was how it was go-
    ing to stay. And I am expected to have any
    respect for education after that?!--Forget it.
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  • Posted by ycandrea 4 years, 11 months ago
    WOW! Reading this story literally brought me to tears! The love of a father for his daughter being ridiculed for being brilliant and the total disregard of the "school system" for a brilliant mind. Cheers to this little girl and her father!
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  • Posted by craigerb 4 years, 11 months ago
    The daughter missed 9! to 9! to 9!.
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    • Posted by Steven-Wells 4 years, 11 months ago
      Nonsense. The problem specified three digits, NOT 3 digits combined with some number of arbitrary written operators, such as the factorial function. Admittedly, factorials increase faster than exponentials, but that wasn't the question. The exponentiation character used in the examples is only an artifice for displaying the positional notation of superscripts, which this forum does not permit as a valid format.
      As a curiosity, I'm testing whether the character set here accepts the extended-ASCII square and cube characters (0178 and 0179), as in 4² or 5³. Well, yes, it does.
      As to allowing arbitrary operators, I create the arbitrary (trivial) function [Goofus-malarkus(999)] which multiplies 999 by Aleph-sub2, which is the second transfinite cardinality. Multiplying any positive integer by that quantity produces the identical number-of-curves-in-space infinity.
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  • Posted by $ 4 years, 11 months ago
    By the way, I showed this story to one of my employees who had probably been much like that little girl when she was growing up and her answer was: What base?

    The digit sequence 999 can be made arbitrarily large by selecting large bases. So, maybe 999 is the right answer after all!
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    • Posted by freedomforall 4 years, 11 months ago
      Larger base would use a different symbol than 9 for the highest single digit number, so 999 is not correct. The larger base is the largest answer I have heard so far, although choosing a larger base is always possible. So is this actually an undefinable answer?
      Also 9^ 99 is greater than 9^9^9, so the answer in the tale should have been shown incorrect by the instructors if they actually cared and thought about it.
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      • Posted by $ 4 years, 11 months ago
        That depends on the meaning of 'digit'. When representing a hexadecimal number we use both letters and digits. Admittedly that's a fuzzy definition but if you define 'digits' as 0-9, then the largest hexadecimal value you can make with three digits is 999.

        I may be biased in this because every time I write a hexadecimal parsing routine I treat digits and letters separately.

        And isn't 9^9^9 evaluated as 9^(9^9)? The author deals with 9^99 in the comments.
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        • Posted by freedomforall 4 years, 11 months ago
          Yes, you are right on the 9^(9^9).
          Still smaller than higher base ops, but greater than 9^99.
          As you said, in hex the higher digits are A-F. Still digits since they represent numbers imo.
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          • Posted by Steven-Wells 4 years, 11 months ago
            If decimal digits are implied (which I think is the case), then the 9s win. If Hexadecimal is accepted, the F^F^F would be larger, but that leads to the next idea, representing bases larger than 16. Do we stop at base 36 that would exhaust the standard English alphabet for single digits.
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  • Posted by Herb7734 4 years, 11 months ago
    This type of thinking has been going on in the education field for as long as I've been around. I went through the same kind of thing when I had to defend my son over an essay he turned in, in the form of an 8mm movie. Educators seem to hate innovation.
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  • Posted by CircuitGuy 4 years, 11 months ago
    "how he battled the entire educational system over his daughter's answer."
    I get what he's saying, but IMHO he's fighting the wrong battle and being annoying little $hit. I admire his daughter for thinking "outside-the-box". If we open the question to operators, what about using the factorial operator? In the article he appears to do the operation on the right first? Is this correct precedence? Thinking of creative ways to get larger numbers using operators is just the kind of nerd discussion I'd be interested in.

    Making it a state case to get her score changed comes of as smart a$$. Then demanding they mark all other answers incorrect (b/c you said, only one answer can be correct. you said it! you said it!) is venturing into dumb a$$. So I find the actual discussion interesting, but the pissing contest with the school really stupid. Just focus on the math, science, and life, and ignore the minutia of testing scoring rules.
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    • Posted by $ 4 years, 11 months ago
      But the question didn't have operators, so you can't legitimately do that. (BTW have you done the numbers between 1 and 100 with exactly 4 4s and operators?)

      Wikipedia says that you evaluate the exponents top down, which is what he did. Unfortunately that line has a "citation needed" annotation so make of it what you wish.

      I agree that requiring everyone to have their answer marked wrong was a bit rude, although technically they were. The inherent problem is the "one right answer" which is clearly a rather important part of math but the application gets fuzzy when testing someone to determine if they understand what's been taught.

      I had an encounter with a grade school teacher who asked what was 1 minus 2. I confidently wrote down -1 and was told I was wrong. At my incredulous request for the 'right' answer I was told it was 0 because "we hadn't been taught negative numbers". I didn't buy that argument then and I don't buy it now. Math is math whether you've been taught it or not!
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      • Posted by Steven-Wells 4 years, 11 months ago
        I am reminded of my own experiences in math edjimukashun.

        In grade school, we had this problem: There is a set of identical boxes to move by truck. The truck can hold 4 boxes at a time. How many trips does it take to move 18 boxes. I had quite an argument with the teacher, who insisted that the answer was 4½. I said that I understood simple division and fractions, but the right answer is 5. The truck requires a full trip to move the last 2 boxes. There is no such thing as half a trip. But she wouldn't admit that it took 5 trips.

        Then in high school algebra, we were asked for the factors of a²-b². They are (a+b) (a-b). Then the teacher asked if you can factor a²+b². The rest of the class said no, and I said yes. The teacher turned to me and said, "You can't factor that," and I told her, "Sure you can. (a+ib) and (a-ib)." She gave me one of her angled glances and said, "We're not using complex numbers in this class."
        It wasn't a battle. We had a mutual understanding that my answer was right but I was being a math wise-ass, going beyond the scope of what most of the rest of the class could deal with.
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        • Posted by ewv 4 years, 11 months ago
          Half a trip means the truck goes half the distance of a full trip. The problem asked how many trips to deliver the boxes. It didn't say the truck had to return. After 4½ trips all the boxes have been delivered.
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          • Posted by Steven-Wells 4 years, 11 months ago
            To conserve space, I left the explicit round trip nature as a mere presumption in this forum. Sorry, I couldn't quote the whole thing word-for-word from my failing memory of that 1959 (or so) classroom question, but round trips were intended. The point was the difference between the teacher’s accepted arithmetic answer and its failure to match the reality of the word problem.

            If you want a word problem with impossible quantities (unless you scale the fractional quantities to integers) but an arithmetically sound result, try my favorite IFAQ (InFrequently Asked Question) from an Isaac Asimov story:
            If 1½ chickens lay 1½ eggs in 1½ days, how many eggs do 9 chickens lay in 9 days?
            Figuring how to set that up the first time is the hard part. The arithmetic involved is very simple.
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          • Posted by Steven-Wells 4 years, 11 months ago
            Actually, I did make a sort of half-trip delivery of cargo once.
            My friend/former coworker had stashed all his spare belongings in my New Jersey basement along with furniture and a grand piano elsewhere in the house. He moved to Denver and called asking for help to find a company to move his belongings quickly. All the pro movers were extremely pricy. I told him I’d do the packing and moving if he paid for a one-way Ryder truck rental, my travel expenses, one day of my lost salary, and airfare home. I packed the truck and drove the one-way trip. He got his stuff intact in three days for less than half of the best offer from the pros, who would have taken weeks.
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          • Posted by $ blarman 4 years, 11 months ago
            It's a question of definitions. If a trip is defined as "there and back again" (apologies to J.R.R. Tolkien) or a full trip then a 1/2 unit is not a trip. Thus 5 trips would be necessary. Alternatively, one could state the answer as 4 trips (full there and back agains) and a one-way delivery. The other answer would be 9 trips with a trip defined as merely one leg.

            I can tell you that none of the people in my trucking department would answer 4-1/2!
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      • Posted by CircuitGuy 4 years, 11 months ago
        "the 'right' answer I was told it was 0 because 'we hadn't been taught negative numbers'."
        That's really bad b/c the teacher is just patently wrong.

        "But the question didn't have operators, so you can't legitimately do that."
        I was considering the exponent as an operator, one that we commonly denote with the superscript. So in my mind if the question does not allow the factorial operator, it does not allow exponents either.
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        • Posted by $ 4 years, 11 months ago
          Is the exponent an operator? We denote it via position and often, but not always size. But then, even the number 999 (base 10) uses positional arrangement to denote meaning.

          One thing this entire discussion shows is that the idea of there being one right answer isn't as solid as we normally believe with math.
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          • Posted by CircuitGuy 4 years, 11 months ago
            "Is the exponent an operator? We denote it via position and often"
            I often use the caret (^) when typing. If it's a a natural anitlog, e^[large expression], on paper I write exp(large expression).

            I agree that teaching math as something where you plug and chug and get the one and only answer is wrong. More and more those problems are being solved by computers. In some ways our schools still teach kids to get on the train track that leads from one level of education to the next and then to a stable job. My 7-y/o's teacher told him this narrative. I told him maybe that was true when this teacher was a young woman, but it's less and less true today and it won't be true when you grow up.
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  • Posted by $ allosaur 4 years, 11 months ago
    Loved that quote at the end of the article by Mark Twain: "I never let schooling interfere with my education."
    That inspires me to write: "Never again let Common Core interfere with your child's education."
    Feel free to use that or even to improve on it.
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  • Posted by ProfChuck 4 years, 11 months ago
    Common core is like saying everyone must wear a size 9 shoe. Standardization is fine when it comes to nut and bolt sizes but when it comes to the human intellect it gets in the way of thinking. Conformity might be fine for social insects but human beings are not bugs.
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    • Posted by $ 4 years, 11 months ago
      In theory, the idea of having a set of things that everyone who goes through the education system knows as a "common core" of information does not seem like a bad idea.

      Where it seems to have gone off the rails is in trying to dictate how the information is to be taught instead of saying "x graders should be able to add and subtract", they give these absurd methods for adding and subtracting.
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    • Posted by $ MichaelAarethun 4 years, 11 months ago
      Common Core professes to set certain minimal standards for attaining different grade and later degree levels. It only works in education systems which have standards. Which neatly leaves out most if not all of the Common Core Crowd
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  • Posted by $ MichaelAarethun 4 years, 11 months ago
    But according to many almost the whole planet it 's the number of years in the second millennium. Coupled with giving zero a value of 1. The mathematical proof is the same formula that describes the group in question. X+Y=Zero using Zero to replace Millennial.

    387,420,489. nine to the ninth my calculator gave up

    Asking Google resulted in the comment the number is so large it would take more space than this forum allows.

    I suspect his daughter was entirely correct and has somehow evaded being part of the x+y=z generation.
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