What does it mean when we feel we "understand" something?

On facebook, I'm in a group for people who use Beast Academy (even though I'm not using it), because Beast Academy fascinates me. I love most of what they do.

A parent today posted that she was confused about the BA way of multiplying 59*59. They have you draw a 60 by 60 box, and then take off one row (of 60) and one column (which is now 59). Your box is now 59 by 59, and its area is 60*60 - 60 - 59. Cool.

She wasn't seeing it, so she taught her kid the standard algorithm. Lots of people were giving her flak for that. (We each do our best, so I don't see why folks would jump on her.) She replied to them that she thought learning it multiple ways was a good thing.

I wrote: "Sure it's great to do things multiple ways, but does he really understand the algorithm you showed him? (Do you really understand why it works?) I think that's why you're getting pushback here."

She said they both understood it. I replied that I'd have trouble explaining to a young kid why you "put a 0". She wrote: "I just told him we put it to show that the one number is done. I don’t know if it’s accurate but he understood it. I don’t really remember it ever being explained in school."

So what she originally meant when she said he understood it, was that he could follow the steps and get it right. Not that he understood why it worked.

I think this is common with math.  People think 'understand' means the same as 'can follow the steps'. But I'm afraid that doing math without really seeing why each step makes sense is part of why a lot of people don't like math. It's surely why we easily forget how to do those things.

Here's an article by Richard Skemp, written back in 1978, about why the deeper understanding, which he calls "relational understanding" is a better way to approach math. (He calls being able to follow the steps "instrumental understanding".) I wrote about this topic and this article ten years ago here, but people's ideas about math haven't changed much in that time.

Of course, this parent can still explain to her son why the standard algorithm works, so she hasn't somehow wrecked the beauty of Beast Academy, as some people seemed to feel. And that's what got me writing - I want to see how well I can explain the standard algorithm.

I figure that the standard algorithm packs in a lot with as little writing as possible. (Maybe when we didn't have calculators, and had to do lots of by-hand multiplication, writing as little as possible was considered an important goal for the way we write things down?) So I figured that it needs to be unpacked a little. That's what I tried to do here.

 

The first calculation is adding up all 4 areas. The one to the far right is the standard algorithm. The first number in the standard algorithm (531) is the 81 and the first 450 added together (with carrying), and the second number (2950) is the other 450 and the 2500 added together. It's surely as little writing as possible, but it hides so much! Does my unpacking on the left help?

It all makes sense to me, but the Beast way feels more fun. (And I don't have to write anything that way. I can hold it all in my head!) What do you think?

Prepping for Fall, Calc II: Lovely Arc Length Example

I'll be teaching Calc II for the first time in a few years. This is my first time starting out online with it. So I'm preparing my Canvas shell and thinking about how I want to explain each topic in Canvas. (I know the material well enough that I didn't have to prep this much when we were in person.) The extra prep before we start is so much work, but today it feels totally worthwhile.

 

 

For arc length I was excited to use "crinkle crankle walls" as an example. Isn't that a pretty wall? And you can actually use fewer bricks this way than for a straight wall, because one layer of bricks here is stronger than it would be straight (so the straight wall would need extra bricks for support). I'm thinking we'll try to prove that assertion in my Calc II class.

 

It turns out that arc length uses an integral which often has no "elementary solution", meaning there is no anti-derivative using the functions we are familiar with. 

 

The arc length for y=sin x is...

 

 

 

And this has no "elementary solution".

 

 

I often tell my students that we study infinite series to solve the integrals with no easier solution, but I just realized that that won't work here. (Can't do a square root of an infinite series!) 

 

Ok, no problem. I'm also teaching numerical integration. So I made a google sheet to do Simpson's method, and it turns out beautifully!! (Beautifully meaning that my answer matched the answers on Math SE that people explained in ways that were above my head. I don't know a thing about "elliptical integrals".)

 

I still need to remember how to explain Simpson's rule, but I'll get that back easily enough. 

 

If this wall follows a sine wave, then for 6.28 feet (2π feet) of straight distance covered, it has a length of 7.64 feet. That's just over 20% extra length. (Now to think with my students about whether that's better than the straight wall with supports.)

Technology Woes and Cheers: Venn diagram edition

I'm writing questions for my Discrete Math course that will be available to my students (and others) through MyOpenMath, a free online homework system. I'm not very good at programming in their environment, but I'm learning. The cool thing about MyOpenMath is that it uses random numbers in the questions so that each student might get a (slightly) different question.

I wanted a way to ask, for a random Venn diagram: What is the set notation for this?

First, I needed a way to make lots of Venn diagrams, all pretty, and all in the same style. I searched the internet for a free online Venn diagram maker. Nothing right showed up. I looked at over a dozen sites. Many wanted me to sign in. That should not be necessary and I skipped those. None of the others were even close to what I wanted, which is pretty simple. Really?! Isn't this something lots of people would want? 

I asked about it on Math Educators Stack Exchange. Within hours, Cameron Williams posted an answer. He made it on desmos for me. (How sweet is that?! Amazing.) I know desmos well, so I was able to modify his version to be exactly what I wanted, in less time than I had already spent searching. (I suggest you go play with it - it's lovely.) And, if you want orange shading instead of blue, it's very easy to modify this to get exactly what you want.

Then I made 17 screenshots of various combinations of the basic regions, and named them based on the set notation. So "(A un B) int not C.jpg" is the filename for ...

 

Next I went back to MyOpenMath, and wrote most of my multiple choice problem. I'm still stuck on how to get it to display a randomly chosen image file. I think the folks at the help forum there will help me out on that. Once I finish fixing it, I'll edit this post to show the question. MyOpenMath allows attached videos to explain how to answer the questions. I think I might do a video for this one. 

So if you want a free online Venn diagram maker, it's here.  I don't know how to help google move this up in the searches so people can find it. Do you?

Math Teachers at Play (aka Playful Math Education, Blog Carnival #157)

 

 

 

About once a year, I sign up to host this long-running blog carnival. Ever since Google Reader was snatched away, blogs seem to have fewer readers and less activity. Mine certainly has straggled along in recent years. (I guess I needed a very long rest after finishing my big book.) Today, I'm looking forward to exploring the new ideas I'll find online and gather here.

 

We start with cool facts about 157, and a puzzle...

 

Cool Little Facts

• 157 is the 37th prime number. (37 is prime too.)
• 157 is the largest known prime p for which ${\displaystyle {\frac {p^{p}+1}{p+1}}}$ is also prime (see OEIS: A056826).
  157 is a palindromic number in bases 7 (313₇) and 12 (111₁₂).
• 157 is the largest odd integer that cannot be expressed as the sum of four distinct nonzero squares with greatest common divisor 1. 
• 157 is the smallest three-digit prime that produces five other primes by changing only its first digit: 257, 457, 557, 757, and 857. [Opao] 
• 157 is the largest rating on the Saffir-Simpson Hurricane Wind Scale occurs at sustained winds of 157 mph or higher. 
• If we use the English alphabet code a = 1, b = 2, c = 3, … , z = 26, then número primo = 157. 

 

Puzzle

How many 3-digit numbers can we find where the last digit equals 2 times the first digit plus 1 times the second digit? 157 is one answer. How would you find the others without tediously checking each 3-digit number? (I use a spreadsheet when I want enough data to see patterns, but I worked hard to get the digits apart. Once you find the first few answers by hand, you might see the pattern...)

[Solution at bottom.]

Not Just Blogs...

I'm working on another book, much smaller this time. Althea and the Mystery of the Imaginary Numbers should be ready sometime next year. Since I'm working on a book, I've been thinking a lot about what makes a fun mathy book. It needs a good storyline. It needs interesting math. And if it's for young kids, it needs lovely illustration. 

There's a prize for good mathy books, called the Mathical Book Prize. It started in 2015 and doesn't seem to include small publishers like Natural Math (my publisher), so some of my favorites are missing. I think my favorite book on their list might be the picture book Which One Doesn't Belong, by math blogger Christopher Danielson.

Here are a few of my favorites that aren't on their list:

Quack and Count, by Keith Baker (for ages 2 to 7), a board book good for the youngest child who will sit and listen to a story. And it stays good because it's so luscious. Great illustrations, fun rhythm and rhyme, cute story, and good mathematics. 7 ducklings are enjoying themselves in every combination. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” (And then 5 plus 2, 4 plus 3, 3 plus 4, and so on.) It would be great to have a book like this for each number, showing all the number pairs that make it.

How Hungry Are You? by Donna Jo Napoli and Richard Tchen (for ages 3 to 12), on equal sharing. The picnic starts with just two friends, rabbit is bringing 12 sandwiches and frog is bringing the bug juice. Monkey wants to come, "My mom just made cookies. I could take a dozen." They figure out how much of each goody each friend will get. In the end, there are 13 of them, and the sharing becomes more complicated. One of the delights of this book is the little icons showing who’s talking. It would make a good impromptu play. [There are lots of good books on equal sharing. Another lovely one is The Doorbell Rang, by Pat Hutchins.]

The Cat in Numberland, by Ivar Ekeland (for ages 5 to adult), starts when Zero knocks on the door of the Hotel Infinity. He’d like a room, but they’re all full (with the number One in Room One, and so on). Turns out that’s no problem. The cat who lives in the lobby gets confused - if the hotel is full, how can the numbers make room for zero just by all moving up one room? Things get worse when the fractions come to visit. This story is charming enough to entertain young children, and deep enough to intrigue anyone. Are you ready to learn about infinity with your 5 year-old?

 

The Man Who Counted, by Malba Tahan (for ages 6 to adult), was written in Brazil, and set in the Middle East. We follow the adventures of Beremiz, an accomplished mathematical problem-solver. He uses math to settle disputes, solve riddles and mysteries, and entertain his hosts. The series of 34 adventures, each with a math puzzle, is reminiscent of the Arabian Nights. If you read one chapter a night, your audience will be begging for more – and isn’t that the way it should be?

 

Carry On, Mr. Bowditch, by Jean Lee Latham (for ages 7 to adult), is a slightly fictionalized account of the life of Nathaniel Bowditch, who loved math, but had to leave school when his family needed his help. He was indentured to a ship chandlery for 9 years. Although that dashed his hopes of someday going to Harvard to study math, it was the right place to learn the mathematics behind navigation. When he finally went to sea, he invented a new way to ‘do a lunar’, and spent endless hours correcting errors in the tables used for navigation. Bowditch’s book, the American Practical Navigator, first published in 1902, is still regularly updated, and is carried on U.S. naval vessels to this day. 

 

And coming very soon ... Denise Gaskins' 2nd edition of Word Problems from Literature. (She'll be using kickstarter to raise some funds to get this out the door. Crowdfunding is how tiny publishers make it work!)

...And Now the Blogs (mostly geometry)

Denise Gaskins' How to Make Time for Exploration, in which Denise considers the benefits of Michelle's "Minimalist Math" curriculum, used along with games and books.

 

John Golden's Art, Math, and Geogebra Project, in which John has created a way for you to change a Kandinsky box to be new combinations of colors. Fun.

 

Daniel Scher's Euclid Walks the Plank on Geometric Construction, in which Daniel explores helping students to see the power of circles in building equal length line segments, using Geometer's Sketchpad for his online experiments. Once again, you get to play with the geometry.

Sam Shah on 3D Printing, in which Sam shares lots of cool 3D printing projects but decides they aren't really helping his students learn math. Do you have any 3D printing projects that help your students learn math?

Joann Sandford's Play, Persist, Prove on thinking about the angles in polygons. Can you use pattern blocks to prove what the angles are?

I adore Catriona Schearer's geometry puzzles, which she posts on twitter and elsewhere. Here's a video of her talking about them. (I recommend starting at about 8:30. They wait for participants and talk about Mathigon first.) Here's a lovely puzzle of hers. The big triangle that holds all the others is also isosceles. Find it on her twitter feed, and you'll see lots and lots of thoughts about it.

One more way to play with geometry ... this site gamifies geometric construction. I love it. 

 

Do you want more info on this blog carnival, or would you like to read old carnival posts? Denise Gaskins has got you covered.

 

 

 

 

Puzzle solution: There are 8 of these starting with 1: 113, 124, ..., 179, then 6 starting with 2, up to 2 starting with 4, for a total of 20.

Logic Puzzle, Supposedly from Einstein...

 ... but there's no evidence for that. The puzzle originally had folks smoking cigarettes. Yuck. I've changed that to eating candy.

The situation:

• There are 5 houses in five different colors.
• In each house lives a person with a different nationality.
• These five people drink a certain beverage, eat a certain candy, and keep a certain pet.
• No one has the same pet, eats the same kind of candy, or drinks the same beverage.  

 

The question is: Who owns the fish?

 

 

Hints:

• the Brit lives in the red house
• the Swede has a dog
• the Dane drinks tea
• the green house is on the left of the white house
• the green house's owner drinks coffee
• the person who snarfs M&Ms has birds
• the owner of the yellow house loves peanut butter cups
• the person living in the center house drinks milk
• the Norwegian lives in the first house
• the person who adores Heath bars lives next to the one who keeps cats
• the person who has a horse lives next to the peanut butter cup lover
• the person who eats Snickers bars drinks beer
• the German eats Almond Joys
• the Norwegian lives next to the blue house
• the person who eats Heath bars has a neighbor who drinks water

 

[There is one thing that's unclear: Is "the first house" the one on the left of the bunch? I assumed that. Apparently, you can assume that it's on the right end, and according to Wikipedia, you'll get the same answer. I haven't explored that.]

Still learning, after all these years...

This semester I'm teaching Calculus I and Linear Algebra. In each class, I've had a moment of discovery in the past week or so.

 

Calculus: Derivatives from Graphs

In calculus, I work with them on what the derivative graph of a function would look like, given just the graph of the function. So if the graph of f is this ...

... then what would f' look like? The activity (with 8 different graphs) went as it usually does. 

• Step 1: Find where the slope is 0, and give f' a value of 0 at that x.
• Step 2: Where the slopes of f are positive, highlight positive values for f' (and similarly for negative slopes). (Actually, the highlighting was new. I usually just draw dotted lines.)
  
• Step 3: Draw a curve that connects it all.

We had an absolute value curve and discussed where the derivative is undefined. (Which I marked with vertical red lines.) 

 

And then we got to this one ...

I said that w' looked like this ...

And a student asked how I knew the lines were straight.  Hmm, do I know that? "I'm not sure. Let's see..."

 

I thought about the curve given for w and said it looked like a bunch of parabola shapes (which I know have straight line derivatives), ... or like the absolute value of sine. I decided this was a fascinating question, and put both on desmos.

The red is y = |sin(π/2*x)|, and the blue is y = -(x+1) ² + 1 and y = -(x-1) ² + 1. To me, it looks like the original graph of w could be either one. But the derivative is the straight line segments only if w came from parabolas. If it came from a sine wave, then the derivative is curved (coming as it does from cosine). Using orange for the derivative of the sine graph and purple for the derivative of the parabolas graph, I got this in desmos...

Very different look to the derivatives, even though the original w could have been either of the original functions I put onto desmos. Fascinating!

 

 

Linear Algebra: Pivots vs Free Variables

We are using some fabulous activities from the Inquiry-Oriented Linear Algebra project, along with our textbook, Linear Algebra and Its Applications, by David Lay (we're using the 4th edition). We had just done part 3 of the Magic Carpet project the day before, and I was summarizing. We were talking about the span of a set of 3 vectors in ℝ³, and saw that the span made a plane through the origin. This was because there were 2 pivot columns. And then a student asked, "But don't we use the number of free variables to decide whether we have a line or a plane?"

 

To me that felt like a very deep question for a student to be asking this early in the semester. I said I'd answer the next day, since we were almost out of time. The next day I said, "We looked at the pivots because we were asking about span, which is all the linear combinations of the column vectors. Until we started considering span, we more typically asked about all the solutions to a set of equations, which is a different sort of question. For that, we look at how many free variables to determine if all our solutions create a line or a plane (or something more)."

I have never had a student ask a question like this, and was quite intrigued. I told them we'd explore somewhat similar questions in our 3rd unit (chapter 4 of Lay), when we will explore column space and null space. Once again, I was fascinated.

I've been teaching for over 30 years. I know calculus I inside and out. I've taught linear algebra often enough to feel like I'm a pretty solid expert on the basics. (I'd love to have more expertise on where this class might lead them.) Even so, I learn new things each semester. Even teaching beginning algebra, I have repeatedly seen it from a new perspective when prodded by some unique question a student was asking.

Yay for student questions.