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The Numberphile infinite series video

I was having a fun conversation about the Numberphile 1 + 2 + 3 + . . . . = -1/12 video on twitter yesterday, but the limitations of twitter were making the conversation difficult.  So I decided to spend a little time today gathering my thoughts on the video, and perhaps more to the point, the video’s potential impact on the public perception of math.  It is proving to be more difficult than I expected  to pin down my own thoughts, but I’ve spent all the time I can, so here goes nothing!

**Update** – February 10th, 2014 – Ed Frenkel has an amazing 10 minute video interview about his reaction to the Numberphile video.  Wonderful.  I put this interview at the top of my post here because he says much of what I wanted to but far more eloquently than I could have:

** End update **

It almost feels as though there must be some sort of uncertainty principle for popular math presentations -> rigor * popularity =  constant.   The more rigor, the less popularity, and the more popularity, the less the rigor will have to be.

Who knows if that uncertainty principle is true or not, but even if it is true, I’m not sure it would bother me all that much.  It would mean that occasionally there will be popular presentations of interesting math that fall short on rigor.   Maybe the difficulty that I’m having gathering all of my thoughts is simply because  the Numberphile video falling short on rigor doesn’t bother me all that much.    I do not believe that people who are out to tell stories about interesting or unusual math are out intentionally  trying to deceive anyone.  If the cost of bringing math to the masses is an occasional angry lecture from the internet about things like analytic continuation,  then so be it.

Essentially everything that could possibly be written about the Numberphile video is covered in this wonderful blog post from Aperiodical:
http://aperiodical.com/2014/01/an-infinite-series-of-blog-posts-which-sums-to-minus-a-twelfth/#more-11423

For the purest of the pure math people, the two Terry Tao pieces tell the story.  Evelyn Lamb’s post is probably the best place (by far) to start if you don’t live and breathe math everyday.   If the video made  you cringe, then Cathy O’Neil’s post will probably make you stand up and cheer.

Rather than rehashing all of the math, I want to focus more on why I liked the video despite the flaws, and also talk through my reaction to two other popular math videos for comparison.

Let’s start with a fun one – Vi Hart’s “Wind and Mr. Ug” :

I wanted to start with this one because it is one of the most brilliant, if not the most brilliant, presentation of math for the masses that I have ever seen.  The first time I saw it I thought she should win a genius grant.  The only difficulty that I have with this video is that I don’t really know what you should do with people (kids mostly) after watching it.  Sure, you can make a few Moebius strips – and I’ve done that every time I’ve watched this video with kids – but that seems to be about it.  It is a beautiful video, but it floats away like a balloon after it ends.  Truthfully, I’m  not at all sure how I’d use it in a class if I was still teaching.   The one and only thing I’ve seen that used this video in a clever way is hardly math-related at all.  It is still fun though, so check out this  great parody video that tells the story from Mr. Ug’s perspective:

Next up is a video I don’t particularly like – Khan Academy’s video about the Golden Ratio and the moon:

For me, the example in this video is so contrived that I have a hard time seeing how students  would find it interesting.  Unlike the two videos above that I love (and watched all the way through before posting them to this article),  I struggle to make it even 30 seconds in to this one.  There’s so much beautiful math behind the golden ratio, why in the world you’d want to spend time on example like this is beyond me.  While I am probably a bigger fan of Khan Academy than most folks, and happily use their exercises with my kids from time to time, I worry that videos like this make math seem lifeless.   These contrived ideas make my blood boil, and so before I get too far down the road of a full-blown rant, let me stop and simply say that I wrote about it here:

https://mikesmathpage.wordpress.com/2013/12/12/the-golden-ratio-jumping-the-shark/

Finally, the Numberphile video.

As I write my post here, the video has been viewed just over 1.3 million times.  Wow.

Yes there are mathematical flaws, and yes some articles have point out more flaws than I had noticed myself the first time I saw it, but I liked it and still like it despite the criticism.  I watched it with my kids who screamed at the screen that the equation was wrong (especially my 7 year old!).  I asked them about it again today at breakfast.  My 7 year old just repeated that it was all wrong.  My 10 year old said that it was just a bunch of physics nonsense (ha!).  At least their emotional reactions stuck with them.

If I was still teaching college or high school kids I would definitely watch the video with them.   Spending a day or two with students talking through some of the ideas in this video would, I think, be absolutely amazing.   There are so many different, and so many interesting paths to go down to help understand this crazy looking sum.

There are a couple of reasons that It didn’t really bother me that they were cutting corners on the math details.  First, many of those loose ends / errors are very interesting things to talk about in their own right and would be great things to talk through with students after watching the video.  Many of the blog articles explain in varying amounts of detail, how to correct these errors.    Second, I’m sure that a lot of physicists think about the world exactly the way that these guys in the video do, and I believe that they were trying to explain the rather odd result here as best they could without getting too caught up in ideas of analytic continuation or unusual interpretations of infinite sums.

A lot of the criticism reminds me of the criticism I heard directed at a new theory from Seiberg and Witten back in the 90s.  They had found a completely new way of thinking about some ideas in physics that had amazing applications in pure math.  Some mathematicians were worried about the lack of rigor.  Others ran with the theory and some big problems that had been unsolved for 100’s of years fell like dominoes.  I remember attending a few standing room only lectures by Cliff Taubes at Harvard.  I’ll never forget the excitement and buzz in the math world then, and I’ll also  never forget the healthy skepticism either – mathematicians crave rigor.

Physicists aren’t nearly as worried about rigor, though.  For example, here’s a write up on the Numberphile video from a physics blog:

http://physicsbuzz.physicscentral.com/2014/01/does-1234-112.html?m=1

Two of the items emphasized in this blog are (1) the infinite sum result is well known going back to Euler and Ramanujan, and (2) it gives the right answers in Quantum Electro Dynamics (QED) so as strange as the bizarre some of the series seems, there must be something there.  I’m sure there are some physicists who would want more rigor, but my guess is that those two points of emphasis would carry the day most of the time in the physics community.

Away from the mathematical rigor, one other very important point of focus was mentioned by Patrick Honner in our twitter conversation yesterday:

I have a couple of reactions.  First,  if you want to talk about fun and interesting math, I do not think you can ever completely eliminate a this problem.   If you are worried that people won’t get the math, you can report the proof of Fermat’s Last Theorem, or note that Harald Helfgott proved the weak Goldbach conjecture, but you can never give any details.  With regard to the specific topic of this video, covering analytic continuation or non-standard techniques used in infinite sums would alleviate the problem only because no one would watch the video.  Both of the topics are too technical for inclusion in a 10 minute presentation to a general audience.

Second, push back against ideas in science isn’t new or unusual and I hope that a large part of the message to the public about math is that it is ok to ask questions and ok to not understand.  Sometimes the pushback can be terribly unhealthy – see the story of Georg Cantor presented in “The Mystery of Aleph” for example.   But it can be healthy, too.   “The Infinity Puzzle” gives an absolutely fascinating (and a little controversial) description of how ideas in quantum physics have evolved over time.  Gary Kasparov’s series “My Great Predecessors” provides an astonishing presentation of the evolution ideas in chess.    Of the 1.3 million people who have viewed the Numberphile video, there are probably at least 1 million who thought “I don’t get this.”   There were many resources that came to light for people looking to understand it, which I think is great.  I really hope that these resources found they way to anyone whose reaction to the video was similar to the reaction Mr. Honner is worried about.

Third,  I do share Mr. Honner’s concern in general, and worry that a lot of popular math presentations end up being pitched in ways that are not always the best.  One that I saw recently really struck me because the author has done, and I’m sure will continue to do, more for math in the US than just about anyone.  If you look at some of the promotional material for Jordan Ellenberg’s new book “How not to be Wrong” you’ll find that “Ellenberg learned algebra at the age of 8 and got a perfect score on his Math SATs as a 12 year old.”

http://www.penguincatalogue.co.uk/hi/press/title.html?catalogueId=257&imprintId=1166&titleId=19456

Of all the accomplishments he’s had, and all of the amazing things he’s done, I wouldn’t have even dreamed about putting his age 12 SAT score near the top.  I worry about a subtle message to kids that reinforces the idea that math is simply about innate ability – you are either a math person or you aren’t, and you might know by age 12 either way.  I worry a lot about the reaction of someone who thinks “I don’t get math” to this message.

Finally,  if you are  looking to find new and interesting and certainly non-controversial ways to pitch fun math, look no further than . . . . Numberphile!  Check out their interview with Ed Frenkel:

Frenkel’s quote at the end really struck me:  “What if I told you there is this beautiful world out there and you don’t even have to travel anywhere to find it.  It’s right at your fingertips . . . . this is the coolest stuff in the world.”

At this point there are many people out there working in various different ways to communicate fun and interesting math to the public.  While those people might have differing views about Numberphile’s infinite series video, I’m sure that group broadly agrees with Frenkel’s position.  His note at the end inspired me to do a series about 4 dimensional spheres with my kids (see my blog post right before this one), and hopefully other people who agree with Frenkel will continue to put new and fun math resources out for public consumption.  Can’t wait for the next public math controversy 🙂

4 Dimensional Spheres

I hope that everyone has seen the nice video that Numberphile and Ed Frenkel did about why people hate math:

As I write this blog post the video has about 130,000 hits, so I guess there’s still a few views to go before everyone has seen it.  Too bad.

The statement that really struck me in the video comes near the end (around 8:00).  Talking about the beauty of math, Frenkel says – “What if I told you there is this beautiful world out there and you don’t even have to travel anywhere to find it.  It’s right at your fingertips . . . . this is the coolest stuff in the world.”
The day before Numberphile posted the video I had done a fun project with my kids about circles and spheres.  We found how to calculate that the area of a circle was $\pi * R^2$ (two ways, actually!) and also how to find that the volume of a sphere was $\frac{4}{3} \pi * R^3.$    That talk with my kids had been inspired by a paper that Stephen Strogatz had posted earlier in the week that discussed the volume of a sphere in any dimension.  Pretty neat stuff for sure, but I stopped with just the circle and sphere with my kids.

After hearing Frenkel’s message at the end of the video, I changed my mind and plan to spend the next week talking with them about the 4 dimensional sphere.  They seemed pretty excited about the idea, and I’m super excited to try it so we started tonight.

The first thing that I wanted to talk about was some basics about circles.  They remembered some of the discussion from the weekend  ( it was sort of neat to see that they remembered the methods more than they remembered the results) and we quickly reviewed the rest.

After the review I backed up – what is a circle in the first place?  How can you extend that definition to a sphere?  What is a one dimensional version of a circle?    All fun things to discuss.

Finally I showed how we could create a 2 dimensional circle from a bunch of 1 dimensional circles and then build a sphere out of a bunch of 2 dimensional circles.   Of course, this was what we’d done over the weekend, but it probably didn’t hurt for them to see it again since we’ll use the same method to build up our 4 dimensional sphere.

All in all, it seemed like a nice start to our little project.

to be continued  (Jan 20, 2014)

Jan 21, 2014

Thought I’d be shoveling snow tonight so we moved on to our next talk about 4D spheres this morning.  My goal today was to help them get a slightly better understanding of how we move from zero dimensions to 4 dimensions.    To help with the geometry we built some models using our Zometool set.  The models were of squares and cubes rather than circles and spheres, but that just gave us two examples in the higher dimensions.  I took advantage of the two different shapes in each dimension to remind them of how we calculated the area and volume for the circle and sphere and to show that we could use the same method for the square and cube.    Seemed like the had a good time and they spent about 30 minutes after me made the movie building other stuff with the Zometool set.    Haven’t decided what we’ll talk about tomorrow, but I’m happy with how these two talks have gone.

to be continued (Jan 21, 2014)

Jan 22, 2014

I decided that the next step would be reviewing two methods that we’d talked about previously for finding the area of a circle and the volume of a sphere ( see this blog post:  https://mikesmathpage.wordpress.com/2014/01/18/showing-the-kids-about-the-area-of-a-circle/) .  I settled on this approach because I wanted the kids to see how you could extend the concepts in these methods even though you didn’t necessarily know exactly what a 4d Sphere looked like.  Unfortunately this talk didn’t go nearly as well as I was hoping and I feel that I could have tied the points together much better if we went through it again.  Oh well, teaching lesson learned (hopefully!).

The main idea I’m trying to emphasize is breaking up a problem into smaller problems that you already know how to solve.  The specific example in the problems at hand is chopping a circle up into rectangles and chopping a sphere up into discs.  We extend that idea into chopping a 4 dimensional sphere up into spherical discs.  As we found with the circle, this method for finding the volume of a 4D sphere leaves us with a complicated sum involving square roots, and we’ll use Wolfram Alpha to help us understand that sum a little better tomorrow.

So, although my explanations in this talk were a little clumsy, I hope that the main ideas did come across and I’m excited to move on to understanding the new sum that will tell us the volume of the 4D sphere.  There’s a big surprise waiting for them in that sum!

to be continued (Jan. 22, 2014)

We took a break from shoveling out this morning to finish off the problem of finding the area of a 4 dimensional sphere.  After a quick review of what we’d done the last few days, we took a final look at the equation for the volume of one of the slices of the 4d sphere and jumped over to Wolfram Alpha to help us with the sum.  I didn’t want to dwell too much on how to evaluate the sum anyway, but using Wolfram Alpha had also had a special purpose in the lesson.  Since the sum we were looking at has a value of $\pi / 2$, I was wondering if either of the kids would be able to recognize the number.  Turns out that my oldest son actually did guess the right number after I asked him a few questions.  My youngest son thought it was pretty neat that we found a value for the sum, and also really cool that the volume of the 4d sphere with radius 1 was $\pi^2 / 2.$

Anyway, that’s the end of the main journey.  This exercise turned out to be one of the most enjoyable math talks that I’ve had with the boys.  I’m really happy that we were able to make it all the way to this neat result.

Cathy O’Neil and Numberphile have two really great posts about math today.

O’Neil first:

Don’t know if WordPress will preserve the link from Twitter or not, so just in case here a direct link to the post:

http://mathbabe.org/2014/01/19/billionaire-money-in-mathematics/

One point that really struck me is this paragraph:

“My suggestion is that we should think about representing ourselves in this PR campaign, if we have one to wage, and we should focus efforts on things that would improve NSF funding instead of getting us addicted to private funding. And it should be a community conversation where everyone participates who cares enough.”

Next is a great video interview of Ed Frenkel by Numberphile on the subject of why people hate math:

This fantastic interview is here:

This whole video resonates with me – I wish we could put the whole world on pause for 10 minutes for everyone to watch it!

Maybe I’m must looking more carefully these days, but I’m happy to see a growing number of great public advocates for math.  Cathy O’Neil, Ed Frenkel, and Numberphile are among my favorites.

Showing the kids about the area of a circle

This week’s fun little twitter math coincidence involved circles and spheres.

Last Sunday Patrick Honner tweeted about Cavalieri’s principle:

and during the week Stephen Strogatz linked to a really interesting paper in American Scientist about n-dimensional spheres:

The combination made me want to show the boys  how to find the volume and area formulas for the sphere and the circle.  Going over all of the geometric ideas involved here with kids would be bit of a challenge, but it sure seemed like there was a fun lesson hiding in here somewhere.   My main goal was communicating the idea of reducing a new and difficult problem to an easier problem that you already know how to solve.  The secondary goal was showing that this strategy doesn’t always work out as well as you’d hoped, and luckily an observation from my older son actually sent us down that path  (see the 3rd video).

Our first conversation was about the area of a circle.  The method for finding the area that I wanted to show first was chopping up the circle into wedges and then re-assembling these wedges into something that looks a lot like a rectangle.  Although the pure mathematical details behind why this method are beyond the scope of what I was trying to communicate, the method itself, I think, is something that kids can understand and enjoy.

From here we moved on to talk about the volume of a sphere.  The approach we took here is a little different – instead of chopping up into “circular” pieces and assembling something sort of rectangular, we approximated the sphere with a pile of disks.   As I thought about how to present this material to the boys this week, I thought it was really interesting that this method works well for the sphere but not so well for the circle.

I started by showing them our Tower of Hanoi puzzle so that they could see how disks could approximate a cone.  Having that puzzle right in front of them seemed like it would help them understand that approximating the sphere by disks is possible.  I tried to not get too bogged down in the formulas for summing up the volume of the disks since that wasn’t really the main point I was trying to convey and was happy to just present the result of the sum quickly.

After we turned off the camera, my younger son said that he thought it was really cool that Archimedes had used the Tower of Hanoi to find the volume of a sphere.  Perfect!

Also after we turned off the camera, my older son said that he thought the method we just used to find the volume of the sphere could be used to find the area of a circle.  He then drew a picture of what he meant on the board.  His reaction made me really happy since that’s exactly what I’d planned to talk about next.  Of course this method will lead to the right area formula for the circle, but working through the math turns out to be quite a bit more complicated than it is for the sphere.

So, I let my son lead off the third video with his idea.  We go through the same math and quickly encounter some square roots that won’t go away.  Yuck.  After writing down a few terms of the sum we are looking at, we jump over to Wolfram Alpha.  A little bit of playing around with Wolfram Alpha shows that the sum seems like it might converge to $\pi.$

All in all, a nice morning with the boys.  A fun next step would be showing them that the volume of the 4 dimensional sphere is $\pi^2 R^4 / 2,$ though that might require another week or two of pondering on my end!

Powers of Pi and Unsolved problems.

Steven Strogatz tweeted about a really interesting paper yesterday:

http://www.americanscientist.org/libraries/documents/201110101628308738-2011-11CompSciHayes.pdf

The paper discusses a pretty simple question – what is the volume of a unit sphere in n dimensions?  The answer has at least one really cool surprise, and the whole paper is a great read.

I can’t remember ever thinking about these volumes before and was pleasantly surprised to see that these spherical volumes had increasing powers of  $\pi$ in their formulas.   That result reminded me of another famous problem whose solution also has powers of $\pi$ – summing inverse powers of integers.

In the 1700s Euler found a closed form for the sum of the reciprocals of the integers raised to any positive even integer.  For example:

$\sum_{1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}$ and $\sum_{1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}.$

He was unable to find a sum for any of the odd powers, though, and that problem remains unsolved even today.    Finding a closed form for the some of the inverse cubes is a problem that absolutely fascinated me as a kid, and I guess it remains in the back of my mind even today.  My assumption was  that there was some undiscovered connection between geometry and this sum that would provide an answer to the problem.  Before seeing the paper about the volumes of higher dimensional spheres, I don’t remember ever previously seeing a geometric result with higher powers of $\pi$ (though I’ve been away from academic math for 15 years, so maybe my memory is going . . . .)  The paper definitely made me wonder if there was some relationship between these higher dimensional volumes and the sums that Euler solved.  Any relationship would be especially interesting since the volumes of the odd dimensional spheres have closed forms, too.     Some more reading on this subject is definitely in order!

Finally, if I’m writing about $\pi$, I have to mention another fun little closed form solution that I first saw in one of my courses in college.  It has always had a special place in my heart because it was such an incredible surprise to me that I made the decision right then and there that I would major in math:

Fun memories!

Evolution of my teaching and the surprising (to me) influence of Twitter

I was extrodinarily lucky to have an outstanding math teacher in high school.  Mr. Waterman set the bar about as high as it could be set and taught a generation of kids in Omaha about the beauty of math.  I stayed in touch with him after graduation and did as best as I could to approach teaching the same way he did.  He was also incredibly influential in helping me improve in teaching my own kids and provided me with dozens, and probably hundreds, of really helpful ideas for how improve the little videos we do.     With every new topic we cover, the first thing I try to think through is how he would teach it and hope to find an approach that would make him proud.

http://www.omaha.com/article/20130929/NEWS/130928683

My first cut at standing in front of a class was in a program at MIT called Interphase.  I taught calculus in this program for 6 years both as an undergraduate and graduate student under the watchful eye of Arthur Mattuck.  He was the hardest of all hard asses when it came to views about teaching and my style didn’t connect with his at all.  And I mean at all. He gave me great feedback nonetheless, and put the fact that he could see that the kids were connecting with me and learning in front of the fact that he didn’t like my teaching style.    I was also lucky to have Mike Keynes, who is now a math professor at American University, teaching with me.  We learned a lot from each other.

In graduate school the teaching program was supervised by Susan Parker, who was and is still, beloved by her students.  She, like Mattuck, had quite a different approach to teaching than I did, but helped me become a much better in the classroom by the time I graduated.  It is always fun to go back and touch base with her when I’m up in Boston.

My two years at the University of Minnesota after graduate school were by far the most fun I’ve had teaching.  I taught a few college courses, but the real fun came from a special program that the University offered for local kids called the University of Minnesota Talented Youth in Mathematics program.  I worked with some sensational kids during the two years I was there.  One, Allison Gilmore, is now a postdoc in math at UCLA.  I’ve stayed in touch with a lot of these “kids” (who are now close to their 30s . . .) and it is so cool to see what they are all doing now.

One thing that was particularly fun with this group was the variety of interests they had in math.  Many, not surprisingly, were math contest kids just like I had been.  Helping them along with the various contest was an absolute blast.  Others were much more interested in pure math and I was able to get them involved in some pretty neat projects.  I did a reading course in Massey’s Algebraic Topology with Allison, for example, and now she does research in low dimensional topology.  Seriously . . . how freaking cool is that!!

Another valuable experience there was working with two other people – Cindy Kaus and Doug Shaw – who joined the faculty there about the same time that I did.  They both had completely different interests in math than I did (Cindy had been an engineer before getting her math PhD, for example) and connected with the kids in ways that were really different than how I did.  There was always something to learn.  Doug’s now at Northern Iowa and Cindy’s just won a Fulbright award and is spending this semester teaching and reseaching math in the Seychelles – fun!

Having left academia in 1999, I haven’t really had any new people to talk math with in a while.  As I said, I got a lot of great feedback from Mr. Waterman when I started doing the math videos on line, but I still felt that I wasn’t doing as good a job as I could.  It was much more difficult than I thought it would be to teach really basic math to my kids.

Somehow or other I stumbled on twitter and ran across an almost overwhelming about of information and ideas about math.  It is actually pretty difficult to sort through all of it, but after a while I found a few voices that really have struck a chord.  One is Patrick Honner who, because of his incredible and infectious love of math, seems very much like a younger version of Mr. Waterman.  Although he teaches high school level courses, I’ve been able to take a few of the ideas that he’s posted about and turn them into fun lessons for the boys.

And then there is Fawn Nguyen.  She teaches middle school in California and has the most creative ideas for teaching math that I’ve ever seen.  Even better, she’s constantly sharing all of her incredible ideas on Twitter and on her various websites.  It is remarkable to me that someone who lives 3000 miles away can have such a positive impact on my teaching, but I guess that’s the power of the internet!

I was thinking about some of her visual methods as I was getting ready to teach my youngest son about adding fractions today.   The approach I’d taken with my older son a few years ago was mainly computational, which is certainly consistent with how I think about the world.  Seeing Fawn’s work since then opened my mind to many new ways of thinking about teaching this material, though, and this morning we made this little video as a starting point about adding fractions:

I’ve been lucky to have so many great colleagues and mentors over the years, and I’m happy with how my own teaching is evolving because of their influence.  I think Mr. Waterman would be happy about the evolution too, and that makes me really happy, too.

The area inside of a parabola

About a week ago my older son asked me a really fun question about parabolas -> in the Cartesian plane, is there more area inside a parabola or outside of the parabola?   His hypothesis was that since both of the areas were infinite, the two areas would actually be the same.

As excited as I was by the question, talking through the answer isn’t as simple as it seems.  First, playing around with infinity produces all sorts of unusual results.   See, for example, Bertrand’s Paradox (which I accidentally called “Russell’s paradox” in the 5th video):

http://www.cut-the-knot.org/bertrand.shtml

Second, there are all sorts of little details you can either skip over or dive way, way, way into along the way, so finding the right set of ideas to bring together to talk about this problem was proving to be hard.  Luckily there are folks on Twitter who can think about teaching math way better than me, and this time Earl Samuelson (@earlsamuelson) provided exactly the idea I was looking for :

So, with that idea in place, we began talking.  The first talk was just recapping the problem itself and talking through a couple of non-intuitive things about infinity.  I use the example of comparing the number of positive integers to the number of positive even integers to show that some aspects of infinity are a little strange:

In the next talk I brought up Earl Samuelson’s idea and looked at the area inside of y = | x | since that area is a little bit easier to get your arms around. I always tell my kids that a really important step in problem solving is trying to start with a problem that is similar to, but a little bit easier than the problem you are trying to solve.  Thanks again to Earl for reminding me of my own lesson!

Next we start down the path of finding the area inside of a parabola.  We begin by looking at the various pieces of the area we need to add up to figure out the area outside of the parabola (and I sort of botch the geometry at the beginning, but that was fortunately a minor point).  That leads us pretty naturally to the problem of finding a way to calculate the area under the parabola.  That, of course,  leads to the basic idea of a Riemann sum.

So, having touched on the concept of a Riemann sum in the last talk, I actually really struggled to think about what to do next.  The problem with really rolling up your sleeves and digging into Riemann sums is that you are heading down a black hole – so much material and no easy place to stop.  I’m not trying to really cover calculus anyway, so I just decided to show the geometry and save the details for a few more years!  Outside  of the video, though, we did play around on a Riemann sum app we found online just to see how the process works.

As an aside, in thinking about how to talk through Riemann sums, I read through the treatment that Spivak gives in his Calculus text.  As with just about everything of his that I’ve read, I was really impressed with his presentation.  Particularly the end of the section where he says simply:

“The moral of this tale is that anything which looks like a good approximation to an integral really is . . . .”

Anyway, on to the next talk (and thankfully I realized I’d gotten the geometry wrong in the 3rd video before we made this one):

Now with the Riemann sum talk out of the way, all we had to do was add up the areas.  So, off we went:

And that’s the story.  As I mentioned in the beginning of this post, playing around with infinity can produce odd results, and looking at finite areas that aren’t squares might lead to a different result  (take rectangles centered at (0,0) that are 2x wide and 2x^2 tall, for example) but I didn’t really want to let those sorts of complexities get in the way of the discussion here.

All in all talking through the answer to my son’s question was really fun.  Thanks again to Earl Samuelson for helping me see how to put these talks together.

As always, I’m happy that I have the time and flexibility to have talks like these with my kids.

Pascal’s Triangle and Powers of 11

The most difficult thing about teaching my kids, by far, has been that I have no experience at all teaching elementary math.  When a concept is difficult for either of them to understand, quite often I struggle to work out exactly what it is that they struggling to understanding.  But we muddle along.

One obvious consequence (and sometimes it is a “feature” and other times a “bug” !!) is that I have no idea at all about the accepted ways to teach some of the most basic subjects.  Fairly often I just let them figure it out the elementary stuff on their own.  Basic arithmetic is a good example of a subject where my older son came up with his own methods (which have nothing to do with borrowing and carrying).  Since his ideas were perfectly fine mathematically, I just ran with ran with them.   Here are three examples that illustrate his method:

(1)  356 + 672 = 900 + 120 + 8 = 1028,

(2)  532 – 384 = 200 – 50 – 2 = 148

(3) 25 * 13 = 200 + 60 + 50 + 15 = 325

I think the main disadvantage of this way of doing arithmetic is that it is a little slow, but there are at least two nice advantages.  First, the approach highlights place value and therefore made it very easy to talk about arithmetic in other bases.  Second, this method looks pretty similar to the way we normally do arithmetic in algebra, so learning to multiply polynomials, for example, was not particularly difficult.

Yesterday I got a nice surprise when we stumbled on a new problem where this arithmetic method added a lot of value. We were doing some review work and one of the questions was simply to find the cube root of 1,331.  My son told me that he new the cube root of 1,331 was 11 because 1,331 was a row of Pascal’s triangle.

I love opportunities to talk math that come out of the blue, and this was as good an opportunity as any.  I’m not actually sure where the connection between powers of 11 and Pascal’s triangle came from since we haven’t talked about Pascal’s triangle in a long time.  However, as luck would have it, we’d spent the last couple of months talking about polynomials, so we were primed for a fun discussion.  It turned out to be even better than I’d hoped!

The first question I asked him was what he thought 11^4 was.  He drew out Pascal’s triangle and said he thought that 11^4 would be 14,641.  Fine, but the next row of Pascal’s triangle is 1 5 10 10 5 1, so what would 11^5 be?    Since the method of finding powers of 11 using Pascal’s triangle now appears to break down, he proceeded to calculate:

14,641 * 11 = 146410 + 14,641 = 100,000 + 50,000 + 10,000 + 1,000 + 50 + 1 = 161,051.

My plan, of course, why the connection to Pascal’s triangle has disappeared, but his unusual method of addition meant that the coefficients of Pascal’s triangle were right there on the board!  Ha ha, the joke was on me.  We revisited the calculation this morning:

Following that discussion this morning, we spent a few minutes connecting polynomials to Pascal’s triangle and showing why the powers of 11 are hiding inside the triangle.  Definitely a fun and surprising weekend of math!!

For me moments like these have always been the best part of teaching.  As I’ve said many times, I’m glad that I have the time and flexibility to teach my kids.

Another great problem from Fawn Nguyen (2 of infinity)

Earlier this week Fawn Nguyen posted a nice conjecture from one of her students on twitter:

I love Fawn’s approach to teaching kids math and try to blatantly steal incorporate her work into what I’m doing with my kids.  This problem had so many possibilities and I put it in my back pocket for the weekend.
The reason that I didn’t use it right away is the fraction / decimal component seemed like it would be too distracting for my 7 year old, so I wanted to take a day or two to think through how to present it.  There was, however, a similar problem that avoided the fractions that we played with right way -> multiplying 3 consecutive integers.  Here, instead of being nearly a square, you get a product that is nearly a cube:

We had a lot of fun building these shapes and talking about the pattern.  In retrospect, I wish I would have asked about 0 x 1 x 2 as well.

As an aside, building these shapes got me thinking a little more about connecting geometry and arithmetic and the next day we had a really fun talk related to the geometry of imaginary and non-commutative “numbers” :

Anyway, we circled back to Fawn’s problem this morning.  I’d toyed around with the idea of modifying the problem to talk about the Arithmetic Mean / Geometric Mean inequality, but thought that would be a little too ambitious.  Finally I decided that a better lesson than AM / GM, as well as the easy way to avoid the complexity of decimals, was to talk about multiplying integers that differed by 2 rather than 1.  We talked through the original problem, moved on to the slightly modified problem, and then talked through some of the algebra.  Made for a really fun morning.  Thanks for another great lesson, Fawn!

A little fun and a little math wtih Rubik’s cubes

Sometime last year my kids became fascinated by Rubik’s cubes.  Not sure why or how it happened, but once they started playing around a little they were hooked.  So much so that learning more about how to solve the cubes seemed like a fun topic to include as part of the school year, so we’ve been studying some of the 2×2 and 3×3 speed solving techniques for fun since September.   Even though I’m not practicing the speed solving with the boys, I’ve gotten a little hooked, too 🙂

Following a few folks on twitter also led to some interesting Rubik’s cube related reading in the last year.  Christopher D. Long (@octonion) tweeted about the book “Adventures in Group Theory:  Rubik’s Cube, Merlin’s Machine & Other Mathematical Toys.”  Definitely a fun read if you are into math, though it’ll be a while before I can pull much from it for the boys.

I also ran across Cathy O’Neil’s (@mathbabedotorg) old post about math contests :

http://mathbabe.org/2011/07/17/math-contests-kind-of-suck/

This comment really struck me  – “I have never been particularly fast at working out the details of something from the conceptual understanding (for example, it takes me a long time to solve a 7x7x7 Rubik’s cube) but it turns out the Rubik’s cube doesn’t mind. And in fact mathematics in real life isn’t a timed tests- the idea that you need to be original and creative really quickly is just a silly, arbitrary way to select for talent.”  (as an aside, you should definitely follow her blog and follow her on twitter – you’ll not find a more interesting blog.)  I agree with Cathy O’Neil’s point that there’s not anything special about solving the cubes fast.  The kids seem to like it and enjoy learning the techniques, but it is mostly just a matter of practice.  That said, the world record solve times (~5.5 seconds for solving the 3×3, for example) really are  mind blowing.

So, what can kids learn from these cubes?

There is quite a bit of interesting math related to the cube solving algorithms (see the book mentioned above).  A simple introduction to these algorithms probably has some benefit, but I’m aiming a little lower right now.

One interesting advanced topic is parity.  This position on the 3x3x3 is impossible to solve:

You can not create a position with just one middle reversed with legal moves.  In order to make this postion, you have to take the cube apart.

However, this position on the 4x4x4 cube, which seems pretty similar to the picture above, is solvable:

At least for my (very, very very slow) solving technique, figuring out how to solve the 4x4x4 from this position was the final obstacle to overcome in learning how to solve the larger cubes.

For my younger son, it turned out that the cubes were also fun tools for learning about topics like fractions:

ratios:

and exponents:

** Update **  Imaginary and non-commutative numbers!!

It isn’t hard to believe that kids will be more excited about learning when they are having fun, but it great to see that excitement in practice.    Of course, it has also been really fun for me to use the Rubik’s cubes to help teach a bunch of different math topics.  Maybe one day we’ll even be able to replicate something like this 🙂