# This is how they're teaching fractions? **URL:** \ **Category:** The BBQ Pit\ **Created:** [November 11, 2010, 2:12am UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264 "2010-11-11T02:12:25Z")\ **Posts on this page:** 20\ **Page:** 4
**Author:** ![Nava](https://avatars.discourse-cdn.com/v4/letter/n/da6949/32.png) [@Nava](https://boards.straightdope.com/u/Nava)\ **Post date:** [November 12, 2010, 7:15am UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/61 "2010-11-12T07:15:50Z")
> [@Jorge\_Burrito](#): > > Cross multiplication is taking the recipricol and multiplying, you just didn’t realize it :p. The difference being that it’s one step, not two. The result is the same; the process is not. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [November 12, 2010, 11:55am UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/62 "2010-11-12T11:55:25Z")
> [@Nava](#): > > The difference being that it’s one step, not two. The result is the same; the process is not. Well, then you’d probably hate the proof that was always used in our textbooks. ```auto a/b ÷ c/d = a/b = a/b * d/c c/d [del]c[/del]/[del]d[/del] * [del]d[/del]/[del]c[/del] ``` ---
**Author:** ![Nava](https://avatars.discourse-cdn.com/v4/letter/n/da6949/32.png) [@Nava](https://boards.straightdope.com/u/Nava)\ **Post date:** [November 12, 2010, 1:12pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/63 "2010-11-12T13:12:12Z")
… why would I hate that proof? And what’s that proof got to do with “1) take the reciprocal, 2) multiply”? That proof doesn’t involve the reciprocal. The process with the reciprocal would be a/b ÷ c/d = a/b \* d/c = (a_d)/(b_c) Your proof proves that division is crossed multiplication without invoking the reciprocal at all. Do you think I’d hate it because it has more than one step? Proofs always involve more steps than what they’re proving. Did you re-invent the formula to solve quadratic equations every time you had to solve one? Help me here, because I’m thoroughly confused. ---
**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:** [November 12, 2010, 2:32pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/64 "2010-11-12T14:32:20Z")
> [@Nava](#): > > That proof doesn’t involve the reciprocal. 😕 Then what’s that “d/c” thing I see in it? For what it’s worth, if you take a theoretical, axiomatic approach, division is **defined** to be multiplication by the reciprocal. But that’s not really relevant to how you’d teach 7th graders to do things. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [November 12, 2010, 2:56pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/65 "2010-11-12T14:56:41Z")
> [@Thudlow\_Boink](#): > > 😕 Then what’s that “d/c” thing I see in it? > For what it’s worth, if you take a theoretical, axiomatic approach, division is **defined** to be multiplication by the reciprocal. But that’s not really relevant to how you’d teach 7th graders to do things. I dunno, I remember being told the dividing by a fraction was equal to multiplying by the fraction’s reciprocal, without being offered any kind of formal proof. Since it always worked, I never questioned it. BigT’s elegant little demo might have helped, but it wasn’t necessary. ---
**Author:** ![constanze](https://avatars.discourse-cdn.com/v4/letter/c/f9ae1b/32.png) [@constanze](https://boards.straightdope.com/u/constanze)\ **Post date:** [November 12, 2010, 5:05pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/66 "2010-11-12T17:05:08Z")
> [@AWB](#): > > One big problem I see is that lessons today have to be all touchy-feely, everybody-have-fun, nobody-gets-left-behind sorts of affairs. Going to stations around the room, having worksheet/puzzles that tell corny puns, give-them-multiple-choice-so-they’ll-have-a-25%-chance-of-getting-it-right. We had to memorize multiplication tables, unit circle chart (trig), and rules of fractions. Now, those things have to be posted on the walls somewhere for the students to refer to at any time. Couple problems here: 1. Positive re-inforcment (I assume that’s what you mean with touchy-feely stuff) has been shown endlessly to be better for learning (and education general) than both negative re-inforcement and mixed re-inforcment. So creating a positive atmosphere instead of belittling kids is good psychology/ pedagogy. 2. Children going around stations, solving problems at their own pace is an idea from [Montessori](http://en.wikipedia.org/wiki/Montessori_method), which has been proven endlessly to be better for learning than frontal-only style. Children who learn at their own speed have a better chance at understanding things than those who have to learn in one class group. 3. Children parroting mulitplication tables **don’t** understand math better. They get another “Black Box”, and if they slip up, they are capable of saying that 4x4 is 64 without realizing why that might be odd. The problem is that too many strategies are perceived by the children as black box: put the problem in one end, crank the handle, out comes an answer which is submitted to the oracle (Teacher) for checking. Realizing what’s really going on, checking your own work and so on would be a better step towards understanding of math. 4. All proven pedagogic and psychological methods and tools can, like any other tools, be useless or harmful in the hands of dumb people (teachers in that case). That doesn’t invalidate the tools. 5. I’m against multiple-choice tests, because even with hard answers (very very similar so you have to know the right answer exactly) and punishment for wrong answers (to prevent guessing) it still requires less active knowledge than open-end questions. My private impression however was always that multiple-choice was introduced by teachers not to make things easier for dumb students, but to make grading easier because of the test inflation (A test each friday in my Chem. class, for example). We don’t have multiple choice tests in school here, and I think people would protest at that. I suggest for further reading the books of [John Holt](http://en.wikipedia.org/wiki/John_Holt_%28educator%29), who taught 5th graders Math, made notes and commented on them. His biggest problem was the fear of failure that the kids were installed with - they didn’t want to look dumb (or think of themselves as dumb), so they took no risk trying to understand things or speaking up. One example was - he was supposed to teach certain Math concepts, but realized that his 5th graders didn’t have a full concept of numbers and division, so he went back to teach them the basics before going on - on how division works: He used [coloured wooden rods](http://en.wikipedia.org/wiki/Cuisenaire_rods) and paper cups to give division problems. (The lenght in cm responds to the numerical value, but the children can simply go by colour). Although these children, back in the 50s, had memorized the multiplication tables, they never thought of using them. Instead they changed the big rods to the white ones (1) and distributed them among the cups until all were used up, and counted. Holt didn’t make them memorize more or shout at them; he realized that the children literally had not fully grasped what numbers really meant, and concluded to let them play with the rods as long as each child needed, until they hit the quicker method themselves. Then, by discovering it themselves, it was something that would stick much more firmly in their brain than some black box that the teacher had told them “to make their life easier” when it wasn’t obvious to the children that it did. And because they would grasp the principle the manual way, they would understand why 64 is much more than 16 and thus 4 x 4 can’t be such a big number. Another anecdote: when I was in 7th class, we got a small booklet with lots and lots of multiplication problems. We had to do 10 or so each day. Until we had done scores, the teacher showed us, first with a numerical example and then general with “a and b” the binomic formulas, and how we could solve the rest of the problems in the booklet much quicker using this method than the long route. We all saw and believed that this would make things easier. So you have to teach always “how” and “why” together. Only why without the necessary practise doesn’t get far; only how without why results in parrots without brains. ---
**Author:** ![Shot\_From\_Guns](https://avatars.discourse-cdn.com/v4/letter/s/ccd318/32.png) [@Shot\_From\_Guns](https://boards.straightdope.com/u/Shot_From_Guns)\ **Post date:** [November 12, 2010, 5:14pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/67 "2010-11-12T17:14:49Z")
> [@constanze](#): > > Positive re-inforcment (I assume that’s what you mean with touchy-feely stuff) has been shown endlessly to be better for learning (and education general) than both negative re-inforcement and mixed re-inforcment. Positive reinforcement for actually doing good work is good. Constant positive reinforcement for no reason–gee you’re just such a special unique snowflake and everyone is a winner–is detrimental. ---
**Author:** ![Kiber](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kiber/32/317_2.png) [@Kiber](https://boards.straightdope.com/u/Kiber)\ **Post date:** [November 12, 2010, 5:30pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/68 "2010-11-12T17:30:02Z")
Just to add another data point - my 8 year old 3rd grader is doing fractions right now. Not multiplying them yet - but adding, subtracting, finding common denominators, comparing, etc. In general, she’s doing stuff 1 to 2 years earlier than I learned stuff as a kid. (Not math - but last week, she also had a test which included opportunity cost. Opportunity cost? Hell - I didn’t learn THAT until college!!!) ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [November 12, 2010, 5:41pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/69 "2010-11-12T17:41:26Z")
> [@Kiber](#): > > (Not math - but last week, she also had a test which included opportunity cost. Opportunity cost? Hell - I didn’t learn THAT until college!!!) And I’m sure the delay caused you to forgo numerous potentially advantageous situations that would have improved your economic position. ---
**Author:** ![constanze](https://avatars.discourse-cdn.com/v4/letter/c/f9ae1b/32.png) [@constanze](https://boards.straightdope.com/u/constanze)\ **Post date:** [November 12, 2010, 6:01pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/70 "2010-11-12T18:01:23Z")
> [@Shot\_From\_Guns](#): > > Positive reinforcement for actually doing good work is good. Constant positive reinforcement for no reason–gee you’re just such a special unique snowflake and everyone is a winner–is detrimental. That’s not positive re-inforcment, though, constant praise for no specific achievement is a tool misused by ignorant people. ---
**Author:** ![Shot\_From\_Guns](https://avatars.discourse-cdn.com/v4/letter/s/ccd318/32.png) [@Shot\_From\_Guns](https://boards.straightdope.com/u/Shot_From_Guns)\ **Post date:** [November 12, 2010, 6:07pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/71 "2010-11-12T18:07:03Z")
> [@constanze](#): > > That’s not positive re-inforcment, though, constant praise for no specific achievement is a tool misused by ignorant people. Yes, but the post you were responding to specified “touchy-feely, everybody-have-fun, nobody-gets-left-behind sorts of affairs.” Which, to me, says “constant meaningless positive comments” versus \*true \*positive reinforcement. ---
**Author:** ![Johnny\_Angel](https://avatars.discourse-cdn.com/v4/letter/j/d26b3c/32.png) [@Johnny\_Angel](https://boards.straightdope.com/u/Johnny_Angel)\ **Post date:** [November 12, 2010, 7:10pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/72 "2010-11-12T19:10:27Z")
> [@friedo](#): > > Why _do_ they insist on deradicalizing the denominator, anyway? I had an algebra teacher once with a big stick up her ass about that, and I never saw the point. x = 3/sqrt(5) is just as good an answer as x = 3( sqrt(5)/5 ). Easier to type, too. My calculus teacher told the class that the gospel about not leaving the radical in the denominator is still around because on a slide rule you saved yourself a lot of headaches trying to divide by 2, say, rather than SQRT(2). The word hadn’t gotten around to high school teachers that we use calculators now. That was 20 years ago. Do they still do this? ---
**Author:** ![Dr.Drake](https://avatars.discourse-cdn.com/v4/letter/d/ad7895/32.png) [@Dr.Drake](https://boards.straightdope.com/u/Dr.Drake)\ **Post date:** [November 12, 2010, 7:53pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/73 "2010-11-12T19:53:13Z")
> [@Johnny\_Angel](#): > > My calculus teacher told the class that the gospel about not leaving the radical in the denominator is still around because on a slide rule you saved yourself a lot of headaches trying to divide by 2, say, rather than SQRT(2). The word hadn’t gotten around to high school teachers that we use calculators now. That was 20 years ago. Do they still do this? A lot of education is like this: the teachers do not learn the why, only the how, so they continue to teach by passing on lore, just as they were taught. Another example from the humanities is typing two spaces after a period. This comes from the days of typewriters with monospaced fonts, and has been unnecessary since word processing came along. I imagine most of us had U.S. history with a lot of material that had not been fact-checked or critically evaluated in decades. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 12, 2010, 8:47pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/74 "2010-11-12T20:47:25Z")
> [@Johnny\_Angel](#): > > My calculus teacher told the class that the gospel about not leaving the radical in the denominator is still around because on a slide rule you saved yourself a lot of headaches trying to divide by 2, say, rather than SQRT(2). The word hadn’t gotten around to high school teachers that we use calculators now. That was 20 years ago. Do they still do this? Yup, they still do it, and it sticks really hard. I certainly was instructed to do so in my youth, and in my experience, the kids in college right now have all been inculcated with it as well. (I was also taught to use two spaces at the end of a sentence, though I never have.) ---
**Author:** ![Johnny\_Angel](https://avatars.discourse-cdn.com/v4/letter/j/d26b3c/32.png) [@Johnny\_Angel](https://boards.straightdope.com/u/Johnny_Angel)\ **Post date:** [November 12, 2010, 10:20pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/75 "2010-11-12T22:20:18Z")
> [@Indistinguishable](#): > > (I was also taught to use two spaces at the end of a sentence, though I never have.) I still do, in fact. I know damned well that HTML filters generally reduce it to one space, but it’s a combination of reflex and the fact that it looks wrong to have one space while I’m typing even if I don’t even notice one space in just reading. Thanks, IBM Selectric. ---
**Author:** ![Blaster\_Master](https://avatars.discourse-cdn.com/v4/letter/b/cab0a1/32.png) [@Blaster\_Master](https://boards.straightdope.com/u/Blaster_Master)\ **Post date:** [November 12, 2010, 10:27pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/76 "2010-11-12T22:27:28Z")
> [@constanze](#): > > So you have to teach always “how” and “why” together. Only why without the necessary practise doesn’t get far; only how without why results in parrots without brains. This is a wonderful post, and I agree with it, but its also exactly why I think the methodology described in the OP is poor. If students are taught to always find a common demoninator, they’re parrotting unnecessary steps, which goes to show they don’t really understand the ins and outs of why. They’re over-simplifying the methodology, so rather than understanding a few different ways to do things, they just end up with a simpler, and more tedious, black box. I remember when I learned out to divide fractions and my teacher was very cautious about teaching us to multiple by the reciprocal, I don’t remember exactly how she explained it, but I remember it made sense and I quickly learned how to “cross multiply” on my own, because I figured out I could just flip it over in my head. And this is exactly why it seems so many people are bad at math, because they never understood why, and so it was a huge mess of memorizing formulae and algorithms and trying to figure out which ones apply where rather than just understand what’s going on. In college, I loved how math teachers would “teach” by just doing the proof, then doing a few examples; having seen the proof, the “why” always made sense, so even a complicated algorithm was manageable. I know true proofs don’t work in elementary school, because they rely on concepts they’re not ready for, but the least they can do is offer some intuitive reasoning into what’s going on, rather than just giving them the multiplication tabels and telling them to memorize them. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 12, 2010, 10:51pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/77 "2010-11-12T22:51:56Z")
I’m not sure “true proofs” do rely on concepts elementary schoolchildren aren’t ready for (but I may be more liberal in what I consider a “true proof” than what others have in mind). Certainly, I remember being exposed to many proofs as an elementary schoolchild (though often only outside of elementary school instruction); e.g., that multiplication of natural numbers is commutative because if you have an A x B box of marbles, and turn your head, it becomes a B x A box of marbles, or that multiples of 3 have digits which add up to multiples of 3 because 10 has remainder 1 when you divide by 3, or that the angles in a triangle add up to 180 degrees because of some simple diagram [the way I’d explain it now is that if you imagine a long arrow starting out aligned with one side of a triangle, and consecutively pivot it around each of the three corners, it ends up pointing in the opposite direction from how it started; that is, the direction of the stick changes from AB to CB to CA to BA]. Whether it’s worth the effort to try to instill such understanding in children of all the math we expect them to learn is another question. But I don’t think it’s beyond them to understand the math they work with; it just takes some effort to go through the requisite dialogue to build that understanding. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 12, 2010, 11:02pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/78 "2010-11-12T23:02:39Z")
Which is, I think, possibly the major difficulty; teaching mathematics to the level of genuine understanding of what is going on and not simple competency at playing an opaque game probably really requires a lot of individualized attention and two-way conversation (particularly, I would imagine, with small children), which there perhaps simply isn’t realistically time to pursue. ---
**Author:** ![Capt.Ridley\_s\_Shooting\_Party](https://avatars.discourse-cdn.com/v4/letter/c/cc9497/32.png) [@Capt.Ridley\_s\_Shooting\_Party](https://boards.straightdope.com/u/Capt.Ridley_s_Shooting_Party)\ **Post date:** [November 12, 2010, 11:54pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/79 "2010-11-12T23:54:59Z")
> [@Indistinguishable](#): > > I’m not sure “true proofs” do rely on concepts elementary schoolchildren aren’t ready for (but I may be more liberal in what I consider a “true proof” than what others have in mind). Certainly, I remember being exposed to many proofs as an elementary schoolchild (though often only outside of elementary school instruction); e.g., that multiplication of natural numbers is commutative because if you have an A x B box of marbles, and turn your head, it becomes a B x A box of marbles, or that multiples of 3 have digits which add up to multiples of 3 because 10 has remainder 1 when you divide by 3, or that the angles in a triangle add up to 180 degrees because of some simple diagram [the way I’d explain it now is that if you imagine a long arrow starting out aligned with one side of a triangle, and consecutively pivot it around each of the three corners, it ends up pointing in the opposite direction from how it started; that is, the direction of the stick changes from AB to CB to CA to BA]. [Proofs without words](http://books.google.it/books?id=Kx2cjyzTIYkC&printsec=frontcover&dq=proofs+without+words&source=bl&ots=YOdKQbd9kn&sig=YTlOSnLO1mf3cqs9YpfPX4cuts4&hl=en&ei=5NLdTNnYLdGbOr_njJgP&sa=X&oi=book_result&ct=result&resnum=7&ved=0CCwQ6AEwBg#v=onepage&q&f=false) is a good source of immediately obvious proofs like these. ---
**Author:** ![AWB](https://avatars.discourse-cdn.com/v4/letter/a/bb73d2/32.png) [@AWB](https://boards.straightdope.com/u/AWB)\ **Post date:** [November 13, 2010, 5:21pm UTC](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264/80 "2010-11-13T17:21:13Z")
> [@rachelellogram](#): > > Seventh graders are not old enough to be taking shortcuts yet. > > Their method sucks, but teaching them shortcuts to the method they’re learning is just going to cause confusion. Most of the process of math for me was learning how to do things the long hard way and then learning why/the shortcut later. I wouldn’t find this method confusing, but I was also G/T in math starting around 2nd grade. > > Being realistic, you’re just a sub at this school and have no power to change their curriculum (although complaining about it is cathartic). If you don’t like it, don’t sub for math. Or sub at another school. Y’know, I take serious offense to the “just a sub” comment. Substitute teachers are just random adults that come in and “babysit”. Many of us are certified teachers, who just aren’t hired because of budgetary reasons. (I myself taught for 4 years before moving [certified to teach secondary mathematics in two states], and have just been sidetracked by the school districts cutting back because of economic issues.) We’re investigated by _every_ school district (fingerprints, background checks, etc.). We’re expected to follow a teacher’s lesson plans that are left (95% of the time), or come up with reasonable alternatives if nothing is left. And it helps a great deal if we’re certified in the subject to which were subbing. So in finding a methodology that seems to be confusing the students more than the method I was raised and trained in, it’s my professional responsibility to assess that and see why it’s being touted as _the_ way to teach a particular topic. As to not subbing for math classes: it’s very trying to sub for subjects that aren’t your forte. Handing out worksheets, enforcing a “quiet reading time”, or monitoring 6 classes watching the same movie as many times in a single day (ask me about the fall of the Roman Empire sometime) are very taxing on one’s attention. And attention to the students is the prime reason I’m there. As my OP said, it was the “central office” that dictated this bizarre method of teaching fraction operations. That covers the entire school district: 7 high schools, 14 middle schools, and 40-some elementaries. I already sub for 3 contiguous school districts, but the one I’m talking about is the one that provides about 80% of my assignments. Statistics show that a typical public school child will be supervised by substitute teachers for one full school year of their 13 years in school. Be thankful that those substitutes are held to a standard equal to the teachers that they fill in for. [Previous page](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264.md?page=3) [Next page](https://boards.straightdope.com/t/this-is-how-theyre-teaching-fractions/560264.md?page=5)