I don't think the student's writing is available anywhere, but here is what Paul Lockhart said about it in his "A Mathematician’s Lament" [1]:
> To be fair, I did paraphrase the proof considerably. The original was quite a bit more convoluted, and contained a lot of unnecessary verbiage (as well as spelling and grammatical errors). But I think I got the feeling of it across. And these defects were all to the good; they gave me something to do as a teacher. I was able to point out several stylistic and logical problems, and the student was then able to improve the argument. For instance, I wasn’t completely happy with the bit about both diagonals being diameters— I didn’t think that was entirely obvious— but that only meant there was more to think about and more understanding to be gained from the situation. And in fact the student was able to fill in this gap quite nicely: “Since the triangle got rotated halfway around the circle, the tip must end up exactly opposite from where it started. That’s why the diagonal of the box is a diameter.” So a great project and a beautiful piece of mathematics. I’m not sure who was more proud, the student or myself. This is exactly the kind of experience I want my students to have.
People seem to forget a lot that child math prodigies are being trained by parents to be math geniuses instead of enjoying being a kid. I had a 12 year old in my freshman CS class and he was the most miserable unhappy kid I've ever seen in my entire life.
Edit: Pretty good points in the replies. Probably a lot of latent jealousy on my side in this. I should have thought on this more before commenting.
What's the take away you want from this comment? Advancement of humanity, feeling bad for missed childhood fulfillments, some mix? Did the child actually meaningfully contribute anything to your class? Were they a real person? Right now there's a lot we've "forgotten" about child geniuses that we should all know from our interactions from them ?
As someone who was enrolled in university starting from 9th class - i agree that this is not what children should be made to do. Adults were projecting weird ideas of success onto us.
When I was tutoring HS kids (as a grad student myself), we also had afternoon practices for the kids that wanted to compete in the math olympiads.
Truth be told, some of the kids loved it - but those were the kids that were really passionate about math, for the sake of math.
But you also had a bunch of kids that seemingly hated every second of it. So why did they do it?
- Some had pushy parents that wanted them to excel in extracurricular activities. Think typical tiger parenting...
- Some had big ambitions about certain schools, and felt that they had to compete on a national level in something, in order to stand out in the selection/application process.
Granted, this was over 15 years ago. A couple of years ago I checked up on some of the kids (LinkedIn), and the most passionate kids were either now math/physics Ph.Ds, or Ph.D track.
A counterpoint: I was accelerated by three year levels, and it was the best thing that could have happened to me. Otherwise I would have been intensely bored in high school for six years straight, with all the problems that would inevitably have brought.
Of course, this depends so much on the person in question. Acceleration can be great for some people and awful for others. But it’s not about ‘enjoying being a kid’ — that’s something I hear a lot, but I enjoyed myself a lot more in the higher grades where I was actually learning material I found interesting.
Can confirm: My parents made me go through the grades one at a time, despite the principal's recommendation that I start in 1st grade, move to 2nd mid-semester, do 3rd grade in the spring, and do two a year after that. It was intensely boring. I can see why they thought a bunch of teens and young adults wouldn't want to be friends with a much younger child, but I've always wondered if they would have been more willing to overlook the age difference than the other young children were willing to overlook the fact that the only thing we had in common was our age.
Ironically, I found it decent preparation for life, though, which seems to be occasional bursts of furious activity punctuated by long stretches of repetition, which could easily become boring. I ended up becoming very good at taking responsibility for making my own life interesting.
> I ended up becoming very good at taking responsibility for making my own life interesting.
Oh yes… I also spent considerable time teaching myself interesting topics. It’s a good skill to have. But I was still bored a lot of the time, and being able to do the interesting stuff in school too was a much better experience for me.
Ultimately I've had to get to a point where I realize I'm grateful for the life I live now, and accelerating through school would have changed a lot of it, such that I probably wouldn't have met some of the people most important to me, so I'm grateful for it in the long run. But I don't know that I'll ever stop wondering, and it's not a comparative gratitude, because I'm sure my life could have become beautiful like this in other ways. It's more a practice of radical acceptance with gratitude.
I will say one of the things that always made me laugh was that at each milestone, like finishing elementary school and starting middle school, my parents would try to tell me now it would be different, because I was at a higher level. I remember pointing out to them all the same doofuses from the previous years were moving up right along with me, and it wasn't just that I was ahead of them, but that I was faster, so this didn't seem to help. I never did understand why they thought that would work!
It is unfortunate that this is not a core principle of education.
The best school system I ever attended rigorously divided classes between social and academic --- for social classes (gym, health, social studies, homeroom) one attended at one's grade level, while for academic classes, (English, other languages, science, math) one worked at one's grade level (but with a cap of 4 grade levels through 8th grade, so a 4th grader couldn't take high school classes) --- after 8th grade this cap was removed, and students could take any classes which they could qualify for academically. To facilitate this, some teachers were also accredited as faculty at a nearby college, and if necessary arrangements were made either for college professors to come to the school and teach, or students travelled to the college.
Most students graduated with at least a little college credit, and many received a college degree along with their high school diploma --- until the Mississippi State Supreme Court decided that it was illegal because it conferred an unseemly advantage on some students with no corresponding compensation for students who were unable to avail themselves of it academically.
> The best school system I ever attended rigorously divided classes between social and academic
This is such a great idea! And it sounds like it works well in practise, too. I’ve often pondered the idea of separating classes by ability rather than age, and this is a really practical method of achieving it.
It worked quite well --- helped that it was a small, cohesive school system with all grades in the same building/complex.
A similar idea is to group children by reading level which makes classes function more smoothly since students finish reading assignments in similar periods of time.
Ironically I think a version of this had to happen in the one-room schoolhouses of yore. If you're the only teacher, you can't afford to split everyone by age. You kind of have to roll people into whatever lessons they can handle and do your best.
It was "innovative," in a way I think we're coming to regret, to suggest that all 8-year-olds (for example) must be at the same level across every subject, and none of them should ever be taught alongside a group of mostly 6- or 10-year-olds.
The public school system near Columbus Air Force Base and I attended for 3rd and 4th grades (taking 7th and 8th grade English and science courses) in the '74 and '75 school years.
For reading, worked through a box of SRA booklets in a couple of weeks time and then went to the library for the rest of the year --- mostly I was bored and coasted until my classes caught up, and even then, I was able to get decent grades w/ minimal effort.
Of course this all came to a head when I was a junior and the school system couldn't find a teacher for Calculus for myself and the couple of other students who wanted it which pretty much killed my college prospects, so aced the ASVAB, DLPT, and EDPT and enlisted.
It wasn't too far off from that --- my understanding of the court case was that since some students got free college out of it, the tort demanded that free college tuition be afforded to those students who only graduated with a high school diploma.
When I was in kindergarten, I was one of a handful of kids who were invited to come back to the school after lunch (this was in the days of half-day kindergarten being the norm) to learn math with the first graders. I got kicked out for being disruptive.
My mom, annoyed at not having her afternoons free, gave me math learning materials to keep me busy. Since she wasn‘t a teacher, she didn’t really know what she was doing and I ended up entering first grade reading at a fifth-grade level and doing math at a third-grade level.
I really should have been skipped ahead at least a year, if not two. The first-grade teacher had me teach the gifted kids in the back of the room while she taught the rest of the class, the second-grade teacher didn’t really know what to do with me and I was bored out of my skull as a result (first and second grade swapped between the two teachers at lunch time with the first-grade teacher focusing on reading and writing, the second-grade teacher focusing on math and science).
One of the last times my parents moved, they sent me this big box they had of papers from my childhood.
One of the ones that fascinated me was dated from when I was in 2nd grade. It was a big printout with rows and columns. Some of them contained numbers, but most of them just said "PHS."
It was an oddly folded poster-size thing, and once I finally got it open, I found out the numbers were grade levels, and "PHS" stood for "Post High School."
I had been 8 when I sat for this test. My parents had staunchly refused to accelerate my education.
And they continued to refuse for ten. more. years.
I might recommend that parents not send their children proof if they went out of their way to stymie them.
I was handed math textbooks a few years above my grade when I was in 5th grade, and it kept me interested longer. I think I overall preferred that, to being moved. Socially I was definitely not ready for being moved into higher grades etc. In highschool my maths teachers mostly let me do my own thing, other than asking if I'd be interested in helping others on occasion, and frankly, that was probably more valuable to me than advancing faster in the subjects. Not trying to contradict you, btw., just another option.
This is completely valid. Like I said, acceleration isn’t right for everyone! I also started out by reading more advanced maths textbooks, and it’s a great thing to do for students where acceleration isn’t a good option.
The post may not make it clear that this was a student of Paul Lockhart's whose main aim is not make maths fun. This is a good teacher, not parental training.
It was also similar to how I taught my home educated kids - do fun stuff. It worked very well. Both love maths (and hugely enjoyed studying in general). My older daughter is doing a degree apprenticeship in electronic and electrical engineering[1] and the younger one got a 9 in IGCSE maths[2] and plans to do A level maths and further maths[3]. The younger one hated maths while she was in school, which is common, and which is the problem Lockhart was trying to solve.
This is not about pushing kids or making them prodigies, it is about making them enjoy what they do, which leads to higher achievement and and happiness.
That said, not all prodigies are miserable. Ganesh Sittampalam who broke the UK record for youngest graduate taught himself for fun until he got into university!
[1] UK term for degree paid for by employer done while working (and getting paid). Good for parental wallet!
> No parent can “train” a kid to be a math genius.
While that's true, there's lots of parents who don't believe it, or don't care if the end result isn't actual mathematical genius as long as it's some sort of distinction. (I'm in the happy position where my parents encouraged me to pursue the paths that I enjoyed, which wound up with some acceleration, but my getting in the local newspaper meant that we were besieged by the kind of parents who do just want to push their kids towards some pre-determined "success.")
While I might quibble with the parenting technique because I believe an interest in useful topics can be cultivated, I'm not going to knock those parents for developing the ability to do hard work. The rest of the world is better off with these kids becoming competent professionals due to parental control than it would be if they became wastrels due to parental neglect or encouragement to pursue useless hobbies. In the US at least, the much larger problem is the latter.
> While I might quibble with the parenting technique because I believe an interest in useful topics can be cultivated, I'm not going to knock those parents for developing the ability to do hard work. The rest of the world is better off with these kids becoming competent professionals due to parental control than it would be if they became wastrels due to parental neglect or encouragement to pursue useless hobbies. In the US at least, the much larger problem is the latter.
I think that's a very optimistic view of the outcome of this sort of thing, although it could be that I just didn't describe the parental behavior well. This wasn't parenting telling their kids "hard work is worth it!", but parents who decided on the specific honors that their child would achieve, and would permit no deviation from the path, as a result of which most of these kids either burned out before achieving their goal, or rebelled, usually in self-destructive ways, as soon as they were given a little freedom from parental control.
Different parenting styles work for different people, but for me (as a child in the US in the '80s) I think that my parents did the best possible thing, which was to instill in me lessons about the value and importance of hard work, but to make it clear that, in the end, the decision about whether I would pursue those values was up to me. It probably helped that, despite the many other advantages I enjoyed, my family was insufficiently well off to represent a guaranteed financial fallback for me, so that I knew that, one way or the other, I'd have to make my way in the world—which made it particularly appealing to me to be able to find a way to earn my keep by working hard for something I loved.
My son who is just finishingh fourth grade has been demanding harder math problems than he gets in school. He learned how to calculate square roots by hand from a YouTube video and enjoys doing these problems on his own, checking his answers against a calculator. When I pick him up from school, when we’re walking home, he demands that I teach him algebra. I’ve taught him how to solve basic linear equations (ax + b = cx + d) and also I have him take a number and try to find all the sums and differences of pair-wise factors (I haven’t told him why but people who remember high school algebra will recognize this as a skill essential to factoring quadratics) We’re going to learn Trachtenberg Speed Math over the summer. He also enjoys coding and I remember being astonished watching him—without guidance—figure out how to rewrite a program he’d done in Scratch using Python (although right now JavaScript has become his favorite language).
I’ve not pushed him into any of this (I really would prefer that he not pursue programming as a career as I fear that we’re heading to a dark ages of software development, but he can do what he likes).
His twin sister, meanwhile, has been demanding that I teach her social studies so I’ve been doing my best to give her bits and pieces of history and government from my memory. She doesn’t demand the math, but I’ve noticed that she is paying attention during the algebra on the walk lessons as she’ll occasionally jump in with an answer herself.
My purely selfish side wouldn’t object to them skipping a grade as my ex-wife seems committed to them doing K–8 in private schools (even though our local public schools are quite good except at the junior high level, but 6–8 is hell at any school) and skipping a grade would save me a year’s tuition expenses.
I recommend you find a copy of Lenchner's book Creative Problem Solving in School Mathematics (2nd ed.). Lenchner ran math olympiads for late elementary school / middle school for decades, and does a good job of covering a wide variety of useful strategies for approaching nontrivial word problems (i.e. real math problems, not just arithmetic drills). But you don't have to be interested in contests per se to get a lot from the book. In my opinion that's going to be more pedagogically valuable and more fun than Trachtenberg. Here's an Internet Archive scan https://archive.org/details/creativeproblems0000lenc
Your kid would also probably really enjoy The Number Devil, a breezy novel about a kid who gets trapped in his dreams with a devil who won't stop telling him about elementary number theory, using various dream-world props.
If you want more problems after that, there are a lot of great puzzles in Kordemsky's The Moscow Puzzleshttps://archive.org/details/moscowpuzzles3590000kord_m9a0, or you could take a look at Fomin, Genkin, & Itenberg's Mathematical Circles: Russian Experience which your kid could probably just about handle.
There are also some great puzzles and games explained at the website of the Julia Robinson Math Festival, https://jrmf.org which can be played with fairly basic easily available materials.
Proving that a parallelogram with equal diagonals is a rectangle is an exercise in itself; I’d prove it through Thales’ theorem, myself...
Also, Lockhart's Lament is from 2002, so this post probably needs a (2002). It is very unlikely that the proof was new. It was certainly new to the seventh-grader, and a great result at that.
> Proving that a parallelogram with equal diagonals is a rectangle is an exercise in itself; I’d prove it through Thales’ theorem, myself..
IIRC you prove it via angles.
a) The sum of angles of any triangle is pi.
b) The sum of angles of any quadrilateral is 2*pi.
c) Since you rotate the triangle, opposite angles end up adding together. Because of the above, a bit of reasoning around symmetry shows that when diagonals are equal they can only add to pi/2 and that the other ones can only be pi/2. Any other angle leads to a contradiction.
I mentioned angle values but (again IIRC) this can all be proven with compass and ruler.
As the article mentions, in this particular case it's simpler: You have a point on the triangle that is also on the circle. Now you rotate the triangle 180 degrees, with the point following the circle. The point inherently must end up exactly opposite the original point, and so the diagonal formed between the old and new point must be the diameter of the circle.
This proves that the parallelogram diagonals, both being diameters of the same circle, are equal, akho was pointing out that there's a leap from that to "the parallelogram is a rectangle", which can be proven with Thales.
But then, the article is about laying out a proof of Thales so if it's using a lemma that leveraged Thales itself that's circular logic right there and so we're in trouble.
One other way to prove Thales is by computing angles, but then you've got yourself a proven Thales so the point of the article is moot.
I didn't read that very carefully, but proving that a parallelogram with equal diagonals is a rectangle is obvious enough via triangle congruency that I suspect they were just taught it without the proof. It's not a particularly interesting detail.
Um… small correction. Sum of angles in triangle is 180 not pi (right-angle triangle is 90 + 45 + 45). Similarly for quadrilateral it’s 360 not 180 (a canonical being rectangles which is 4 90 degree angles).
It is, but Thales is what this article attempts to prove (a).
By the point you've proven Thales with isosceles (b), proving that a parallelogram with equal diagonals is a rectangle thanks to Thales(b) in order to go forward along the Thales(a) proof is circular logic.
The net change in direction around a triangle, quadrilateral, or any simple closed curve is a full rotation, i.e., 2π.
The sum of the interior angle and the signed change in direction around a vertex of a polygon is π (consider the case of a "vertex" with interior angle π).
So for a polygon with interior angles θ₁, θ₂, …, θₙ,
Proving that a parallelogram ABCD with equal diagonals is a rectangle is an exercise, but not too hard. You can make two different triangles ABC and ABD with the same parallelogram base, where one has second side BC and third side (a diagonal) AC, and the other has second side AD and third side (the other diagonal) BD. These triangles are SSS congruent, therefore must have congruent angles (Euclid I.8) ABC and BAD. This gives you a parallelogram with 4 congruent angles.
Nice.
You can alternatively prove the negative: suppose it's a parallelogram with non-right angles, assume angle at A is "sharper". Thus, you can create two right angle triangles: AB'C with the base extended beyond B (= B') where AC is one diagonal and A'BD where A' is "below" D and BD is the other diagonal. As AB'C is larger than A'BD (because A' is within AB and B' is outside while the height is the same) AC can't be the same length as BD.
Thus, no parallelogram with non-right angles exists that has equal length diagonals.
+1, would also enjoy a book recommendation. I can recommend a fun app - Euclidea :)
My favorite book on the topic, unfortunately only in Slovak, is: "Matematici, ja a ty" (mathematicians, you and me). I don't think there is a translation. I also had a university professor who had a similar style - not dry proofs, but "Here is how Archimedes thought about proofs" - which is related to the comment above: if A is not smaller than B and B is not smaller than A, they need to be equal.
Breaking proofs of trickier theorems up into a few parts and hiding the details of each part behind another theorem whose proof is self-contained and easy to follow (even better if it's "obviously true") is the basic idea of mathematics. Trying to make the top-level proof very clearly demonstrate, without a lot of bookkeeping or head scratching, that the theorem must be true is a wonderful goal.
In my opinion it's a lot more obvious that a parallelogram with 2 equal diagonals must be a rectangle than that an inscribed angle intersecting a circle at ends of a diameter must be a right angle.
Whether this proof is complete depends on what was known before. Was the lemma introduced by a teacher? was it not actually proven, and the student missed the hole in their proof? these are different teachable moments.
I think the two facts, by themselves, are roughly equivalent in difficulty. Moving from Thales to equal diagonals parallelogram requires drawing an extra circle; moving from parallelogram facts to Thales requires either the construction in the article, or continuing the missing radius to get the same rectangle.
I don't understand why you are invested in nitpicking the work of an anonymous 7th grader from decades ago.
Clearly, filling in all of the details of any formal proof in Greek style is going to require a careful knowledge of the axioms in use and some list of previously proven theorems which are allowed to be used without re-proof. Depending on which theorems are at hand already, one or another proof might be shorter or longer or more or less obvious. When trying to construct a whole mathematical treatise, the best order for the theorems so that each proof depends only on previously proven theorems is a tricky choice involving some trade-offs between pedagogical goals and concision of individual proofs.
I don't think the note "Since the triangle got rotated halfway around the circle, the tip must end up exactly opposite from where it started. That's why the diagonal of the box is a diameter." is needed, nor do you need to show the diagonals of the parallelogram are equally long.
Rather, it suffices to simply state, the only parellelogram that is inscribed in a circle is a rectangle.
For the less rigorous, that statement is obviously true. For complete rigor, it is sufficient to argue that the center of the circle, combined with the vertices of the parallelogram forms an isoceles triangle. So the centre of the circle must lie on the bisector all edges of a parallelogram. But on a non-rectangle parellelogram the bisectors of opposite edges never intersect.
If your informal proof is to draw a picture and say "it's obviously true by lookikg at it", then you don't need a parallelogram at all. Just draw the diametric triangle and look at it.
His student's "proof" is an illustration, not a proof.
There's no way to know if its circular or simply unfounded, since it is purely an appeal to intuition. That's only a part of mathematics. The correct thing to do, mathematically, is to validate the intuition by formatting it as a proper proof, based on non-circular axioms and theorems. In the paper itself he admits that his student's work was incoherent and needed him to rewrite it.
He's right that 2 column geometry proofs are ugly, and could be presented better.
This has been known for centuries.
For children and for starting out, the pictures are great. But for mathematics, pictures are extremely limiting.
2D and 3D are notgod models for N-Dimemsions. ("Spiky balls" , for example).
Mathematics is far, far more powerful than human eyes. The amazing thing about geometry is that the whole thing works without pictures! A blind person can be a great geometer, because geometry is axiomatizable. Meanwhile, Euclid's Elements, while an incredible achievement in its day, is not well-founded, relying on unstated axioms.
I think it's fairer to say that his point may not have been what you expected it to be. As a teacher myself, trust me, if a student comes to you and says "I came up with a proof!" and you say "no, you see, what you have is an intuitive explanation that can possibly be turned into a proof," then all that will happen is that that student will not be interested any more in exploring, or at least will not be interested any more in sharing their explorations with you. At that point, you have both lost.
Lockhart is well aware of the standard of mathematical proof, and knows that, all else aside, this theorem is not in want of proof. His focus is on the fact that, if we want mathematics to remain a live profession, then we must improve our pedagogy, and help to train students who enjoy and want to pursue mathematics—even if it means occasionally accepting less than maximally rigorous mathematics from a seventh grader.
While I respect your point about rigour I should say one has to be careful not least because the way the subject is presented in textbooks are most often backwards -- axioms are really an end not the starting point.
Here's another view: Euclidean geometry is euclidean because the underlying transformation group (the group of those transformations which preserve what we want to preserve -- in this case the metric) is the euclidean group (the semi-direct product of the orthogonal group and translations). This is the symmetry that encodes our intuition -- the same intuition the kid is using to prove the above result. If we were to change the underlying space to the real projective space instead of R^2, and instead of choosing to preserve the metric we choose incidence and cross-ratio, we'd get a different group (GL(3,R)) and different geometry, viz. projective geometry.
This is an ancient dialectic that runs within mathematics -- embodied in modern math by Hilbert on one side (the formalist) and Poincare on the other.
> Since the triangle got turned completely around, the sides of the box must be parallel, so it makes a parallelogram. But it can't be a slanted box because both of its diagonals are diameters of the circle, so they're equal, which means it must be an actual rectangle.
I'd be careful with such "visual proofs", even more so if accompanied by such handwavy reasoning. Eg, do we know that both diagonals are diameters? Do we know that a parallelogram with equal diagonals is a rectangle? While in this case things do work out nicely, I'd say this is almost more luck than a real proof - it's easy to mistakenly "prove" stuff like Pi=4 with similar reasoning. I believe 3B1B even has a video on the topic.
This must be true, because the diagonals are both straight lines that go through the centre and are bound by the edges, so it follows they must be equal to the diameter of the circle by definition.
> Do we know that a parallelogram with equal diagonals is a rectangle?
As another commenter points out, this is a theorem you can reach for, but proving it by itself is a bit more of a task.
Yes.
So, here when we rotate the triangle, we are essentially rotating each of the endpoints. For each endpoint, we rotate it by 180 degrees around the line segment joining the endpoint and the center. This by definition will result in a new position for each endpoint that creates a chord (as the two endpoints lie on the circle) and passes through the center (we rotated around it). A chord that passes through the center is by definition a diameter.
Thanks. Thinking about it, another way of looking at it is to construct the rotated triangle by drawing a line from each point through the center to where it intersects with the circle on the other side. This is obviously a 180' rotation, but by construction we explicitly know the diagonal goes through the center.
It maybe more clear if you visualize the reflection as a pair of perpendicular reflections, first across the diameter (which is also across the center) and then internally reflecting the diameter (which is again also across the center.)
Two reflections with a common fixed point make a rotation around that fixed point (angle of reflection is double the angle between the reflection axes.).
Two perpendicular reflections make a 180 degree rotation around the intersection of the axes of rotation.
I think the point is less to provide a completely proper proof of Thales Theorem, and more to demonstrate the fundamental principle of what a proof is (an argument to back-up a seemingly intractable statement), and how one might construct one (use concepts which we already understand, ex. rectangles, to create some plausible reasoning). Yes it involves some bad habits (relying primarily on visual intuition), but you've got to start somewhere. Moreover, the deficiencies of the example become the motivators for the next example ("so in the last example we did X, but that has problem A, so now we try Y").
ok, but if it's for pedagogical purposes, then it's even more important to point out that you're not done here (which the article misses).
also, but this is purely personal preference, I don't think geometrical constructions are a good way to introduce someone to proofs. it's easy to fall into traps of circular reasoning (as in this example) or outright wrong arguments, it's not clear when you've done enough work, and it doesn't generalize well to other problems (what is the general strategy here - "draw more stuff and hope you see something"?). personally, I'd much rather have an introduction to proofs eg through something like Induction - it's very clear when you're 'done', there's no debate about whether it's a 'real' proof, much less risk of falling into traps, it's closer to university-level math and, most importantly, it's a versatile general tool that is easy to apply to new problems (and easy to see where it can be applied)
The normal approach to prove Thale's theorem should be induced from the property of central angle being twice of an inscribed angle that subtends the same arc. Since a diameter has central angle of 180 degrees, its corresponding inscribed angle should be half of 180, that is 90 degrees.
Your second sentence denies the first sentence. The proof of the Inscribed angle theorem does not need Thale's Theorem, and it is stronger than Thale's Theorem.
The corners of the "parallelogram" in the diagram don't touch the circle at the top or bottom. So, those two corners wouldn't be right angles, but instead would be slightly obtuse - rather like this comment!
One of the illustrations in the article is off by a few pixels. Which of course has no bearing on the text, and so pointing it out above was intentionally obtuse, presumably for the sake of the pun.
BTW it's also quite direct using vectors: the legs of the triangle are the sum and difference of radius vectors. Take their dot product, distribute it, it's zero because radii are the same length.
Thales and Pythagoras are quasi-mythical figures, and we don't actually know anything concrete about any mathematical accomplishments they might have had, which are all apocryphal and date from many centuries after their deaths. Greek deductive mathematics per se dates from at least a century after Thales' time, while many of the basic facts about Euclidean geometry were understood in ancient Egypt and Mesopotamia long before him, and it is most likely that Thales himself never did any of the mathematical or scientific things attributed to him.
Viktor Blåsjö's speculation that ancient Greeks began with the same insight as Lockhart's 7th grade student is plausible but is not backed by any evidence whatsoever. (This is an insight that many people have had over the centuries, certainly including anyone deeply investigating cyclic quadrilaterals, but also probably plenty masons or metalworkers working with circles and right angles, etc.)
> sum and difference of radius vectors
This is a nice one.
Another way to use Thales' theorem in characterizing a circle, without involving the center point, is to start with one point P on a circle and a vector d which is a diameter from that point to the antipodal point. Then the vector v from P to any other point Q on the circle satisfies v² = v · d, or equivalently v · (v − d) = 0.
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[ 2.7 ms ] story [ 194 ms ] threadGreat mathematicians tend to start young. I don't question it, I'm just curious how the student phrased it.
> To be fair, I did paraphrase the proof considerably. The original was quite a bit more convoluted, and contained a lot of unnecessary verbiage (as well as spelling and grammatical errors). But I think I got the feeling of it across. And these defects were all to the good; they gave me something to do as a teacher. I was able to point out several stylistic and logical problems, and the student was then able to improve the argument. For instance, I wasn’t completely happy with the bit about both diagonals being diameters— I didn’t think that was entirely obvious— but that only meant there was more to think about and more understanding to be gained from the situation. And in fact the student was able to fill in this gap quite nicely: “Since the triangle got rotated halfway around the circle, the tip must end up exactly opposite from where it started. That’s why the diagonal of the box is a diameter.” So a great project and a beautiful piece of mathematics. I’m not sure who was more proud, the student or myself. This is exactly the kind of experience I want my students to have.
[1] https://fermatslibrary.com/s/a-mathematicians-lament
Edit: Pretty good points in the replies. Probably a lot of latent jealousy on my side in this. I should have thought on this more before commenting.
It may be more likely but not impossible.
Truth be told, some of the kids loved it - but those were the kids that were really passionate about math, for the sake of math.
But you also had a bunch of kids that seemingly hated every second of it. So why did they do it?
- Some had pushy parents that wanted them to excel in extracurricular activities. Think typical tiger parenting...
- Some had big ambitions about certain schools, and felt that they had to compete on a national level in something, in order to stand out in the selection/application process.
Granted, this was over 15 years ago. A couple of years ago I checked up on some of the kids (LinkedIn), and the most passionate kids were either now math/physics Ph.Ds, or Ph.D track.
Of course, this depends so much on the person in question. Acceleration can be great for some people and awful for others. But it’s not about ‘enjoying being a kid’ — that’s something I hear a lot, but I enjoyed myself a lot more in the higher grades where I was actually learning material I found interesting.
Ironically, I found it decent preparation for life, though, which seems to be occasional bursts of furious activity punctuated by long stretches of repetition, which could easily become boring. I ended up becoming very good at taking responsibility for making my own life interesting.
Oh yes… I also spent considerable time teaching myself interesting topics. It’s a good skill to have. But I was still bored a lot of the time, and being able to do the interesting stuff in school too was a much better experience for me.
I will say one of the things that always made me laugh was that at each milestone, like finishing elementary school and starting middle school, my parents would try to tell me now it would be different, because I was at a higher level. I remember pointing out to them all the same doofuses from the previous years were moving up right along with me, and it wasn't just that I was ahead of them, but that I was faster, so this didn't seem to help. I never did understand why they thought that would work!
The best school system I ever attended rigorously divided classes between social and academic --- for social classes (gym, health, social studies, homeroom) one attended at one's grade level, while for academic classes, (English, other languages, science, math) one worked at one's grade level (but with a cap of 4 grade levels through 8th grade, so a 4th grader couldn't take high school classes) --- after 8th grade this cap was removed, and students could take any classes which they could qualify for academically. To facilitate this, some teachers were also accredited as faculty at a nearby college, and if necessary arrangements were made either for college professors to come to the school and teach, or students travelled to the college.
Most students graduated with at least a little college credit, and many received a college degree along with their high school diploma --- until the Mississippi State Supreme Court decided that it was illegal because it conferred an unseemly advantage on some students with no corresponding compensation for students who were unable to avail themselves of it academically.
This is such a great idea! And it sounds like it works well in practise, too. I’ve often pondered the idea of separating classes by ability rather than age, and this is a really practical method of achieving it.
A similar idea is to group children by reading level which makes classes function more smoothly since students finish reading assignments in similar periods of time.
It was "innovative," in a way I think we're coming to regret, to suggest that all 8-year-olds (for example) must be at the same level across every subject, and none of them should ever be taught alongside a group of mostly 6- or 10-year-olds.
Of course this all came to a head when I was a junior and the school system couldn't find a teacher for Calculus for myself and the couple of other students who wanted it which pretty much killed my college prospects, so aced the ASVAB, DLPT, and EDPT and enlisted.
My mom, annoyed at not having her afternoons free, gave me math learning materials to keep me busy. Since she wasn‘t a teacher, she didn’t really know what she was doing and I ended up entering first grade reading at a fifth-grade level and doing math at a third-grade level.
I really should have been skipped ahead at least a year, if not two. The first-grade teacher had me teach the gifted kids in the back of the room while she taught the rest of the class, the second-grade teacher didn’t really know what to do with me and I was bored out of my skull as a result (first and second grade swapped between the two teachers at lunch time with the first-grade teacher focusing on reading and writing, the second-grade teacher focusing on math and science).
One of the ones that fascinated me was dated from when I was in 2nd grade. It was a big printout with rows and columns. Some of them contained numbers, but most of them just said "PHS."
It was an oddly folded poster-size thing, and once I finally got it open, I found out the numbers were grade levels, and "PHS" stood for "Post High School."
I had been 8 when I sat for this test. My parents had staunchly refused to accelerate my education.
And they continued to refuse for ten. more. years.
I might recommend that parents not send their children proof if they went out of their way to stymie them.
And you avoided being bullied ? Nice!
It was also similar to how I taught my home educated kids - do fun stuff. It worked very well. Both love maths (and hugely enjoyed studying in general). My older daughter is doing a degree apprenticeship in electronic and electrical engineering[1] and the younger one got a 9 in IGCSE maths[2] and plans to do A level maths and further maths[3]. The younger one hated maths while she was in school, which is common, and which is the problem Lockhart was trying to solve.
This is not about pushing kids or making them prodigies, it is about making them enjoy what they do, which leads to higher achievement and and happiness.
That said, not all prodigies are miserable. Ganesh Sittampalam who broke the UK record for youngest graduate taught himself for fun until he got into university!
[1] UK term for degree paid for by employer done while working (and getting paid). Good for parental wallet!
[2] UK exams sat at 16. 9 is a top grade.
[3] Closest American thing is APs.
While that's true, there's lots of parents who don't believe it, or don't care if the end result isn't actual mathematical genius as long as it's some sort of distinction. (I'm in the happy position where my parents encouraged me to pursue the paths that I enjoyed, which wound up with some acceleration, but my getting in the local newspaper meant that we were besieged by the kind of parents who do just want to push their kids towards some pre-determined "success.")
I think that's a very optimistic view of the outcome of this sort of thing, although it could be that I just didn't describe the parental behavior well. This wasn't parenting telling their kids "hard work is worth it!", but parents who decided on the specific honors that their child would achieve, and would permit no deviation from the path, as a result of which most of these kids either burned out before achieving their goal, or rebelled, usually in self-destructive ways, as soon as they were given a little freedom from parental control.
Different parenting styles work for different people, but for me (as a child in the US in the '80s) I think that my parents did the best possible thing, which was to instill in me lessons about the value and importance of hard work, but to make it clear that, in the end, the decision about whether I would pursue those values was up to me. It probably helped that, despite the many other advantages I enjoyed, my family was insufficiently well off to represent a guaranteed financial fallback for me, so that I knew that, one way or the other, I'd have to make my way in the world—which made it particularly appealing to me to be able to find a way to earn my keep by working hard for something I loved.
I’ve not pushed him into any of this (I really would prefer that he not pursue programming as a career as I fear that we’re heading to a dark ages of software development, but he can do what he likes).
His twin sister, meanwhile, has been demanding that I teach her social studies so I’ve been doing my best to give her bits and pieces of history and government from my memory. She doesn’t demand the math, but I’ve noticed that she is paying attention during the algebra on the walk lessons as she’ll occasionally jump in with an answer herself.
My purely selfish side wouldn’t object to them skipping a grade as my ex-wife seems committed to them doing K–8 in private schools (even though our local public schools are quite good except at the junior high level, but 6–8 is hell at any school) and skipping a grade would save me a year’s tuition expenses.
Your kid would also probably really enjoy The Number Devil, a breezy novel about a kid who gets trapped in his dreams with a devil who won't stop telling him about elementary number theory, using various dream-world props.
If you want more problems after that, there are a lot of great puzzles in Kordemsky's The Moscow Puzzles https://archive.org/details/moscowpuzzles3590000kord_m9a0, or you could take a look at Fomin, Genkin, & Itenberg's Mathematical Circles: Russian Experience which your kid could probably just about handle.
There are also some great puzzles and games explained at the website of the Julia Robinson Math Festival, https://jrmf.org which can be played with fairly basic easily available materials.
Also, Lockhart's Lament is from 2002, so this post probably needs a (2002). It is very unlikely that the proof was new. It was certainly new to the seventh-grader, and a great result at that.
IIRC you prove it via angles.
a) The sum of angles of any triangle is pi.
b) The sum of angles of any quadrilateral is 2*pi.
c) Since you rotate the triangle, opposite angles end up adding together. Because of the above, a bit of reasoning around symmetry shows that when diagonals are equal they can only add to pi/2 and that the other ones can only be pi/2. Any other angle leads to a contradiction.
I mentioned angle values but (again IIRC) this can all be proven with compass and ruler.
But then, the article is about laying out a proof of Thales so if it's using a lemma that leveraged Thales itself that's circular logic right there and so we're in trouble.
One other way to prove Thales is by computing angles, but then you've got yourself a proven Thales so the point of the article is moot.
By the point you've proven Thales with isosceles (b), proving that a parallelogram with equal diagonals is a rectangle thanks to Thales(b) in order to go forward along the Thales(a) proof is circular logic.
The net change in direction around a triangle, quadrilateral, or any simple closed curve is a full rotation, i.e., 2π.
The sum of the interior angle and the signed change in direction around a vertex of a polygon is π (consider the case of a "vertex" with interior angle π).
So for a polygon with interior angles θ₁, θ₂, …, θₙ,
2π = (π - θ₁) + (π - θ₂) + ⋯ + (π - θₙ),
therefore
θ₁ + θ₂ + ⋯ + θₙ = nπ - 2π = (n-2)π.
Thus, no parallelogram with non-right angles exists that has equal length diagonals.
My favorite book on the topic, unfortunately only in Slovak, is: "Matematici, ja a ty" (mathematicians, you and me). I don't think there is a translation. I also had a university professor who had a similar style - not dry proofs, but "Here is how Archimedes thought about proofs" - which is related to the comment above: if A is not smaller than B and B is not smaller than A, they need to be equal.
In my opinion it's a lot more obvious that a parallelogram with 2 equal diagonals must be a rectangle than that an inscribed angle intersecting a circle at ends of a diameter must be a right angle.
I think the two facts, by themselves, are roughly equivalent in difficulty. Moving from Thales to equal diagonals parallelogram requires drawing an extra circle; moving from parallelogram facts to Thales requires either the construction in the article, or continuing the missing radius to get the same rectangle.
Clearly, filling in all of the details of any formal proof in Greek style is going to require a careful knowledge of the axioms in use and some list of previously proven theorems which are allowed to be used without re-proof. Depending on which theorems are at hand already, one or another proof might be shorter or longer or more or less obvious. When trying to construct a whole mathematical treatise, the best order for the theorems so that each proof depends only on previously proven theorems is a tricky choice involving some trade-offs between pedagogical goals and concision of individual proofs.
But all of that is missing the point.
(But then you need to know that parallelogram diagonals split each other in half, …)
Rather, it suffices to simply state, the only parellelogram that is inscribed in a circle is a rectangle.
For the less rigorous, that statement is obviously true. For complete rigor, it is sufficient to argue that the center of the circle, combined with the vertices of the parallelogram forms an isoceles triangle. So the centre of the circle must lie on the bisector all edges of a parallelogram. But on a non-rectangle parellelogram the bisectors of opposite edges never intersect.
His student's "proof" is an illustration, not a proof. There's no way to know if its circular or simply unfounded, since it is purely an appeal to intuition. That's only a part of mathematics. The correct thing to do, mathematically, is to validate the intuition by formatting it as a proper proof, based on non-circular axioms and theorems. In the paper itself he admits that his student's work was incoherent and needed him to rewrite it.
He's right that 2 column geometry proofs are ugly, and could be presented better. This has been known for centuries.
https://www.c82.net/euclid/en/book3/#prop31
For children and for starting out, the pictures are great. But for mathematics, pictures are extremely limiting. 2D and 3D are notgod models for N-Dimemsions. ("Spiky balls" , for example).
Mathematics is far, far more powerful than human eyes. The amazing thing about geometry is that the whole thing works without pictures! A blind person can be a great geometer, because geometry is axiomatizable. Meanwhile, Euclid's Elements, while an incredible achievement in its day, is not well-founded, relying on unstated axioms.
I think it's fairer to say that his point may not have been what you expected it to be. As a teacher myself, trust me, if a student comes to you and says "I came up with a proof!" and you say "no, you see, what you have is an intuitive explanation that can possibly be turned into a proof," then all that will happen is that that student will not be interested any more in exploring, or at least will not be interested any more in sharing their explorations with you. At that point, you have both lost.
Lockhart is well aware of the standard of mathematical proof, and knows that, all else aside, this theorem is not in want of proof. His focus is on the fact that, if we want mathematics to remain a live profession, then we must improve our pedagogy, and help to train students who enjoy and want to pursue mathematics—even if it means occasionally accepting less than maximally rigorous mathematics from a seventh grader.
Here's another view: Euclidean geometry is euclidean because the underlying transformation group (the group of those transformations which preserve what we want to preserve -- in this case the metric) is the euclidean group (the semi-direct product of the orthogonal group and translations). This is the symmetry that encodes our intuition -- the same intuition the kid is using to prove the above result. If we were to change the underlying space to the real projective space instead of R^2, and instead of choosing to preserve the metric we choose incidence and cross-ratio, we'd get a different group (GL(3,R)) and different geometry, viz. projective geometry.
This is an ancient dialectic that runs within mathematics -- embodied in modern math by Hilbert on one side (the formalist) and Poincare on the other.
I'd be careful with such "visual proofs", even more so if accompanied by such handwavy reasoning. Eg, do we know that both diagonals are diameters? Do we know that a parallelogram with equal diagonals is a rectangle? While in this case things do work out nicely, I'd say this is almost more luck than a real proof - it's easy to mistakenly "prove" stuff like Pi=4 with similar reasoning. I believe 3B1B even has a video on the topic.
This must be true, because the diagonals are both straight lines that go through the centre and are bound by the edges, so it follows they must be equal to the diameter of the circle by definition.
> Do we know that a parallelogram with equal diagonals is a rectangle?
As another commenter points out, this is a theorem you can reach for, but proving it by itself is a bit more of a task.
How do we know the 2nd diagonal goes through the center ? Is it because of the construction by rotation ?
Two reflections with a common fixed point make a rotation around that fixed point (angle of reflection is double the angle between the reflection axes.). Two perpendicular reflections make a 180 degree rotation around the intersection of the axes of rotation.
Your pi=4 example has more defects than defects in symmetry arguments.
also, but this is purely personal preference, I don't think geometrical constructions are a good way to introduce someone to proofs. it's easy to fall into traps of circular reasoning (as in this example) or outright wrong arguments, it's not clear when you've done enough work, and it doesn't generalize well to other problems (what is the general strategy here - "draw more stuff and hope you see something"?). personally, I'd much rather have an introduction to proofs eg through something like Induction - it's very clear when you're 'done', there's no debate about whether it's a 'real' proof, much less risk of falling into traps, it's closer to university-level math and, most importantly, it's a versatile general tool that is easy to apply to new problems (and easy to see where it can be applied)
You can misuse induction (all horses are the same color, heap of sand cannot exist) and algebra (1 = 2 via division by zero)
Thales's Theroem is a simpler, easier to prove (as in OP), less powerful statement than the inscribed angle theorem.
https://www.c82.net/euclid/en/book3/#prop31
BTW it's also quite direct using vectors: the legs of the triangle are the sum and difference of radius vectors. Take their dot product, distribute it, it's zero because radii are the same length.
Viktor Blåsjö's speculation that ancient Greeks began with the same insight as Lockhart's 7th grade student is plausible but is not backed by any evidence whatsoever. (This is an insight that many people have had over the centuries, certainly including anyone deeply investigating cyclic quadrilaterals, but also probably plenty masons or metalworkers working with circles and right angles, etc.)
> sum and difference of radius vectors
This is a nice one.
Another way to use Thales' theorem in characterizing a circle, without involving the center point, is to start with one point P on a circle and a vector d which is a diameter from that point to the antipodal point. Then the vector v from P to any other point Q on the circle satisfies v² = v · d, or equivalently v · (v − d) = 0.
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