# Common Core math questions / difficulty **URL:** \ **Category:** Factual Questions\ **Created:** [May 5, 2014, 4:43pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723 "2014-05-05T16:43:29Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 6, 2014, 5:03pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/41 "2014-05-06T17:03:47Z")
> [@](#): > > Quoth **Pai325** : > > I retired the year before they started Common Core, thank goodness. It’s not so much what they want the kids to know, it’s how the teachers are required to teach it. Can you cite an example from the Common Core standards of the teachers being required to teach in a way you disagree with? ---
**Author:** ![Really\_Not\_All\_That\_Bright](https://avatars.discourse-cdn.com/v4/letter/r/e8c25b/32.png) [@Really\_Not\_All\_That\_Bright](https://boards.straightdope.com/u/Really_Not_All_That_Bright)\ **Post date:** [May 6, 2014, 5:21pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/42 "2014-05-06T17:21:04Z")
> [@Shodan](#): > > [See for yourself](http://www.policymic.com/articles/87477/could-you-ace-a-6th-grade-common-core-math-test). > > Regards, > Shodan I don’t see anything objectionable about those questions. I got #6 wrong because I wasn’t paying attention. ---
**Author:** ![Pai325](https://avatars.discourse-cdn.com/v4/letter/p/c68b51/32.png) [@Pai325](https://boards.straightdope.com/u/Pai325)\ **Post date:** [May 6, 2014, 5:27pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/43 "2014-05-06T17:27:47Z")
> [@Chronos](#): > > Can you cite an example from the Common Core standards of the teachers being required to teach in a way you disagree with? No cites, just what the teachers still at school tell me. Because I was retiring, I didn’t have to go to any I services or trainings., so I don’t know if it’s Common Core or the school requiring this. I’m guessing its a combination, because one of the teachers in my building was the one pushing the adoption of Common Core, and has very strong ideas about how things should be taught. With the retirements that year he became the senior member of our four person math department. ---
**Author:** ![Really\_Not\_All\_That\_Bright](https://avatars.discourse-cdn.com/v4/letter/r/e8c25b/32.png) [@Really\_Not\_All\_That\_Bright](https://boards.straightdope.com/u/Really_Not_All_That_Bright)\ **Post date:** [May 6, 2014, 5:36pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/44 "2014-05-06T17:36:13Z")
I don’t think it tells us anything that teachers are averse to a change in the way something is taught; regardless of its merits, it requires the _teachers_ to learn new stuff, not just the students. ---
**Author:** ![Shodan](https://avatars.discourse-cdn.com/v4/letter/s/9f8e36/32.png) [@Shodan](https://boards.straightdope.com/u/Shodan)\ **Post date:** [May 6, 2014, 5:36pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/45 "2014-05-06T17:36:14Z")
I got #8 wrong, because I have forgotten being taught about ranges and means and such. Presuming that Common Core would have taught the concept recently enough to be as fresh in my mind as dividing by fractions, I might have gotten that question as well. Maybe the idea proposed by the OP, that there are questions on the Common Core that adults generally would not get, is because there are concepts taught nowadays that were given much shorter shrift than in years gone by. So, like the “range” question, old people like me would not remember being taught how to solve it. Regards, Shodan ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [May 6, 2014, 5:49pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/46 "2014-05-06T17:49:22Z")
> [@Voyager](#): > > Based on my kids, both of whom were very good in math, and me, (ditto) having a fraction in the denominator is the least intuitive part of division… Yes, quite likely… > [@](#): > > , so I think models would help. That’s where I disagree. Dividing up boxes into fractions and counting them _is_ nuts. It’s not a useful, real-world way of thinking about it. Whereas teaching kids that dividing by a fraction is the same as multiplying by the inverted fraction is a very useful rule that can be applied in all sorts of situations. E.g. 3 ÷ ¾ = \_\_\_\_ Is it more useful to have people waste ages dividing three boxes into quarters and counting them up, or to teach them that this means the same as 3 x 4/3 to which the answer is immediately obvious? What about if you’re dividing, say, 12 by 5/9? Are they really going to sit there dividing a dozen boxes into 9 and then count them all up? ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [May 6, 2014, 5:56pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/47 "2014-05-06T17:56:21Z")
> [@Colophon](#): > > Is it more useful to have people waste ages dividing three boxes into quarters and counting them up, or to teach them that this means the same as > > 3 x 4/3 > > to which the answer is immediately obvious? Why can’t you do both? ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [May 6, 2014, 6:01pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/48 "2014-05-06T18:01:39Z")
> [@Colophon](#): > > That’s where I disagree. Dividing up boxes into fractions and counting them _is_ nuts. It’s not a useful, real-world way of thinking about it. I’m guessing you’re not a very visually-oriented person. But, for some people, it’s really helpful to be able to visualize things. And I can’t think of a better way to visualize what’s going on when you divide 5 by 1/8 than the “drawing boxes” method described. ---
**Author:** ![yearofglad](https://avatars.discourse-cdn.com/v4/letter/y/b782af/32.png) [@yearofglad](https://boards.straightdope.com/u/yearofglad)\ **Post date:** [May 6, 2014, 6:07pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/49 "2014-05-06T18:07:26Z")
> [@Pai325](#): > > No cites, just what the teachers still at school tell me. > > Because I was retiring, I didn’t have to go to any I services or trainings., so I don’t know if it’s Common Core or the school requiring this. I’m guessing its a combination, because one of the teachers in my building was the one pushing the adoption of Common Core, and has very strong ideas about how things should be taught. With the retirements that year he became the senior member of our four person math department. Then the teachers at your school are uninformed. Nothing in the Common Core details HOW something is to be taught. The Common Core details WHAT is to be taught. My cite is the Common Core. ---
**Author:** ![yearofglad](https://avatars.discourse-cdn.com/v4/letter/y/b782af/32.png) [@yearofglad](https://boards.straightdope.com/u/yearofglad)\ **Post date:** [May 6, 2014, 6:14pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/50 "2014-05-06T18:14:17Z")
> [@Colophon](#): > > … or to teach them that this means the same as > > 3 x 4/3 > > to which the answer is immediately obvious? Why not just teach them to punch it in a calculator or into Google, then? If the only point is to get the answer, then no understanding of anything is necessary. (And sometimes, the only point is the answer, but typically not when a new topic is being introduced in which case the point is the conceptual understanding.) It’s because conceptual understandings ARE important. Nobody expects anyone to do draw and divide boxes for the rest of their lives. You do it until you get the conceptual understanding. If I asked you to multiply two 8-digit numbers, I wouldn’t expect you to do it by hand. It’s also not something I’d expect a fifth-grader to do by hand. I would, however, expect that the 5th grader: - understands, conceptually, what it means to multiply, and - has already demonstrated the ability to perform multiplication by hand. Example: Most students are taught that to convert a mixed number to an improper fraction, you multiply the denominator and whole number and add the numerator. [4 2/3 = 14/3; you take 3 x 4 and then add 2 and that’s your new numerator.] I can’t count the number of students I’ve explained the CONCEPT behind this rote process to, and I can count reasonably high. [Consider how to write 4 in terms of thirds; how many thirds are there in 1 whole? (3) How many in 4 wholes? (12) How did you get 12 thirds? (I multiplied 4 by 3 - Oh, I see why that works now!)] Concepts are important to help further understanding of things down the road. Sorry, kind of rambling. ---
**Author:** ![Learjeff](https://avatars.discourse-cdn.com/v4/letter/l/94ad74/32.png) [@Learjeff](https://boards.straightdope.com/u/Learjeff)\ **Post date:** [May 6, 2014, 6:28pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/51 "2014-05-06T18:28:56Z")
> [@Terr](#): > > Exactly. This is nuts. > [@Thudlow\_Boink](#): > > Why is it nuts? > > And how would you teach it, if your goal was to help her understand what it means to divide by a fraction? > [@Terr](#): > > I was never taught to “understand what it means to divide by a fraction”. I was taught a simple thing - dividing by a fraction is the same as multiplying by the inverse of a fraction. The lack of “understanding what it means to divide by a fraction” has not hampered me in either my academic studies or professional career. > [@wevets](#): > > Because people learn in different ways. **wevets** beat me to it. I distinctly remember being in 4th grade and hearing about “multiplying fractions” as something we’d learn in 5th grade. I tried to fathom what it might mean, and came up with zippo (so I’m clearly no genius). But as soon as it was introduced in 5th grade with a fairly simple model, it made sense (and also using “of” for “times”, just to get used to it.) The rules for manipulating numbers never helped me until I understood why we were doing those manipulations that way. > [@Hari\_Seldon](#): > > Mean is not defined for infinite sets. Good point! > [@](#): > > The first seven questions were based on common notions and quite easy. The 8th is based on a word I had never heard before, not in that context (the range of a function is the set of values it takes on). Still, I was able to guess it. Me too. I learned that the range of a function is the set of values, such as the values a function can take, contrasted with “domain” as the set of input values. I don’t remember ever hearing “range” used as a scalar in math class, but I was able to guess. > [@Leaffan](#): > > Advanced calculus? Fuuuuuuccckkkkk. Diff EQ is where I went off the rails. I remember leaving the final exam knowing I’d failed, since all I did was parrot stuff similar to what the prof wrote for each of the questions, without getting any actual answers, and with no understanding of what I was doing. I was amazed that I got a C and passed the course. My guess is that if I’d gone on and applied it, I’d actually have learned it, in the next class. I think that happens a lot! > [@Colophon](#): > > That’s where I disagree. Dividing up boxes into fractions and counting them _is_ nuts. It’s not a useful, real-world way of thinking about it. It may be nuts to you, but different people learn differently. Using models like this is good for teaching, but IMHO shouldn’t be on the independent exams of what students are required to learn. No harm putting questions like this on quizzes to encourage kids to try to make sense of it. But on the major tests, it should be correct answers that matter, not the mental model you use. Flexibility for partial credit when showing work, but right answers are right answers regardless of method, assuming no cheating of course. ---
**Author:** ![YamatoTwinkie](https://avatars.discourse-cdn.com/v4/letter/y/cc9497/32.png) [@YamatoTwinkie](https://boards.straightdope.com/u/YamatoTwinkie)\ **Post date:** [May 6, 2014, 6:29pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/52 "2014-05-06T18:29:16Z")
> [@Colophon](#): > > Yes, quite likely… > That’s where I disagree. Dividing up boxes into fractions and counting them _is_ nuts. It’s not a useful, real-world way of thinking about it. Whereas teaching kids that dividing by a fraction is the same as multiplying by the inverted fraction is a very useful rule that can be applied in all sorts of situations. Its certainly more useful if your goal is getting an answer quickly, but basically it does nothing to understand the problem conceptually. All it amounts to is “do this method because it works. Trust me”. Taken to its extreme, might as well just skip the math portion to begin with and just teach the kid how to use a calculator. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [May 6, 2014, 6:38pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/53 "2014-05-06T18:38:05Z")
> [@YamatoTwinkie](#): > > Its certainly more useful if your goal is getting an answer quickly, but basically it does nothing to understand the problem conceptually. All it amounts to is “do this method because it works. Trust me”. Taken to its extreme, might as well just skip the math portion to begin with and just teach the kid how to use a calculator. I disagree with that. The important concept to learn is surely that dividing something by a quarter is the same as multiplying it by four: if you’re splitting it into portions that are four times smaller than “one” then you will have four times as many. You don’t have to draw pictures to understand that, if you’ve progressed beyond the level of drawing two apples and then drawing two more apples to make four apples. > [@yearofglad](#): > > Why not just teach them to punch it in a calculator or into Google, then? If the only point is to get the answer, then no understanding of anything is necessary. (And sometimes, the only point is the answer, but typically not when a new topic is being introduced in which case the point is the conceptual understanding.) > > It’s because conceptual understandings ARE important. Nobody expects anyone to do draw and divide boxes for the rest of their lives. You do it until you get the conceptual understanding. But this is nothing like “just punching it into a calculator”. It is teaching the important rule that division by a fraction is the same as multiplication by the reciprocal of the fraction. It’s a basic rule. Teaching people to divide boxes into quarters and count them isn’t teaching them any kind of conceptual understanding at all, it’s just wasting time and ink. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 6, 2014, 7:11pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/54 "2014-05-06T19:11:54Z")
> [@Shodan](#): > > [See for yourself](http://www.policymic.com/articles/87477/could-you-ace-a-6th-grade-common-core-math-test). For what it’s worth, my opinion is that 2, 3, 5, 6, and 7 are reasonably straightforward arithmetic (/geometry, in the case of 7), and I doubt anyone would have major complaints about them. 1 is a somewhat pointless “Rephrase this ordinary language in the notation I want” question, and any difficulty with 4 is similarly mostly “Can you keep your head straight as I phrase things in a deliberately confusing manner?”. 8, however, is on a whole other level of complexity from all the other questions, in terms of the reasoning expected from the student. Not that that’s necessarily a bad thing. I would, however, prefer that the correct choice be strictly less than 4 instead (or, alternatively, the question rephrased to explicitly state that the “data set” in question takes the form of a finite collection of (equally weighted) values), to avoid worrying about such things as “Well, if I consider a dartboard where every spot is worth 12 points except the _exact_ center is worth 4 points, then the range and mean score are…”. (You may object that no child thinks this way, but I object that people can think all kinds of ways. You might as well make a robust question instead of defending a fragile one.) Overall, the sample test questions seem fine. That having been said, the concern is not just over the test questions, but over the entire influence of Common Core on the education process. It doesn’t matter how reasonable the test questions are in themselves if the effect is still to corrupt the curriculum and teaching methods. (For example, the linked article says “the famously confusing math problems floating around the web are the result of a bad curriculum trying to comply with the standards”. But this isn’t necessarily a total absolving of Common Core; we might well wonder if there is something about the program, no matter how well-intentioned, which leads naturally to this consequence). I think, re: education, people are very fond of saying “More standards better! You don’t want to reduce standards and make children stupider, do you? No, you want to increase standards and make them smarter!”, as though any policy introducing such standards was inarguably, tautologously justified. But standards, in themselves, don’t do anything. We still have to ask ourselves what we ultimately hope to achieve with this system and then check empirically whether it is actually achieving that. (And weigh the gains and opportunity costs against other approaches which may be even more effective at accomplishing our underyling goals.) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 6, 2014, 7:30pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/55 "2014-05-06T19:30:34Z")
> [@Indistinguishable](#): > > But standards, in themselves, don’t do anything. (For otherwise, we could simply declare “It shall be standard that everyone be proficient in both the theory and application of calculus, physics, computer science, and, oh, let’s say chemical engineering, fluent in two languages, free from poverty, drug addiction, or chronic health problems…”, and be done with it. It’s meaningless to note that a program calls for high standards; the question is what system of feedback, incentives, etc., it proposes to achieve those standards, and whether that system actually (as in, according to empirical evidence) succeeds at doing so) ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [May 6, 2014, 7:36pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/56 "2014-05-06T19:36:21Z")
When I read the OP, I remembered soooooooo many elementary teachers I know who got frustrated with advanced arithmetic (think pre-algebra), couldn’t solve simple problems themselves and would in some cases refuse to teach math. Do you think that might be contributing to the problem? ---
**Author:** ![Dogzilla](https://avatars.discourse-cdn.com/v4/letter/d/e480ec/32.png) [@Dogzilla](https://boards.straightdope.com/u/Dogzilla)\ **Post date:** [May 6, 2014, 7:41pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/57 "2014-05-06T19:41:25Z")
> [@Jas09](#): > > Here ya go **Terr** : **[IXL | Divide whole numbers by unit fractions using models | 6th grade math](http://www.ixl.com/math/grade-6/divide-by-fractions-with-models)** > > You’re supposed to divide each box into 8ths and then count the small boxes, I think. Apparently inverting and multiplying is out these days. Actually, as a spatial learner, this is the only way I can manage to understand abstract mathematical concepts. (To me, all math is abstract. ;)) I was great at fractions because of that pie. Mmmm… cherry pie. Look, a squirrel! ---
**Author:** ![AlsoNamedBort](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/alsonamedbort/32/3213_2.png) [@AlsoNamedBort](https://boards.straightdope.com/u/AlsoNamedBort)\ **Post date:** [May 6, 2014, 7:45pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/58 "2014-05-06T19:45:10Z")
> [@Jas09](#): > > Here ya go **Terr** : **[IXL | Divide whole numbers by unit fractions using models | 6th grade math](http://www.ixl.com/math/grade-6/divide-by-fractions-with-models)** > > You’re supposed to divide each box into 8ths and then count the small boxes, I think. Apparently inverting and multiplying is out these days. Actually looking at the link, it looks like the next two sections are directly about inverting and multiplying. ---
**Author:** ![Voyager](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/voyager/32/133_2.png) [@Voyager](https://boards.straightdope.com/u/Voyager)\ **Post date:** [May 6, 2014, 7:46pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/59 "2014-05-06T19:46:28Z")
> [@Colophon](#): > > That’s where I disagree. Dividing up boxes into fractions and counting them _is_ nuts. It’s not a useful, real-world way of thinking about it. Whereas teaching kids that dividing by a fraction is the same as multiplying by the inverted fraction is a very useful rule that can be applied in all sorts of situations. > > E.g. > > 3 ÷ ¾ = \_\_\_\_ It is exactly the real world way of thinking about it - just not the efficient way. When you write it like 3 ÷ ¾ = it is easy to see how to do the transformation. But if it is written ## 3 ¾ it is not so clear. > [@](#): > > Is it more useful to have people waste ages dividing three boxes into quarters and counting them up, or to teach them that this means the same as Anyone getting the concept won’t be spending a lot of time counting. I/m not visual at all, and I think in terms of algorithms, but the boxes really show the concept quite well. ---
**Author:** ![yellowjacketcoder](https://avatars.discourse-cdn.com/v4/letter/y/4bbf92/32.png) [@yellowjacketcoder](https://boards.straightdope.com/u/yellowjacketcoder)\ **Post date:** [May 6, 2014, 7:47pm UTC](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723/60 "2014-05-06T19:47:39Z")
> [@Colophon](#): > > I disagree with that. The important concept to learn is surely that dividing something by a quarter is the same as multiplying it by four: if you’re splitting it into portions that are four times smaller than “one” then you will have four times as many. You don’t have to draw pictures to understand that, if you’ve progressed beyond the level of drawing two apples and then drawing two more apples to make four apples. > > But this is nothing like “just punching it into a calculator”. It is teaching the important rule that division by a fraction is the same as multiplication by the reciprocal of the fraction. It’s a basic rule. Teaching people to divide boxes into quarters and count them isn’t teaching them any kind of conceptual understanding at all, it’s just wasting time and ink. See, I have to disagree with this. Humans in general at bad at remembering what it was like before they internalized a particular skill, and how to explain to someone that doesn’t understand that skill. Sure, dividing is the same as inverting and multiplying. It’s also the same result when diving boxes into smaller boxes. Some methods work for some kids and some work for others. Why is it so bad to teach something both ways, so all kids can understand it? [Previous page](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723.md?page=2) [Next page](https://boards.straightdope.com/t/common-core-math-questions-difficulty/687723.md?page=4)