Category Archives: Math on the web

I’m back

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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!

What does it mean to truly prove something?

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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 GĂśdel 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!

Catch yourself up on the world of origami

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Have you been doing other things and failed to notice the origami world evolve without you? Have you fallen asleep and been left behind? If you want to get caught up on what you’ve been missing in the world of origami, I suggest you visit Hannah’s origami blog, A Soul Made of Paper. She’ll have you caught up in no time.

I especially like giving her blog a shout-out because Hannah is a student of mine. She often comes by my classroom to show me her latest paper creations. I like origami, and I’ve dabbled in it–stuck my toe in the stream, if you will–but Hannah is like a scuba diver in the origami world. She’s loves modular origami, but she’s also great at the artsy curved creations (like origami roses), textures, and tessellations.

If you haven’t clicked over to her blog yet, here are a few more pictures to whet your appetite.

Go get lost at A Soul Made of Paper. Maybe you can join her other 3000 followers on tumblr :-).

*All of the above are Hannah’s pieces and Hannah’s photos.

Looking back on 299 random walks

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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!

What does a point on the normal distribution represent?

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Here’s another Quora answer I’m reposting here. This is the question, followed by my answer.

What does the value of a point on the normal distribution actually represent, if anything?

 

It’s important to note the difference between discrete and continuous random variables as we answer this question. Though naming conventions vary, I think most mathematicians would agree that a discrete random variable has a Probability Mass Function (PMF) and a continuous random variable has a Probability Density Function (PDF).

The words mass and density go a long way in helping to capture the difference between discrete and continuous random variables. For a discrete random variable, the PMF evaluated at a certain x gives the probability of x. For a continuous random variable, the PDF at a certain x does not give the probability at all, it gives the density. (As advertised!)

So what is the probability that a continuous random variable takes on a certain value? For example, assume a certain type of fish has length X that is normally distributed with mean 22 cm and standard deviation 1.6 cm. What is the probability of selecting a fish exactly 26 cm long? That is, what is P(X=26)?


The answer, for any continuous random variable, is zero. More formally, if X is a continuous random variable with support \mathcal{S}, then P(X=x)=0 for all x\in\mathcal{S}.

For the fish problem, this actually does make sense. Think about it. You pull a fish out of the water which you claim is 26 cm long. But is it really 26 cm long? Exactly 26 cm long? Like 26.00000… cm long? With what precision did you make that measurement? This should explain why the probability is zero.

If instead you want to ask about the probability of getting a fish between 25.995 and 26.005 cm long, that’s perfectly fine, and you’ll get a positive answer for the probability (it’s a small answer :-).

Let’s return to the words mass and density for a second. Think about what those words mean in a physics context. Imagine having a point mass–this is in an ideal case–then the mass of that point is defined by a discrete function. In reality, though, we have density functions that assign a density to each point in an object.

Think about a 1-dimmensional rod with density function \rho(x)=x, x\in (0,10). What is the mass of this rod at x=5? Of course, the answer is zero! This should make intuitive sense. Of course, we can get meaningful answers to questions like: What is the mass of the rod between x=5 and x=6? The answer is \int_5^6 xdx=5.5.

Does the physical understanding of mass vs density clear things up for you?

How do you expand âˆš(a+b)?

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This is a question that was recently asked on Quora:

it’s easy to expand
(a+b)^2 = a^2+2ab+b^2 or
(a+b)^3=a^3+3ab^2+3a^2b+b^3
or some other (a+b)^n but what about (a+b)^{1/2} aka. \sqrt{a+b}

Here’s my answer:

Just have Wolfram|Alpha do it for you :-).

But if you were on a desert island without access to Wolfram Alpha, here’s how you might think it through:

Are you already comfortable with the Binomial Theorem? Here it is again, but stated in a particular way that I think we’ll like.

\left(x+1\right)^r=1+rx+\frac{r(r-1)}{2!}x^2+\frac{r(r-1)(r-2)}{3!}x^3+\cdots

Look at it and make sure you understand it, and verify that it really is equivalent to the formulation of the Binomial Theorem you know.

Now, for the big trick. It turns out the above statement holds true not for just r=1,2,3,\ldots but for all real r. The only catch is that this often results in an infinite series. (These series results can also be obtained by Taylor expansion.)

In particular, it works for r=1/2:

\left(x+1\right)^{1/2}=1+\frac{1}{2}x+\frac{1/2(1/2-1)}{2!}x^2+\cdots

\left(x+1\right)^{1/2}=1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3+\cdots

Now, rewriting your original expression (a + b)^{\frac{1}{2}} as \sqrt{b}\left(a/b+1\right)^{1/2} gives

\sqrt{b}\left(1+\frac{1}{2}\left(\frac{a}{b}\right)-\frac{1}{8}\left(\frac{a}{b}\right)^2+\frac{1}{16}\left(\frac{a}{b}\right)^3+\cdots\right)

=\sqrt{b}+\frac{a}{2\sqrt{b}}-\frac{a^2}{8b^{3/2}}+\frac{a^3}{16b^{5/2}}+\cdots

which is the same result Wolfram Alpha will spit back.

Hope that helps!

In Defense of Calculus

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In the following article, I expand and clarify my arguments that first appeared in this post.

A colleague recently sent me another article (thanks Doug) claiming that Statistics should replace Calculus as the most important math class for high school students.

Which peak to climb? (CCL, click on image for source)

The argument usually goes: Most kids won’t use Calculus. Statistics is more useful.

As you might know already, I disagree that the most important reason for teaching math is because it is useful. I don’t disagree that math is useful. Math is not just useful, but essential for STEM careers. So “usefulness” is certainly one reason for teaching math. But I don’t think it’s the most important reason for teaching math.

The most important reason for teaching math is because it is beautiful and eternal. Math is the single place in school where students can find deductive certainty and eternal truth. Even when human activity ceases, math will persist. When we study math, we tap into something bigger than ourselves. We taste the divine!

We are teaching students to think deductively—like a mathematician would. This is such an important area of knowledge for students to explore. They need to know what it means to prove something. A proof provides a kind of truth that is unattainable in other subjects, even the hard sciences. At best, the scientific method is still just guesses compared to math.

This is the most important thing we pass on to our students. Though some will, most of our students will not directly use the math we teach. This is actually true about every subject in high school. Most students will not remember the details of The Great Gatsby or remember the chemical formula for Ammonium Nitrate. But we do hope they learn the bigger skills: analyzing text and thinking scientifically. In math, the “bigger skills” are the ones I outlined above—proof, logic, reasoning, argumentation, problem solving. They can always look up the formulas.

Math is a subject that stands on its own and it is not the servant of other subjects. If we treat math as simply a subject that serves other subjects by providing useful formulas, we turn math into magic. We don’t need to defend math in this way. It stands on its own!

Calculus = The Mona Lisa

If students can take both Statistics and Calculus, that is ideal. But if I had to choose one, I would pick Calculus. The development of “the Calculus” is one of the great achievements of mankind and it’s a real crime to go through life never having been exposed to it. Can you imagine never having seen The Mona Lisa? Calculus is like the Mona Lisa of mathematics :-).

Probability questions from Tanton

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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

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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. 🙂

MAA Distinguished Lecture Series

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If you live in the DC area and you like math, you have no excuse! Come to the MAA Distinguished Lecture Series.

These are one-hour talks, complete with refreshments, all for free due to the generous sponsorship of the NSA. The talks are at the Carriage House, at the MAA headquarters near Dupont Circle.

Here are some of the great talks that are on the schedule in the next few months (I’m especially excited to hear Francis Su on May 14th).

I’ve been to many of these lectures and always enjoyed them. Robert Ghrist‘s lecture was out of this world (here’s the recap, but no video, audio, or slides yet) and was so very accessible and entertaining, despite the abstract nature of his expertise–algebraic topology.

And that’s the wonderful thing about all these talks: Even though these are very bright mathematicians, they go out of their way to give lectures that engage a broad audience.

Here’s another great one from William Dunham, who spoke about Newton (Dunham is probably the world’s leading expert on Newton’s letters). Recap here, and a short youtube clip here:

(full  talk also available)

So, if you’re a DC mathophile, stop by sometime. I’ll see you there!