Tuesday, May 9, 2017

1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13, ...

Sequences are ordered progressions of numbers or other objects, as in the examples

1, 2, 4, 8, 16, …


A, B, AA, BB, AAA, BBB, …. 

In school mathematics, sequences are often called "patterns." I don't like that usage, because it's terribly limiting. Mathematical patterns can be found in visual designs, crystals, the multiplication table, or some totality of facts…patterns aren't just about sequences! You'll often hear people say, "Mathematics is the study of patterns." They don't mean, "Mathematics is the study of sequences."

My colleague William McCallum has a useful dictum: "Patterns are a tool, not a topic." For example, patterns in the multiplication table could be used as a tool for teaching about the properties of operations; patterns in the sequence 1, 2, 4, 8, 16 could be used as a tool for teaching about exponential functions. Whatever work is done with sequences at a given grade ought to transcend 'patternology' to intentionally build up students' strengths in the most important mathematical topics at each grade level.


Lately in idle moments I'll try to pass the time by thinking up a sequence of numbers that isn't yet included in the Online Encyclopedia of Integer Sequences (OEIS). It's a fun game—like trying to think of a notable topic that isn't in Wikipedia.

This was my first attempt:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 30, 41, 50, 61, 70, 81, 90, 111, 200, ...

The rule of this sequence is that, beginning with zero, each successive number must share no digits with the previous number—and be as small as possible subject to that constraint. For example, the number after 200 will be 311, because 311 is the smallest number greater than 200 that doesn't have 2 or 0 as a digit.

This is sequence A030283 in OEIS.

To create the sequence whose initial terms are in the title of this post, I considered a sequence of regular polygons with sides of unit length (triangle, square, pentagon, hexagon, etc.). Choosing a standard position for the polygons (centered at the origin), and also choosing a standard orientation (with the top edge horizontal), I counted how many grid points (m, n) were inside the polygon or on its boundary. (Here m and n are integers.)

Here are some diagrams showing the first eleven values 1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13:

My sequence doesn't appear to be in OEIS. There's a partial match with A219844, but the two sequences aren't the same.


When you look at the numbers 1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13, what do you notice? Will the feature you noticed hold true for the entire sequence, out to infinity?

Feel free to write any observations, conjectures or arguments down in the comments. Also, feel free to put a sequence of your own in the comments!


My daughter and I talked a bit about sequences during Saturday School a few weeks ago. That morning, my daughter was writing in a workbook she'd brought home from school. Looking across the table, I noticed that whoever had written the workbook had misunderstood 4.OA.5, a standard that involves sequences. In an effort to rescue the math, I quickly sketched the following problem:

Start with 256. At each step, divide by 2. Repeat forever!

a) Show 4 steps of the pattern (sequence)

b) Prove or disprove: every number in this pattern (sequence) is a whole number.

(As you can see, I took this opportunity to reinforce fractions and calculation of quotients. Also, note that the rule for the sequence is given. This is a math problem, not an  IQ test.)

By making the starting number a large power of 2, I had hoped to lure her into believing that all of the numbers in the sequence would be whole numbers. That gambit didn't work, however—she saw from the outset that the values would eventually drop below 1, and she proceeded to show this by generating sufficiently many values. (Actually, more than enough values! I then drew her attention to the second of the two arguments below.)

My daughter had fun solving this, and we enjoyed talking about it. What's cooler than infinity? Done right, I think 4.OA.5 can prompt some very good mathematics. I expect it has also helped publishers and school systems hang onto old-style, non-aligned "pattern" work in the curriculum.

Thursday, May 4, 2017

My Own Struggles With Grammar

Articles on grammar seldom feature the author's own mistakes or shortcomings. The genre typically delivers advice, not confession. But in order to balance my other posts on the topic, I thought I'd list some areas where I tend to struggle.

Words That Give Me Trouble
  • Embarrass is hard for me to spell (one r or two?).
  • There's a dead neuron in my brain where the proper spelling of maybe is supposed to be stored: sometimes I write mabey the first time!
  • Are dubious and doubtful synonyms? I can't tell.
  • I only recently learned the distinction between masterful (done in a domineering way) and masterly (done in the way an expert would do it). Almost nobody marks you down for using the words interchangeably, and the third definition of masterful in the American Heritage 4th Edition makes masterful and masterly synonymous. However, the conceptual distinction itself is real, and I generally favor making distinctions so I'll give this one a try.
  • The word publicly just looks wrong to me! We don't write mysticly, franticly, or graphicly…. It should just be publically, a spelling I see in online writing from time to time.

Rules I Refuse To Follow 

  • Some of the classical rules (or "rules") of grammar. I split infinitives, occasionally end sentences with prepositions, and often use who for whom. (On these and some other matters I'm with Steven Pinker.) 
  • The rule about capitalizing words in titles. It's absurdly intricate…there's an entire website devoted to figuring it out! It also leads to typographic ugliness, as in

In the early years of this blog, I tried to respect the classical rule; an example would be "Reverie on the Principle of Equivalence" from 2007. Later I gave up and adopted the simple rule of capitalizing every word in the title. See for example the recent post "In Honor Of William Wootters On The Occasion Of His Retirement." This might be wrong, but even so I prefer the simplicity of my rule.

Emerging Usage I Have Embraced

  • Internet abbreviations. In informal email I'll use IIRC, BRB, nvm, and others.
  • They as singular pronoun. A few years ago I began losing patience with the construction "he or she." In email, I started abbreviating this to "s/he." Finally I gave up and started allowing myself to use they as a singular pronoun. (If I can easily revise it away then I do, but I don't work as hard at that anymore.) The big news in copy editing so far in 2017 is that the Associated Press style guide will now allow for limited use of the singular theyHere is Grammar Girl with the story.

Tuesday, May 2, 2017

Revisiting My Case Against Trump

In my post just before the election, "Why I'm With Her," I offered four reasons to vote for Hillary Clinton. I don't do a lot of overt politicking on this blog—my own politics are complicated anyway—but in this case, I was definitely writing to persuade. I was hoping to convert just one voter from non-Clinton to Clinton, and I was hoping to convince just one wavering Clinton voter to show up on election day. I forwarded the post to some friends and relatives, which led to some good conversations.

I wasn't enthusiastic about voting for Clinton, but I swallowed my reservations and titled the post with affirmative language, rather than more grudging language along the lines of "Why We Mustn't Elect Donald Trump." An anti-Trump title would have described my feelings better, but I didn't think that a negative message would bring a dissatisfied person out to vote or persuade a third-party voter to pull the lever for Clinton instead.

In writing the piece, I worked to craft a set of points that would be almost inarguable. I wanted there to be no good comeback to these items. I judged, for example, that even a Clinton hater would have to grant that Clinton's temperament is preferable to Donald Trump's.

After the election, I showed my list of reasons to a friend—a Republican who didn't vote for Trump or Clinton. My friend agreed with all of the points except one, the one about the alt-right vs. the Marxist Left. If my audience had been Republicans in particular, then I wouldn't have included that point.

Of course, one can agree with all of the individual points and still not believe that they add up to a yes vote for Clinton. I tried to make important points, and briefly underline their importance. Still, a person might have reasons of their own that they feel are more important. Some of the most persuasive essays I read during the campaign aired pro-Trump arguments and countered them directly. (Here was an example.)

Nobody can know how Hillary Clinton would be governing today, so there's no way to know whether my arguments were "right." Conversely, documenting the evidence against Trump in each of these categories would be a big job that I can't undertake. But below I list the reasons once again. They still seem right to me—perhaps even more than they did a hundred-odd days ago.

One writer I follow appears to be rethinking things. Andrew Sullivan recently published a thought-provoking piece titled "Maybe America Wasn't Crazy to Elect Donald Trump." Sullivan is a longtime Clinton critic who voted for her anyway and was as attuned as anybody to the downsides of Donald Trump. He argues now however that it might have been worse had Clinton been elected:
I still would never have risked putting this menacing clown into the Oval Office. But in the long run, if catastrophe doesn’t strike, it might even be better for the future health of our politics that Clinton is not president. 
Read Sullivan's case here.


1. She has a better temperament. The President's job is stressful, so I think temperament matters. Donald Trump seems excitable and impulsive, which in a President makes me nervous. Hillary Clinton seems like a calmer, more deliberate person.

2. Her base is more manageable. The alt-right is nasty enough as a fringe movement. If Trump wins, the alt-right will have a seat at the table in policy discussions.

Bad things would also happen if the Marxist left were to grow in power, but the situation isn't symmetrical. The relationship between Clinton and progressive activists isn't nearly as cozy as the relationship between Trump and the alt-right. And the Republicans in Congress won't roll over for the Marxist left the way they will for the alt-right.

3. She is more fact-based and can listen better. I don't like all of Hillary's policies, and I especially don't like her interventionist instincts. But I think she is capable of weighing the arguments of critics and capable of adjusting her plans in response to facts. Donald Trump can't take any criticism, and facts don't inform his plans at any stage.

4. Criticism of her seems overblown, while criticism of him is just. Matt Yglesias's take on the email thing is pretty much how I look at it, and for every bad trait of Hillary's, Trump has the same trait except worse (12). The record of Trump's bad character spans decades. He is a small, insecure person with a cruel streak and a terrible lusting after power for its own sake. Lots of people are trying to explain how it happened—but however it happened, this year the Republican party nominated a candidate who deserves to lose.