Infinitesimal Land, Or How Infinitesimals Make Calculus Proofs More Natural

I'm writing the 4th book in the series of Althea's MathMysteries, Althea and the Mysteries of Calculus. (I've written the first two, and they still need more beta readers before we publish them. If you're interested, email me at altheasmathmysteries on gmail. The 3rd isn't written yet.) 

I taught calculus almost every semester for 28 years, and then I retired and started writing more. But writing this book is making me think about how I could have taught calculus better. I had already departed from the textbook in many ways, but I'm seeing today how chained to it I was, at least once in a while.

I taught the conventional proof for the derivatives of sine and cosine. I made a (very dense) four-page handout, walking students through the steps necessary. This involves a bit of review of trig, and some geometry, and some sophisticated use of inequalities. It's a lot. I'll guess now that maybe one student a year could really follow all that.

I knew there was another way, and I briefly showed my students the cool way to see this geometrically, if you allow for infinitesimals. But for some reason, I would stumble when explaining that way. So I never threw out the conventional proof.

This post is all about limit-based proofs versus infinitesimal-based proofs in a beginning course in calculus. It will help to know a little of the history. 

1. In the late 1600s, Newton and Leibniz put together what we now call calculus. Newton spoke of fluxions when he wanted to discuss infinitely small quantities. But what the heck is an infinitely small quantity?! Calculus was a huge boon to science, and there was much anxiety about whether its foundations were logically sound. 
2. It took other mathematicians desperately seeking a solution to this issue 150 years (early 1800s) to develop a logically sound foundation for calculus. That tells you how hard a problem it was. And their solution was so logically complicated that you have to be a lover of logic problems to actually get it. Their solution was limits. Back to that in a moment. First let's move another 150 years forward. 
3. In 1960, Abraham Robinson developed a number system called hyperreals (very carefully, one logial step at a time). And in that number system, infinitesimals are fine. But the textbooks had been written long ago, and no one was going to throw out those limits. (So sad.)

Here's a definition (written almost like a poem, because that helps):

limit (x -> c) f(x)=L (read this as "the limit, as x approaches c, of f(x) is L")

means

for any epsilon > 0

there exists a delta > 0

such that

if | x-c | < delta

then | f(x)-L | < epsilon  

Yeah, right. I'm guessing that bit sounds perfectly alien to 80% of the people who read my blog. (Does anyone still read blogs?) And it would be over 99% if you all hadn't already self-selected as math lovers.

 

It might be hard to understand how that solved the problem of infinitely small quantities. But it does. You want an example? Ok. 

limit (x -> 2) (x^2 - 4) / (x - 2) = 4. We can see that by multiplying out the top and factoring out x-2. If we don't do that, we have 0/0, which some of us like to call indeterminate. We know that 0/0 is undefined, but what we really want to know is what happens when x is infinitely close to 2. And then we don't have 0/0, we actually have x+2. And 2+2=4, so we have an answer. And that answer actually tells the slope of the y=x^2 parabola when x=2. That's one of the things calculus can do, and the limit stuff makes sure that it works.

But the limit stuff involves lots of hard algebra. And it's often hard to see where a step in the reasoning came from. So let's look at how infinitesimals solve this problem. That four-page handout for sine and cosine derivatives becomes a one-page diagram with a bit of explanation. I got the online handout, by Alex Alemi, years ago. [It just took me well over an hour to find it tonight online. Does he not see this as valuable? I found other stuff he's written much more quickly. Hmm.]

If you go look at that handout, it might take you a bit, but the connections between sine and cosine and their derivatives make visual sense once you get it. After writing some dialogue between Althea and her friends, as they try to figure out what's going on, I wrote their product rule adventures the next day, and then today they blasted through the quotient rule. (In the conventional proofs, both of these have non-intuitive algebra steps.) After their adventures with the sine and cosine derivatives, Aiden starts to talk about their time in Infinitesimal Land.

Excerpt from Althea and the Mysteries of Calculus

Perhaps I need to study a good calculus through infinitesimals textbook before finishing Althea's story. Any recommendations?

 

 

 

 

 

Writing Althea

I just (re)learned how different reading on a screen and reading on paper are for me. I wrote this scene about two years ago, and have probably re-read it over twenty times. Last night in the tub, reading my new copy of my manuscript (thanks, Lulu!), I realized there was a big problem.

Althea is 13, and has a younger brother(Rudy) and two moms.

All my characters feel pretty real to me. All but one are completely made up. The one is 'Mom'. She's a better version of me.

Here's the scene. I wonder if you can guess what the problem is.

Mom walked in then and listened for a bit. “It’s fun listening to you all talk about those. When I first started playing with these ten years ago, I struggled with a bunch of them, even though I had already been teaching math for over twenty years. What I usually teach is more related to algebra than geometry, so I still felt like a beginner.”

 

Aiden looked really surprised at that.

 

Well, Althea's mom is about 35. (And I'm 68.) With the ten and twenty years she mentions, she had to start teaching at about 5 years old! Oops. I guess I'll change that to 'five years ago', and 'almost ten years'.

Lots of advance readers have read this passage, and didn't mention that 'Mom' sounds pretty old to have a 13-year-old daughter. I guess that's a hard one to catch.

Farzanah and the 17 Camels

My publisher, Natural Math, uses crowdfunding to support each book they publish. I'm excited about this book.

Check it out, and contribute (which is really the same as making an advance order). I think you'll enjoy it.

Farzanah and the 17 Camels celebrates the excitement and the rewards of solving a challenging and intriguing math problem. Set against the backdrop of the ancient Silk Road, with bustling markets, stunning carpets, fun characters, and camels, the story draws readers into the magic of Farzanah's surroundings. 

As Farzanah searches for an unusual approach, a way of solving the problem that no one else could think of, she follows the wise advice of her mother: 

"My dear Farzanah, don't be discouraged,” said Mama. “Sometimes, being stuck is exactly where you need to be. I find the best thing I can do is to step away. I free my mind to think about other things. It is in that space that the magic happens. I am able to look at things from a different perspective. With wait time and wishful thinking comes the solution.”

 

Join the crowdfunding campaign here.

 

 

Playful Math Carnival #174 (the June, July, and now August edition): On Fractions & Division

 

The Playful Math Carnival is a collection of blog posts and articles from around the internet, putting lots of goodies in one place for your enjoyment. The theme for this issue is fractions and division. Why are division and fractions so much harder than what came before? And how can we explore them in playful, delightful, engaging ways? This carnival includes lots of perspectives, and approaches the topic from many levels, elementary to college.

 

 

A puzzle for 174:  What are all the factors of 174? Learning how to find factors goes hand in hand with division and fractions. It's easy to see that 2 is a factor of 174. Can you see any others before you divide by 2? There's a "trick" for 3 (and 9), but everything in math has a reason. Do you know why that "trick" works? I see that 1 plus 7 plus 4 is 12, and I know that 3 goes into 12. Why would that tell me something about whether or not 3 is a factor of 174? [Solutions at end. Hint: It's got to do with 10 being 9 plus 1.]

Before I started teaching, I had no idea that fractions might be hard. Part of what makes fractions difficult for some students is how many meanings fractions can have: a fraction of one whole, a fraction of some collection, a fraction of a measurement, etc.

 

My own troubles with division came from a slight case of (undiagnosed) dyslexia.  Why is it that we write  a / b, but then we have b going into a, with the numbers in the opposite order? The way we write it made no sense to me. And I got confused if the numbers were big ones. Because of this challenge for me, I learned one of my first problem-solving lessons: Make a simpler problem with the same structure.  If I saw 158 ÷ 79, I could think to myself, "That's like 6 ÷ 3." And then I knew what to do - find out how many 79s in 158. Aha, it's 2, just like 6 ÷ 3!

I used to hate the words divisor and dividend. I could not keep them straight. And I still don't know which is which (but if I care, the internet is my friend). And I, my friends, am a math professor. I tell my students often that my bad memory has helped me learn math, because I always tried to make sense out of it, instead of memorizing.

 

My personal favorite division issue now is why division by 0 is undefined. I wrote about it in my forthcoming book, Althea and the Mysteries of Triangles, Circles, and Pi. I'll share that passage at the end of this post. It's written at about high school level. We can go even higher level with the math and explore 0 / 0, an important concept for calculus that took mathematicians 150 years to come to terms with, which I did in a post a few years back.

 

 

John Golden is the Math Hombre. 

• He wrote Divide and Conquer in 2010. It's still golden. 
• John loves games and works with future teachers. This post, Fraction Reaction, written mostly by one of his students, really gives you a feel for how anyone can create a new game. 
• When we make fractals, we can explore what fraction of the area is shaded. Here's something John made in geogebra to allow students to play with fractals.
  

Denise Gaskins writes at Let's Play Math!  

• John and Denise have something in common ...  What is it? Games, of course! Check out  a perennial favorite from Denise: The Game That's Worth 1,000 Worksheets. (It's just variations on the card game of war, but 'just' is entirely the wrong word for how much you can do with that! And there is a Fractions War.)
• Her Dividing Fractions article helps you see how to think about dividing fractions so that you're not tempted to fall into using a procedure that might make no sense to you. 
• Besides writing articles, collecting math games, and publishing books with all of that goodness, Denise has also written a lovely series of stories in which Alexandria Jones solves some sort of problem her archeologist dad, Dr. Fibonacci Jones, encounters. Here's a small taste, where the two of them are puzzling out Egyptian Fractions. 
• If that story entices you to want to learn more, David Reimer has written a lovely book, Count Like An Egyptian. (Used copies are available at biblio.com.)

Who the heck is Professor Smudge? 

• Here's what they say in their twitter bio: "Wrote Maths Medicine. Rumoured to be called Sigi, after Sigismund (hello surds, hello pi) Arbuthnot, and to have personated Dietmar Küchemann on occasions." Hmm, that's a bit mysterious. Anyway, I found some good looking fraction puzzles on their twitter feed, and you can probably find lots more. 
• Here's one:
  
  
• And here's another:

 

Shayla Heavner (aka SJ Bennett) created MathBait, and is the author of Marcos the Great and the History of Numberville.  She brings us two factoring games.

• Emoji Mystery asks students to use some logic to help them find the factors of two numbers. Scroll down this page until you see it.  
• A digital version of the Taxman game, one of my favorite factoring games (again, scroll down to find it), is also on her wealth of pages.

 

Maria Droujkova, founder of Natural Math (my publisher), is conducting a crowdfunding campaign for a lovely book, Farzanah and the 17 Camels, by Dr. Sue Looney, which tells the story of an ancient math puzzle. One part of that puzzle asks: How can we possibly give one heir half of the 17 camels? Join that campaign here (your donation is basically an advance order of the book).

 

Do you want more?! The Ontario Math Links blog is updated weekly. Browse to your heart's content. (That's where I found Professor Smudge.)

 

 

The Math Teachers at Play Blog Carnival was created in 2009. Its name changed to Playful Math Carnival along the way, and it's been going strong for 15 years! (15 years online feels like a century anywhere else.) Links to all past posts available here. I used to include dozens of bloggers in my posts. This one only includes 5 people. (When Google evilly got rid of Google Reader, it really devastated the "math blogosphere".) If you have written something you think we'd like to see, please add a comment.

Puzzle Solutions:

The factors of 174 are 1, 2, 3, 6, 29,  58, 87, and 174. (There are 8 of them. Do all numbers have an even number of factors, or do some have an odd number of factors? Which are which?)

Understanding that factoring "trick" for 3 and 9: Add the digits of your number. If 3 or 9 goes into the sum, then it goes into the original. Why? Let's consider 174. The sum of the digits is 12, and 3 goes into 12. Hmm. 174 means 1*100 + 7*10 + 4, and that can be written 1*(99+1) + 7* (9+1) + 4. If I distribute, I get 1*99 + 1 + 7*9 +7 + 4. 99 and 9 are multiples of 3. So we have 1*99 + 7*9 + (1+7+4). Each term is a multiple of 3. The last term is that sum of the digits we looked at. After reading this, could you explain to someone else why the 3 and 9 factoring "tricks" work?

 

 

 

p.s Here's that ...

Sneak Preview from Althea and the Mysteries of Triangles, Circles, and Pi:

Sofia nods. “I messed up. I had 1 over 0, so I wrote 0 for my final answer. I’m not really sure why it’s supposed to be undefined instead. Can you explain that? It did feel kind of tricky to me.”

 

Mom says, “That’s a great question. And to answer it, we actually need to go back to some basics. The problem is that division doesn’t always work. It turns out that dividing by 0 doesn’t make sense. But to see why, we have to go back and look at how we define division."

She starts writing on the whiteboard and explaining at the same time. “We know that 6 over 2 is 3, because 2 times 3 is 6. I want to take that relationship and write it in a more generic way. I’m going to use T for top, B for bottom, and A for answer. When I was younger, I think I had trouble remembering numerator and denominator. That might be why I like saying top and bottom. Or maybe I just like shorter words. Anyway, now I can look at multiplication to help me think about weird division problems.

 

 

I look at Sofia. It seems like she’s deep in thought. Kiara’s taking notes, even though she seemed to know this. I think Mom has shown me this before.

 

“So 0 over 5 equals A becomes 5 times A equals 0. So what’s A?”

 

Sofia says, “It can only be 0.”

 

Mom nods. “And there are other good ways to think about this one. But for the one that tripped you up, this is the only way I know of to make it really make sense. So now 5 over 0 equals A. What does that become?”

 

Aiden starts to talk, but Sofia gives him a look. She says, “I’m the one who doesn’t get this, so let me try.  It turns into 0 times A equals 5. But Miss Annie, you can’t get 5. If you have 0 times anything, you’ll get 0.”

 

Mom nods and waits.

 

Sofia continues, “So there is no A that works in this one, and that means there’s no A for the first one. So it has no answer, and that’s why they say undefined?”

 

Mom nods again.

 

Kiara says, “Whoa! I just knew it was supposed to be undefined, but I definitely did not know why. And until this moment, I would not have known I was missing something.”

 

Aiden is nodding too. “I always knew one of those was undefined, but sometimes I mix up which one is which. I don’t think I’ll have that problem anymore.”

 

Sofia looks at him. “So you didn’t get it either?”

 

Aiden says, “I had number 5 right, but I think it was a lucky guess.”

Free Online Math Circle Has a Few Spots Open Still

 

We picked a time. We're meeting for nine weeks, each Saturday from March 2 to April 27, for an hour, at 3pm PT / 6pm ET. We still have a few spots open. We'll be playing with Triangles, Circles, and Pi, along with the fictional Althea and her friends. Participants will get an introduction to geometry, proof, and trigonometry.

 

I'm writing a new book series, Althea's Math Mysteries. In four young adult novels, Althea and her friends explore some of the mysteries of mathematics. The first two books are nearing publication, and the second book needs folks to test it out. In Althea and the Mysteries of Triangles, Circles, and Pi, Althea and friends, with the help of Althea’s mom, explore geometry and proof in order to then learn the basics of trigonometry.

 

We'd like to find some eager math students to join us in an online math circle, led by me, to explore along with Althea and her friends. Students will participate in 9 weeks of lively small-group sessions: in part a deep and friendly math course, and also a unique book club, with the author refining the story based on student reactions.

 

Do you know any students who enjoy math, know a bit of algebra, and would enjoy "user testing” Althea and the Mysteries of Triangles, Circles, and Pi? We're looking for a few more young people to try out the activities in this book together. 

Commitments: 

• Attend 9 weeks of 60-minute live online sessions from March 2 to April 27, each Saturday at 3pm PT / 6pm ET.
•  Read and comment on 1 to 3 chapters of the book each week. 
• Keep an informal math journal during this time.

For all who stay the course: 

• You'll learn the foundations of geometry and trigonometry (and will get a certificate for completing the course).
• You'll get a signed copy of the published book.
•  Your name or alias will appear in the book's acknowledgements, and you will receive a letter of appreciation for your help with this STEM project. (If you’d like a letter of recommendation later, we will be happy to write one for you.)
•  You’ll get to build community with math friends and mentors.

 

 

Interested? Please email me at mathanthologyeditor@gmail.com for more information, or to sign up.

Openings Now in Free Online Math Circle

 

Join an online math circle for students ages 12 to 15 in March and April, exploring geometry, proof, and the basics of trigonometry.

 

As most of you know, I'm writing a new book series. In four young adult novels, Althea and her friends will be exploring some of the mysteries of mathematics. The first two books are nearing publication at Natural Math.

In Althea and the Mysteries of Triangles, Circles, and Pi, Althea and friends, with the help of Althea’s mom, explore geometry and proof in order to then learn the basics of trigonometry. My publisher and I would like to find some eager math students to join me in an online math circle, exploring some math mysteries along with Althea and her friends. Participants will join lively small-group sessions: in part a deep and friendly math course, and also a unique book club, allowing me to refine the story based on student reactions.

Althea and the Mysteries of Triangles, Circles, and Pi is a fictional story set in the present, in which the characters discuss math, with Mom throwing in a few true stories from the past. Like The Number Devil and Math Girls, this book gives you more the more you put into it by doing the math yourself.

Do you know any students who enjoy math, know a bit of algebra, and would enjoy user testing Althea and the Mysteries of Triangles, Circles, and Pi? We’re looking for 5 to 8 young people to try out the activities in this book together. If interested, please add your name and information here.

 

Commitments:

• Attend 9 weeks of 60-minute live online sessions in March and April. Times to be determined, most likely 4 p.m. EST / 1 p.m. PST, on Saturday or a weekday (whichever works for more students).
• Read and comment on 1 to 3 chapters of the book each week.
• Keep an informal math journal during this time.

 

For all who stay the course:

• You’ll learn the foundations of geometry and trigonometry (and will get a certificate for completing the course).
• You’ll get a signed copy of the published book.
• Your name or alias will appear in the book’s acknowledgements, and you will receive a letter of appreciation for your help with this STEM project. (If you’d like a letter of recommendation later, we will be happy to write one for you.)
• You’ll get to build community with math friends and mentors.

 

The Storytellers of Math

Anyone here reading knows that I'm working on my series of young adult novels with math at the center - Althea's Math Mysteries.

 

But did you know that this drive to tell math stories is growing among budding storytellers across the lands? 

Sue in California (me!) is writing Althea's Math Mysteries. Four of them!

Shayla (aka SK Bennett) in New Mexico is writing the next book after Marco the Great and the History of Numberville. (I'm loving this one. I'm so glad there will be another.)

Sarah in Washington has written some wonderful fairy tales about physics and math. I'm reading Newton's Laws: A Fairy Tale right now.  (Currently free.)

And of course there are about a dozen lovely stories from the authors who work with Natural Math.

Who else is out there, writing tales of mathjoy that I haven't discovered yet?!

Illustrating Althea

I'm not much good at drawing, but most of the illustrations in Althea and the Mysteries of Triangles, Circles, and Pi are math work. I can do those. So I've put my own illustrations into the manuscript as placeholders. There will be a professional illustrator, later.

This past week, I was looking for where more illustrations are needed. I decided Althea would draw a map of California while thinking about their summer trips. So I drew it. Their home is in Berkeley, they go to camp in Quincy, and they're planning a trip to San Diego to visit Legoland (because her younger brother Rudy would love that, and their moms met in San Diego).

I had fun drawing the map. First I downloaded a map of California into goodnotes, as the template for my document. Then I outlined it, and then changed the template to make the original fancy map go away. Finally, I got to add the places of interest to Althea.

As one friend on facebook pointed out, it would help to make the line weights different for the outline versus the routes she's imagining. The professional illustrator can either take care of that, or show me how it will work best for the published book.

Althea's Math Mysteries

 

This blog may not be as active as it used to be, but it's a good way for me to remember some things. My first post about my Althea stories was in September of 2019, so I've been working on the first two books in this series for four years now. I'm hoping we'll be able to publish them in about a year. 

I have a very hopeful timeline that puts publication in October. But we all know that things never go as well as we hope. (And I'm wishing we could do it just a bit faster than that, so they'd come out in time for Math Storytelling Day, September 25, Maria's and my birthday.)

I have pretty complete drafts done of Althea and the Mystery of the Imaginary Numbers and Althea and the Mysteries of Triangles, Circles, and Pi. Soon I'll be asking for folks to read the manuscripts and comment on them. (Email me at mathanthologyeditor@gmail.com if you'd like to be one of our readers.) After that's done, we'll do our usual (Natural Math publishing's usual) crowdfunding campaign. And then there will be illustration, copy editing, proofreading, page layout, and books!

Here's my mock-ups of the covers, and lots of information that goes with the books.

At a few points, I've really wanted to see what these would look like as actual paperback books. lulu.com made that easy. Two books cost me under $25. I've done that 3 times, while I've polished up the books. What you see in the photo above is me holding the 3rd printed draft copy of Althea and the Mysteries of Triangles, Circles, and Pi.

In other news, I retired on May 20 from my full-time job teaching math at a community college. Teaching online was way too much work, and less satisfaction than teaching in person. I'm still covid-cautious, so I also didn't want to go back to teaching in person. Retirement has been wonderful. I'm working hard on the books, visiting Michigan where I help my dad (who's 90), cleaning up my house a bit, and working on my yard. 

All that was plenty for about the first five months. When I noticed that I sometimes felt like I had nothing to do, I posted in a Beast Academy group on facebook that I was thinking of offering a class. Someone suggested that I apply for a position with AOPS. (Art of Problem Solving is an amazing online resource, providing great math textbooks and classes, and they wrote the fabulous Beast Academy curriculum.) I did that, and I'll start teaching for them soon!

I will definitely be blogging more over the next year, to let anyone still reading my blog know what's up with these books. (If you're out there reading this, I'd love to hear from you.)

Playful Math Blog Carnival #166

This blog carnival has been around for 14 years. Almost every month for 14 years, someone has added a post to this collection. That's quite a long life for an internet phenomenon. (Congratulations, Denise, for keeping this going!) If you'd like to see any of the previous posts in this series, check them out here. For many years, blogs were a big part of my time online. But not so much lately.

When our number (166 now) was in the 20s, 30s, or 40s, I'd make sure to have that many links. Nope, I don't have time to find 166 great links (and you'd get tired just looking through them). But they're out there. I have learned so much from bunny hopping around the web of math bloggers over the years. And even though blogs aren't the popular thing now, most of the old ones are still out there, waiting for you to find them and get excited.

 

The 166 puzzle: It turns out that 166 is a 'centered triangular number'. If you start with a dot, and then you put a triangle around that, and a bigger one around that, etc, you get up to 166. How many triangles did you use?

 

I have just run out of envelopes. How should I make myself one? (a puzzle from Fawn Nguyen) What shape of paper will you use?

 

 

Online Mathy games

• The Dailies: When wordle came out, there were a number of mathy imitators, including nerdle, numberle (like nerdle, but you can do longer or shorter equations), mathler, numble, summle, and lots more. Most of them have vanished. More recently, Beast Academy added their All Ten game, which I'm hooked on.What's your favorite?
  
• Games Before Class (from Math Hombre, not all are math, hence "before class")
• Target Ten (from Denise Gaskins)
• Factors and Multiples Game (from nrich, I got 48 as my longest possible chain of factors and multiples. Can you beat me?)
• Do you know of other goodies? Let us know in the comments.
  

Geometry Puzzles

 

Find the blue area. 

This one stumped me (no trig required).

 

 

 This one's a lot easier.

 

 

 

 

 

 

What fraction is shaded? Catriona Shearer (@Cshearer41) made this, along with gobs more, mostly pretty challenging, which she posts on twitter. And here's a collection of over 300 of them.

 

Beyond the Games & Puzzles

• Balance Fractions (from Math for Love) looks fun.
  
• I must be on a fractions kick. This stood out for me, too. Fractions on Grids (from Henri Picciotto).
  
• In the series of young adult math novels I'm writing, circles come up in the 2nd book. An early reader pointed out that I might be mixing up circles and disks, so I re-wrote a few phrases. Here are some lovely circle lessons (from Sara VanDerWerf), where she helps students distinguish between those two ideas.
  

 

 

 

Searching for more? Some good hashtags are:  #MTBoS #ITeachMath  #Elemmathchat #MSmathchat

 

 

 

If you've seen some good math pedagogy out in the wilds of the internet, add a comment.