A unique book ! .. some really unique concepts ..! Sir, You really are making the doors of mathematical reality open to the students who really love mathematics !..
Wow! thanks. I love this. I teach math and am definitely going to check out your books. Students usually just want the 'how'; not the 'why'. But the 'why' always stays with you.
This is clear if we use vectors: If the quadrilateral's vertices are a, b, c, d, then the midpoints are a/2 + b/2, b/2 + c/2, c/2 + d/2, d/2 + a/2, and the vectors connecting them are a/2 - c/2, b/2 - d/2, c/2 - a/2, d/2 - b/2; they form a parallelogram. What's a more elegant way?
Spoiler alert, my solution to the parallelogram question. Imagine shrinking one of the sides of the four-sided shape, to a point. That gives you a triangle, with two of the sides of the inscribed shape flush with the edges of the triangle. Then it is easy to show that the inscribed shape is a parallelogram by an argument involving similar triangles.
any two intersecting lines will form angles that add up to 360deg and the opposing angles will be of same degs. Process threw this one more time to make the sides of your inscribed quadrilateralsand threw the resulting symmetry the resulting figure has to be a parallelogram
All art can be studied through a mathematical lens, and its beauty often arises from unseen mathematical relationships that we experience but don't necessarily understand. Some art is seen as beautiful because it breaks our expectation for those mathematical relationships.
I seems the reason it's a parallelogram is because there's always two pairs of lines, each with some degree of non-parallel-ness (divergence/convergence). And each par is connected to the other pair, in the same way, with straight lines, in a 2D space (paper) therefore each pair must have a measure of convergence/divergence that is 180°. Why? Because if you cross two straight lines on a piece of paper the total of the angles of those two lines must be 360°. In other words, cross two lines. If one angle is theta then the adjacent angle is 180°-theta. So when you draw two lines that aren't crossed and then two other lines that aren't crossed the corners, the connection, is where they cross. Each pair of lines is a 180° complement to the other pair ... their 'convergence' is going to be proportion to other pair of lines making them average, become parallel ... a parallelogram.
I’m following your work for some time now and it’s amazing how the Lament put everything I thought but couldn’t say, in perfect words. My question is this. You have this amazing statement how we should teach kids math and though I’m using your examples (wonderfully I must add), I don’t have much to work with. Please write a book “lament - how to, with examples”.
Hi there! Since this video was recorded, Paul Lockhart has published his next book, Arithmetic, which you might find helpful. Read more about it: www.amazon.com/Arithmetic-Paul-Lockhart/dp/0674972236/
@@fahimchowdhury4477 Apologies for the delay in responding. If you email your query to digitalsupport_hup@harvard.edu, we would be happy to forward it along to Mr. Lockhart.
I love this video! I have been reading Measurement and felt like I might be missing something. Watching this video definitely gave me a better idea of how to approach this text.
Sorry for the misunderstanding but I meant that at the guy you were talking about, not the person in this video. He was saying how math wasn't art and I was replying that, that guy didn't understand art as math is art.
Hello there, I'm a mathematics student and have been thinking about this same issue for a while. I'm seriously considering making a long term project of completely re-writing the curriculum for uk mathematics. The more you study maths, the more you realise the creative mindset necessary for real mathematics could be taught and integrated into study from as young as year 1.
One further point; math isn't always a purposeful study. Like Dr. Lockhart just demonstrated math can sometimes just be pattern searching. Yes, sometimes these patterns can turn out to be useful, but that's not the point. The point is to find something interesting or "beautiful" and if the expression of beautiful things, regardless of its media, isn't "art" then I'm not sure if they've been careful with their definitions.
Now imagine slowly untightening the side from a point, by moving one of the corners, to make a four-sided shape. Then it is clear that both of the centre points inscribed in the four sided shape, that are moving, are moving at half the velocity at which the corner is moving. Since they are moving at the same velocity, the lines of the inscribed shape remain parallel, and you have a parallelogram. postimg.org/image/5xhizg2vj/
Claiming mathematics is about crunching numbers and combining rules in novel ways, is like claiming art is about producing ink and combining photoshop filters.
Pardon us for the persecution, but you've touched on something that we all feel passionately about. That said, it seems like you're arguing from an incomplete view of mathematics. Theorems can be built in more than just the mathematical deduction you're used to. If you're not completely turned off by the subject I would recommend looking up what some of the open problems of mathematics are and how solutions to some have been found. Or just google "beautiful proofs".
You totally misunderstood what I was replying to. ACaffeineAddict was replying to someone who said math wasn't art and I was replying to ACaffeineAddict's comment to that guy, not the guy in the video. At the time it made sense but the top comment changed too quickly so now it looks like I'm dissing math.
The two *_greatest_* and *_most beautiful_* brainchildren of the human brain that blow our minds are math and music, especially classical and especially Beethoven’s. 🎵🎵 🎶 💕 ☮ 🌎 🌌
I even try to write my own sort of classical music melodies, actually. And have little success with it. Unlike math... I like math, but in math I have to put big efforts to understand anything, because I almost forgot even how to do arithmetic since I left the school (I had low grades). So now I have to look for some basic tutorials in order to "refresh" my arithmetic knowledge and continue reading Paul Lockheart's book later. Would be nice to have someone near, a nice sincere person, to go through this book together. Sorry for I wrote this yammering thing.
Please stop expressing your opinion as fact. You say math isn't art, but to some people it is. Math gives us a way to explain the happenings in our universe and some people find that beautiful, hence math is art. You undermined your own argument at "Bleck."
The amount of ambiguous and abstract poetry in his speech made me lose interest. He's expressing wonder, which is great and all, but in the end he isn't really talking about anything. There is nothing to actually learn from his speech. This feels too much like a glamor piece.
Paul-ie, although the respect I have for you about some of the things you've said in Lochart's lament, it is obvious to me (as I previously doubted) that you're weak in math conceptual understanding. The fact that you mention in the video can amaze a layperson, but not a mathematician. There is a simple explanation for the question you asked. The joint midpoints of any quadrilateral form a parallelogram because any such quadrilateral has only 2 diagonals. And then additionally because in any triangle, the line that joins the midpoints of two sides is parallel with the third side of the triangle. That's all. Mystery solved!
What books or resources do you recommend which you feel are superior to that of Paul Lockhart? Any input from yourself would be highly appreciated! Thanking you in anticipation!
+Richard Cane It's an entry-level problem which introduces the student to the economy and elegance of mathematics (as you demonstrated in your concise description of the solution) a selling point if you will. Solutions _ought_ to be obvious after the fact but non-intuitive up to the fact of their revelation. The mathematical difficulty just keeps going up and the pursiut of soluble problems becomes its own art - the mathematician a practiioner of this 'art'. (What's the 3 dimensional analogue of this problem? Now I'd claim that _is_ tall challenge even for yourself to purue!)
Interested, but nothing like taking a sequence of four Fourier sums, and producing a Pythagorean Triad from them. Or taking the sides of that triangle as radii and producing a set of polygons from them.
I’m reading his book measurement, one of the best math books I read
I've just discovered it amongst others tonight. Waiting on the download now!
I've just discovered it amongst others tonight. Waiting on the download now!
Could you please list other great books on math you've read?
I love how excited he is
Paul is a great teacher. He taught me how to play Go.
He taught me to play Go too! Where was it that you learned?
@@amarigonzalez4771 Saint ann's school in brooklyn
A unique book ! .. some really unique concepts ..! Sir, You really are making the doors of mathematical reality open to the students who really love mathematics !..
Is he the Master himself??? Measurement is the one (one of) best book ever on mathematics.
Wow! thanks. I love this. I teach math and am definitely going to check out your books. Students usually just want the 'how'; not the 'why'. But the 'why' always stays with you.
This is clear if we use vectors: If the quadrilateral's vertices are a, b, c, d, then the midpoints are a/2 + b/2, b/2 + c/2, c/2 + d/2, d/2 + a/2, and the vectors connecting them are a/2 - c/2, b/2 - d/2, c/2 - a/2, d/2 - b/2; they form a parallelogram. What's a more elegant way?
Spoiler alert, my solution to the parallelogram question.
Imagine shrinking one of the sides of the four-sided shape, to a point. That gives you a triangle, with two of the sides of the inscribed shape flush with the edges of the triangle. Then it is easy to show that the inscribed shape is a parallelogram by an argument involving similar triangles.
any two intersecting lines will form angles that add up to 360deg and the opposing angles will be of same degs. Process threw this one more time to make the sides of your inscribed quadrilateralsand threw the resulting symmetry the resulting figure has to be a parallelogram
Read Jo Boaler's "What's Math Got To Do With It?" She's also teaching a free course online right now on the Stanford website. :)
All art can be studied through a mathematical lens, and its beauty often arises from unseen mathematical relationships that we experience but don't necessarily understand.
Some art is seen as beautiful because it breaks our expectation for those mathematical relationships.
I seems the reason it's a parallelogram is because there's always two pairs of lines, each with some degree of non-parallel-ness (divergence/convergence). And each par is connected to the other pair, in the same way, with straight lines, in a 2D space (paper) therefore each pair must have a measure of convergence/divergence that is 180°.
Why? Because if you cross two straight lines on a piece of paper the total of the angles of those two lines must be 360°.
In other words, cross two lines. If one angle is theta then the adjacent angle is 180°-theta. So when you draw two lines that aren't crossed and then two other lines that aren't crossed the corners, the connection, is where they cross.
Each pair of lines is a 180° complement to the other pair ... their 'convergence' is going to be proportion to other pair of lines making them average, become parallel ... a parallelogram.
Absolutely amazing video!
I’m following your work for some time now and it’s amazing how the Lament put everything I thought but couldn’t say, in perfect words.
My question is this. You have this amazing statement how we should teach kids math and though I’m using your examples (wonderfully I must add), I don’t have much to work with. Please write a book “lament - how to, with examples”.
Hi there! Since this video was recorded, Paul Lockhart has published his next book, Arithmetic, which you might find helpful. Read more about it: www.amazon.com/Arithmetic-Paul-Lockhart/dp/0674972236/
@@harvardupress How can I find his email address?
@@fahimchowdhury4477 Apologies for the delay in responding. If you email your query to digitalsupport_hup@harvard.edu, we would be happy to forward it along to Mr. Lockhart.
I love this video! I have been reading Measurement and felt like I might be missing something. Watching this video definitely gave me a better idea of how to approach this text.
The 4 sides of the inner shape are parallel to the diagonals of the shape.
In absence of to-scale drawing, you'd have to prove this as well
Sorry for the misunderstanding but I meant that at the guy you were talking about, not the person in this video. He was saying how math wasn't art and I was replying that, that guy didn't understand art as math is art.
Hello there, I'm a mathematics student and have been thinking about this same issue for a while. I'm seriously considering making a long term project of completely re-writing the curriculum for uk mathematics. The more you study maths, the more you realise the creative mindset necessary for real mathematics could be taught and integrated into study from as young as year 1.
One further point; math isn't always a purposeful study. Like Dr. Lockhart just demonstrated math can sometimes just be pattern searching. Yes, sometimes these patterns can turn out to be useful, but that's not the point. The point is to find something interesting or "beautiful" and if the expression of beautiful things, regardless of its media, isn't "art" then I'm not sure if they've been careful with their definitions.
Now imagine slowly untightening the side from a point, by moving one of the corners, to make a four-sided shape. Then it is clear that both of the centre points inscribed in the four sided shape, that are moving, are moving at half the velocity at which the corner is moving. Since they are moving at the same velocity, the lines of the inscribed shape remain parallel, and you have a parallelogram.
postimg.org/image/5xhizg2vj/
What's the name of theorem that's tell you that the line always parallel ???
How do you think about mathematical questions and puzzles?
Claiming mathematics is about crunching numbers and combining rules in novel ways, is like claiming art is about producing ink and combining photoshop filters.
I have a solution of the first one parallelogram question which is correct
Pardon us for the persecution, but you've touched on something that we all feel passionately about. That said, it seems like you're arguing from an incomplete view of mathematics. Theorems can be built in more than just the mathematical deduction you're used to. If you're not completely turned off by the subject I would recommend looking up what some of the open problems of mathematics are and how solutions to some have been found. Or just google "beautiful proofs".
You totally misunderstood what I was replying to. ACaffeineAddict was replying to someone who said math wasn't art and I was replying to ACaffeineAddict's comment to that guy, not the guy in the video. At the time it made sense but the top comment changed too quickly so now it looks like I'm dissing math.
Suggestion:
You should cap your marker and stop inhaling its fumes before mind blown.
to which I would reply, what's your point? your statement doesn't negate the fact that there is an art to math.
I like math
Great!... That was the very same property I showed in my first video... :D
The two *_greatest_* and *_most beautiful_* brainchildren of the human brain that blow our minds are math and music, especially classical and especially Beethoven’s. 🎵🎵 🎶
💕 ☮ 🌎 🌌
Me too like Beethoven's music the most.
I even try to write my own sort of classical music melodies, actually. And have little success with it. Unlike math... I like math, but in math I have to put big efforts to understand anything, because I almost forgot even how to do arithmetic since I left the school (I had low grades). So now I have to look for some basic tutorials in order to "refresh" my arithmetic knowledge and continue reading Paul Lockheart's book later. Would be nice to have someone near, a nice sincere person, to go through this book together. Sorry for I wrote this yammering thing.
@@flowerflower8061 1 In TH-cam search bar, type "basic arithmetic."
2 Try Krista King. th-cam.com/channels/UDlvPp1MlnegYXOXzj7DEQ.html. Good luck! 🤓
@@totalfreedom45 Thank you... I hope my little understanding of English language will be enough to get that.
@@flowerflower8061 There are many courses on TH-cam to learn American English. Try this one: th-cam.com/channels/lPyOwXLnSMejFdLvJXjA5A.html.
Consider a concave 4 sided figure
I am the first subscriber
THIS is how you mathematics. Precisel- oops, there is no precise way to do mathematics.
Please stop expressing your opinion as fact. You say math isn't art, but to some people it is. Math gives us a way to explain the happenings in our universe and some people find that beautiful, hence math is art. You undermined your own argument at "Bleck."
Why do all people who teach math look like him
well said. lockhart's lament, anyone?
worrydream dotcom/refs/Lockhart-MathematiciansLament dot pdf
No, it just went over your head.
The amount of ambiguous and abstract poetry in his speech made me lose interest. He's expressing wonder, which is great and all, but in the end he isn't really talking about anything. There is nothing to actually learn from his speech. This feels too much like a glamor piece.
9.42 is not a real number , it's a wannabe number that should be trying harder.
Lefty!
I'm pretty sure you've never "tasted" mathematics.
eurrghhhh, so american.
It's because the world is flat
Paul-ie, although the respect I have for you about some of the things you've said in Lochart's lament, it is obvious to me (as I previously doubted) that you're weak in math conceptual understanding. The fact that you mention in the video can amaze a layperson, but not a mathematician. There is a simple explanation for the question you asked. The joint midpoints of any quadrilateral form a parallelogram because any such quadrilateral has only 2 diagonals. And then additionally because in any triangle, the line that joins the midpoints of two sides is parallel with the third side of the triangle. That's all. Mystery solved!
What books or resources do you recommend which you feel are superior to that of Paul Lockhart?
Any input from yourself would be highly appreciated!
Thanking you in anticipation!
+Richard Cane It's an entry-level problem which introduces the student to the economy and elegance of mathematics (as you demonstrated in your concise description of the solution) a selling point if you will. Solutions _ought_ to be obvious after the fact but non-intuitive up to the fact of their revelation. The mathematical difficulty just keeps going up and the pursiut of soluble problems becomes its own art - the mathematician a practiioner of this 'art'. (What's the 3 dimensional analogue of this problem? Now I'd claim that _is_ tall challenge even for yourself to purue!)
Interested, but nothing like taking a sequence of four Fourier sums, and producing a Pythagorean Triad from them.
Or taking the sides of that triangle as radii and producing a set of polygons from them.
He has a PhD in math from Columbia, and has taught at major universities such as Brown & UC Santa Cruz.
Wow, you are dense.
what about a square