Unsolved Problems in Math Class

Credit: Randall Munroe. CC BY-NC 2.5

A few years ago, I directed a high school summer math program. Half the day was devoted to exploring the delights of modular arithmetic—we ended the summer with a cake decorated with Fermat’s Little theorem!—and half to learning to program in Python, with number theory questions as motivation. One Friday afternoon, we included these questions in the programming part of the day.

Goldbach conjecture: Any even number larger than 2 is the sum of two prime numbers.

  • Is there a counterexample to this conjecture for an even number less than 10,000
  • Prove this conjecture.

Collatz conjecture: Choose some number a0.
Define an by a
n=3an-1+1 if an-1 is odd or an-1/2 if an-1 is even.
Then an will be 1 for some n.

  • Is there a counterexample to this conjecture for a0<10,000?
  • Prove this conjecture.

Perhaps it was a tiny bit evil to give these longstanding open problems to high school students without warning them, but it was a lot of fun to watch them come up with programs to search for counterexamples and brainstorm about ways of approaching the proofs. (And yes, we did eventually tell them the questions were still open. We didn’t want to ruin their weekends completely!)

Math teachers Annie Perkins and Sam Shah have written about the benefits of exposing kids to advanced math concepts early rather than waiting until they’ve mastered all the easy stuff. If you too would like to torture your students kindle your students’ curiosity and challenge their intuition, with unsolved math problems, there are lots of places to go for inspiration.

The MathPickle site (tagline: “Put your students in a pickle!”) has puzzles organized by grade level, board game suggestions, and a blog. I’ve seen this site mentioned in a few places, including a discussion on Dan Meyer’s blog.

Lior Patcher has a list of suggestions for how to use unsolved problems in K-12 classrooms at his computational biology blog Bits of DNA. I was especially excited to see a question involving Namibia’s mysterious “fairy circles,” circular patches of bare ground surrounded by vegetation. It’s nice to see some modeling and applied math get some love there. Why should number theory have all the fun?

Mike Lawler often discusses advanced and unsolved problems with his kids, and the Collatz conjecture has made several appearances on his blog. In his most recent post on the topic, his kids make music with John Conway’s “amusical” variation of the problem. (As a violist, I’m delighted that one of them does so in alto clef!)

Ben Braun writes about using unsolved problems in his college math classes at the AMS math education blog On Teaching and Learning Mathematics. He highlights some of the benefits for his students, including mindset shifts away from answer-getting and toward seeing failure as part of mathematical productivity.

It goes without saying that your students probably won’t solve the Goldbach conjecture or get a definitive answer about fairy circles in one or two class periods, but you never know. Some open problems might end up being easy to solve, or at least easier than we might think. The Gödel’s Lost Letter and P=NP blog has a fun post about open problems with short solutions.

Posted in Math Education | Tagged , | 4 Comments

Just In Time For The Holidays

When it comes time to breaking bread on the holidays, just remember that 58% of people in the room is more than half the people in the room. Image via Flickr CC Satya Murthy.

Well, I’ve done you a favor and shielded you from these juicy mathematical and political morsels until after Thanksgiving. A recent NPR/PBSNewshour/Marist poll showed that 58% of people were not looking forward to discussing politics at their holiday table, while 31% said they were looking forward to it. Now, I know this isn’t really how these things work, but my preferred reading of this is the following: if you are one of the 31% who is excited to talk about politics while eating your turkey, look to your left, and look to your right. Neither of those people want to talk to you.

So I’ve spared you from being that guy. But spare no more! I have some mathematical and political news I want to talk about!

Nate Silver, editor-in-chief of FiveThirtyEight finished his 11 part series The Real Story of 2016, examining the misunderstood polling, flawed mathematical modeling, biased human intuition, and journalistic errors that went into the coverage and execution of the 2016 election. I just finally read through the whole thing today and it is an eye-opening interrogation of the intersection of math and media. Silver gives a thorough analysis of what went wrong in the Electoral College math for Clinton and what went right for Trump.

In part VII, Clinton’s Groundgame Didn’t Cost Her The Election, Silver provides some compelling data from after-the-fact regression analysis to point out the unavoidable importance of demographics in election outcomes. We can pick apart the ground strategies and die speculating that if-only-she’d-gone-to-Wisconsin, but Silver makes a pretty clear case that it would have made almost no difference.

In the final installment, The Media Has A Probability Problem, Silver talks about how easy it is for journalists and consumers to misinterpret probabilities. If one candidate has an 85% chance of winning, that just means that the modelers have spun out all likely scenarios and in 85% of possible scenarios the candidate will win. Which means she still loses in 15% of them. And this is totally different from polling at 85. And we mathematicians obviously get that, but I think when people glance at infographics those numbers can be quickly and easily conflated and internalized in the wrong way.

Understanding what numbers mean and why they work is so important and sometimes so difficult.

Since we’re already in the political mode, if you haven’t read it yet, I implore you to go read Cathy O’Neil’s Weapons of Math Destruction. It’s a trove of infuriating examples of data being used and abused to the detriment of democracy and the citizens of the United States. If you can’t see yourself reading an entire book in the next few weeks (I know, it gets hectic, but come on, treat yo self) then at least read O’Neil’s recent New York Times oped about the looming specter of big data and algorithms.

So, I’m sorry I didn’t share this with you sooner so that you could simultaneously dazzle your Aunt TeeTee with your pumpkin pie recipe and all the facts she so desperately craves. The good news is, you now have all of this in your back pocket so that you can be sure to win every conversation you have at every holiday party in the next month.

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Application Advice for Students, Job-Seekers, and Recommendation Letter Writers

I really didn’t know what I was doing when I applied for graduate school, and I am thankful for the assistance of the professors at my undergraduate university who helped me and the luck that got me into a few schools, including one that ended up being a good fit for me. But I could have used more help at all stages of the process.

If you’re in a similar position, Susannah Shoemaker’s posts on the subject for the AMS Graduate Student blog might be useful. In her first post, she suggests starting with some introspection about why you want to go to grad school. I particularly saw myself in that part of the post. She writes, “If I’m being completely honest, I applied to grad school in part because it was a well-defined, familiar path: more schooling (a known quantity), followed by a career in academia, which I imagined would be full of fulfilling teaching interactions and, importantly, blissfully free of business-wear and rigid 9 am start times.” In the second post, she focuses on crafting a good statement of purpose and advocating for yourself once the acceptances start coming in.

A few years later, you may be on the market for an academic job. Zsuzsanna Dancso, who writes the blog Math, Chocolate & Circus has some advice academic job applications and interviews. I appreciated the way she framed her advice. (Also, I kind of want a pizza now.)

I know you’ll probably be applying for a hundred jobs, but this is advice on how to get one of the few you actually want. I’m going to tell you to put in a lot of effort, and you should focus that effort where it counts. When I was applying widely, I had this rule of thumb: if I’d rather have delivered pizza in Toronto, I would not apply. Imagine living in your favourite location working in a boring unremarkable job; this is a thing you are allowed to do. As you’re writing a cover letter for a tenure track position that will make you strictly less happy, remind yourself that it’s ok not to.

It’s not just applicants who can use advice. Sexism in academic letters of recommendation has been a hot topic for several years, but I still hear about sexist letters making their way to the eyeballs of hiring committee members. Recommendation letter writers, you want to avoid embarrassing yourself and hurting applicants’ chances by accidentally writing a sexist letter. The Astrobetter blog has some letter writing advice for avoiding unconscious sexism. There’s also a handy poster (pdf) from the University of Arizona commission on the status of women. This online gender bias calculator will take the content of your letter and point out gendered words to you. It’s a blunt tool, but it should help you fend off some glaring problems. There is some more general advice about writing academic letters from the Science Professor blog and Inside Higher Ed. If you’ve got some recommendation letter writer’s block, Natalia Lukina has a list of words and phrases that might help you get the process flowing.

If you end up staying in academia (by no means the only successful career trajectory; see this post for information about non-academic math jobs), you may one day find yourself with the awesome responsibility of advising students yourself. At the Computational Complexity blog, Lance Fortnow has some advice about advising.

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Bees and Bombs

hue strip, courtesy of David Whyte.

His name is Dave, he used to do physics, now he makes GIFs. Dave is known on Twitter as beesandbombs and he has a Tumblr of the same name.

David Whyte studied theoretical physics as an undergraduate at Trinity College in Dublin, Ireland. He went on to get his PhD in physics, studying foams and soap bubbles. It was at this point that Whyte says he started messing around with math and was really taken by the powerful visualizations in mathematics. On math, Whyte says, “it was the visually accessible courses that grabbed my attention. As soon as I couldn’t understand it visually it eluded my grasp.”

Inspired by the blog dvdp, Whyte decided to take his mathematical visualization game to the next level by making mathematically inspired GIFs. He creates them using an open-source sketchbook software called Processing. The software uses Processing (the language), which is inspired by BASIC and Logo, and is intended to be a first programming language for people at the intersection of visual arts and technology. I’ve never used it myself, but according to Whyte, it was pretty easy to use straight out of the box.

What’s not easy, is using the software to make kickass GIFs. In making his GIFs, Whyte has a few standard tricks he turns to. He says, “find something simple and duplicate it on a grid so that timing depends on placement on the grid. Also offsetting a simple motion based on position.” This week, I’ve been particularly mesmerized by polygon laps, below.

polygon laps, courtesy of David Whyte.

One that he said mathematicians might like to consider is weaving stars, below. It’s all built out of explicit translations and rotations. It may looking like mysterious infinite void, if you look under the hood it’s just linear algebra!

weaving stars, courtesy of David Whyte.

I’ve been looking at Whyte’s GIFs for awhile now, but since I talked to him I’ve started looking at them with a totally different eye. What are the explicit linear transformations that would give me that? And can I ask that on a linear algebra final?

two squares / four triangles. Courtesy of David Whyte

Another of Whyte’s favorite visual tricks is finding clever ways to morph one object into another, like in two squares/four triangles above (which totally makes me think of all the interesting square to triangle disection problems). Whyte says he usually has an idea of what he might want to do, and then he sketches the whole thing out on paper before trying to code it.

You can step up your own GIF game by check out 17 Mathematical GIFs That Are Deeply Soothing by other people. You can follow Whyte on Twitter @beesandbombs and you can be a patron of the arts by supporting him on Patreon.

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Math Education Researchers Deserve Respect

In what has become sadly routine, right-wing news sites started publishing inflammatory articles about a professor whose work they don’t like about two weeks ago. (I am not linking to their stories in this post because they contribute to this scholar’s harassment.) In this case, it was a math education professor at UIUC, Rochelle Gutiérrez. Her faculty profile begins, “Dr Gutiérrez’ scholarship focuses on equity issues in mathematics education, paying particular attention to how race, class, and language affect teaching and learning.” She is being harassed because in a chapter of a book about math education, she wrote that mathematics teaching can reinforce white supremacy and that math has a level of unearned privilege in society, just like whiteness does.

I understand why some mathematicians have had negative reactions towards these statements. Is she calling us racist? We’re good people! We’re not racists. Math isn’t white. By saying that it is tangled up in whiteness, she’s showing us that she’s the Real Racist! One of my best mathematician friends is black…

If someone wandered into an algebra seminar and said everything in it was wrong because a group doesn’t have to have inverses because any collection of anything is a group, mathematicians wouldn’t take their criticism seriously. They would (kindly, I hope) explain that mathematicians use a particular technical definition for that word and that the person needed to learn a little more about the foundations of the subject before lobbing criticism at seminar speakers.

Most mathematicians do not have much training in education or social science research. We don’t always know the terminology, assumptions, or methods in those fields. It’s arrogant to assume we can swoop in and understand education researchers’ work better than the researchers themselves do, especially when our understanding is based on a few inflammatory news articles. Mathematicians absolutely should participate in discussions about math education research and practices, but we should do so with humility and a willingness to do some background research.

Instead of a knee-jerk reaction about Gutiérrez’ work, what if we (and by we I mostly mean mathematicians who are who are not from racial or ethnic groups underrepresented in math or who feel defensive about the idea of white privilege in math) started with the assumption that she has thought about and studied these questions for a long time and probably isn’t a quack? What if we started by assuming people who study math education know more about it than research mathematicians who don’t study math education, that they, like mathematicians, are experts in their fields? Once we do that, how can we learn more about Gutiérrez’ work and what can we do to help her and other scholars who become the targets of harassment campaigns?

I’m glad you asked! In the weeks since Dr. Gutiérrez became a target, many math and math education bloggers have blogged and tweeted (using the hashtag #IStandWithRochelle) about race and equity in math education, particularly Dr. Gutiérrez’ work. Here are some of the articles I have seen. I urge you to read them with an open mind and, even if you disagree with them in the end, try to contribute to the discussion about these issues without feeding into the harassment machine.

Get up to speed on #IStandWithRochelle with this post from the newly launched Equity Mathematics Education blog, which also shared this statement of support for Dr. Gutiérrez from Deborah Ball, math education researcher and president of the American Education Research Association. At the AMS inclusion/exclusion blog, Brian Katz also wrote about this incident in his post Complicit Function Theorem. (His post includes a link to an earlier i/e post about Gutiérrez’ work.)

Math teachers and math education professors who have been influenced by Dr. Gutiérrez’ work have written in support of her. Jennifer Dao worked with Dr. Gutiérrez starting as an undergraduate and writes, “This is the professor who embraces the beauty of mathematics, strives for equity in mathematics education, and recognizes the politics surrounding the teaching of mathematics.” Jose Vilson reminds us that math was never neutral. Matt Felton-Koestler defines some terms and writes a about math and white privilege in a post on his blog. Trevor Warburton writes that “Dr. Gutiérrez’s influence on my work is without equal” and describes some of his reflections about whiteness in math education and her generosity to him when he was starting his dissertation work in math education.

As I wrote at the top of this post, harassment campaigns like this are becoming more and more common. Inside Higher Ed published an article about how to support academics under attack two years ago, and it is still relevant.

Posted in Math Education | Tagged , , | 3 Comments

A Not Too Mathy Math Blog

Lauren Miller’s favorite number is 23. “I really liked being 23, that was the year I decided to become a mathematician,” Miller told me over burgers and beers in Claremont, California this week. After taking a circuitous route through education that took her through costume design, working as an optician, and eventually landing her in a mathematics program, Miller is now a mathematician, librarian, lady-hacker, and blogger at Life By Number.

Life By Number blogger Lauren Miller, photo courtesy of Lauren Miller.

Life by Number is a blog, in the words of Miller, “for people who find math fun.” Whether that’s the students she’s teaching this semester at Lindenwood University, the great math teachers of the #MTBoS, or her own mother, Miller’s goal is to bring math to the level where people are. “I started writing it for me,” she says, “but then I also want to the share interesting things I’ve found, the amazing literary resources that are out there!”

She covers a range of topics, but working on the staff of the St. Louis Public Library, her book reviews are a particularly unique feature to her blog. Written in a chatty and comfortable style, they make me want to hit the stacks and nestle up to a good mathematical biography. She discusses the relative merits of several children’s books on Ada Lovelace (I know, can you believe there are several of them? I’ve been hanging out in the wrong Dewey decimal numbers) as well as some books on fellow mathematicians.

Miller also has a keen interested in math history. In one post she gives a brief history of the axiom of choice, tracing the AC from Cantor to Cohen. As a mathematician who likes to sweep this sort of thing under the rug all the time, it’s fun to see a glimpse at the drama behind it all. Miller places the axiom of choice in context, she says, viewing it in terms of the main players’ “attitudes towards others as well as in communication standards of the time.” It is fun to trace an idea from 19th century Germany all the way to New Jersey.

I met Miller at a Women In Sage workshop where she’s been working on a project in dynamical systems, a topic she studied while completing her Master’s degree at St. Louis University. Miller has been involved in other initiatives for women in programming, including a partnership with Girls Who Code at the the St. Louis public library where she is the adult research and community outreach coordinator. She’s also participated in — and blogged about! — her experiences in Google’s Summer of Code.

Check out the blog, and let Miller know your favorite number. Mine is 7, which apparently makes me not very unique . I just like that it’s prime, congruent to 3 mod 4, and big but not too big. Also, it was at age 7 that I first decided I wanted to be an engineer and dressed up as one for career day. I wore a sensible knee-length skirt and carried physics books around all day. Some of that came true.

Posted in History of Mathematics, Math Education | Tagged , , , , | 2 Comments

Un-Junking your Charts

Junk Charts is a blog by Kaiser Fung, who describes himself as “the Web’s first data visualization critic.” People have been criticizing and prescribing solutions for misleading data visualization for a long time. (How to Lie With Statistics was first published in 1954, when a gallon of gas was 22 cents, a movie ticket was 70 cents, and the average new house was $10,250.00.) I don’t know whether Fung was literally the first to do it on the Web, but his blog has been around for over a decade and has an extensive archive of interesting posts for your perusal.

A found graph displays the density of cats in the vicinity of each part of the park bench. Credit: Evelyn Lamb

When I first saw the title Junk Charts, I assumed it would be a blog that pointed out and made fun of bizarre and misleading graphs and charts. That’s all good fun, but this blog generally takes a less adversarial approach. Fung often examines data visualizations that are pretty good and shows how he would make them even more effective. For example, a recent post shows his suggested tweaks for a Washington Post graphic about voter polarization. The original graphic isn’t ugly or misleading, but the new one makes certain statistics jump out more readily.

Kaiser Fung’s redesigned display of information about how the two parties have diverged in the past few decades. Credit: Kaiser Fung. CC BY-NC-SA 3.0

Some posts start with less successful original material, such as this post discussing a flawed chart about politician approval ratings. On Pi Day 2014, Fung started the #onelesspie initiative to replace pie charts with better charts. (Except when they are self-descriptive, pie charts are mostly bad. Embrace non-pies!) The #onelesspie posts in later years have been entertaining.

The only good pie chart is a self-descriptive one. Credit: Randall Munroe, xkcd. CC BY-NC 2.5.

I don’t have much experience creating data visualizations or working with statistics, so I’ve enjoyed the perspective Fung brings in Junk Charts. Synergistically, while I was writing this post, the Information Is Beautiful website unveiled their Information is Beautiful Awards longlist for this year, which has lots and lots of interesting and visually arresting data displays. I can use my gradually developing chart sense when public voting opens later this month.

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Hacking Cracking & Packing

The original gerrymander courtesy of Wikimedia Commons.

Sometimes the boundaries of voting districts can look really suspicious. If you’ve ever seen Illinois’ 4th Congressional District, you know what I mean. Sometimes there are good reasons for this; communities with common interests may want to vote together. But sometimes the reasons are bad; partisan politicians might be cracking and packing certain demographics. That is, cracking up certain demographic groups and scattering them through the districts and then packing all of the remainder into one often strangely shaped (perhaps dragon shaped?) district to minimize their votes. These are the classic tools of gerrymandering.

In the state of Wisconsin this practice has got particularly bad, and last week the supreme court heard oral arguments in the case of Gill v. Whitford. This case seeks to determine exactly how bad the partisan map rigging in Wisconsin is, and hopefully the outcome will be some sort of consensus on how to recognize and rectify the systematic disenfranchisement that comes with hardcore gerrymandering.

Jordan Ellenberg, a resident of Wisconsin, wrote about the case for The New York Times and explained why this might be of interest to us as mathematicians.

Over the summer, the Metric Geometry and Gerrymandering Group at Tufts University led by Moon Duchin, ran a summer camp where participants developed tools for detecting and understanding gerrymandering. As Duchin often points out, sometimes a weird looking district looks weird for a reason, so it’s important to find out why things look the way they do.

A few reasonable measures have been proposed. One being used in the Wisconsin case is called the Efficiency Gap, and it measures the net wasted votes as a share of the total votes in the state. Wasted votes are all votes cast for the losing candidate and all extra votes for the winning candidate, beyond what was needed to win. But as Olivia Watch illustrates in this graphic explainer for The Nib, efficiency gap can’t tell the whole story.

Another way to measure gerrymandering is to consider, in some systematic way, all possible redistricting schemes in a given state and compare them to what is being used. If the one being used is a significant outlier compared to the others, then it’s probably gerrymandered. A paper that recently appeared on the Arxiv uses this method to expose the badness of Wisconsin’s districting.

Recognizing and measuring gerrymandering is of course a totally different task from actually redrawing the lines in a fair and unbiased way. There are many schemes for this, but my favorite one to explain to strangers (and then watch them get all in a lather) is the shortest split line algorithm. It sort of disregards human interests, but hey, it makes really good looking maps!

This will not be the first time that math has been used to fairly distribute representation. Apparently Thomas Jefferson devised a sensible algorithm for assigning seats in congress by state. I imagine Jefferson might have something to contribute to our current discussion.

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The arXiv, Curated

The arXiv: a mathematician’s favorite preprint server and semiproductive procrastination enabler. Don’t get a morning newspaper? You can enjoy your breakfast over the arXiv submissions for your favorite area of math. Stuck on that lemma? Might as well surf on over and see if you missed any important breakthroughs in your field. The arXiv contains multitudes, and that’s exactly what the website arxivist.com aims to help you with.

Anton Lukyanenko, a mathematician at George Mason University, started arxivist while he was a postdoc at the University of Michigan to help people sort through the flood of arXiv submissions and find papers that might be of interest to them. There can be dozens of arXiv submissions every day in any given sub-area, and a mathematician (/physicist/computer scientist/quantitative biologist/etc.) who browses through a few of them can end up overwhelmed. He says he found himself checking for certain keywords and author names and wondered if the process could be automated. After setting it up for himself, he decided to work on making a public version of the website. “It was fun to do a hands-on project alongside with my much more theoretical research,” he wrote in an email.

To use arxivist, you sign in using your Google account and start rating papers by giving the site’s suggestions thumbs-up or thumbs-down. The more you rate, the better the site’s recommendations will become. If you’d like, you can sign up for a daily email with some suggestions for the day. The site is currently in beta, but it has worked smoothly for me so far.

I’ve had fun rating papers on arxivist and seeing the suggestions respond to my ratings, but my favorite part is probably the “arXiv catch of the day” from the arxivist Facebook page. As the name suggests, it’s a fun paper to peruse every day. If you enjoy commutative mathematical phrases as much as I do, “Quasihyperbolic geodesics are hyperbolic quasi-geodesics” should be right up your alley.

Lukyanenko says there are some improvements to the user interface and algorithms in the works, but right now he’s interested in building the number of users and hearing what features are most appealing to them.

“It’s been really cool seeing the arxivist spread,” says Lukyanenko. “It’s also been exciting (and a bit terrifying) to hear that some of my more senior colleagues are using the system.”

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Exploding Dots For Global Math Week

A man learns math by exploding dots. Image courtesy of Francisco Barberis via FlickrCC

If you hang around the #MTBoS long enough you can’t help but notice something called exploding dots. Today in a quite moment I took some time to dig in, and I am not disappointed.

Exploding dots is the focus project of Global Math Week, happening Oct. 10-17, 2017. Kind of like Hour of Code, the aim of the Global Math Project is to get hundreds of thousands of people all over the world doing math together at the same time. The architect behind exploding dots is MAA mathematician-at-large James Tanton who hosts G’day Math a blog full of problems, lessons ideas, and mathematical essays.

Today I watched this video of Tanton presenting exploding dots to a general audience. First off, he is a mesmerizing lecturer and the video is worth watching if just as pedagogy inspo. But on top of that, this thing he presents is just so much fun. The idea behind exploding dots is a simple visual representation of base 2, base 10, and eventually base x. The basic idea is that you fill a row of boxes with dots, and the dots represent 1, x, x2 and so on depending which box they lie in. If one box gets full, the dots explode, and move over to the next box. Using this representation, Tanton builds arithmetic from the ground up, starting with addition and multiplication, and then adding in subtraction and division — even polynomial long division!

I won’t say any more about the precise details, since it’s better to explain visually. Just go watch the video.

One thing I appreciate immensely about his presentation is that he’s very clear that the point of this isn’t to get answers (we can easily do multiplication on our iPhones), but rather it’s figuring out how to develop a system intuitively and rationally to make it do what you want.

The only thing that is just niggling my brain a little bit is the problem with convergence of power series. In the video, Tanton shows how easy it is to use his exploding dots method to write


giving an infinite power series representation to the rational function 1/(1-x). But this really gives the impression that you can put any number in for x and the equality holds, like for example, x=2. But this would give


in other words


which of course can’t be true, and I imagine it wouldn’t take long for a clever student to pick up on that. I understand that Tanton is presenting this as an elementary alternative to the usual presentation, which is great. But I’m just curious if there’s some obvious way in his construction to see when a function is actually going to be equal to its power series representation. I’m sure the internet will let me know.

You can sign up to be part of Global Math Project and check out all of the great lessons and resources for you, your students, your children, or any unsuspecting friends who are foolish enough to go to happy hour with you on October 10th.

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