I’m back

Hey everyone.

I took a two year hiatus from blogging. Life got busy and I let the blog slide. I’m sorry.

But I’m back, and my New Year’s Resolution for 2017 is to post at least once a month!

new-year_resolutions_list

Here’s what I’ve been up to over the last two years:

  • Twitter. When people ask why I haven’t blogged, I say “twitter ate my blog.” It’s true. Twitter keeps feeding me brilliant things to read, engaging me in wonderful conversations, and providing the amazing fellowship of the MTBoS.
  • James Key. I consistently receive mathematical distractions from my colleague and friend, James, who has a revolutionary view on math education and a keen love for geometry. This won’t be the last time I mention his work. Go check out his blog and let’s start the revolution.

    with my nerdy friends named James

    with my nerdy friends named James

  • My Masters. I finally finished my 5-year long masters program at Johns Hopkins. I now have a MS in Applied and Computational Mathematics…whatever that means!
  • Life. My wife and I had our second daughter, Heidi. We’re super involved in our church. I tutor two nights a week. Sue me for having a life! 🙂
family photo

family photo

  • New curriculum. In our district, like many others, we’ve been rolling out new Common Core aligned curriculum. This has been good for our district, but also a monumental chore. I’m a huge fan of the new math standards, and I’d love to chat with you about the positive transitions that come with the CCSS.
  • Curriculum development. I’ve been working with our district, helping review curriculum, write assessments, and I even helped James Key make some video resources for teachers.
  • Books. Here are a few I’ve read in the last few months: The Joy of x, Mathematical Mindsets, The Mathematical Tourist, Principles to Actions
  • Math Newsletters. Do you get the newsletters from Chris Smith or James Tanton (did you know he’s pushing three essays on us these days?). Email these guys and they’ll put you on their mailing list immediately.
  • Growing. I’ve grown a lot as a teacher in the last two years. For example, my desks are finally in groups. See?
my classroom

my classroom

  • Pi day puzzle hunt! Two years ago we started a new annual tradition. To correspond with the “big” pi-day back in 2015, we launched a giant puzzle hunt that involves dozens of teams of players in a multi-day scavenger hunt. Each year we outdo ourselves. Check out some of the puzzles we’ve done in the last two years.
  • Quora. This question/answer site is awesome, but careful. You’ll be on the site and an hour later you’ll look up and wonder what happened. Here are some of the answers I’ve written recently, most of which are math-related. I know, I know, I should have been pouring that energy into blog posts. I promise I won’t do it again.
  • National Math Festival. Two years ago we had the first ever National Math Festival on the mall in DC. It was a huge success. I helped coordinate volunteers for MoMATH and I’ll be doing it again this year. See you downtown on April 22!
famous mathematicians you might run into at the National Math Festival

famous mathematicians you might run into at the National Math Festival

Now you’ll hopefully find me more regularly hanging out here on my blog. I have some posts in mind that I think you’ll like, and I also invited my colleague Will Rose to write some guest posts here on the blog. Please give him a warm welcome.

Thanks for all the love and comments on recent posts. Be assured that Random Walks is back in business!

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What does it mean to truly prove something?

Let me point you to the following recent blog post from Prof Keith Devlin, entitled “What is a proof, really?”

After a lifetime in professional mathematics, during which I have read a lot of proofs, created some of my own, assisted others in creating theirs, and reviewed a fair number for research journals, the one thing I am sure of is that the definition of proof you will find in a book on mathematical logic or see on the board in a college level introductory pure mathematics class doesn’t come close to the reality.

For sure, I have never in my life seen a proof that truly fits the standard definition. Nor has anyone else.

The usual maneuver by which mathematicians leverage that formal notion to capture the arguments they, and all their colleagues, regard as proofs is to say a proof is a finite sequence of assertions that could be filled in to become one of those formal structures.

It’s not a bad approach if the goal is to give someone a general idea of what a proof is. The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?

(more)

Click the link to read the rest of the article. Also read the comments below the article to see what conversation has already been generated.

I won’t be shy in saying that I disagree with Keith Devlin. Maybe I misunderstand the subtle nuance of his argument. Maybe I haven’t done enough advanced mathematics. Please help me understand.

Devlin says that proofs created by the mathematical community (on the blackboard, and in journals) are informal and non-rigorous. I think we all agree with him on this point.

But the main point of his article seems to be that these proofs are non-rigorous and can never be made rigorous. That is, he’s suggesting that there could be holes in the logic of even the most vetted & time-tested proofs. He says that these proofs need to be filled in at a granular level, from first principles. Devlin writes, “no one has ever carried out that filling-in process.”

The trouble is, there is a whole mathematical community devoted to this filling-in process. Many high-level results have been rigorously proven going all the way back to first principles. That’s the entire goal of the metamath project. If you haven’t ever stumbled on this site, it will blow your mind. Click on the previous link, but don’t get too lost. Come back and read the rest of my post!

I’ve reread his blog post multiple times, and the articles he linked to. And I just can’t figure out what he could possibly mean by this. It sounds like Devlin thoroughly understands what the metamath project is all about, and he’s very familiar with proof-checking and mathematical logic. So he definitely isn’t writing his post out of ignorance–he’s a smart guy! Again, I ask, can anyone help me understand?

I know that a statement is only proven true relative to the axioms of the formal system. If you change your axioms, different results arise (like changing Euclid’s Fifth Postulate or removing the Axiom of Choice). And I’ve read enough about Gödel to understand the limits of formal systems. As mathematicians, we choose to make our formal systems consistent at the expense of completeness.

Is Devlin referring to one of these things?

I don’t usually make posts that are so confrontational. My apologies! I didn’t really want to post this to my blog. I would have much rather had this conversation in the comments section of Devlin’s blog. I posted two comments but neither one was approved. I gather that many other comments were censored as well.

Here’s the comment I left on his blog, which still hasn’t shown up. (I also left one small comment saying something similar.)

Prof. Devlin,

You said you got a number of comments like Steven’s. Can you approve those comments for public viewing? (one of those comments was mine!)

I think Steven’s comment has less to do with computer *generated* proofs as it does with computer *checked* proofs, like those produced by the http://us.metamath.org/ community.

There’s a big difference between the proof of the Four Color Theorem, which doesn’t really pass our “elegance” test, and the proof of e^{i\pi}=-1 which can be found here: http://us.metamath.org/mpegif/efipi.html

A proof like the one I just linked to is done by humans, but is so rigorous that it can be *checked* by a computer. For me, it satisfies both my hunger for truth AND my hunger to understand *why* the statement is true.

I don’t understand how the metamath project doesn’t meet your criteria for the filling in process. I’ll quote you again, “The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?”

What is the metamath project, if not the “filling in” process?

John

If anyone wants to continue this conversation here at my blog, uncensored, please feel free to contribute below :-). Maybe Keith Devlin will even stop by!

Looking back on 299 random walks

This is my 300th post and I’m feeling all nostalgic. Here are some of the popular threads that have appeared on my blog over the last few years. If you’ve missed them, now’s your chance to check them out:

Thanks for randomly walking with me over these last few years (though, some say it’s a “drunken walk” 🙂 ). Either way, I’ll raise a glass to another 300 posts!

Random Walks Mural

I’ve been meaning to give the back wall of my classroom a makeover for a while. This summer I finally found some time to tackle the big project. I took down all the decorations and posters. I fixed up the wall and painted it a nice tan color. Then, I let loose the randomness!

and some added, inspirational, text :-)I struggled with what the new mural would be–I’ve thought about it over the last few years. I considered doing some kind of fractal like the Mandelbrot Set. But it should have been obvious, given the name of my blog!! What you see in the picture above is three two-dimensional random walks in green, blue, and red. In the limiting case, one gets Brownian motion:

Brownian motion of a yellow particle in a gas. (CCL)

I honestly didn’t know what it was going to look like until I did it. I generated it as I went, rolling a die to determine the direction I would go each time. I weighted the left and right directions because of the shape of the wall (1,2=right; 3,4=left; 5=up; 6=down). For more details about the process of making it, here’s a documentary-style youtube video that explains all:

Actually, I lied–it doesn’t tell “all.” If you really want to know more of my thought process and some of the math behind what I did, watch the Extended Edition video which has way more mathematical commentary from me. I’ve also posted the time lapse footage of the individual green, blue, and red. Just for fun, here’s an animated random walk with 25,000 iterations:

Wikipedia, Creative Commons License

A two-dimensional random walk with 25,000 iterations. Click the image for an animated version! (CCL)

I think the mural turned out pretty well! It was scary to be permanently marking my walls, not knowing where each path would take me, or how it would end up looking. At first I thought I would only do ONE random walk. However, the first random walk (in blue) went off the ceiling so I stopped. And then I decided to add two more random walks.

In retrospect, it actually makes complete sense. I teach three different courses (Algebra 2, Precalculus, and Calculus) and I’ve always associated with each of theses courses a “class color”–green, blue, and red, respectively. I use the class color to label their bins, to write their objective and homework on the board, and many other things.

The phrase “Where will mathematics take you?” was also a last-minute addition, if you can believe it. There just happened to be a big space between the blue and red random walks and it was begging for attention.

good question!What a good question for our students. The random walks provide an interesting analogy for the classroom. I’d like to say I’m always organized in my teaching. But some of the richest conversations come from a “random walk” into unexpected territory when interesting questions are raised.

Speaking of interesting questions that are raised, here are a few:

  • Can you figure out how many iterations occurred after looking at a “finished” random walk? Or perhaps a better question: What’s the probability that there were more than n iterations if we see m line segments in the random walk?
  • Given probabilities p_1, p_2, p_3, p_4 of going in the four cardinal directions, can we predict how wide and how high the random walk will grow after n iterations? Can we provide confidence intervals? (might be nice to share this info with the mural creator!)
  • After looking at a few random walks, can we detect any bias in a die? How many random walks would want to see in order to confidently claim that a die is biased in favor of “up” or “left”…etc?

Some of the questions are easy, some are hard. If you love this stuff, you might be interested in taking a few courses in Stochastic Processes. Any other questions you can think of?

Where will math take you this coming academic year? Welcome back everyone!

Probability questions from Tanton

Confession: I still haven’t figured out how to use twitter. (Feel free to follow me @mrchasemath, though!) I always feel like I’m drinking from a fire hose when I get on the site–I can’t keep up with the twitter feed, so I don’t even try.

But when I do, I love seeing what people are posting. Here’s a great math problem from James Tanton. He always has such interesting problems!

Feel free to work it out yourself. It’s a fun problem! Here are my tweets that answer the question (can you follow my work?):

It’s hard to do math with 140 characters! 🙂

Here’s his follow-up question which has still gone unanswered. My approach to the first problem won’t work here, and I want to avoid brute-forcing it. (Reminds me of my last post!) Any ideas?

Let us know in the comments…or tweet @jamestanton!

Four ways to compute a probability

I have a guest blog post that appears on the White Group Mathematics blog here. (My first guest post!) Here’s a taste:

One thing I love about math, and particularly combinatorics and probability, is the fact that many methods exist for solving the same problem.

Each method may have its advantages. The advantage might be conceptual (as in “this makes most sense to me”) or the advantage might be computational (as in “this is the fastest way to do it”).

Discussing the merits of different methods is exactly what math class is for!

For example, check out this typical probability question that could appear in a Precalculus course:

The Texas Ranger pitching staff has 5 right-handers and 8 left-handers. If 2 pitchers are selected at random to warm up, what is the probability that at least one of them is a right-hander?

In fact, it’s one I use in my own Precalculus course and it generated a great class discussion. In teaching it this past year, I ended up showing students four ways to do the problem this year! Here they are…

For the epic conclusion of this post, visit White Group Mathematics. 🙂

Math on Quora

quora iconI may not have been very active on my blog recently (sorry for the three-month hiatus), but it’s not because I haven’t been actively doing math. And in fact, I’ve also found other outlets to share about math.

Have you used Quora yet?

Quora, at least in principle, is a grown-up version of yahoo answers. It’s like stackoverflow, but more philosophical and less technical. You’ll (usually) find thoughtful questions and thoughtful answers. Like most question-answer sites, you can ‘up-vote’ an answer, so the best answers generally appear at the top of the feed.

The best part about Quora is that it somehow attracts really high quality respondents, including: Ashton Kutcher, Jimmy Wales, Jermey Lin, and even Barack Obama. Many other mayors, famous athletes, CEOs, and the like, seem to darken the halls of Quora. For a list of famous folks on Quora, check out this Quora question (how meta!).

Also contributing quality answers is none other than me. It’s still a new space for me, but I’ve made my foray into Quora in a few small ways. Check out the following questions for which I’ve contributed answers, and give me some up-votes, or start a comment battle with me or something :-).

And here are a few posts where my comments appear: