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Math Education

Israeli 10th-Grader Discovers Elegant Geometry Theorem 173

An anonymous reader writes with a report that: Tamar Barbi, a 10th grade student living in Hod Hasharon, Israel, discovered that the theorem she was using to solve one of the problems on her geometry homework didn't actually exist. With the help of her teacher and mathematicians, she wrote up a proof for the theorem, which helps provide new and more elegant proofs for many other mathematical theorems. Posters at Hacker News have some skeptical words about the theorem's novelty, but also about the phrasing of the news report, which seems to omit some crucial words.
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Israeli 10th-Grader Discovers Elegant Geometry Theorem

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  • They probably would have marked the answer on her homework as wrong because she didn't use the Common Core government approved method of solving the problem.

    • by Sarten-X ( 1102295 ) on Sunday March 13, 2016 @10:12PM (#51691575) Homepage

      Never mind that that's not actually how Common Core guidelines work, but hey... it's the current target of hate, and we've got two minutes to spare...

      • Re: (Score:3, Interesting)

        by Anonymous Coward

        You'd believe that, but you'd be wrong, see, Common Core Math was designed as a collection of best practices for teaching mathematics. However, the fucking idiots didn't realize that, and made it into a federal standard that must be taught and tested to, so now we must test children on their ability to alorithmically apply a teaching tool towards computations. It's the most amazingly fucking stupid thing we've done against education in the last 30 years.

        • by Sarten-X ( 1102295 ) on Sunday March 13, 2016 @11:01PM (#51691745) Homepage

          Last I checked, the actual standard doesn't actually include any testing standards or teaching methods. It's really pretty loose for a standard (though my engineering bias rears its ugly head here).

          Rather, the actual standard [corestandards.org] says what concepts must be taught at what grade levels... and that's about it. There are some examples and the set of minimal facts to be understood, but it doesn't prescribe any curriculum, and it doesn't say how to evaluate students' progress toward that basic comprehension.

          It's also not a "federal standard". States are adopting it on their own, and if your state has chosen to legislate partucular testing methods to ensure compliance, that's your legislators' fault, not Common Core.

          From what I've seen (from association with a highly-regarded educator's college), Common Core is a great step forward. Previously, every state had their own standard, so a Louisiana high-school student, for (a fictitious, as I've forgotten all states' relative rankings) example, might be far behind a similar Oregon student in mathematics, but still meet their state's standards. For high-achieving students who relocated and were then told that their education wasn't good enough for their new location, it was devastating. For students transferring the other way, they'd often end up skipping grades, leaving holes in their understanding that wouldn't appear until later, when the curriculum assumes a particular concept was covered.

          Common Core has actually done the impossible: It is being adopted as a One True Standard to gauge a student's understanding, based on a set of concepts, rather than a district's particular placement test. Well-written tests against Common Core can also indicate whether a student has understood the concepts adequately for their grade level, based on real-world needs, rather than the opinions of a teacher who hasn't seen business needs in the past decade.

          • by WarJolt ( 990309 )

            It matters how it's applied.

            Let's suppose the standard will shape the qualification process for instructors, leading to instructors who spend far too much time in their own careers learning a standard instead of learning real math and how to teach it. This may lead to a circumstance where the instructor can't understand a method that the student used to solve a math problem. I fear an overall decline in teaching quality. I hope this isn't happening, but when it comes to children's education you can't blame

            • Let's suppose the standard will shape the qualification process for instructors, leading to instructors who spend far too much time in their own careers learning a standard instead of learning real math and how to teach it

              Based on what the post you replied to said, I don't see how this could possibly be the case.

              The "standard" only dictates what must be taught at a certain grade level, and not how it should be taught. From the teacher's perspective it's exactly the same as before since the school or state typically sets the curriculum requirements anyway.
              =Smidge=

          • by houghi ( 78078 ) on Monday March 14, 2016 @09:21AM (#51692887)

            States are adopting it on their own

            Is this the same adoption as the states that can decide on the drinking age, but if it is below 21, they loose a lot of money on roads?

            This does not mean that I am for or against states or the governement deciding what the law is. It is just that is seems like childish behaviour and pointing fingers I would expect from a 5 year old.

          • by CimmerianX ( 2478270 ) on Monday March 14, 2016 @09:58AM (#51693095)

            I'll tell you this... my 5th grader asked for help on his homework consisting of dividing 2 and 3 digit numbers.

            So we worked through all the problems together.

            He got a 0 on the homework even though all the answers were correct.

            When I went in to see the teacher about it, she said that we used long division and not the new math method of solving the problem. Thus he got a 0 even though all the answers were correct and my kid now knew how to do the work after I showed him the method I was taught.

            Stupid as far as I am concerned.

            • Re: (Score:2, Insightful)

              by Anonymous Coward

              What is stupid is you. You thought the homework was about getting the right answer. The homework ( as is ALL homework, until maybe Grad School ) is about validating that the student is understanding the concepts presented.
              What you did, using long division, got him the answer, but it did not teach him how to do proper grouping or estimation skills. Your same attitude should have told him to use a calculator... because he would get the answer right.
              These are building blocks for the future. Of course these met

            • And your point would be? There have always been crappy teachers.

            • Let me try and give and example to show why you are wrong: Imagine your kid where learning multiplication methods and had a homework with the simple multiplication 345*10, and he answered all the assignment by doing repetitive sums (345+345+345+345+345+345+345+345+345+345=3450). In this case wouldn't you agree with the teacher to fail the student's assignment? After all, the objective of the assignment is not to give the result to the teacher, the teacher already knows the answer, instead the objective is t

          • Common Core has actually done the impossible: It is being adopted as a One True Standard to gauge a student's understanding, based on a set of concepts, rather than a district's particular placement test...

            You seem to think that's a good thing.

          • Despite Gates’ and others’ assurances that the Common Core reform was “state-led,” the former director of the Race to the Top competitive grant program, and outgoing U.S. Education Secretary Arne Duncan’s chief of staff, recently admitted the federal government had “forced” full support for adoption of the Common Core standards from each state by requiring its governor, chief state school officer, and head of the state board of education to sign off on the grant app

        • by dywolf ( 2673597 )

          no.
          no test requirements
          no actual curriculum either.

          CC is merely a set of standard goals that students should be able to achieve.
          actually getting to achieve that is something entirely different, and up to the individual states/schools and the publishers they should to purchase.

      • No, common core is best saved for the full ten minutes hate.
    • by guruevi ( 827432 )

      In an American high school you don't have to prove anything, you just have to tick the right boxes.

      • In an American high school you don't have to prove anything

        In Russian high school, proof anythings you.

        OK, it's gonna need a bit more work, but it's a start, it's a start. Probably need to work in a Putin reference somewhere.

    • by rs1n ( 1867908 ) on Sunday March 13, 2016 @10:44PM (#51691707)
      Do you realize that the common core is nothing but a set of standards as far as what students should be able to achieve at various levels? It does not dictate how teachers are supposed to teach the standards. That is left completely up to the teachers. The problem is that private companies are taking advantage of the fact that there currently is a lack of teaching materials that address the common core. Then to compound the problem are teachers who are often not specialists in their own area. I have taught an entire class of future math teachers, and most of them chose that profession because 1) they will always be in demand and 2) because they like to work with kids -- neither of which necessarily result in strong math teachers. (In fact, most of them would probably never become great math teachers, to be perfectly honest.) Anyway, your beef with the common core lies with the companies trying to cash in on the teaching materials void.
      • Do you realize that the common core is nothing but a set of standards as far as what students should be able to achieve at various levels?

        I do. Question is whether all (or even most) teachers do, as well.

      • by tsqr ( 808554 )

        Do you realize that the common core is nothing but a set of standards as far as what students should be able to achieve at various levels? It does not dictate how teachers are supposed to teach the standards.

        Yes, I do. I also realize that when many implementations of a standard are clearly defective, there may actually be something wrong with the standard.

        • How many implementations are there out there? I've only read of one big corp trying to impose its version of the standard. When we have several, and they're all bad, then we'll talk.

          • by tsqr ( 808554 )

            How many implementations are there out there? I've only read of one big corp trying to impose its version of the standard. When we have several, and they're all bad, then we'll talk.

            A quick search turned up this [apexlearning.com],this [scholastic.com], this [inspiration.com], this [coreknowledge.org], and this [teachingchannel.org]. But wait, those are more or less commercial offerings. It seems that individual states [uen.org], districts [lausd.net], and schools [k12.wi.us] are rolling their own implementations as well.

            • Thank you; I hadn't heard of them. Are all, or most, of these implementations bad? If so, is there something else (like NCLB) going on?

              • by tsqr ( 808554 )

                Thank you; I hadn't heard of them. Are all, or most, of these implementations bad? If so, is there something else (like NCLB) going on?

                I honestly can't say whether they're all bad. I can say I've seen examples of implementations that I personally consider pretty tragic; particularly in math, where methods presented for solving simple algebraic equations fail when coefficients aren't integers, where addition and subtraction are called 'put together' and 'take apart' and where terms like 'zero pairs' (aka, 'additive inverses' - two numbers whose sum is zero) seem to be preferred over traditional concepts like associative, distributive, and c

    • by Roger W Moore ( 538166 ) on Sunday March 13, 2016 @11:35PM (#51691815) Journal
      Actually if the theorem is exactly as the article states then it should have been marked wrong because it is wrong:

      According to the new "Three Radii Theorem," if three or more lines extend from a single point to the edge of a circle, then the point is the center of the circle and the straight lines are the radii.

      I think what they meant to say was three lines of equal length in which case this just defines three points on a circle which is of course enough to uniquely define it. It also only works in two dimensions otherwise the point does not have to be the centre. This is the sort of geometric proof problem we used to get at secondary school. Have standards really fallen so incredibly far that this is noteworthy now let alone publishable? If so me and my old schoolmates can probably rustle up quite a few more "theorems" for publication in the journal of bleeding obvious mathematics.

      • Segment. The word you are looking for is segment.
      • You're right. The lines have to be equal. Otherwise, next time I need to find the center of a circle all I'd have to do is pick any point within the circle.

        Draw a circle. Pick a point anywhere within the circle. Now draw three lines from that point to the edge of the circle. According to how it's stated in the article, you've chosen the exact center of the circle. Fat chance of that! In reality, though, your chances of finding the center that way are too low to even talk about.
    • For all the wrong reasons, you're right. Stop self medicating.
    • Actually, having found a similar therom when I was in HS geometry, I can tell you that in the US they would of spent the time to try an publish it only to have academic journal editors figure out it wasn't novel at all and make allegations of plagiarism...
  • Moral (Score:5, Insightful)

    by Deadstick ( 535032 ) on Sunday March 13, 2016 @10:11PM (#51691567)

    Don't try to learn about math from news media.

  • by Sarten-X ( 1102295 ) on Sunday March 13, 2016 @10:16PM (#51691595) Homepage

    Even if the proof isn't novel, or if there's some glaring error, Israeli secondary-school students now have a champion for a while, who found something interesting. That student in particular has a vested interest in a particular area of her field, and hopefully that will grow into a later expertise, and ultimately significant contributions to human knowledge.

    Faults and all, this is how mankind progresses... Stumbling forward one mistake at a time.

    • by superwiz ( 655733 ) on Sunday March 13, 2016 @10:23PM (#51691625) Journal
      As the MIT discussion (linked in the slashdot summary) shows, it's actually in the Elements. But the theorem was not in the textbook used by the school and the student did stumble on it on her own. Good for her.
    • The Ycombinator is apparently the link with all the info rather than the "news" source. I mistakenly said it was "MIT". Her proof is original. The theorem is not.
    • by Anonymous Coward

      Israeli secondary-school students now have a champion for a while,

      No. If anything, their mathematical secondary-school level mathematical champions are the IMO participants and so on who worked hard enough to hone their math skills to the point where they can prove this while blacked out. The theorem she proved is the type of thing I used give college freshmen on their first or second in-class 5-minute pop quiz back when I used to teach a proof writing class. A class that is taught at a level every IMOer or AMC10/AMC12 regular (or the Israeli equivalent) is far beyond. Sh

    • by aliquis ( 678370 )

      In the actual article it says she want to do theater ..

      Also if you know the radius can't you just put down a compass (weird name in English) along the edge and draw a part of a circle inside the circle and then put it down somewhere else and repeat that and see where they meet?

      Isn't the third line only needed to not put the center along the edge of the circle rather than the middle?

      Or if you have a straight angle with 45 degrees marked onto it hold that towards the circle and mark out the 45 degrees and the

    • Besides, give her a break -- after all, math *is* hard.

    • That student in particular has a vested interest in a particular area of her field, and hopefully that will grow into a later expertise, and ultimately significant contributions to human knowledge.

      Not so much. If you'd read the article, you'd have seen this:

      Barbi remains unexcited. She is involved with theater arts, studies acting, plays the piano and the guitar, sings, and dances.

      "I don't think math will become my profession. I hope to work in theater arts," Barbi says.

    • That student in particular has a vested interest in a particular area of her field, and hopefully that will grow into a later expertise, and ultimately significant contributions to human knowledge.

      Well actually, from TFA:

      Barbi remains unexcited. She is involved with theater arts, studies acting, plays the piano and the guitar, sings, and dances.

      "I don't think math will become my profession. I hope to work in theater arts," Barbi says.

  • The linked news article misses the key features of the line segments: "of equal length". The "theorem" as mentioned in the news article is patently false.
    • The linked news article misses the key features of the line segments: "of equal length". The "theorem" as mentioned in the news article is patently false.

      Not only that, the three line-segments must intersect the edge of the circle at three distinct points. That may seem like nitpicking, but precise language is important in mathematics.

    • by AgNO3 ( 878843 )
      HEEEEYYYYY, This is slashdot, you can't patent "false" here. Not only are we anti-patent, Our whole site is previous art on "false."
  • Feelgood story about how smart israeli kids are, same as egyptian stories bubbling up from time to time...

    This is what she "invented".
    http://www.mathopenref.com/const3pointcircle.html

  • Non-invention (Score:4, Insightful)

    by Sigma 7 ( 266129 ) on Sunday March 13, 2016 @10:53PM (#51691733)

    Tamar Barbi, a 10th grade student living in Hod Hasharon, Israel, discovered that the theorem she was using to solve one of the problems on her geometry homework didn't actually exist.

    Okay, the article says:

    According to the new "Three Radii Theorem," if three or more lines extend from a single point to the edge of a circle, then the point is the center of the circle and the straight lines are the radii.

    That's a definition, not a theorem. Even if you're generous enough to fix the wording, it's been proven centuries ago [clarku.edu]. If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, then the point taken is the center of the circle.

    Not to mention that the article doesn't actually give the proof, and is simply a "yay, new invention by youngster" fluff.

    Posters at Hacker News have some skeptical words about the theorem's novelty

    And if you need to include that in the blurb, it's perhaps a good reason the article itself is garbage, especially when the topmost comment shows exactly why it's wrong.

  • How is this novel, or a theorem or anything for that matter? What is the definition of a circle - a line, all points on which lay an equal distance from a single point, being the circle's center. All lines connecting the center are of same length, and are radius(es) of that same circle. This is a definition, not a theorem. 3 lines with a common ending, define 3 points in space. Every circle can be defined either by a center coordinates and radius, or by coordinates of 3 points, laying on the ark line. This

    • by I4ko ( 695382 )
      Agreed
    • For all points on a line to be equidistant from the center of a circle, an infinite number of line segments of equal length must extend from the circle's center to the circle's edge. If the circle *is* a circle, then any three line segments of equal length extending from one point to three distinct points on the circle's edge are extended from the circle's center.

  • It's precisely why three points unambiguously define a unique circle that passes through each of them. Obviously the center of the circle must be equidistant form all of them.
  • when I was in the 9th grade. Part of learning math. Good for Tamar that she's likes math enough to play with it, but must've been a slow news day in Israel.
  • This algorithm was described by Euclid (the Greek) in his book "Euclids Elements" about 2300 years ago. She used it to do homework, was asked to make a proof by her teacher, which she did (with help). And why everyone is getting excited is somehow odd, especially the one from MIT who should know better (and should know the algorithm already or at least know of it. It makes for good internet theater though.

  • by fph il quozientatore ( 971015 ) on Monday March 14, 2016 @03:15AM (#51692175) Homepage
    I don't know what she came up with, but a possible proof is a one-liner: draw another circle with center in the given point and radius equal to the length of the three given line segments. This circle intersects the existing one in three points (the endpoints of the segments), hence they must coincide (because of https://proofwiki.org/wiki/Two... [proofwiki.org]).
  • by urdak ( 457938 ) on Monday March 14, 2016 @04:45AM (#51692309)

    It's sad how stupid reporters report wrong "news", the error gets repeated all over the Internet, and finally lands in Slashdot whose editor didn't know the original news report was wrong.

    The 16-year-old girl did not invent a previously-unknown theorem. What she did is to re-invent a theorem which Euclid already listed and proved over two thousand years ago (http://aleph0.clarku.edu/~djoyce/elements/bookIII/propIII9.html). But Euclid listed hundreds of theorems, most have simple and basic proofs, and most of them are never specifically taught. In this case, the girl was not taught this theorem, but she thought that she could have used such a theorem in her homework, so she went about proving it (with help from her teacher, who was also not familiar with Euclid's mention of this theorem).

    The girl's proof is different Euclid's, but still very simple and elementary, and is in no way a profound addition to Mathematics. But this girl is still admirable, in that she had the creativity and resourcefulness to imagine a "new" (to her) theorem, and to go around proving here - rather than sticking to the "cheat sheet" of theorems she was taught in class. This girl definitely deserves an A in her math class, but not worldwide mention on news classes.

    Of course, it's not her fault, but rather that of the reporters who blew this story out of proportions, and reported this stuff as a new theorem, a breakthrough, or other irrelevant adjectives - without checking the validity of this "story" with any Mathematician worth his salt. This "story" should never have made headlines, and definitely not slashdot. But the girl still deserves praise, and of course an A :-)

  • "if three or more lines extend from a single point to the edge of a circle, then the point is the center of the circle and the straight lines are the radii". Could someone reformulate this in english to give it some meaning ?
  • Can anyone give the actual theorem as formulated by Tamar?
    Because what she found sounds obvious, the proof is well within reach of a relatively gifted 10th-grader helped by a teacher and isn't new. In itself, nothing impressive.
    The interesting part would be if she found some particularly clever way of solving the problem of if her proof shows some particularly deep understanding of maths.

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